Merge branch 'master' of https://gitee.com/agyz/obsidian_backup

# Conflicts: # .obsidian/copilot-index-ef6af226357d1e8b24528ffe939abb69.json # .obsidian/plugins/copilot/data.json # .obsidian/workspace.json # 多体+耦合求解器/数据结构讨论.md
2025-01-15 13:59:43 +08:00 · 2025-01-15 13:59:43 +08:00 · 1094191210
commit 1094191210
parent bc25d467b6 43e7b08163
421 changed files with 1696651 additions and 32 deletions
--- a/.DS_Store
+++ b/.DS_Store
--- a/.gitignore
+++ b/.gitignore
@ -23,3 +23,10 @@ bin-release/
 # Ignore Smart Environment folder
 .smart-env
 .DS_Store
+workspace.json
+
+
+
+
+# Ignore Smart Environment folder
+.smart-env
--- a/.obsidian/community-plugins.json
+++ b/.obsidian/community-plugins.json
@ -1,4 +1,5 @@
 [
  "copilot",
-  "obsidian-git"
+  "obsidian-git",
+  "smart-connections"
 ]
--- a/.obsidian/copilot-index-ef6af226357d1e8b24528ffe939abb69.json
+++ b/.obsidian/copilot-index-ef6af226357d1e8b24528ffe939abb69.json
--- a/.obsidian/plugins/copilot/data.json
+++ b/.obsidian/plugins/copilot/data.json
@ -13,12 +13,16 @@
  "googleApiKey": "",
  "openRouterAiApiKey": "",
  "defaultChainType": "llm_chain",
+<<<<<<< HEAD
  "defaultModelKey": "gemma2:latest|ollama",
+=======
+  "defaultModelKey": "phi4:latest|ollama",
+>>>>>>> 43e7b08163ff98d031f47a55e6d8abfb871edb14
  "embeddingModelKey": "nomic-embed-text|ollama",
  "temperature": 0.1,
  "maxTokens": 1000,
  "contextTurns": 15,
-  "userSystemPrompt": "",
+  "userSystemPrompt": "翻译成中文",
  "openAIProxyBaseUrl": "",
  "openAIEmbeddingProxyBaseUrl": "",
  "stream": true,
@ -43,21 +47,27 @@
      "provider": "openai",
      "enabled": true,
      "isBuiltIn": true,
-      "core": true
+      "core": true,
+      "baseUrl": "",
+      "apiKey": ""
    },
    {
      "name": "gpt-4o-mini",
      "provider": "openai",
      "enabled": true,
      "isBuiltIn": true,
-      "core": true
+      "core": true,
+      "baseUrl": "",
+      "apiKey": ""
    },
    {
      "name": "claude-3-5-sonnet-latest",
      "provider": "anthropic",
      "enabled": true,
      "isBuiltIn": true,
-      "core": true
+      "core": true,
+      "baseUrl": "",
+      "apiKey": ""
    },
    {
      "name": "llama3.2:latest",
@ -78,6 +88,16 @@
      "isBuiltIn": false,
      "enableCors": true,
      "isEmbeddingModel": false
+    },
+    {
+      "name": "phi4:latest",
+      "provider": "ollama",
+      "baseUrl": "https://possibly-engaged-filly.ngrok-free.app",
+      "apiKey": "",
+      "enabled": true,
+      "isBuiltIn": false,
+      "enableCors": true,
+      "isEmbeddingModel": false
    }
  ],
  "activeEmbeddingModels": [
--- a/.obsidian/plugins/obsidian-git/data.json
+++ b/.obsidian/plugins/obsidian-git/data.json
@ -4,7 +4,7 @@
  "autoSaveInterval": 0,
  "autoPushInterval": 0,
  "autoPullInterval": 0,
-  "autoPullOnBoot": false,
+  "autoPullOnBoot": true,
  "disablePush": false,
  "pullBeforePush": true,
  "disablePopups": false,
@ -14,8 +14,8 @@
  "updateSubmodules": false,
  "syncMethod": "merge",
  "customMessageOnAutoBackup": false,
-  "autoBackupAfterFileChange": false,
-  "treeStructure": true,
+  "autoBackupAfterFileChange": true,
+  "treeStructure": false,
  "refreshSourceControl": true,
  "basePath": "",
  "differentIntervalCommitAndPush": false,
--- a/.obsidian/plugins/smart-connections/data.json
+++ b/.obsidian/plugins/smart-connections/data.json
@ -0,0 +1,25 @@
+{
+  "new_user": false,
+  "legacy_transformers": false,
+  "enable_mobile": true,
+  "actions": {
+    "lookup": true
+  },
+  "smart_notices": {
+    "muted": {
+      "no_entity": true
+    }
+  },
+  "system_prompts_folder": "smart prompts",
+  "smart_chat_folder": "smart-chats",
+  "smart_chat_folder_last": "smart-chats",
+  "chat_model_platform_key": "open_router",
+  "open_router": {},
+  "api_key": "",
+  "excluded_headings": "",
+  "folder_exclusions": "smart-chats",
+  "show_full_path": false,
+  "expanded_view": true,
+  "language": "en",
+  "version": "2.3.47"
+}
--- a/.obsidian/plugins/smart-connections/main.js
+++ b/.obsidian/plugins/smart-connections/main.js
--- a/.obsidian/plugins/smart-connections/manifest.json
+++ b/.obsidian/plugins/smart-connections/manifest.json
@ -0,0 +1,10 @@
+{
+  "id": "smart-connections",
+  "name": "Smart Connections",
+  "author": "Brian Petro",
+  "description": "Find links to similar notes using artificial intelligence from OpenAI.",
+  "minAppVersion": "1.1.0",
+  "authorUrl": "https://wfhbrian.com",
+  "isDesktopOnly": false,
+  "version": "2.3.47"
+}
--- a/.obsidian/plugins/smart-connections/styles.css
+++ b/.obsidian/plugins/smart-connections/styles.css
@ -0,0 +1,968 @@
+/* deprecated positioning, use bottom bar instead */
+.view-content > .sc-brand {
+  position: fixed;
+  bottom: 0;
+  right: 0;
+  background-color: var(--titlebar-background);
+}
+
+.sc-brand {
+  > svg,
+  > p {
+    display: inline;
+    margin: 0 0.1rem 0 0.3rem;
+    color: var(--text-muted);
+    font-size: var(--font-smallest);
+    line-height: 1;
+    height: 0.88rem;
+    width: auto;
+  }
+
+  > p > a {
+    color: var(--text-muted);
+  }
+}
+
+.sc-list {
+  .tree-item-self {
+    cursor: pointer;
+
+    small {
+      color: var(--color-gray-40);
+    }
+  }
+
+  > .sc-collapsed ul {
+    display: none;
+  }
+
+  > .sc-collapsed span svg {
+    transform: rotate(-90deg);
+  }
+
+  > :not(.sc-collapsed) span svg {
+    transform: rotate(0deg);
+  }
+
+  > div {
+    span svg {
+      height: auto;
+      margin: auto 0.5em auto 0;
+      flex: none;
+    }
+
+    > span {
+      display: inline-flex;
+      width: 100%;
+      padding-left: 0;
+    }
+
+    ul {
+      margin: 0;
+      padding-left: 1.3rem;
+    }
+
+    > a {
+      display: block;
+    }
+
+    > ul > li {
+      display: block;
+    }
+  }
+  .sc-result {
+    > ul {
+      list-style: none;
+      padding-left: 0;
+    }
+  }
+
+  .sc-result.sc-result-plaintext {
+    font-size: var(--font-ui-smaller);
+    line-height: var(--line-height-tight);
+    background-color: var(--search-result-background);
+    border-radius: var(--radius-s);
+    overflow: hidden;
+    margin: var(--size-4-1) 0 var(--size-4-2);
+    color: var(--text-muted);
+    box-shadow: 0 0 0 1px var(--background-modifier-border);
+
+    & > * li {
+      cursor: var(--cursor);
+      position: relative;
+      padding: var(--size-4-2) var(--size-4-5) var(--size-4-2) var(--size-4-3);
+      white-space: pre-wrap;
+      width: 100%;
+      border-bottom: 1px solid var(--background-modifier-border);
+    }
+  }
+
+  .sc-result:not(.sc-result-plaintext) {
+    cursor: pointer;
+    padding: var(--nav-item-padding);
+    padding-left: 10px;
+    margin-bottom: 1px;
+    align-items: baseline;
+    border-radius: var(--radius-s);
+    font-weight: var(--nav-item-weight);
+
+    &:hover {
+      color: var(--nav-item-color-active);
+      background-color: var(--nav-item-background-active);
+      font-weight: var(--nav-item-weight-active);
+    }
+
+    span {
+      color: var(--h5-color);
+    }
+
+    small {
+      color: var(--h5-color);
+      font-size: 0.8rem;
+      font-weight: 500;
+    }
+
+    p {
+      margin-top: 0.3rem;
+      margin-bottom: 0.3rem;
+    }
+
+    ul > li {
+      h1 {
+        font-size: 1.3rem;
+      }
+
+      h2 {
+        font-size: 1.25rem;
+      }
+
+      h3 {
+        font-size: 1.2rem;
+      }
+
+      h4 {
+        font-size: 1.15rem;
+      }
+
+      h5 {
+        font-size: 1.1rem;
+      }
+
+      h6 {
+        font-size: 1.05rem;
+      }
+
+      h1,
+      h2,
+      h3,
+      h4,
+      h5,
+      h6 {
+        margin-block-start: calc(var(--p-spacing)/2);
+        margin-block-end: calc(var(--p-spacing)/2);
+      }
+    }
+  }
+} /* end .sc-list */
+
+/* Only on right sidebar */
+.mod-right-split .sc-list .sc-result {
+  font-size: var(--font-text-size);
+  font-size: 0.88rem;
+}
+
+.sc-top-bar {
+  display: flex;
+  width: 100%;
+  justify-content: end;
+
+  .search-input-container {
+    width: calc(100% - var(--size-4-8));
+    margin: 4px auto;
+  }
+
+  .sc-context {
+    color: var(--nav-item-color);
+    font-size: var(--nav-item-size);
+    margin: 0.5em 0.5em 1em;
+    width: 100%;
+  }
+}
+
+/* Chat */
+.sc-chat-container {
+  display: flex;
+  flex-direction: column;
+  align-items: center;
+  justify-content: flex-end;
+  height: 100%;
+
+  .sc-top-bar-container {
+    align-self: flex-end;
+    display: flex;
+    width: 100%;
+
+    .sc-chat-name-input {
+      flex-grow: 1;
+    }
+  }
+
+  .sc-thread {
+    display: flex;
+    flex-direction: column;
+    align-items: flex-start;
+    height: 100%;
+    width: 100%;
+    overflow: hidden;
+    user-select: text;
+    overflow-y: auto;
+
+    .sc-message-container {
+      border: 1px solid var(--divider-color);
+      border-radius: 10px;
+      margin: 0.5rem 0;
+      display: flex;
+      flex-direction: column;
+      align-items: flex-start;
+      width: 100%;
+      height: 100%;
+      overflow-y: auto;
+      background-color: var(--background-primary-alt);
+
+      .sc-message {
+        max-width: 90ch;
+        width: 90%;
+        margin: 10px;
+        padding: 10px;
+        border-radius: 1.5rem;
+        word-break: break-word;
+
+        &.user {
+          align-self: flex-end;
+          color: var(--text-normal);
+          background-color: var(--background-primary);
+        }
+
+        &.assistant,
+        &.system {
+          background-color: var(--background-primary-alt);
+          color: var(--text-normal);
+        }
+
+        .sc-message-content {
+          margin: 0;
+          padding: 1rem;
+
+          > * p {
+            margin: 0;
+          }
+        }
+      }
+    }
+
+    .sc-chat-form {
+      display: flex;
+      padding: 0 10px 1rem 0;
+      width: 100%;
+      max-height: 50%;
+
+      .sc-chat-input {
+        flex-grow: 1;
+        padding: 0.88rem;
+        border: none;
+        border-radius: 1.5rem;
+        resize: none;
+        height: auto;
+        min-height: 4.2lh;
+        max-height: 100%;
+        background-color: var(--background-primary);
+        color: var(--text-normal);
+        margin-right: -2.7rem;
+        padding-right: 3rem;
+      }
+
+      .sc-btn-container {
+        width: 2rem;
+        height: 2rem;
+        margin: auto;
+
+        > button.send-button {
+          border-radius: 99999px;
+          cursor: pointer;
+          outline: 2px solid transparent;
+          padding: 0;
+          outline-offset: 2px;
+          background: none !important;
+
+          &:focus-visible {
+            outline-color: var(--text-faint);
+            box-shadow: none;
+          }
+
+          > svg {
+            > circle {
+              fill: var(--text-faint);
+            }
+            > path {
+              fill: var(--background-primary);
+            }
+          }
+        }
+      }
+    }
+  }
+
+  #settings {
+    display: flex;
+    flex-direction: column;
+    max-width: 100%;
+    width: 100%;
+  }
+}
+
+.sc-system {
+  align-self: center;
+  font-style: italic;
+  color: var(--text-faint);
+}
+
+.sc-msg-button {
+  cursor: pointer;
+  float: right;
+  margin-left: 5px;
+  opacity: 0.8;
+
+  &.cycle-branch {
+    float: left;
+    display: flex;
+  }
+
+  &:hover {
+    opacity: 1;
+  }
+}
+
+#sc-abort-button {
+  cursor: pointer;
+  padding: 10px;
+  border-radius: 5px;
+
+  &:hover {
+    background-color: var(--background-primary);
+  }
+}
+
+.sc-change {
+  .sc-change-footer {
+    display: flex;
+    justify-content: space-between;
+    align-items: center;
+
+    & > * {
+      display: inline-block;
+      color: var(--text-muted);
+      font-size: var(--font-smallest);
+    }
+
+    .sc-brand {
+      position: unset;
+      background-color: var(--background-primary);
+    }
+  }
+
+  .sc-variation {
+    --new-content-color: hsl(124, 75%, 55%, 0.13);
+    --old-content-color: hsl(3, 43%, 47%, 0.13);
+    display: flex;
+    align-items: flex-end;
+    justify-content: space-between;
+    margin: 0 -300px;
+    padding: 0 300px;
+
+    &:has(.new-content) {
+      background-color: var(--new-content-color);
+
+      > button {
+        background-color: var(--new-content-color) !important;
+      }
+    }
+
+    &:has(.old-content) {
+      background-color: var(--old-content-color);
+
+      > button {
+        background-color: var(--old-content-color) !important;
+      }
+    }
+
+    > div {
+      > *:last-child {
+        margin-bottom: 0;
+      }
+    }
+
+    .old-content {
+      color: var(--text-muted);
+      text-decoration: line-through;
+
+      > *:first-child {
+        margin-top: 0;
+      }
+    }
+  }
+}
+
+.markdown-source-view.mod-cm6 .cm-embed-block:not(.cm-table-widget):hover:has(.sc-change) {
+  overflow: unset;
+  box-shadow: unset;
+  cursor: unset;
+}
+
+.notice .sc-notice-actions {
+  display: flex;
+  justify-content: space-between;
+  flex-direction: row-reverse;
+}
+
+.sc-chat-container {
+  #settings {
+    display: flex;
+    flex-direction: column;
+    max-width: 100%;
+    width: 100%;
+  }
+
+  .sc-config-error-notice {
+    display: flex;
+    align-items: center;
+    justify-content: space-between;
+    padding: 10px;
+    background-color: #ffcccc;
+    border: 1px solid #ff0000;
+    border-radius: 5px;
+    margin: 10px 0;
+    font-size: 14px;
+    font-weight: bold;
+    color: #ff0000;
+    width: 100%;
+
+    span {
+      flex-grow: 1;
+    }
+
+    button {
+      margin-left: 10px;
+    }
+  }
+}
+
+.sc-supporters {
+  max-height: 37ch;
+  border: 1px solid var(--h1-color);
+  border-radius: 10px;
+  padding: 10px;
+  margin-left: -10px;
+  box-shadow: 0 1px 0 1px var(--shadow-300);
+  overflow: auto;
+  text-wrap: balance;
+
+  > p {
+    margin-top: 0;
+    margin-bottom: 5px;
+  }
+
+  > ul {
+    margin-top: 5px;
+  }
+
+  > * li {
+    margin-top: 0.37em;
+  }
+}
+
+/* settings */
+.vertical-tab-content:has(> .sc-supporters) {
+  .setting-item-control {
+    > select {
+      max-width: 500px;
+    }
+  }
+}
+
+.sc-bottom-bar {
+  position: fixed;
+  bottom: 0;
+  right: 0;
+  left: 0;
+  width: 100%;
+  display: flex;
+  justify-content: space-between;
+  align-items: center;
+  background-color: var(--titlebar-background);
+  padding: 0 0.5rem;
+
+  .sc-brand {
+    flex-shrink: 0;
+  }
+
+  .sc-context {
+    flex-grow: 1;
+    font-size: var(--font-smallest);
+    overflow: hidden;
+    text-overflow: ellipsis;
+    white-space: nowrap;
+  }
+}
+
+.setting-component[data-setting*="/"][data-setting*="api_key"] {
+  display: none;
+}
+
+.setting-component[data-setting*="gpu"]:not([data-setting*="/"]) {
+  display: none;
+}
+
+/* SINCE COMPONENT PATTERN SETTINGS */
+.setting-component[data-setting="is_obsidian_vault"] {
+  display: none;
+}
+
+.setting-component[data-setting="smart_change.active"] {
+  display: none;
+}
+
+.group-header {
+  display: flex;
+  text-wrap: nowrap;
+  flex-wrap: wrap;
+  align-items: baseline;
+
+  > h2 {
+    width: 100%;
+    margin-bottom: 0;
+  }
+
+  > * {
+    flex-grow: 1;
+    margin-bottom: 10px;
+  }
+}
+
+/* SMART CHAT v2 */
+.sc-context-list {
+  list-style: none;
+  margin: 0;
+  padding: 0 1rem 1rem;
+  display: none;
+  flex-direction: column;
+  gap: 0.5rem;
+}
+
+.sc-context-header[aria-expanded="true"] + .sc-context-list {
+  display: flex;
+}
+
+.sc-context-header[aria-expanded="false"] + .sc-context-list {
+  display: none;
+}
+
+.sc-context-toggle-icon {
+  margin-left: 0.5rem;
+  transition: transform 0.3s ease;
+}
+
+.sc-context-header[aria-expanded="true"] .sc-context-toggle-icon {
+  transform: rotate(180deg);
+}
+
+.sc-context-container {
+  border: 1px solid var(--divider-color);
+  border-radius: 10px;
+  margin: 0.5rem 0;
+  background-color: var(--background-primary-alt);
+  overflow: auto;
+  max-width: 95%;
+  margin: 10px;
+  flex-shrink: 0;
+}
+
+.sc-context-header {
+  display: flex;
+  justify-content: space-between;
+  align-items: center;
+  padding: 0.88rem 1rem;
+  background-color: var(--background-primary);
+  color: var(--text-muted);
+  font-weight: var(--font-medium);
+  cursor: pointer;
+  user-select: none;
+
+  &:hover {
+    background-color: var(--background-primary-hover);
+  }
+
+  &:focus {
+    outline: 2px solid var(--text-muted);
+    outline-offset: 2px;
+  }
+}
+
+.sc-context-item {
+  padding: 0.5rem;
+  border-radius: var(--radius-s);
+  background-color: var(--background-secondary);
+  color: var(--text-normal);
+  display: flex;
+  justify-content: space-between;
+  align-items: baseline;
+  font-size: var(--font-smallest);
+
+  &:hover {
+    background-color: var(--background-secondary-hover);
+  }
+}
+
+.sc-context-item-path {
+  font-weight: var(--font-medium);
+  text-overflow: ellipsis;
+  overflow: hidden;
+  white-space: nowrap;
+  max-width: 70%;
+}
+
+.sc-context-item-score {
+  font-size: var(--font-small);
+  color: var(--color-gray-40);
+}
+
+/* Tool Calls Component Styles */
+.sc-tool-calls-container {
+  border: 1px solid var(--divider-color);
+  border-radius: 10px;
+  margin: 0.5rem 0;
+  background-color: var(--background-primary-alt);
+  overflow: auto;
+  max-width: 95%;
+  margin: 10px;
+  flex-shrink: 0;
+}
+
+.sc-tool-call {
+  margin-bottom: 0.5rem;
+
+  &:last-child {
+    margin-bottom: 0;
+  }
+}
+
+.sc-tool-call-header {
+  display: flex;
+  justify-content: space-between;
+  align-items: center;
+  padding: 0.88rem 1rem;
+  background-color: var(--background-primary);
+  color: var(--text-muted);
+  font-weight: var(--font-medium);
+  cursor: pointer;
+  user-select: none;
+
+  &:hover {
+    background-color: var(--background-primary-hover);
+  }
+
+  &:focus {
+    outline: 2px solid var(--text-muted);
+    outline-offset: 2px;
+  }
+
+  &[aria-expanded="true"] .sc-tool-call-toggle-icon {
+    transform: rotate(180deg);
+  }
+}
+
+.sc-tool-call-content {
+  padding: 0.5rem 1rem;
+  background-color: var(--background-secondary);
+  font-size: var(--font-smallest);
+
+  pre {
+    margin: 0;
+    white-space: pre-wrap;
+    word-break: break-word;
+  }
+
+  code {
+    font-family: var(--font-monospace);
+  }
+}
+
+/* Hide content when collapsed */
+.sc-tool-call-header[aria-expanded="false"] + .sc-tool-call-content {
+  display: none;
+}
+
+/* Show content when expanded */
+.sc-tool-call-header[aria-expanded="true"] + .sc-tool-call-content {
+  display: block;
+}
+
+/* System Message Styles */
+.sc-system-message-container {
+  margin: 1rem 0;
+  border: 1px solid var(--background-modifier-border);
+  border-radius: 6px;
+  background: var(--background-secondary);
+  flex-shrink: 0;
+}
+
+.sc-system-message-header {
+  display: flex;
+  justify-content: space-between;
+  align-items: center;
+  padding: 0.75rem 1rem;
+  cursor: pointer;
+  font-weight: 500;
+  border-bottom: 1px solid transparent;
+  transition: background-color 0.2s ease;
+
+  &:hover {
+    background: var(--background-modifier-hover);
+  }
+
+  span {
+    display: flex;
+    align-items: center;
+    gap: 0.5rem;
+  }
+
+  .sc-system-message-toggle-icon {
+    transition: transform 0.2s ease;
+  }
+
+  &[aria-expanded="true"] {
+    border-bottom-color: var(--background-modifier-border);
+  }
+}
+
+.sc-system-message-content {
+  padding: 1rem;
+  position: relative;
+  background: var(--background-primary);
+  border-bottom-left-radius: 6px;
+  border-bottom-right-radius: 6px;
+
+  .sc-system-message-text {
+    font-family: var(--font-monospace);
+    white-space: pre-wrap;
+    word-break: break-word;
+    margin-right: 2rem;
+  }
+
+  .sc-system-message-copy {
+    position: absolute;
+    top: 1rem;
+    right: 1rem;
+    padding: 0.4rem;
+    background: transparent;
+    border: none;
+    cursor: pointer;
+    opacity: 0.6;
+    transition: opacity 0.2s ease;
+
+    &:hover {
+      opacity: 1;
+    }
+
+    &.sc-copied {
+      color: var(--text-accent);
+    }
+  }
+}
+
+.sc-chat-container {
+  .smart-chat-overlay {
+    position: absolute;
+    top: 0;
+    left: 0;
+    right: 0;
+    bottom: 0;
+    background-color: var(--background-primary-alt);
+    z-index: 100;
+    overflow: auto;
+
+    .smart-chat-overlay-header {
+      display: flex;
+      justify-content: flex-end;
+    }
+  }
+}
+
+.sc-typing-indicator {
+  display: flex;
+  align-items: center;
+  padding: 12px 16px;
+  display: none;
+
+  &.visible {
+    display: flex;
+  }
+
+  .sc-typing-dots {
+    display: flex;
+    gap: 4px;
+
+    .sc-typing-dot {
+      width: 8px;
+      height: 8px;
+      background: var(--text-muted);
+      border-radius: 50%;
+      animation: typing-bounce 1.4s infinite ease-in-out;
+
+      &:nth-child(1) {
+        animation-delay: 0s;
+      }
+
+      &:nth-child(2) {
+        animation-delay: 0.2s;
+      }
+
+      &:nth-child(3) {
+        animation-delay: 0.4s;
+      }
+    }
+  }
+
+}
+/* keyframes must be at root level */
+@keyframes typing-bounce {
+  0%,
+  80%,
+  100% {
+    transform: scale(0.6);
+  }
+  40% {
+    transform: scale(1);
+  }
+}
+
+/* Smart Directories */
+.sg-top-bar {
+  display: flex;
+  justify-content: flex-end;
+}
+.sg-list {
+  > .sg-collapsed .sg-directory-header svg {
+    transform: rotate(-90deg);
+    transition: transform 0.15s ease;
+  }
+
+  > :not(.sg-collapsed) .sg-directory-header svg {
+    transform: rotate(0deg);
+    transition: transform 0.15s ease;
+  }
+
+  .sg-directory-header {
+    cursor: pointer;
+    user-select: none;
+
+    &:hover {
+      background-color: var(--nav-item-background-hover);
+    }
+  }
+}
+
+/* Add this to handle content visibility */
+.sg-directory-item.sg-collapsed .sg-directory-content {
+  display: none;
+}
+
+/* Lookup */
+#sc-lookup-view {
+  .sc-container {
+    .sc-textarea-container {
+      display: flex;
+      padding: 0 10px 1rem 0;
+      width: 100%;
+      max-height: 50%;
+
+      > textarea {
+        flex-grow: 1;
+        padding: 0.88rem;
+        border: none;
+        border-radius: 1.5rem;
+        resize: none;
+        height: auto;
+        min-height: 4.2lh;
+        max-height: 100%;
+        background-color: var(--background-primary);
+        color: var(--text-normal);
+        margin-right: -2.7rem;
+        padding-right: 3rem;
+      }
+      .sc-textarea-btn-container {
+        width: 2rem;
+        height: 2rem;
+        margin: auto;
+
+        > button.send-button {
+          border-radius: 99999px;
+          cursor: pointer;
+          outline: 2px solid transparent;
+          padding: 0;
+          outline-offset: 2px;
+          background: none !important;
+
+          &:focus-visible {
+            outline-color: var(--text-faint);
+            box-shadow: none;
+          }
+
+          > svg {
+            > circle {
+              fill: var(--text-faint);
+            }
+            > path {
+              fill: var(--background-primary);
+            }
+          }
+        }
+      }
+    }
+  }
+}
+
+/* Side Panes */
+div.workspace-leaf-content[data-type^="smart-"] {
+  > .view-content {
+    display: flex;
+    flex-direction: column;
+
+    .sc-overlay {
+      background-color: var(--background-primary-alt);
+      max-width: 100%;
+      max-height: 42%;
+
+      &:has(.setting-component) {
+        overflow-y: auto;
+        padding: var(--size-4-4);
+        flex: none;
+
+        .setting-item {
+          flex-direction: column;
+          align-items: flex-start;
+        }
+      }
+    }
+  }
+}
+
+.sc-pro {
+  .sc-settings {
+    padding: var(--size-4-4);
+
+    textarea {
+      min-height: 4rem;
+      flex-grow: 1;
+    }
+  }
+}
--- a/.obsidian/workspace.json
+++ b/.obsidian/workspace.json
@ -4,24 +4,103 @@
    "type": "split",
    "children": [
      {
+<<<<<<< HEAD
        "id": "6bfcfecbb751351a",
        "type": "tabs",
        "children": [
          {
            "id": "5bdaac52da6d5967",
+=======
+        "id": "ec0d65b5f47f4a2a",
+        "type": "tabs",
+        "children": [
+          {
+            "id": "28741259da40d4c3",
+            "type": "leaf",
+            "state": {
+              "type": "canvas",
+              "state": {
+                "file": "多体+耦合求解器/理论框架.canvas",
+                "viewState": {
+                  "x": 5,
+                  "y": 94,
+                  "zoom": 0
+                }
+              },
+              "icon": "lucide-layout-dashboard",
+              "title": "理论框架"
+            }
+          },
+          {
+            "id": "850e7478b6a3e8f9",
+>>>>>>> 43e7b08163ff98d031f47a55e6d8abfb871edb14
            "type": "leaf",
            "state": {
              "type": "markdown",
              "state": {
+<<<<<<< HEAD
                "file": "多体+耦合求解器/数据结构讨论.md",
+=======
+                "file": "多体+耦合求解器/Kane方法.md",
+>>>>>>> 43e7b08163ff98d031f47a55e6d8abfb871edb14
                "mode": "source",
                "source": false
              },
              "icon": "lucide-file",
+<<<<<<< HEAD
              "title": "数据结构讨论"
            }
          }
        ]
+=======
+              "title": "Kane方法"
+            }
+          },
+          {
+            "id": "c9060a50b8bee62c",
+            "type": "leaf",
+            "state": {
+              "type": "markdown",
+              "state": {
+                "file": "力学书籍/Kane-Dynamics-Theory-Applications/auto/Kane-Dynamics-Theory-Applications.md",
+                "mode": "source",
+                "source": false
+              },
+              "icon": "lucide-file",
+              "title": "Kane-Dynamics-Theory-Applications"
+            }
+          },
+          {
+            "id": "0f7e37da67e92e6a",
+            "type": "leaf",
+            "state": {
+              "type": "markdown",
+              "state": {
+                "file": "力学书籍/Kane-Dynamics-Theory-Applications/auto/Kane-dynamics-theory翻译.md",
+                "mode": "source",
+                "source": false
+              },
+              "icon": "lucide-file",
+              "title": "Kane-dynamics-theory翻译"
+            }
+          },
+          {
+            "id": "12fc6b4d0f447553",
+            "type": "leaf",
+            "state": {
+              "type": "markdown",
+              "state": {
+                "file": "copilot-conversations/#_2.9_CONFIGURATION_CONSTRAINTS_The_configuration_of_a_set_S@20250114_160735.md",
+                "mode": "source",
+                "source": false
+              },
+              "icon": "lucide-file",
+              "title": "#_2.9_CONFIGURATION_CONSTRAINTS_The_configuration_of_a_set_S@20250114_160735"
+            }
+          }
+        ],
+        "currentTab": 2
+>>>>>>> 43e7b08163ff98d031f47a55e6d8abfb871edb14
      }
    ],
    "direction": "vertical"
@ -72,12 +151,27 @@
              "icon": "lucide-bookmark",
              "title": "书签"
            }
+          },
+          {
+            "id": "f96b12d12e968e83",
+            "type": "leaf",
+            "state": {
+              "type": "copilot-chat-view",
+              "state": {},
+              "icon": "message-square",
+              "title": "Copilot"
+            }
          }
-        ]
+        ],
+        "currentTab": 3
      }
    ],
    "direction": "horizontal",
+<<<<<<< HEAD
    "width": 332.5
+=======
+    "width": 447.5
+>>>>>>> 43e7b08163ff98d031f47a55e6d8abfb871edb14
  },
  "right": {
    "id": "de2dec4e906755e6",
@ -93,6 +187,7 @@
            "state": {
              "type": "backlink",
              "state": {
+                "file": "力学书籍/Kane-Dynamics-Theory-Applications/auto/Kane-Dynamics-Theory-Applications.md",
                "collapseAll": false,
                "extraContext": false,
                "sortOrder": "alphabetical",
@ -102,7 +197,7 @@
                "unlinkedCollapsed": true
              },
              "icon": "links-coming-in",
-              "title": "反向链接"
+              "title": "Kane-Dynamics-Theory-Applications 的反向链接列表"
            }
          },
          {
@ -111,11 +206,12 @@
            "state": {
              "type": "outgoing-link",
              "state": {
+                "file": "力学书籍/Kane-Dynamics-Theory-Applications/auto/Kane-Dynamics-Theory-Applications.md",
                "linksCollapsed": false,
                "unlinkedCollapsed": true
              },
              "icon": "links-going-out",
-              "title": "出链"
+              "title": "Kane-Dynamics-Theory-Applications 的出链列表"
            }
          },
          {
@ -137,10 +233,10 @@
            "state": {
              "type": "outline",
              "state": {
-                "file": "多体求解器编写/多体+水动 platform+tower debug.md"
+                "file": "力学书籍/Kane-Dynamics-Theory-Applications/auto/Kane-Dynamics-Theory-Applications.md"
              },
              "icon": "lucide-list",
-              "title": "多体+水动 platform+tower debug 的大纲"
+              "title": "Kane-Dynamics-Theory-Applications 的大纲"
            }
          },
          {
@ -152,13 +248,37 @@
              "icon": "git-pull-request",
              "title": "Source Control"
            }
+          },
+          {
+            "id": "162ba1965655cb5e",
+            "type": "leaf",
+            "state": {
+              "type": "smart-connections-view",
+              "state": {},
+              "icon": "smart-connections",
+              "title": "Smart Connections"
+            }
+          },
+          {
+            "id": "69aa49226b362e40",
+            "type": "leaf",
+            "state": {
+              "type": "smart-lookup-view",
+              "state": {},
+              "icon": "search",
+              "title": "Smart Lookup"
+            }
          }
        ],
        "currentTab": 4
      }
    ],
    "direction": "horizontal",
+<<<<<<< HEAD
    "width": 507.5
+=======
+    "width": 625.5
+>>>>>>> 43e7b08163ff98d031f47a55e6d8abfb871edb14
  },
  "left-ribbon": {
    "hiddenItems": {
@ -169,9 +289,12 @@
      "templates:Insert template": false,
      "command-palette:Open command palette": false,
      "copilot:Open Copilot Chat": false,
-      "obsidian-git:Open Git source control": false
+      "obsidian-git:Open Git source control": false,
+      "smart-connections:Open: View Smart Connections": false,
+      "smart-connections:Open: Smart Chat Conversation": false
    }
  },
+<<<<<<< HEAD
  "active": "5bdaac52da6d5967",
  "lastOpenFiles": [
    "conflict-files-obsidian-git.md",
@ -207,5 +330,57 @@
    "Ai/ollama 设置模型上下文大小.md",
    "Ai/deepSeek Api Key.md",
    "多体求解器编写/多体+水动 platform+tower debug.md"
+=======
+  "active": "8080c9209794d082",
+  "lastOpenFiles": [
+    "多体+耦合求解器/Kane方法.md",
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+    "多体+耦合求解器/理论框架.canvas",
+    "力学书籍/Kane-Dynamics-Theory-Applications/auto/Kane-Dynamics-Theory-Applications_origin.pdf",
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+    "力学书籍/Kane-Dynamics-Theory-Applications/auto/Kane-dynamics-theory翻译.md",
+    "copilot-conversations/#_2.9_CONFIGURATION_CONSTRAINTS_The_configuration_of_a_set_S@20250114_160735.md",
+    "力学书籍/input/计算多体系统动力学 (洪嘉振著, 洪嘉振, 1944-) (Z-Library).pdf",
+    "多体+耦合求解器/动态数组调研.md",
+    "多体+耦合求解器/数据结构讨论.md",
+    "理论力学（II） (哈尔滨工业大学理论力学教研室 编) (Z-Library)^广义坐标.md",
+    "copilot-conversations/The_number_n_of_generalized_coordinates_of_a_set_S@20250115_092015.md",
+    "力学书籍/理论力学Ⅰ(第8版) (哈尔滨工业大学理论力学教研室) (Z-Library)/auto/理论力学Ⅰ(第8版) (哈尔滨工业大学理论力学教研室) (Z-Library).md",
+    "力学书籍/结构力学Ⅰ（基础教程） (龙驭球、包世华、袁驷) (Z-Library)/auto/结构力学Ⅰ（基础教程） (龙驭球、包世华、袁驷) (Z-Library).md",
+    "力学书籍/FASTKinetics/auto/FASTKinetics_origin.pdf",
+    "力学书籍/FASTKinematics/auto/FASTKinematics_origin.pdf",
+    "copilot-conversations/中文回复@20250113_144007.md",
+    "copilot-conversations",
+    "力学书籍/结构动力学 (R. 克拉夫，J. 彭津) (Z-Library)/auto/结构动力学 (R. 克拉夫，J. 彭津) (Z-Library).pdf",
+    "力学书籍/FASTKinetics/auto/FASTKinetics_spans.pdf",
+    "力学书籍/FASTCoordinateSystems/auto/FASTCoordinateSystems_layout.pdf",
+    "力学书籍/FASTKinetics/auto/FASTKinetics_layout.pdf",
+    "力学书籍/FASTKinetics/auto/FASTKinetics.md",
+    "力学书籍/FASTKinematics/auto/FASTKinematics.md",
+    "力学书籍/FASTCoordinateSystems/auto/FASTCoordinateSystems.md",
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+    "力学书籍/FASTMotions/auto/images/0c218afa5aa05229840608cb8327fe4452c4006430a61cd378e0c48bb48f3d00.jpg",
+    "力学书籍/FASTMotions/auto/images/2057057aaac1479c30036b6175385a20823e1b62a204561d7dd450dc0ac8fb7a.jpg",
+    "力学书籍/FASTMotions/auto/images/0613df5529705531bfe85ad3c3f4a4b9e1f0ba2ef16cc0950c42d5628ebf4a1c.jpg",
+    "力学书籍/FASTMotions/auto/images",
+    "力学书籍/FASTMotions/auto/FASTMotions.md",
+    "力学书籍/FASTLoads/auto/images/f94847c77122a3b73ec487fb8cfdd59f3c1b34e6d2350069aef3bfff567fcb13.jpg",
+    "力学书籍/FASTLoads/auto/FASTLoads.md",
+    "力学书籍/Kinematically nonlinear finite element model of a horizontal axis wind turbine. Part 2 Supplement. Inertia matrices and aerodynamic model/auto/Kinematically nonlinear finite element model of a horizontal axis wind turbine. Part 2.md",
+    "力学书籍/Kinematically nonlinear finite element model of a horizontal axis wind turbine/auto/Kinematically nonlinear finite element model of a horizontal axis wind turbine. Part 2.md",
+    "多体调研/sci论文框架.canvas",
+    "力学书籍/材料力学2（第6版） (刘鸿文) (Z-Library)/auto/材料力学2（第6版） (刘鸿文) (Z-Library).md",
+    "InterestingStuffs/剧自动化获取任务+下载素材+制作+上传/程序流程.canvas",
+    "conflict-files-obsidian-git.md",
+    "InterestingStuffs/本地知识库+大模型/ollama 设置模型上下文大小.md",
+    "InterestingStuffs/本地知识库+大模型/deepSeek Api Key.md",
+    "多体求解器debug/多体+气动 转速 debug.md",
+    "多体求解器debug/多体+水动 platform+tower debug.md"
+>>>>>>> 43e7b08163ff98d031f47a55e6d8abfb871edb14
  ]
 }
--- a/InterestingStuffs/剧自动化获取任务+下载素材+制作+上传/程序流程.canvas
+++ b/InterestingStuffs/剧自动化获取任务+下载素材+制作+上传/程序流程.canvas
@ -0,0 +1,29 @@
+{
+	"nodes":[
+		{"id":"8fe0adf04ce0d8c2","type":"text","text":"获取**当天**任务","x":-400,"y":80,"width":250,"height":60},
+		{"id":"b671a458da3e722d","type":"text","text":"任务安排","x":-100,"y":80,"width":250,"height":60},
+		{"id":"4250f79d778b5364","type":"text","text":"星图网页，任务时间开始时间降序排列","x":-400,"y":175,"width":250,"height":60},
+		{"id":"6ebf0df69dc10018","x":-400,"y":360,"width":250,"height":60,"type":"text","text":"第二页信息抓取"},
+		{"id":"c02a7bb0e0e301a7","x":-400,"y":270,"width":250,"height":60,"type":"text","text":"重合剔除"},
+		{"id":"f1a6c858d37f1304","type":"text","text":"例如：每小时发一个剧，每个剧发3个视频","x":-100,"y":360,"width":250,"height":60},
+		{"id":"75a8182ceea075c0","x":-100,"y":270,"width":250,"height":60,"type":"text","text":"做一个从素材数据库删除一个 或添加到完成数据库"},
+		{"id":"1be83a74a3da4208","x":-100,"y":161,"width":250,"height":88,"type":"text","text":"当天任务 按照开始时间 前几个+前天任务后几个 6 4"},
+		{"id":"52a68e10b5ed9ac6","type":"text","text":"自动化过程，不与本机正常使用冲突\n- docker ？ 如何实现不影响\n- 无头模式 （cookie失效，需要显示窗口再登陆）","x":-575,"y":880,"width":345,"height":160},
+		{"id":"548efc71081e36d6","x":-150,"y":880,"width":250,"height":160,"type":"text","text":"辅助功能：\n- 定期删除无用素材"},
+		{"id":"cdd1884827a50f05","x":180,"y":1100,"width":250,"height":60,"type":"text","text":"todo\n过验证码"},
+		{"id":"210b409cc6c143e6","type":"text","text":"Class\n剧名\n剧场\n素材链接\n话题\n封面jpg","x":-400,"y":520,"width":250,"height":190},
+		{"id":"ff0858dd4aa9c151","type":"text","text":"对一个剧首次：\n- 百度网盘素材添加和下载\n- 分析高光时刻，得到能生成几个视频 作为本剧视频数\n\n根据任务安排生成视频","x":440,"y":-17,"width":250,"height":255},
+		{"id":"2c43079140e74206","type":"text","text":"发布过程","x":800,"y":80,"width":250,"height":60},
+		{"id":"a64898985c682307","x":180,"y":860,"width":330,"height":200,"type":"text","text":"改进：\n- 根据完成数据库\n- 监控播放数据\n- 根据播放数据，增加火爆视频的跟发"},
+		{"id":"308ca7cba9848a9a","x":440,"y":-680,"width":520,"height":540,"type":"text","text":"百度网盘里的内容类型\n- .mp4视频+jpg\n- - .mp4视频+jpg+特别版文件夹\n- 好几个版本文件夹 匠心短剧\n\t  - 完整版\n\t  - 无字幕版\n\t  - 无BGM版\n\t  - 封面\n\t    \n\n\n没有视频的 文件名与剧名不对的\n\n删除任务 递补"}
+	],
+	"edges":[
+		{"id":"bdea2d039e30f024","fromNode":"8fe0adf04ce0d8c2","fromSide":"bottom","toNode":"4250f79d778b5364","toSide":"top"},
+		{"id":"b138bf1a3118f584","fromNode":"8fe0adf04ce0d8c2","fromSide":"right","toNode":"b671a458da3e722d","toSide":"left"},
+		{"id":"197d9780b0fdf2ae","fromNode":"ff0858dd4aa9c151","fromSide":"right","toNode":"2c43079140e74206","toSide":"left"},
+		{"id":"2584e6b85dec7b7f","fromNode":"4250f79d778b5364","fromSide":"bottom","toNode":"c02a7bb0e0e301a7","toSide":"top"},
+		{"id":"e30a6d6d69948893","fromNode":"c02a7bb0e0e301a7","fromSide":"bottom","toNode":"6ebf0df69dc10018","toSide":"top"},
+		{"id":"b89c688915356d6e","fromNode":"6ebf0df69dc10018","fromSide":"bottom","toNode":"210b409cc6c143e6","toSide":"top"},
+		{"id":"37313a8638196d3e","fromNode":"b671a458da3e722d","fromSide":"right","toNode":"ff0858dd4aa9c151","toSide":"left"}
+	]
+}
--- a/InterestingStuffs/本地知识库+大模型/deepSeek
+++ b/InterestingStuffs/本地知识库+大模型/deepSeek
--- a/InterestingStuffs/本地知识库+大模型/ollama
+++ b/InterestingStuffs/本地知识库+大模型/ollama
--- a/conflict-files-obsidian-git.md
+++ b/conflict-files-obsidian-git.md
@ -1,17 +0,0 @@
-# Conflicts
-Please resolve them and commit them using the commands `Git: Commit all changes` followed by `Git: Push`
-(This file will automatically be deleted before commit)
-[[#Additional Instructions]] available below file list
-
- Not a file: .obsidian/workspace.json
-
-# Additional Instructions
-I strongly recommend to use "Source mode" for viewing the conflicted files. For simple conflicts, in each file listed above replace every occurrence of the following text blocks with the desired text.
-
-```diff
-<<<<<<< HEAD
-    File changes in local repository
-=======
-    File changes in remote repository
->>>>>>> origin/main
-```
--- a/copilot-conversations/#_2.9_CONFIGURATION_CONSTRAINTS_The_configuration_of_a_set_S@20250114_160735.md
+++ b/copilot-conversations/#_2.9_CONFIGURATION_CONSTRAINTS_The_configuration_of_a_set_S@20250114_160735.md
@ -0,0 +1,90 @@
+---
+epoch: 1736842055308
+modelKey: phi4:latest|ollama
+tags:
+  - copilot-conversation
+---
+
+**user**: # 2.9 CONFIGURATION CONSTRAINTS  
+
+The configuration of a set S of $\pmb{\nu}$ particles $P_{1},...,P_{\nu}$ in a reference frame A is known whenever the position vector of each particle relative to a point fixed in A is known. Thus, v vector quantities, or, equivalently, $\mathbf{3}\upnu$ scalar quantities, are required for the specification of the configuration of S in $\pmb{A}$  
+
+If the motion of $\boldsymbol{s}$ is affected by the presence of bodies that come into contact with one or more of $P_{1},...,P_{\nu}$ , restrictions are imposed on the positions that the affected particles may occupy, and $\pmb{s}$ is said to be subject to configuration constraints;  
+
+an equationexpressing sucharestrictionis calleda holonomicconstraintequation. $\mathbf{a}_{x}$ $\mathbf{a}_{y}$ ， ${\mathfrak{a}}_{z}$ are mutually perpendicular unit vectors fixed in $A$ and $x_{i},y_{i},z_{i},$ called Cartesiancoordinatesof $P_{i}$ in $A$ , are defined as  
+翻译成中文
+[Timestamp: 2025/01/14 16:07:35]
+
+**ai**: # 2.9 配置约束条件  
+
+一组 $\pmb{\nu}$ 粒子 $P_1, \ldots, P_\nu$ 在参考系 A 中的配置是已知的，当每个粒子相对于固定在 A 中的一个点的位置矢量被知道时。因此，需要 v 个矢量量或等价地 $\mathbf{3}\upnu$ 个标量量来指定 S 在 $\pmb{A}$ 中的配置。
+
+如果由于与 $P_1, \ldots, P_\nu$ 中一个或多个粒子接触的物体的存在而影响了 $\boldsymbol{s}$ 的运动，就会对这些受影响的粒子可能占据的位置施加限制，并且说 $\pmb{s}$ 受到配置约束；一个表达此类限制的方程被称为齐次约束方程。$\mathbf{a}_x$、$\mathbf{a}_y$ 和 ${\mathfrak{a}}_z$ 是固定在 $A$ 中且互相垂直的单位矢量，而 $x_i, y_i, z_i$，称为 $P_i$ 在 A 中的笛卡尔坐标，则定义如下：  
+
+---
+
+这段文字描述了粒子系统在参考系中配置的约束条件，并提到了齐次约束方程以及相关的笛卡尔坐标。
+[Timestamp: 2025/01/14 16:08:25]
+
+**user**: # 2.9 CONFIGURATION CONSTRAINTS  
+
+The configuration of a set S of $\pmb{\nu}$ particles $P_{1},...,P_{\nu}$ in a reference frame A is known whenever the position vector of each particle relative to a point fixed in A is known. Thus, v vector quantities, or, equivalently, $\mathbf{3}\upnu$ scalar quantities, are required for the specification of the configuration of S in $\pmb{A}$  
+
+If the motion of $\boldsymbol{s}$ is affected by the presence of bodies that come into contact with one or more of $P_{1},...,P_{\nu}$ , restrictions are imposed on the positions that the affected particles may occupy, and $\pmb{s}$ is said to be subject to configuration constraints;  
+
+an equation expressing such a restriction is called a holonomic constraint equation.If $\mathbf{a}_{x}$ $\mathbf{a}_{y}$ ， ${\mathfrak{a}}_{z}$ are mutually perpendicular unit vectors fixed in $A$ and $x_{i},y_{i},z_{i},$ called Cartesian coordinates of $P_{i}$ in $A$ , are defined as  
+翻译成中文
+[Timestamp: 2025/01/14 16:13:59]
+
+**ai**: # 2.9 配置约束条件  
+
+一组 $\pmb{\nu}$ 粒子 $P_1, \ldots, P_\nu$ 在参考系 A 中的配置是已知的，当每个粒子相对于固定在 A 中的一个点的位置矢量被知道时。因此，需要 v 个矢量量或等价地 $\mathbf{3}\upnu$ 个标量量来指定 S 在 $\pmb{A}$ 中的配置。
+
+如果由于与 $P_1, \ldots, P_\nu$ 中一个或多个粒子接触的物体的存在而影响了 $\boldsymbol{s}$ 的运动，就会对这些受影响的粒子可能占据的位置施加限制，并且说 $\pmb{s}$ 受到配置约束；一个表达此类限制的方程被称为齐次约束方程。如果 $\mathbf{a}_x$、$\mathbf{a}_y$ 和 ${\mathfrak{a}}_z$ 是固定在 $A$ 中且互相垂直的单位矢量，而 $x_i, y_i, z_i$，称为 $P_i$ 在 A 中的笛卡尔坐标，则定义如下：  
+
+---
+
+这段文字解释了粒子系统在参考系中配置的约束条件，并提到了齐次约束方程以及相关的笛卡尔坐标。
+[Timestamp: 2025/01/14 16:14:54]
+
+**user**: where $t$ is the time. Holonomic constraint equations are classified as rheonomic or scleronomic, according to whether the function $f$ does, or does not, contain t explicitly.  
+
+Example Figure2.9.1showstwosmall blocks, $P_{1}$ and $P_{2}$ ,connected by a thin rod $R$ of length $L$ ,and constrained to remain between two parallel panes of glass that are attached to each other, forming a rigid body $B_{\cdot}$ This body is made to rotate at a constant rate $\omega$ about a line $\boldsymbol{Y}$ fixed both in $B$ and in a reference frame $A$ Treating $P_{1}$ and $P_{2}$ as a set ${\cal{S}}$ of two particles, and letting $\mathbf{p}_{1}$ and ${\bf p}_{2}$ be their position vectors relative to the point $o$ shown in Fig.2.9.1, one can express $\mathbf{p}_{1}$ and $\mathfrak{g}_{2}$ as 
+翻译成中文
+[Timestamp: 2025/01/14 16:18:18]
+
+**ai**: 齐次约束方程根据函数 \( f \) 是否显式包含时间 \( t \)，被分类为流体动力学的（rheonomic）或刚性的（scleronomic）。  
+
+示例图 2.9.1 展示了两个小块，\( P_1 \) 和 \( P_2 \)，它们通过一根长度为 \( L \) 的细棒 \( R \) 连接，并被限制在两块平行玻璃板之间移动。这些玻璃板相互连接，形成一个刚体 \( B_{\cdot} \)。该刚体以恒定速率 \( \omega \) 旋转于一条线 \( \boldsymbol{Y} \)，此线固定在 \( B \) 和参考系 \( A \) 中。将 \( P_1 \) 和 \( P_2 \) 视为两个粒子的集合 \( {\cal{S}} \)，并设其相对于图 2.9.1 所示点 \( o \) 的位置矢量分别为 \( \mathbf{p}_1 \) 和 \( \mathfrak{g}_2 \)，则可以表达出 \( \mathbf{p}_1 \) 和 \( \mathfrak{g}_2 \)。  
+
+---
+
+这段文字描述了一个示例，其中两个粒子通过一根细棒连接，并受到约束条件的限制，在旋转刚体中移动。齐次约束方程根据是否包含时间因素进行分类。
+[Timestamp: 2025/01/14 16:19:06]
+
+**user**: When a set $s$ of $v$ particles $P_{1},\dots,P_{v}$ is subject to constraints (see Sec.2.9) represented by $M$ holonomic constraint equations, only  
+
+$$
+\boldsymbol{n}\triangleq\boldsymbol{3}\nu-\boldsymbol{M}
+$$  
+
+of the $3v$ Cartesian coordinates $x_{i},y_{i},z_{i}\,(i=1,\ldots,\nu)$ of S in a reference frame $\pmb{A}$ are independent of each other. Under these circumstances one can express each of $x_{i},y_{i},z_{i}\,(i=1,\ldots,\nu)$ as a single-valued function of the time $t$ and $\pmb{n}$ functions of $t$ say, $q_{1}(t),\ldots,q_{n}(t),$ in such a way that the constraint equations are satisfied identically for all values of $t$ and $q_{1},\ldots,q_{n}$ in a given domain. The quantities $q_{1},\ldots,q_{n}$ are called generalized coordinates for $s$ in $\pmb{A}$  
+
+Example For the set $s$ in the example in Sec. 2.9, $\nu=2$ and $M=3.$ Hence $n=3$ . Three generalized coordinates for $s$ in $\pmb{A}$ may be introduced by expressing $x_{i},y_{i},z_{i}\,(i=1,2)$ as  
+翻译成中文
+[Timestamp: 2025/01/14 16:37:34]
+
+**ai**: 当一个由 \( v \) 个粒子组成的集合 \( s = \{P_1, \dots, P_v\} \) 受到约束（见第 2.9 节）并由 \( M \) 个齐次约束方程表示时，参考系 \( \pmb{A} \) 中的 \( S \) 的 \( 3v \) 个笛卡尔坐标 \( x_i, y_i, z_i\ (i=1,\ldots,v) \) 中只有  
+
+$$
+\boldsymbol{n} \triangleq 3\nu - M  
+$$  
+
+个是相互独立的。在这种情况下，可以将每一个 \( x_i, y_i, z_i\ (i=1,\ldots,v) \) 表示为时间 \( t \) 和 \( n \) 个函数 \( q_1(t), \ldots, q_n(t) \) 的单值函数，使得约束方程对于给定域中的所有 \( t \) 和 \( q_1, \ldots, q_n \) 值都被恒等满足。这些量 \( q_1, \ldots, q_n \) 被称为 \( s \) 在 \( \pmb{A} \) 中的广义坐标。
+
+示例：对于第 2.9 节中的集合 \( s \)，有 \( v = 2 \) 和 \( M = 3 \)。因此，\( n = 3 \)。可以引入三个广义坐标来表示 \( S \) 在 \( \pmb{A} \) 中的 \( x_i, y_i, z_i\ (i=1,2) \)，使得  
+
+---
+
+这段文字解释了在约束条件下，如何通过引入广义坐标来简化粒子系统中独立变量的数量。示例说明了具体情况下的应用。
+[Timestamp: 2025/01/14 16:38:51]
--- a/copilot-conversations/The_number_n_of_generalized_coordinates_of_a_set_S@20250115_092015.md
+++ b/copilot-conversations/The_number_n_of_generalized_coordinates_of_a_set_S@20250115_092015.md
@ -0,0 +1,81 @@
+---
+epoch: 1736904015261
+modelKey: phi4:latest|ollama
+tags:
+  - copilot-conversation
+---
+
+**user**: The number n of generalized coordinates of a set S of $\pmb{\nu}$ particles in a reference frame $\pmb{A}$ (see Sec.2.10) is the smallest number of scalar quantities such that to every assignment of values to these quantities and the time $t$ (within a domain of interest) there corresponds a definite admissible configuration of $s$ in $\pmb{A}$ . Frequently, one can find $\pmb{n}$ by inspection rather than by determining the number $M$ of holonomic constraint equations (see Sec. 2.9) and then subtracting $M$ from $\,\mathfrak{z}_{v}$ For example, suppose that S consists of $\upnu$ particles $P_{1},...,P_{\nu}$ forming a rigid body $\pmb{B}$ that is free to move in A. Then there corresponds a definite admissible configuration of Sin $\pmb{A}$ to every assignment of values to three Cartesian coordinates o one particle of $\pmb{B}$ and three angles that characterize the orientation of $\pmb{B}$ in $\pmb{A}$ Hence, $\pmb{n}=6.$ The same conclusion is obtained formally by letting $\mathfrak{p}_{1},\ldots,\mathfrak{p}_{\nu}$ be the position vectors from a point fixed in $\pmb{A}$ to $P_{1},...,P_{\nu}$ , respectively, and noting that rigidity can be ensured by letting $P_{1},P_{2}$ , and $\boldsymbol{P_{3}}$ be noncollinear particles and requiring (1) that the distances between $\boldsymbol{P}_{1}$ and $P_{2},P_{2}$ and $\boldsymbol{P_{3}}$ , and $\boldsymbol{P_{3}}$ and $\boldsymbol{P}_{1}$ remain constant, so that  
+翻译成中文
+[Timestamp: 2025/01/15 09:20:15]
+
+**ai**: 在参考系 \( \pmb{A} \) 中，一组 \( \pmb{\nu} \) 粒子的广义坐标数 \( n \) 是最小数量的标量量，使得对于这些量和时间 \( t \) 的任何分配（在感兴趣的域内），都对应于参考系 \( \pmb{A} \) 中粒子集合 \( S \) 的一个明确可接受的配置。通常，可以通过观察而非确定 \( M \) 个齐次约束方程的数量（见第2.9节）然后从 \( \,\mathfrak{z}_{v} \) 中减去 \( M \)，来找到 \( n \)。例如，假设集合 \( S \) 包含 \( \upnu \) 个粒子 \( P_{1},...,P_{\nu} \)，它们形成一个在参考系 \( A \) 中自由移动的刚体 \( \pmb{B} \)。那么，对于给定的三个坐标和三个角度（用以描述 \( \pmb{B} \) 在 \( \pmb{A} \) 中的方向），就对应于参考系 \( A \) 中粒子集合 \( S \) 的一个明确可接受的配置。因此，\( n=6 \)。通过正式方法得出相同结论：设 \( \mathfrak{p}_{1},\ldots,\mathfrak{p}_{\nu} \) 是从参考系 \( A \) 中固定点到 \( P_{1},...,P_{\nu} \) 的位置矢量。注意，通过让 \( P_{1},P_{2} \)，和 \( \boldsymbol{P_{3}} \) 为非共线粒子，并要求（1）\( P_{1} \) 和 \( P_{2} \), \( P_{2} \) 和 \( \boldsymbol{P_{3}} \), 以及 \( \boldsymbol{P_{3}} \) 和 \( \boldsymbol{P}_{1} \) 之间的距离保持不变，可以确保刚体性。
+[Timestamp: 2025/01/15 09:21:27]
+
+**user**: # 2.12 GENERALIZED SPEEDS  
+
+As will be seen presently, expressions for angular velocities of rigid bodies and velocities of points of a system S whose configuration in a reference frame $\pmb{A}$ is characterized by $\pmb{n}$ generalized coordinates $q_{1},\ldots,q_{n}$ (see Sec. 2.10) can be brought into particularly advantageous forms through the introduction of $\pmb{n}$ quantities $u_{1},\ldots,u_{n}$ ,called generalized speeds for $s$ in $\pmb{A}$ , these being quantities defined by equations of the form
+翻译成中文
+[Timestamp: 2025/01/15 09:35:56]
+
+**ai**: # 2.12 广义速度  
+
+如将要看到的，通过引入 \( n \) 个量 \( u_{1},\ldots,u_{n} \)，称为参考系 \( \pmb{A} \) 中系统 \( S \) 的广义速度，可以使刚体的角速度和系统 \( S \) 点在参考系 \( \pmb{A} \) 中配置由 \( n \) 个广义坐标 \( q_{1},\ldots,q_{n} \)（见第2.10节）所特征化时的速度表达式形式更为有利。这些量是通过以下形式的方程定义的：
+
+---
+
+在此段落中，广义速度 \( u_1, \ldots, u_n \) 被引入以优化刚体角速度和系统点速度的表达方式，当其配置由广义坐标 \( q_1, \ldots, q_n \) 描述时。
+[Timestamp: 2025/01/15 09:36:41]
+
+**user**: where $Y_{r s}$ and $Z_{\nu}$ are functions of $q_{1},\ldots,q_{n}$ , and the time t. These functions must be chosen such that Eqs. (1) can be solved uniquely for ${\dot{q}}_{1},\dots,{\dot{q}}_{n}$ . Equations (1) are called kinematical differential equations for $\boldsymbol{s}$ in $\pmb{A}$  
+
+Example Letting $\boldsymbol{s}$ be the set of two particles considered in the example in Sec. 2.9, and using as generalized coordinates the quantities $q_{1},q_{2},q_{3}$ indicated in Fig. 2.10.1, one may define three generalized speeds as  
+翻译成中文
+[Timestamp: 2025/01/15 09:38:31]
+
+**ai**: 其中 \( Y_{rs} \) 和 \( Z_{\nu} \) 是 \( q_1,\ldots,q_n \) 和时间 \( t \) 的函数。这些函数必须选择得当，以便方程（1）能够唯一地解出 \( {\dot{q}}_{1},\dots,{\dot{q}}_{n} \)。方程（1）被称为参考系 \( \pmb{A} \) 中系统 \( \boldsymbol{s} \) 的运动学微分方程。
+
+例子：设 \( \boldsymbol{s} \) 是第2.9节中讨论的两个粒子集合，使用图2.10.1中指示的广义坐标量 \( q_1, q_2, q_3 \)，可以定义三个广义速度为：
+
+---
+
+在这段描述中，\( Y_{rs} \) 和 \( Z_{\nu} \) 是依赖于广义坐标和时间的函数，它们需要被选择以确保能够唯一解出广义坐标的导数。方程（1）是系统在参考系中的运动学微分方程。接下来提供了一个例子，说明如何为特定粒子集合定义广义速度。
+[Timestamp: 2025/01/15 09:39:10]
+
+**user**: Here, the third component is a bit more complicated than in Eqs. (13) and (14), but the introduction of generalized speeds has led to a noticeable simplification, nevertheless. The guiding idea in writing Eqs. (7) and (2) was thus to enable one to replace Eqs. (11) and (12), respectively, with expressins having, as nearly as possible, the same simple form as Eq. (10). As for Eq<. (9), their use does not lead to any simplifications since Eq. (10) cannot be simplified further, but they were included to show that the concept of generalized speeds remains applicable even under these circumstances.  
+
+The simplification of an angular velocity expression through the use of generalized speeds can be illustrated by returning to the example in Sec. 2.4. The angular velocity expression recorded in Eq. (2.4.10), while simple in form, is unsuitable for certain purposes because $\mathbf{k}_{2},\,\mathbf{k}_{7}$ ,and ${\bf k}_{3}$ are not mutually perpendicular. To overcome this difficulty, one can let $\bf{k_{4}}$ be a unit vector directed as shown in Fig. 2.4.1 and note that ${\bf k}_{3}$ then is given by  
+翻译成中文
+[Timestamp: 2025/01/15 09:43:17]
+
+**ai**: 在这里，第三个分量比方程（13）和（14）更复杂，但引入广义速度仍然导致了显著的简化。写出方程（7）和（2）的指导思想是为了能够用与方程（10）形式尽可能相似的表达式替代方程（11）和（12）。至于方程（9），它们的使用并不会带来简化，因为方程（10）无法进一步简化，但它们被包括在内是为了展示即使在这些情况下，广义速度的概念仍然适用。
+
+通过使用广义速度来简化角速度表达式可以通过回到第2.4节中的例子进行说明。方程（2.4.10）记录的角速度表达式形式上是简单的，但对于某些目的而言不适用，因为 \( \mathbf{k}_{2},\,\mathbf{k}_{7} \) 和 \( {\bf k}_{3} \) 不互相垂直。要克服这个困难，可以让 \( \bf{k_{4}} \) 是一个指向图2.4.1中所示方向的单位矢量，并注意到此时 \( {\bf k}_{3} \) 则给出为：
+
+---
+
+在这段文字中，尽管引入广义速度后第三个分量变得更复杂，但整体上仍然实现了简化。方程（7）和（2）的目标是用与方程（10）形式相似的表达式替代其他方程。对于方程（9），虽然它们没有带来进一步的简化，但展示了广义速度概念在不同情况下的适用性。接着通过一个例子说明如何使用广义速度简化角速度表达式。
+[Timestamp: 2025/01/15 09:44:35]
+
+**user**: # 2.13 MOTION CONSTRAINTS  
+
+It can occur that, for physical reasons, the generalized speeds $u_{1},\ldots,u_{n}$ for a system S in a reference frame $A$ (see Sec. 2.4) are not independent of each other. In that event, $s$ is said to be subject to motion constraints, and an equation that relates $u_{1},\ldots,u_{n}$ to each other is called a nonholonomic constraint equation.  
+
+When a system S is not subject to motion constraints, then $s$ is said to be a holonomic system possessing n degrees of freedom in $\pmb{A}$ .If $s$ is subject to motion constraints, $s$ is called a nonholonomic system.  
+
+When all nonholonomic constraint equations can be expressed as the $m$ relationships  
+翻译成中文
+[Timestamp: 2025/01/15 10:07:25]
+
+**ai**: # 2.13 运动约束条件
+
+有时，由于物理原因，系统 \( S \) 在参考系 \( A \) 中的广义速度 \( u_1, \ldots, u_n \)（见第2.4节）之间不是独立的。在这种情况下，\( s \) 被称为受到运动约束条件，并且将 \( u_1, \ldots, u_n \) 互相联系起来的方程被称为非全微分（nonholonomic）约束方程。
+
+当系统 \( S \) 不受运动约束时，\( s \) 被称为具有 \( n \) 自由度在参考系 \( \pmb{A} \) 中的全微分（holonomic）系统。如果 \( s \) 受到运动约束，则被称为非全微分（nonholonomic）系统。
+
+当所有非全微分约束方程都可以表示为以下 \( m \) 个关系时：
+
+---
+
+在这段文字中，讨论了系统的广义速度可能因物理原因而不是独立的情况。如果存在这样的依赖性，则称系统受到运动约束，并且相关方程被称为非全微分约束方程。若没有这种约束，系统则是全微分系统。对于受到运动约束的系统，所有约束方程可以用一组关系来表示。
+[Timestamp: 2025/01/15 10:08:27]
--- a/copilot-conversations/中文回复@20250113_144007.md
+++ b/copilot-conversations/中文回复@20250113_144007.md
@ -0,0 +1,88 @@
+---
+epoch: 1736750407428
+modelKey: phi4:latest|ollama
+tags:
+  - copilot-conversation
+---
+
+**ai**: This passage discusses the challenges and solutions in teaching graduate-level dynamics, particularly focusing on the need for a robust notation system to handle complex multi-body systems. 
+
+Here's a breakdown of the key points:
+
+* **Problem with Traditional Dynamics Education:** Graduate students often lack practical experience applying dynamics principles to real-world problems, especially those involving multiple interconnected bodies. This gap stems from a focus on theoretical concepts rather than hands-on problem-solving.
+* **Need for Effective Notation:**  Analyzing complex systems requires clear and unambiguous notation to differentiate between various angular velocities, accelerations, and positions of different bodies within a system. 
+* **Proposed Solution: Consistent Notational System:** The author advocates for a specific notational convention using superscripts to clearly identify the reference frame and the body in question. This system helps avoid confusion when dealing with multiple interconnected rigid bodies.
+
+**Example:**
+
+The passage illustrates this notation with a simple example of two rigid bodies, A and B, connected within a larger system. It demonstrates how superscripts are used to distinguish the angular velocity of each body relative to different reference frames (e.g.,  $\mathbf{\bar{N}_{C D}}\mathbf{B}$ for the angular velocity of B in reference frame N).
+
+* **Emphasis on Practice:** The author stresses the importance of extensive problem-solving for mastering dynamics. The book includes numerous problems with varying levels of difficulty to reinforce theoretical concepts and develop practical skills.
+
+
+In essence, this passage highlights the need for a more practical and notationally robust approach to teaching graduate-level dynamics, bridging the gap between theory and real-world applications. 
+
+[Timestamp: 2025/01/13 14:40:07]
+
+**user**: 中文回复
+[Timestamp: 2025/01/13 14:40:14]
+
+**ai**: 这段文字讨论了在研究生层次上教授动力学所面临的挑战和解决方案，特别强调需要一个强大的符号系统来处理复杂的多个物体系统。
+
+以下是关键点的总结：
+
+* **传统动力学教育的问题:** 研究生通常缺乏将动力学原理应用于实际问题的实践经验，尤其是在涉及多个相互连接的物体的方面。这种差距源于对理论概念的关注而不是动手解决问题。
+* **有效符号的需求:** 分析复杂系统需要清晰、明确的符号来区分不同物体在系统中的各种角速度、加速度和位置。
+* **提出的解决方案：一致的符号系统:** 作者主张使用特定符号约定，利用上标清楚地标识参考系和相关物体。这种系统有助于避免处理多个相互连接刚体时出现的混乱。
+
+**示例:**
+
+这段文字用两个相互连接的刚体 A 和 B 的简单例子来说明这种符号，展示了如何使用上标来区分每个物体相对于不同参考系的角速度（例如，$\mathbf{\bar{N}_{C D}}\mathbf{B}$ 表示 B 在参考系 N 中的角速度）。
+
+* **强调实践:** 作者强调了通过大量解决问题来掌握动力学的必要性。这本书包含了许多难度不同的问题，以巩固理论概念并发展实际技能。
+
+
+总而言之，这段文字强调需要一种更实用且符号上更加稳健的方法来教授研究生层次的动力学，弥合理论与实际应用之间的差距。
+
+
+
+
+[Timestamp: 2025/01/13 14:40:36]
+
+**user**: Each of the seven chapters of this book is divided into sections. A section is identified by two numbers separated by a decimal point, the first number referring to the chapter in which the section appears, and the second identifying the section within the chapter. Thus, the identifier 2.14 refers to the fourteenth section of the second chapter. A section identifier appears at the top of each page.  
+
+Equations are numbered serially within sections. For example, the equations in Secs. 2.14 and 2.15 are numbered (1)-(31) and (1)-(50), respectively. References to an equation may be made both within the section in which the equation appears and in other sections. In the first case, the equation number is cited as a single number; in the second case, the section number is included as part of a threenumber designation. Thus, within Sec. 2.14, Eq. (2) of Sec. 2.14 is referred to as Eq. (2); in Sec. 2.15, the same equation is referred to as Eq. (2.14.2). To locate an equation cited in this manner, one may make use of the section identifiers appearing at the tops of pages.  
+
+Figures appearing in the chapters are numbered so as to identify the sections in which the figures appear. For example,the two figures in Sec. 4.8 are designated Fig. 4.8.1 and Fig. 4.8.2. To avoid confusing these figures with those in the problem sets and in Appendix I, the figure number is preceded by the letter $\mathbf{P}$ in the case of problem set figures, and by the letter A in the case of Appendix I figures. The double number following the letter P refers to the problem statement in which the figure is introduced. For example, Fig. P12.3 is introduced in Problem 12.3. Similarly, Table 3.4.1 is the designation for a table in Sec. 3.4, and Table P14.6.2 is associated with Problem 14.6.  
+翻译成中文
+[Timestamp: 2025/01/13 14:41:39]
+
+**ai**: 这本书的七个章节每个都分为若干小节。一个小节由两个数字用小数点隔开标识，第一个数字表示该小节所在的章节，第二个数字表示该章节中的小节序号。例如，标识 2.14 指的是第二章第十四小节。每个页面的顶部都会显示一个小节标识符。
+
+在每个小节内，公式按顺序编号。例如，2.14 和 2.15 小节中的公式分别编号为 (1)-(31) 和 (1)-(50)。引用一个公式可以在它出现的章节和小节中，以及其他章节中进行。前一种情况下，公式编号仅用单个数字表示；后一种情况下，会将小节编号作为三个数字的一部分包含在内。例如，在 2.14 小节中，2.14 小节的公式 (2) 被称为公式 (2)；而在 2.15 小节中，相同的公式被称为公式 (2.14.2)。要查找以这种方式引用的公式，可以使用页面顶部的章节标识符。
+
+出现在章节中的图像是按编号来标识它们所在的章节。例如，4.8 小节中的两个图像分别标记为 Fig. 4.8.1 和 Fig. 4.8.2。为了避免将这些图像与习题集和附录 I 中的图像混淆，如果是在习题集中出现的图像是用字母 $\mathbf{P}$ 前缀，如果是来自附录 I 的图像是用字母 A 前缀。双重数字跟随字母 P 指的是引入了该图像的问题陈述。例如，Fig. P12.3 是在问题 12.3 中引入的。类似地，Table 3.4.1 是 3.4 小节中的表格，而 Table P14.6.2 与问题 14.6 相关联。
+
+
+
+
+[Timestamp: 2025/01/13 14:42:10]
+
+**user**: # DIFFERENTIATION OF VECTORS  
+
+The discipline of dynamics deals with changes of various kinds, such as changes in the position of a particle in a reference frame, changes in the configuration of a mechanical system, and so forth. To characterize the manner in which some of these changes take place, one employs the differential calculus of vectors, a subject that can be regarded as an extension of material usually taught under the heading of the differential calculus of scalar functions. The extension consists primarily of provisions made to accommodate the fact that reference frames play a central role in connection with many of the vectors of interest in dynamics. For example, let $\pmb{A}$ and $\pmb{B}$ be reference frames moving relative to each other, but having one point $o$ in common at all times, and let $\pmb{P}$ be a point fixed in $\pmb{A}$ , and thus moving in B. Then the velocity of $\pmb{P}$ in $\pmb{A}$ is equal to zero, whereas the velocity of $\pmb{P}$ in $\pmb{B}$ differs from zero. Now, each of these velocities is a time-derivative of the same vector, ${\mathfrak{r}}^{o r}.$ the position vector from $^o$ to $\pmb{P}_{\cdot}$ Hence, it is meaningless to speak simply of the time-derivative of ${\mathsf{r}}^{o P}$ . Clearly, therefore, the calculus used to differentiate vectors must permit one to distinguish between differentiation with respect to a scalar variable in a reference frame $\pmb{A}$ and differentiation with respect to the same variable in a reference frame $\pmb{B}.$  
+
+When working with elementary principles of dynamics, such as Newton's second law or the angular momentum principle, one needs only the ordinary differential calculus of vectors, that is, a theory involving differentiations of vectors with respect to a single scalar variable, generally the time. Consideration of advanced principles of dynamics, such as those presented in later chapters of this book, necessitates, in addition, partial differentiation of vectors with respect to several scalar variables, such as generalized coordinates and generalized speeds. Accordingly, the present chapter is devoted to the exposition of definitions, and consequences of these definitions, needed in the chapters that follow.  
+翻译成中文
+[Timestamp: 2025/01/13 14:47:51]
+
+**ai**: # 向量的微分
+
+动力学这一学科涉及各种变化，例如粒子在参考系中位置的变化、机械系统配置的变化等。为了描述这些变化的方式，我们使用向量微积分，这可以视作对通常教授的标量函数微积分材料的扩展。这种扩展主要是为了适应参考系在动力学中许多感兴趣的向量问题中起着核心作用的事实。例如，设 $\pmb{A}$ 和 $\pmb{B}$ 是彼此相对移动但始终有一个共同点 $o$ 的参考系，并设 $\pmb{P}$ 是固定在 $\pmb{A}$ 中的一点，因而在 $\pmb{B}$ 中是运动的。那么，在 $\pmb{A}$ 中 $\pmb{P}$ 的速度为零，而在 $\pmb{B}$ 中 $\pmb{P}$ 的速度不为零。现在，这两个速度都是相同向量 ${\mathfrak{r}}^{o r}$ 关于时间的导数，即从 $^o$ 到 $\pmb{P}$ 的位置矢量。因此，不能简单地谈论 ${\mathsf{r}}^{o P}$ 关于时间的导数。显然，用来微分向量的计算必须允许我们区分在参考系 $\pmb{A}$ 中关于标量变量的微分和在相同变量中但是参考系 $\pmb{B}$ 的微分。
+
+在应用动力学的基本原理时，如牛顿第二定律或角动量原理，只需要普通向量微积分，即关于单一标量变量（通常是时间）的向量微分理论。考虑到更高级的动力学原理，如本书后续章节中所呈现的，还需对向量进行多个标量变量（例如广义坐标和广义速度）的偏导数。因此，本章专门讨论了在接下来的章节中需要的定义及其推论。
+
+---
+
+如果您有任何关于这一主题的具体问题或需要进一步的解释，请随时告诉我！
+[Timestamp: 2025/01/13 14:49:03]
--- a/力学书籍/FASTCoordinateSystems/auto/FASTCoordinateSystems.md
+++ b/力学书籍/FASTCoordinateSystems/auto/FASTCoordinateSystems.md
@ -0,0 +1,119 @@
+The following are transformation equations defining the angular orientation of each coordinate system inherent in FAST.  
+
+Before providing these, it is useful to discuss the transformation equation relating coordinate system $\pmb{x}$ to coordinate system $X$ where $\pmb{x}$ (with orthogonal axes $x_{I},\ x_{2}$ , and $x_{3}$ ) is the coordinate system resulting from three rotations $(\theta_{\!\scriptscriptstyle I},\theta_{\!\scriptscriptstyle2}$ , and $\theta_{3}$ ) about the orthogonal axes $(\,X_{I},\,X_{2}$ , and $X_{3}$ ) of coordinate system $X$ .  With all rotation angles assumed to be small, the order of rotations does not matter and Euler angles do not need to be used. Instead, what we want, is a transformation equation that is consistent with classical Bernoulli-Euler beam theory (which assumes small rotations). The correct transformation equation is:  
+
+$$
+\begin{array}{r}{\left[\!\!\begin{array}{c}{x_{I}}\\ {x_{2}}\\ {x_{3}}\end{array}\!\!\right]\approx\!\!\underbrace{\left[\!\!\begin{array}{c c c}{I}&{\theta_{3}}&{-\theta_{2}}\\ {-\theta_{3}}&{I}&{\theta_{I}}\\ {\theta_{2}}&{-\theta_{I}}&{I}\end{array}\!\!\right]}_{[\!\!\begin{array}{c}{A}\\ {B}\end{array}\!\!\right]}\!\!\left[X_{I}\right],}\end{array}
+$$  
+
+where $[A]$ is referred to as the Bernoulli-Euler transformation matrix in this work.  The approximation symbol $(\approx)$ is used in place of an equals symbol $(=)$ in the above expression since $[A]$ is not orthonormal, which implies that the resulting $\pmb{x}$ from this expression is not made up of a set of mutually orthogonal axes (all transformation matrices between sets of mutually orthogonal axes must be orthonormal).  So it is evident that in place of $[A]$ , what we want is the closest orthonormal matrix to $[A]$ , which is referred to as $\left[T r a n s M a t\right]$ in this work.  From linear algebra, we know that the closest orthonormal matrix to $[A]$ in the Frobenius Norm sense is:  
+
+$$
+\left[T r a n s M a t\right]{=}\left[U\right]\left[V\right]^{T},
+$$  
+
+where the columns of $\left[U\right]$ contain the eigenvectors of $\left[A\right]\!\!\left[A\right]^{T}$ and the columns of $\big[V\big]$ contain the eigenvectors of $\left[A\right]^{T}\left[A\right]$ .  This result follows directly from the Singular Value Decomposition (SVD) of $[A]$ :  
+
+$$
+[A]\!=\!\!\left[U\right]\!\!\left[\Sigma\right]\!\!\left[V\right]^{\scriptscriptstyle T},
+$$  
+
+where $\left[\varSigma\right]$ is a diagonal matrix containing the singular values of $[A]$ , which are $\sqrt{e i g e n\nu a l u e s\;o f\left[A\right]\left[A\right]^{T}}\;=\sqrt{e i g e n\nu a l u e s\;o f\left[A\right]^{T}\left[A\right]}\;.$  
+
+![](images/b2c0bc757ff696893b956868f5c253bf5635452cfb9931e91b125836026cce64.jpg)  
+
+This was derived symbolically by J. Jonkman by computing $\left[U\right]\!\!\left[V\right]^{T}$ by hand with verification in Mathematica.  
+
+Tower Base / Platform Coordinate System  
+
+![](images/88207e1709e48057eba3e20a48c5537a392d9b45b23b859db0e1ec4e12258df7.jpg)  
+
+Tower Element-Fixed Coordinate System  
+
+![](images/3242641825dce121996aa8b64d069f719111b388bf56b4d23f53af451eb49b9d.jpg)  
+
+Tower-Top / Base Plate Coordinate System  
+
+![](images/e15be66677d4a32e794cee5226c75216651a32afe750bafa3f1e124e9814b9c2.jpg)  
+
+Nacelle / Yaw Coordinate System${\pmb d}_{t}$ cos (qYaw) 0 −sin (qYaw) b${\pmb d}_{2}$ 0 1 0 b${\pmb d}_{3}$ sin (qYaw) 0 cos (qYaw) b  
+
+Rotor-Furl Coordinate System  
+
+<html><body><table><tr><td>-cos +COs rf RFrlSkew)cos rf2 +sin rf3 COS RFrlSkew)sin +sin RFrlTilt)</td><td>RFrlSkew)cos RFrlTilt RFrlSkew)cos COS 9RFrl COS RFrlSkew)cos RFrlTilt -sin RFrlSkew)cos RFrlTilt RFrlTilt 1-cos sin 9RFr! OS RFrlSkew)cos RFrlTilt sin 9RFr! RFrlSkew)cos RFrlTilt cos(qRFrl )- 1] sin RFrlSkew)cos sin 9RFrl RFrlSkew)cos</td><td>RFrlTilt sin RFrlTilt L 1-cos 9RFrl COS RFrlTilt sin -sin sin RFrlTilt )cos qRFrl +sin RFrlTilt +cOS RFrlTilt sin RFrlTilt COS )-1 RFrlTilt sin qRFrl</td><td>RFrlSkew)sin RFrlSkew)cos RFrlTilt )-1] qRFrl RFrlTilt )sin( 9RFrl RFrlSkew)cos RFrlTilt RFrlTilt -1 sin qRFrl d RFrlSkew)cos RFrlTilt sin 9RFrl 1-sin RFrlSkew)cos RFrlTilt COS 9RFrl +sin RFrlSkew)cos RFrlTilt</td></tr></table></body></html>  
+
+Shaft Coordinate System  
+
+$c_{I}$ cos (ShftSkew) cos (ShftTilt) sin (ShftTilt) −sin (ShftSkew) cos (ShftTilt) rf $c_{2}$ cos (ShftSkew) sin (ShftTilt) cos (ShftTilt) sin (ShftSkew) sin (ShftTilt) $c_{3}$ sin (ShftSkew) 0 cos (ShftSkew) rf3  
+
+Azimuth Coordinate System $e_{I}$ 0 0 $c_{I}$ $e_{2}$ 0 cos (qDrTr+qGeAz) sin (qDrTr+qGeAz) $c_{2}$ e 0 −sin (qDrTr+qGeAz) cos (qDrTr+qGeAz)  
+
+Teeter Coordinate System cos (qTeet 0 −sin (qTeet 0 1 0 sin (qTeet) 0 cos (qTeet)  
+
+Hub / Delta-3 Coordinate System  
+
+0   
+$g_{I}$ 0 cos (Delta3) sin( Delta3)   
+$g_{2}$   
+$\pmb{g}_{3}$ 0 −sin( Delta3) cos (Delta3)  
+
+Hub (Prime) Coordinate System  
+
+g'1B1 1 0 0 g1   
+g'2B1 = 0 1 0 g2   
+g B1 3 0 0 1 g3  
+
+The equation for ${\pmb g}^{\,\circ\!B2}$ of blade 2 is similar.  
+
+Coned Coordinate System   
+
+
+<html><body><table><tr><td>B1 B1 2 B1 3</td><td>PreCone (1)] PreCone (1)] sin</td><td>-sin 1 COS</td><td>PreCone ()]] PreCone (1)]</td><td>g B1 1 B1 g 2 g B1 3</td></tr></table></body></html>  
+
+The equation for $i^{B2}$ is similar.  
+
+Blade / Pitched Coordinate System  
+
+![](images/a2a9c533925e493d652a0d69730b87a5df026cde2fee0e7d5f9aaa614a8fa533.jpg)  
+
+The equation for $j^{B2}$ is similar.  
+
+Blade Coordinate System Aligned with Local Structural Axes (not element fixed)  
+
+Lj1B1(r ) cos θSB1(r ) −sinθSB1(r ) iB1   
+Lj2B1 (r ) θSB1(r ) cosθSB1(r ) B sin   
+$\left\lfloor L j_{3}^{B I}\left(r\right)\right\rfloor$ B 0 0  
+
+The equation for $L j^{B2}(r)$ is similar.  
+
+Blade Element-Fixed Coordinate System Aligned with Local Structural Axes  
+
+$$
+\begin{array}{r}{\left[n_{I}^{B I}\left(r\right)\right]}\\ {\left.\left|n_{2}^{B I}\left(r\right)\right\rangle=\left[T r a n s M a t\left(\theta_{I}=\theta_{x}^{B I}\left(r\right),\theta_{2}=\theta_{y}^{B I}\left(r\right),\theta_{3}=0\right)\right]\left[L j_{2}^{B I}\left(r\right)\right\rangle\right.}\\ {\left.\left|n_{3}^{B I}\left(r\right)\right\rangle\right]}\end{array}
+$$  
+
+where,  
+
+![](images/cf71b1f3353b9439b47212c215710ee64d1df808df03c0029378347d198cafb7.jpg)  
+
+The equation for ${\pmb n}^{B2}(r)$ is similar.  
+
+Blade Element-Fixed Coordinate System Used for Calculating and Returning Aerodynamic Loads This coordinate system is coincident with $i^{B I}$ when the blade is undeflected.  
+
+![](images/402bb29e7129df0062d48b3c5d723765784131fb70c72e23f07e9c78b7e694af.jpg)  
+
+The equation for $m^{B2}(r)$ is similar.  
+
+Blade Element-Fixed Coordinate System Aligned with Local Aerodynamic Axes (i.e., chordline) / Trailing Edge Coordinate System  
+
+![](images/b7bec12025e6f227efffe993d420000c18ea244ff8bae921c771471511c7eeae.jpg)  
+
+The equation for $t e^{B2}(r)$ is similar.  
+
+Tail-Furl Coordinate System  
+
+![](images/533bb40e8037410168158086acafc99d91b7370bfc9c854092d68c0b8bfaea2e.jpg)  
+
+Tail Fin Coordinate System  
+
+![](images/326d9d71696178edb5d7849627247735ea5edae14d157ee4fcf3753eb8210851.jpg)  
--- a/力学书籍/FASTCoordinateSystems/auto/FASTCoordinateSystems_origin.pdf
+++ b/力学书籍/FASTCoordinateSystems/auto/FASTCoordinateSystems_origin.pdf
--- a/力学书籍/FASTCoordinateSystems/auto/images/3242641825dce121996aa8b64d069f719111b388bf56b4d23f53af451eb49b9d.jpg
+++ b/力学书籍/FASTCoordinateSystems/auto/images/3242641825dce121996aa8b64d069f719111b388bf56b4d23f53af451eb49b9d.jpg
--- a/力学书籍/FASTCoordinateSystems/auto/images/326d9d71696178edb5d7849627247735ea5edae14d157ee4fcf3753eb8210851.jpg
+++ b/力学书籍/FASTCoordinateSystems/auto/images/326d9d71696178edb5d7849627247735ea5edae14d157ee4fcf3753eb8210851.jpg
--- a/力学书籍/FASTCoordinateSystems/auto/images/402bb29e7129df0062d48b3c5d723765784131fb70c72e23f07e9c78b7e694af.jpg
+++ b/力学书籍/FASTCoordinateSystems/auto/images/402bb29e7129df0062d48b3c5d723765784131fb70c72e23f07e9c78b7e694af.jpg
--- a/力学书籍/FASTCoordinateSystems/auto/images/533bb40e8037410168158086acafc99d91b7370bfc9c854092d68c0b8bfaea2e.jpg
+++ b/力学书籍/FASTCoordinateSystems/auto/images/533bb40e8037410168158086acafc99d91b7370bfc9c854092d68c0b8bfaea2e.jpg
--- a/力学书籍/FASTCoordinateSystems/auto/images/88207e1709e48057eba3e20a48c5537a392d9b45b23b859db0e1ec4e12258df7.jpg
+++ b/力学书籍/FASTCoordinateSystems/auto/images/88207e1709e48057eba3e20a48c5537a392d9b45b23b859db0e1ec4e12258df7.jpg
--- a/力学书籍/FASTCoordinateSystems/auto/images/a25275aa7e7c67a744b5425189c4f767da676d7cf3ad526cc4f418377dae534f.jpg
+++ b/力学书籍/FASTCoordinateSystems/auto/images/a25275aa7e7c67a744b5425189c4f767da676d7cf3ad526cc4f418377dae534f.jpg
--- a/力学书籍/FASTCoordinateSystems/auto/images/a2a9c533925e493d652a0d69730b87a5df026cde2fee0e7d5f9aaa614a8fa533.jpg
+++ b/力学书籍/FASTCoordinateSystems/auto/images/a2a9c533925e493d652a0d69730b87a5df026cde2fee0e7d5f9aaa614a8fa533.jpg
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+++ b/力学书籍/FASTCoordinateSystems/auto/images/b7bec12025e6f227efffe993d420000c18ea244ff8bae921c771471511c7eeae.jpg
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--- a/力学书籍/FASTCoordinateSystems/auto/images/cf71b1f3353b9439b47212c215710ee64d1df808df03c0029378347d198cafb7.jpg
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--- a/力学书籍/FASTKinematics/auto/FASTKinematics.md
+++ b/力学书籍/FASTKinematics/auto/FASTKinematics.md
@ -0,0 +1,229 @@
+There are several points on a 2-bladed turbine: Z (platform reference), Y (platform mass center), T (tower node), O (tower-top / base-plate / yaw bearing mass center), U (nacelle mass center), V (arbitrary point on rotor-furl axis), W (arbitrary point on tail-furl axis), D (center of mass of structure that furls with the rotor [not including rotor]), IMU (nacelle inertial measurement unit), P (teeter pin), SG [shaft strain gage location: i.e., a point on the shaft a distance ShftGagL  towards the nacelle from point P (or point Q for a 3-blader since point P does not exist)], Q (apex of coning angle), C (hub mass center), S1 (blade node for blade 1), S2 (blade node for blade 2), I (tail boom mass center), J (tail fin mass center), and K (tail fin center-of-pressure).  There are also several reference frames: E (earth / inertial), X (platform / tower base), F (tower element body), B (tower-top / base plate), N (nacelle), R (structure that furls with the rotor—generator housing, etc…), L (low speed shaft on rotor end of LSS-compliance), H (hub / rotor), M1 (blade 1 element body), M2 (blade 2 element body), G (fixed in the high speed shaft / generator), and A (tail).  The following are derivations of the position vectors, angular velocities, linear velocities, partial angular velocities, partial linear velocities, angular accelerations, and linear accelerations of all these points on the 2-bladed turbine (point SG’s velocities and accelerations are not derived since they wont be used in the ensuing analysis).  The velocities and accelerations of points on a 3-bladed turbine are very similar.  
+
+![](images/aea47d364c3db886f4957a49550f0ee56aef18eef7d0840dede2a171aa848949.jpg)  
+
+ZO =[qTFA1+qTFA2]a1 + Ptfm Re f+TowerHt 21S1T1F+A(S1TT1SwSr(FTlwerxFLl) eqxT2LFA) 1qT+2SSS12T2F+AS(2TT2SSw(rTFlwerxFLl) exqT2LF) A2qT2+SS 22S+1T2F2AS(1T2TSSw(rTFwlerxFLle) xqLTF) A1qTqSTSF1Aq2TSS 2 u +[qTSS1+qTSS 2]a3  
+
+OU =NacCMxnd1+NacCMznd2−NacCMynd3 rVD=(RFrlCMxn−RFrlPntxn)rf1+(RFrlCMzn−RFrlPntzn)rf2−(RFrlCMyn−RFrlPntyn)rf3 rVIMU= $\big(N c I M U x n-R F r l P n t x n\big)r f_{I}+\big(N c I M U z n-R F r l P n t z n\big)r f_{J}-\big(N c I M U y n-R F r l P n t y n+R F r l P n t z n\big)$ )rf3 rVP= −RFrlPntxnrf1+(Twr2Shft−RFrlPntzn)rf2−(Yaw2Shft−RFrlPntyn)rf3+OverHangc1  
+
+![](images/f0aa977f81135ff9ada5ba5bbff16d69e441e535ce42fa7471ef27b93ae21afb.jpg)  
+
+PSG = ShftGagLc  
+
+PQ= −UndSlingg rQC=HubCMg  
+
+$$
+\begin{array}{r l}&{\Bigr|\int_{I}^{B I}+\Bigr[\psi_{I}^{B I}\left(r\right)q_{B I F I}+\psi_{2}^{B I}\left(r\right)q_{B I F2}+\psi_{3}^{B I}\left(r\right)q_{B I E I}\Bigr]\dot{j}_{2}^{B I}}\\ &{\Bigr.}\\ &{\Bigr.\Bigr.\Bigr.\Bigr.\qquad^{2}+S_{33}^{B I}\left(r\right)q_{B I E I}^{2}+2S_{I2}^{B I}\left(r\right)q_{B I F I}q_{B I F2}+2S_{23}^{B I}\left(r\right)q_{B I F2}q_{B I E I}+2S_{I3}^{B I}\left(r\right)q_{B I F I}q_{B I E I}\Bigr]\Biggr\}.}\end{array}
+$$$$
+\begin{array}{l}{{\displaystyle7.8I F I+\phi_{2}^{B I}\left(r\right)q_{B I F2}+\phi_{3}^{B I}\left(r\right)q_{B I E I}\biggr]\dot{J}_{I}^{B I}+\left[\psi_{I}^{B I}\left(r\right)q_{B I F I}+\psi_{2}^{B I}\left(r\right)q_{B I F I2}+\psi_{3}^{B I}\left(r\right)q_{B I E I}\right]\dot{J}_{I}^{B I}+\cdots}}\\ {{\displaystyle:R a d-\frac{I}{2}\Bigl[S_{I I}^{B I}\left(r\right)q_{B I F I}^{2}+S_{22}^{B I}\left(r\right)q_{B I F2}^{2}+S_{33}^{B I}\left(r\right)q_{B I E I}^{2}+2S_{I2}^{B I}\left(r\right)q_{B I F I}q_{B I F2}+2S_{23}^{B I}\left(r\right)q_{B I F}^{2}\Bigr]\dot{J}_{I}^{B I}\,,}}\end{array}
+$$  
+
+where,  
+
+![](images/ef4f76a950c8f1ce0bea0dba255bdc0dc45aee29c9c4dd7e27c2905433d614c8.jpg)  
+The equation for $r^{\varrho s2}\left(r\right)$ is similar.  
+
+Note limit on r : 0≤r≤TipRad−HubRad=BldFlexL  
+
+$$
+\pmb{r}^{o w}=T F r l P n t x n\pmb{d}_{\jmath}+T F r l P n t z n\pmb{d}_{\jmath}-T F r l P n t y n\pmb{d}_{\jmath}
+$$  
+
+$r^{\prime\prime}=\left(B o o m C M x n-T F r l P n t x n\right)t f_{I}+\left(B o o m C M z n-T F r l P n t z n\right)t f_{J}-\left(B o o m C M y n-T F r u n t z n\right)t f_{I}=0.$ TFrlPntyn)tf3  
+
+$$
+r^{\mu\nu}=\left(T F i n C M x n-T F r l P n t x n\right)t f_{I}+\left(T F i n C M z n-T F r[P n t z n)t f_{J}-\left(T F i n C M y n-T F i n C M z n\right)t f_{I}\right).
+$$  
+
+$$
+r^{W K}=\left(T F i n C P x n-T F r l P n t x n\right)t f_{I}+\left(T F i n C P z n-T F r l P n t z n\right)t f_{2}-\left(T F i n C P y n-T F r u n t z n\right)t f_{I}.
+$$  
+
+Angular Velocities:  
+
+${\}\pmb{\omega}^{X}=\dot{q}_{R}\pmb{\Sigma}_{I}+\dot{q}_{Y}\pmb{\Sigma}_{2}-\dot{q}_{P}\pmb{\Sigma}_{3}$  
+
+EωF( h)=EωX dφ1TSS ( h) qTSS1 + dφ2TSS ( h) qTSS 2 a dφ1 ( h) qTFA1 + dφ2TFA ( h) qTFA2 dh dh dh dh  
+
+dφ1TSS( h) dφ2TSS( h) dφ1TFA( h) dφ2TFA( h) ω ω qTSS1+ qTSS 2 a1 qTFA1+ qTFA2 a3 dh dh dh dh h =TwrFlexL h =TwrFlexL h =TwrFlexL h =TwrFlexL  
+
+$$
+{}^{E}{\pmb\omega}^{N}={}^{E}{\pmb\omega}^{B}+\dot{q}_{\mathrm{{}}\scriptscriptstyle{Y a w}}{\pmb d}_{z}
+$$  
+
+${}^{E}{\pmb\omega}^{R}={}^{E}{\pmb\omega}^{N}+\dot{q}_{R F r l}{\pmb r}{\pmb f}{\pmb\dot{a}}$ where, $r f\!\!a=\!c o s\left(R F r l S k e w\right)\!c o s\left(R F r l T i l t\right)\!d_{I}+s i n\left(R F r l T i l t\right)\!d_{J}-s i n\left(R F r l T i l t\right)\!d_{I}-s i n\left(R F r l T i l t\right)\!d_{I}+s i n\left(R F r l T i l t\right)\!d_{I}.$ lSkew) cos (RFrlTilt)d3  
+
+$$
+{}^{E}{\pmb\omega}^{L}={}^{E}{\pmb\omega}^{R}+\dot{q}_{D r T r}{\pmb c}_{I}+\dot{q}_{G e A z}{\pmb c}_{I}
+$$  
+
+EωM1(r )=EωH dψ1B1(r ) dr qB1F1 +dψ2B1(r ) dr qB1F 2 dψ3B1(r ) dr qB1E1 j1B1 dφ1B1(r )qB1F1+dφ2B1(r ) qB1F 2 +dφ3B1(r ) dr qB1E1 2  
+
+The equation for $^{E}\pmb{\omega}^{M2}\left(r\right)$ is similar.  
+
+Since the generator is attached to the high speed shaft which may or may not rotate in the opposite direction of the low speed shaft and since $q_{G e A z}$ represents the position of the low speed shaft near the entrance of the gearbox,  
+
+$$
+{}^{E}{\pmb\omega}^{G}={}^{E}{\pmb\omega}^{R}+G e n D i r\cdot G B R a t i o\cdot{\dot{q}}_{G e d z}{\pmb{c}}_{I}
+$$  
+
+where, $G e n D i r=\binom{-I}{I}\quad f o r\,\,\,\,G B\,R e\,\nu e r s e=T r u e}\\ {\,\,\,\,G\d{p}\,\,\,\,G B\,R e\,\nu e r s e=F a l s e}$  
+
+# ${}^{E}{\pmb\omega}^{A}={}^{E}{\pmb\omega}^{N}+\dot{q}_{{}_{T F r l}}t{\pmb f}\dot{\pmb u}$ where, tfa=cos (TFrlSkew) cos (TFrlTilt)d1+sin (TFrlTilt)d2−sin (TFrlSkew) cos (TFrlTilt)d3  
+
+# Linear Velocities: ${}^{E}{\pmb{\nu}}^{Z}=\dot{q}_{S g}\bar{\sf z}_{I}+\dot{q}_{H\nu}\bar{\sf z}_{2}-\dot{q}_{S w}\bar{\sf z}_{3}$  
+
+$$
+{}^{E}{\pmb{\nu}}^{Y}={}^{E}{\pmb{\nu}}^{Z}+{}^{E}{\pmb{\omega}}^{X}\times{\pmb{r}}^{Z Y}
+$$  
+
+$$
+{}^{E}{\nu}^{T}\left(h\right)={}^{E}{\nu}^{Z}+{}^{X}{\nu}^{T}\left(h\right)+{}^{E}{\omega}^{X}\times{r}^{Z T}\left(h\right)
+$$  
+
+where,  
+
+XvT( h)=φ1TFA( h) qTFA1+φ2TFA( h) qTFA2a1 S1T1FA( h) qTFA1qTFA1+S2T2FA( h) qTFA2qTFA2+S1T2FA( h)(qTFA1qTFA2+qTFA1qTFA2) +S1T1SS( h) qTSS1qTSS1+S2T2SS( h) qTSS 2qTSS 2+S1T2SS( h)(qTSS1qTSS 2+qTSS1qTSS 2) +φ1TSS( h) qTSS1+φ2TSS( h) qTSS 2a3  
+
+${}^{E}{\nu}^{o}={}^{E}{\nu}^{Z}+{}^{X}{\nu}^{o}+{}^{E}{\omega}^{X}\times{r}^{Z O}$ where,  
+
+XvO=[qTFA1+qTFA2]a1 S1T1FA(TwrFlexL) qTFA1qTFA1+S2T2FA(TwrFlexL) qTFA2qTFA2+S1T2FA(TwrFlexL)(qTFA1qTFA2+qTFA1qTFA2) +S1T1SS(TwrFlexL) qTSS1qTSS1+S2T2SS(TwrFlexL) qTSS 2qTSS 2+S1T2SS(TwrFlexL)(qTSS1qTSS 2+qTSS1qTSS 2) +[qTSS1+qTSS 2]a3  
+
+$$
+\begin{array}{l}{{{^E}_{\nu}{^U}={^E}_{\nu}{^o}+{^E}_{\omega}{^N}\times r^{o U}}}\\ {{{}}}\\ {{{^E}_{\nu}{^V}={^E}_{\nu}{^o}+{^E}_{\omega}{^N}\times r^{o V}}}\\ {{{}}}\\ {{{^E}_{\nu}{^D}={^E}_{\nu}{^V}+{^E}_{\omega}{^R}\times r^{V D}}}\end{array}
+$$  
+
+The equation for $E_{\nu}^{\phantom{\mu\nu}}{}^{m U}$ is similar.  
+
+$$
+{}^{E}{\pmb{\nu}}^{P}={}^{E}{\pmb{\nu}}^{V}+{}^{E}{\pmb{\omega}}^{R}\times{\pmb{r}}^{V P}
+$$  
+
+# EvQ=EvP+EωH×r  
+
+×  
+
+EvS1(r )=EvQ+HvS1(r )+EωH×rQS1(r )  
+
+where,  
+
+HvS1(r )=φ1B1(r ) qB1F1+φ2B1(r ) qB1F 2+φ3B1(r ) qB1E1j1B1+ψ1B1(r ) qB1F1+ψ2B1(r ) qB1F 2+ψ3B1(r ) qB1E1j2B1 S1B11(r ) qB1F1qB1F1+S2B21(r ) qB1F 2qB1F 2+S3B31(r ) qB1E1qB1E1 +S1B21(r )(qB1F1qB1F 2+qB1F1qB1F 2)+S2B31(r )(qB1F 2qB1E1+qB1F2qB1E1)+S1B31(r )(qB1F1qB1E1+qB1F1qB1E1)  
+
+The equation for $\boldsymbol{\varepsilon}_{\boldsymbol{\nu}}^{s_{2}}\left(\boldsymbol{r}\right)$ is similar.  
+
+$$
+\begin{array}{r l}&{^{E}_{\nu}\psi^{\nu}=^{E}_{\nu}\!\!\!\phantom{^{(0)}}^{\!\!E}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 
+$$  
+
+Partial Angular Velocities:  
+
+Recall that: $^{E}\!\omega^{N_{i}}\left(\dot{q},q,t\right)\!=\!\!\left(\sum_{r=l}^{22}\d^{E}\omega_{r}^{N_{i}}\left(q,t\right)\!\dot{q}_{r}\right)\!+\!\;^{E}\!\omega_{t}^{N_{i}}\left(q,t\right)$ for each rigid body $N_{i}$ in the system.  Note that all of the ${}^{E}{\pmb{\omega}}_{t}^{N_{i}}$ terms are zero as will be shown.  
+
+z1 for r=R z3 for r=P EωX= 7 for r=Y 0 otherwise EωtX = 0  
+
+<html><body><table><tr><td rowspan="4">E (h) の + E</td><td>dpi TFA h a3 for r =TFAl</td></tr><tr><td>dh dpi TSS h a1 for r =TSS1</td></tr><tr><td>dh dΦ2 TFA (h) a3 for r=TFA2</td></tr><tr><td>dh dΦ2 TSS h a for r =TSS2 dh otherwise</td></tr><tr><td colspan="2">0 0F (h) =0</td></tr></table></body></html>  
+
+<html><body><table><tr><td rowspan="4">Q</td><td>dpIFA TFA (h) dh</td><td>a3 for r =TFAl h=TwrFlexL</td></tr><tr><td>d$iss T'SS (h) dh</td><td>for r = TSS1 h=TwrFlexL</td></tr><tr><td>dΦ2 TFA (h dh TSS</td><td>a3 for r=TFA2 h=TwrFlexL</td></tr><tr><td>dp? h dh</td><td>a for r =TSS2 h=TwrFlexL</td></tr><tr><td>E =0</td><td>0</td><td>otherwise</td></tr></table></body></html>  
+
+d for r=Yaw   
+E ω ω 0 otherwise   
+E ω  
+
+rfa for r=RFrlω ω0 otherwiseE ω R =0  
+
+![](images/3398e4ae47d978c05d1ac6d8435467b2a54017a3ad095973c8560076ec510976.jpg)  
+
+![](images/6312d612c59210b4b88ca58006215d23618c8c2ae4a8405023d27e07a9523168.jpg)  
+
+![](images/fa88774625bec40d9835cefc14b9dab265ecad758ce4f53a290b13815e357f12.jpg)  
+
+The equations for $^{E}\pmb{\omega}_{r}^{M2}\left(r\right)$ and $^{E}\pmb{\omega}_{t}^{M2}\left(r\right)$ are similar.  
+
+![](images/e31134e876cef5a881092ca70c43cba1fd09da2d674dd9f5d4cac0be697c2b35.jpg)  
+
+Partial Linear Velocities:  
+
+Recall that: $^{E}\nu^{X_{i}}\left(\dot{q},q,t\right)=\left(\sum_{r=I}^{22}\varepsilon_{r}^{\phantom{R}}\left(q,t\right)\dot{q}_{r}\right)+^{E}\nu_{t}^{X_{i}}\left(q,t\right)$ for each point $X_{i}$ in the system.  Note that all of the $\ensuremath{\boldsymbol{\varepsilon}}_{\ensuremath{\boldsymbol{\nu}}_{t}^{X_{i}}}$ terms are zero as will be shown.  
+
+7 for r=Sg z3 for r=Sw for r=Hv 0 otherwise v = 0  
+
+E ωr X ×r ZY for r=4,5,6 0 otherwise = 0  
+
+EvrT( h)=EvrZ+ $\begin{array}{r l r}&{\left\{\begin{array}{l l}{\varepsilon_{\omega}^{x}\times r^{z T}\left(h\right)}&{f o r}&{r=d,5,6}\\ {\phi_{l}^{T R L}\left(h\right)a_{l}-\left[S_{l l}^{T R L}\left(h\right)q_{T E A l}+S_{l2}^{T R L}\left(h\right)q_{T E A l}\right]a_{2}}&{f o r}&{r=T F A l}\end{array}\right.}\\ &{\left\{\begin{array}{l l}{\phi_{l}^{T S S}\left(h\right)a_{3}-\left[S_{l l}^{T S S}\left(h\right)q_{T S S l}+S_{l2}^{T S S}\left(h\right)q_{T S S2}\right]a_{2}}&{f o r}&{r=T S S I}\\ {\phi_{2}^{T R L}\left(h\right)a_{l}-\left[S_{22}^{T R L}\left(h\right)q_{T E A2}+S_{l2}^{T R L}\left(h\right)q_{T E A l}\right]a_{2}}&{f o r}&{r=T F A2}\end{array}\right.}\\ &{\left\}\phi_{2}^{T R L}\left(h\right)a_{l}-\left[S_{22}^{T R L}\left(h\right)q_{T E A2}+S_{l2}^{T R L}\left(h\right)q_{T E A l}\right]a_{2}}&{f o r}&{r=T F A2}\\ &{\phi_{2}^{T S S}\left(h\right)a_{3}-\left[S_{22}^{T S S}\left(h\right)q_{T S S2}+S_{l2}^{T S S}\left(h\right)q_{T S S l}\right]a_{2}}&{f o r}&{r=T S S2}\\ {0}&{o t h e r w i s e}\end{array}\right.}\end{array}$ $\boldsymbol{\varepsilon}_{\boldsymbol{\nu}_{t}^{T}}(h)\!=\!\boldsymbol{\ O}$  
+
+$$
+\begin{array}{r}{\left[\begin{array}{l l l}{\varepsilon_{\omega}^{\scriptscriptstyle X}\times r^{2\theta}}&{f o r}&{r=\mathscr{I},\mathscr{S},\theta}\\ {a_{I}-\Big[S_{I I}^{\scriptscriptstyle T\scriptscriptstyle F\scriptscriptstyle L}\big(T w r F l e x L\big)q_{T r\scriptscriptstyle A I}+S_{I2}^{\scriptscriptstyle T\scriptscriptstyle L}\big(T w r F l e x L\big)q_{T r\scriptscriptstyle A2}\Big]a_{2}}&{f o r}&{r=T F A I}\\ {a_{3}-\Big[S_{I I}^{\scriptscriptstyle T\scriptscriptstyle S\scriptscriptstyle S}\big(T w r F l e x L\big)q_{T r\scriptscriptstyle S\scriptscriptstyle I}+S_{I2}^{\scriptscriptstyle T\scriptscriptstyle S\scriptscriptstyle S}\big(T w r F l e x L\big)q_{T r\scriptscriptstyle S\scriptscriptstyle S2}\Big]a_{2}}&{f o r}&{r=T S S I}\\ {a_{I}-\Big[S_{22}^{\scriptscriptstyle T\scriptscriptstyle E\scriptscriptstyle A}\big(T w r F l e x L\big)q_{T r\scriptscriptstyle A2}+S_{I2}^{\scriptscriptstyle T\scriptscriptstyle E\scriptscriptstyle A}\big(T w r F l e x L\big)q_{T r\scriptscriptstyle A I}\Big]a_{2}}&{f o r}&{r=T F A2}\\ {a_{3}-\Big[S_{22}^{\scriptscriptstyle T\scriptscriptstyle S\scriptscriptstyle S}\big(T w r F l e x L\big)q_{T r\scriptscriptstyle A2}+S_{I2}^{\scriptscriptstyle T\scriptscriptstyle S\scriptscriptstyle S}\big(T w r F l e x L\big)q_{T r\scriptscriptstyle A2}}&{f o r}&{r=T S S2}\\ {\theta}&{o t h e r w i s e}\end{array}\right]}\end{array}
+$$  
+
+ω × OU for r=4,5,,11 otherwise  
+
+OV for r=4,5,,11 ω × otherwise EvV =0  
+
+![](images/e3d6a23ac82c553a6109e1bbd076a589d53454144682502bb72c0663d0760822.jpg)  
+
+The equations for $E_{\nu_{r}}^{\,,\,\,\,\,\mu\nu}$ and EvIMU are similar.  
+
+![](images/afa45461db2a2b95476a24dfe5f2800411c600e2ef02e1410f267618deae6c23.jpg)  
+
+![](images/200f40a4ff42db1bfd8e7b1ddc2568d911234a08e251bb70cdd737ad06440482.jpg)  
+
+$\begin{array}{r l r}&{}&{f o r\ \ r}\\ &{}&{\left\{\phi_{l}^{B I}\left(r\right)j_{l}^{B I}+\psi_{l}^{B I}\left(r\right)j_{2}^{B I}-\left[S_{1l}^{B I}\left(r\right)q_{B I F I}+S_{12}^{B I}\left(r\right)q_{B I F I}+S_{13}^{B I}\left(r\right)q_{B I E I}\right]j_{3}^{B I}\ \ \ f o r\ \ r}\\ &{}&{\left\{\phi_{3}^{B I}\left(r\right)j_{l}^{B I}+\psi_{3}^{B I}\left(r\right)j_{2}^{B I}-\left[S_{33}^{B I}\left(r\right)q_{B I E I}+S_{23}^{B I}\left(r\right)q_{B I F I}+S_{13}^{B I}\left(r\right)q_{B I F I}\right]j_{3}^{B I}\ \ \ f o r\ \ r}\\ &{}&{\left\{\phi_{2}^{B I}\left(r\right)j_{l}^{B I}+\psi_{2}^{B I}\left(r\right)j_{2}^{B I}-\left[S_{22}^{B I}\left(r\right)q_{B I F I}+S_{12}^{B I}\left(r\right)q_{B I F I}+S_{23}^{B I}\left(r\right)q_{B I E I}\right]j_{3}^{B I}\ \ \ f o r\ \ r}\\ &{}&{\left\{\varepsilon_{O H}^{B I}\times r^{Q S I}\left(r\right)\right.}\\ &{}&{\left.\rho t\rightarrow r^{r}\left(r\right)}\\ &{}&{\left.\rho t h e r w i\right.}\end{array}$ =4,5,,14 =B1F1 =B1E1 =B1F2 =Teet ise $\boldsymbol{\varepsilon}_{\boldsymbol{\nu}_{t}^{S I}}\left(\boldsymbol{r}\right)=\boldsymbol{\ O}$  
+
+The equations for $^E_{\nu_{r}^{S2}}\left(r\right)$ and $^E_{\nu_{t}^{S2}}\left(r\right)$ are similar.  
+
+ω ×r OW for r=4,5,,11 0 otherwise E = 0  
+
+![](images/e86624538ba5b2865e2e19027cff55cc7241673860f69c31ddc8f7297c122302.jpg)  
+
+Angular Accelerations:  
+
+Recall that: $^{E}\pmb{\alpha}^{N_{i}}\left(\ddot{q},\dot{q},q,t\right)\!=\!\left(\sum_{r=l}^{22}^{E}\pmb{\omega}_{r}^{N_{i}}\left(q,t\right)\ddot{q}_{r}\right)\!+\!\left[\sum_{r=l}^{22}\frac{d}{d t}\!\left(^{E}\pmb{\omega}_{r}^{N_{i}}\left(q,t\right)\right)\!\dot{q}_{r}\right]\!+\!\frac{d}{d t}\!\left(^{E}\pmb{\omega}_{t}^{N_{i}}\left(q,t\right)\right)\!\left(^{E}\pmb{\omega}_{p}^{N_{i}}\left(q,t\right)\right),$ for each rigid body $N_{i}$ in the system.  Note that the $\frac{d}{d t}\Big(^{E}\omega_{r}^{N_{i}}\Big)$ terms are all vector functions of $\left({\dot{q}},q,t\right)$ and that all of the $\frac{d}{d t}\Big(^{E}\omega_{t}^{N_{i}}\Big)$ terms are zero as will be shown.  
+
+$$
+\begin{array}{l}{\displaystyle\frac{d}{d t}\Big(\sp\varepsilon\omega_{r}^{X}\Big)=O}\\ {\displaystyle\frac{d}{d t}\Big(\sp\varepsilon\omega_{{r}}^{X}\Big)=O}\end{array}
+$$  
+
+$\begin{array}{l l}{\displaystyle\frac{d}{d t}\Big[^{\varepsilon}\pmb{\omega}_{r}^{F}\left(h\right)\Big]\!=\!\left\{\!\!\begin{array}{l l}{\displaystyle\varepsilon_{\pmb{\omega}}^{\varepsilon}\!\times^{\varepsilon}\!\omega_{r}^{F}\left(h\right)}&{\displaystyle f o r\;\;r=7,\delta,...,l O}\\ {\displaystyle\theta}&{o t h e r w i s e}\end{array}\!\right.}\\ {\displaystyle\frac{d}{d t}\Big[^{\varepsilon}\pmb{\omega}_{\prime}^{F}\left(h\right)\Big]=O}\end{array}$   
+$\begin{array}{l}{\displaystyle\frac{d}{d t}\Big({}^{E}{\omega}_{r}^{B}\Big)=\left\{{\!\!\begin{array}{l l}{\displaystyle E_{{\pm}}{\omega}^{X}\times{}^{E}{\omega}_{r}^{B}}&{\displaystyle f o r\ \ r=7,\mathrm{}\mathrm{\it{\mathscr{S}}},\ldots,{\it{\it{10}}}}\\ {\displaystyle O}&{\ o t h e r w i s e}\end{array}}\right.}\\ {\displaystyle\frac{d}{d t}\Big({}^{E}{\omega}_{{\iota}}^{B}\Big)=\cal{O}}\end{array}$ E E N for r=Yaw   
+d E ω Yaw   
+ω ω   
+dt d otherwise   
+d E   
+ωt =0   
+dt  
+
+![](images/9b38d49d69832767897ff11cc45ca36c4165b88b914426af13a479d09b6d4510.jpg)  
+
+The equations for ${\frac{d}{d t}}\Big[^{\,E}\pmb{\omega}_{r}^{{\scriptscriptstyle M}2}\,\big(r\big)\Big]$  and d Eω $\frac{d}{d t}\Big[^{\,E}\pmb{\omega}_{t}^{{M}2}\left(r\right)\Big]$ are similar.  
+
+![](images/c5881e166cabde96bd49b6631c6b951f97f8b82ca05b9ad9c40af4ae957d03ed.jpg)  
+
+Linear Accelerations:  
+
+Recall that: $\varepsilon_{\mathbf{}}\boldsymbol{\alpha}^{X_{i}}\left(\ddot{q},\dot{q},q,t\right)\!=\!\!\left(\sum_{r=l}^{22}\varepsilon_{\nu_{r}^{X_{i}}}\left(q,t\right)\ddot{q}_{r}\right)\!+\!\!\left[\sum_{r=l}^{22}\!\frac{d}{d t}\!\left({^{\varepsilon}\nu_{r}^{X_{i}}}\left(q,t\right)\right)\!\dot{q}_{r}\right]\!+\!\frac{d}{d t}\!\left({^{\varepsilon}\nu_{{^I}}^{X_{i}}}\left(q,t\right)\right)\!\left({^{\varepsilon}\nu_{{^I}}^{X_{i}}}\left(q,t\right)\right)\!+\!\frac{d}{d t}\!\left({^{\varepsilon}\nu_{{^I}}^{X_{i}}}\left(q,t\right)\right)\!+\!{^{\varepsilon}\nu_{{^I}}^{X_{i}}}\left(q,t\right),$ for each point $X_{i}$ in the system.  Note that the ${\frac{d}{d t}}{\Big(}^{E}\nu_{r}^{X_{i}}{\Big)}$ terms are all vector functions of $\left({\dot{q}},q,t\right)$ and that all of the ${\frac{d}{d t}}{\Big(}^{E}\nu_{t}^{X_{i}}{\Big)}$ terms are zero as will be shown.  
+
+$$
+\begin{array}{l}{\displaystyle\frac{d}{d t}\Big(\sp\varepsilon\pmb{\nu}_{r}^{z}\Big)=O}\\ {\displaystyle\frac{d}{d t}\Big(\sp\varepsilon\pmb{\nu}_{{\pmb\nu}}^{z}\Big)=O}\end{array}
+$$  
+
+ω ω × for r=4,5,6 v dt otherwise (EvtY) =0 dt ddt $\iota_{\Psi_{r}^{T}}(h)\!\!\!\int_{=\!\!\!\!\!\!\int_{0}^{M}}\!\left[\!\!\!\begin{array}{l l l}{\varepsilon_{\omega_{r}^{X}\times\left[\!\!\!\begin{array}{l}{x}{\nu}{\nu^{r}\left(h\right)+}^{\varepsilon}\omega^{X}\times r^{Z r}\left(h\right)\!\!\right]}}&{f o r}&{r=d,5,6}\\ {-\left[\!\!\begin{array}{l l l}{S_{11}^{T R A}\left(h\right)\dot{q}_{T r A l}+S_{12}^{T R A}\left(h\right)\dot{q}_{T r A l}]\!\!\right]a_{2}+}&{\varepsilon_{\omega}^{X}\times^{E}\nu_{T R A l}^{T}\left(h\right)}&{f o r}&{r=T R A l}\\ {-\left[\!\!\begin{array}{l l l}{S_{11}^{T S B}\left(h\right)\dot{q}_{T S S1}+S_{12}^{T S B}\left(h\right)\dot{q}_{T S S2}}\end{array}\right]a_{2}+}&{\varepsilon_{\omega}^{X}\times^{E}\nu_{T S S1}^{T}\left(h\right)}&{f o r}&{r=T S S I}\\ {-\left[\!\!\begin{array}{l l l}{S_{22}^{T R A}\left(h\right)\dot{q}_{T R A2}+S_{12}^{T R A}\left(h\right)\dot{q}_{T R A l}\right]a_{2}+}&{\varepsilon_{\omega}^{X}\times^{E}\nu_{T R A2}^{T}\left(h\right)}&{f o r}&{r=T F A2}\\ {-\left[\!\!\begin{array}{l l l}{S_{22}^{T R A}\left(h\right)\dot{q}_{T S2}+S_{12}^{T R A}\left(h\right)\dot{q}_{T R A l}}\end{array}\right]a_{2}+}&{\varepsilon_{\omega}^{X}\times^{E}\nu_{T R S2}^{T}\left(h\right)}&{f o r}&{r=T F A2}\\ {-\left[\!\!\left[S_{22}^{T S S}\left(h\right)\dot{q}_{T S S2}+S_{12}^{T S S}\left(h\right)\dot{q}_{T S S1}\right]a_{2}+}&{\varepsilon_{\omega}^{X}\times^{E}\nu_{T S S2}^{T}\left(h\right)}&{f o r}&{r=T S S2}\\ {0}&{o t h e r w i s e}\end{array}\right]$ d[EvtT(h )]=0  
+
+![](images/7f540e8a0b94d65e0c8e85862cbfffbd9db962a124a331a6b20ff526e17acf02.jpg)  
+
+![](images/f5e77b445b09c605b565937c9931c544ab43aea41659c32cb20fd54ba1f400a3.jpg)  
+
+The equations for $\frac{d}{d t}\big(\varepsilon_{\nu_{r}^{I M U}}\big)$ and $\frac{d}{d t}\big(\boldsymbol{\varepsilon}_{\boldsymbol{\nu}_{t}^{I M U}}\big)$ are similar.  
+
+![](images/3ce1abcaeeff5430134d742b3e324508a553d05c8c0f30117b673678c8672d72.jpg)  
+
+![](images/b46f6d5b32c278d5fa117b282e8e89c83efcfe951642463192021963be2383f9.jpg)  
+
+![](images/a5ccd2c04b4bb71c7215c42e6a3f7b90011a6bd8b375fadde860693f0fd076b8.jpg)  
+
+# dtEvtS1(r )=0  
+
+The equations for $\frac{d}{d t}\Big[^{\varepsilon}\nu_{r}^{s_{2}}(r)\Big]$ and $\frac{d}{d t}\Big[^{\varepsilon}\nu_{t}^{s2}\left(r\right)\Big]$ are similar.  
+
+![](images/b425e4c4b9ef2bc7d01a78777fb711557ad6ceeba0f84bf46ae380326bc9c813.jpg)  
+
+![](images/2d58bc879c460dac3de92bb198a38c050563fc9038b4acdddcdd1b0c5834c1ca.jpg)  
--- a/力学书籍/FASTKinematics/auto/FASTKinematics_origin.pdf
+++ b/力学书籍/FASTKinematics/auto/FASTKinematics_origin.pdf
--- a/力学书籍/FASTKinematics/auto/images/200f40a4ff42db1bfd8e7b1ddc2568d911234a08e251bb70cdd737ad06440482.jpg
+++ b/力学书籍/FASTKinematics/auto/images/200f40a4ff42db1bfd8e7b1ddc2568d911234a08e251bb70cdd737ad06440482.jpg
--- a/力学书籍/FASTKinematics/auto/images/2277d15a0b56afcd7524f8a378e1de9a8ca839f48c68db2aebeacb21a1ed8e85.jpg
+++ b/力学书籍/FASTKinematics/auto/images/2277d15a0b56afcd7524f8a378e1de9a8ca839f48c68db2aebeacb21a1ed8e85.jpg
--- a/力学书籍/FASTKinematics/auto/images/2d58bc879c460dac3de92bb198a38c050563fc9038b4acdddcdd1b0c5834c1ca.jpg
+++ b/力学书籍/FASTKinematics/auto/images/2d58bc879c460dac3de92bb198a38c050563fc9038b4acdddcdd1b0c5834c1ca.jpg
--- a/力学书籍/FASTKinematics/auto/images/3398e4ae47d978c05d1ac6d8435467b2a54017a3ad095973c8560076ec510976.jpg
+++ b/力学书籍/FASTKinematics/auto/images/3398e4ae47d978c05d1ac6d8435467b2a54017a3ad095973c8560076ec510976.jpg
--- a/力学书籍/FASTKinematics/auto/images/3ce1abcaeeff5430134d742b3e324508a553d05c8c0f30117b673678c8672d72.jpg
+++ b/力学书籍/FASTKinematics/auto/images/3ce1abcaeeff5430134d742b3e324508a553d05c8c0f30117b673678c8672d72.jpg
--- a/力学书籍/FASTKinematics/auto/images/6312d612c59210b4b88ca58006215d23618c8c2ae4a8405023d27e07a9523168.jpg
+++ b/力学书籍/FASTKinematics/auto/images/6312d612c59210b4b88ca58006215d23618c8c2ae4a8405023d27e07a9523168.jpg
--- a/力学书籍/FASTKinematics/auto/images/7f540e8a0b94d65e0c8e85862cbfffbd9db962a124a331a6b20ff526e17acf02.jpg
+++ b/力学书籍/FASTKinematics/auto/images/7f540e8a0b94d65e0c8e85862cbfffbd9db962a124a331a6b20ff526e17acf02.jpg
--- a/力学书籍/FASTKinematics/auto/images/9b38d49d69832767897ff11cc45ca36c4165b88b914426af13a479d09b6d4510.jpg
+++ b/力学书籍/FASTKinematics/auto/images/9b38d49d69832767897ff11cc45ca36c4165b88b914426af13a479d09b6d4510.jpg
--- a/力学书籍/FASTKinematics/auto/images/a593b4bb09ff3bcc0aa64f9fe4cad48bc1ea831df66f0ed67ed5622ce3ff0a5e.jpg
+++ b/力学书籍/FASTKinematics/auto/images/a593b4bb09ff3bcc0aa64f9fe4cad48bc1ea831df66f0ed67ed5622ce3ff0a5e.jpg
--- a/力学书籍/FASTKinematics/auto/images/a5ccd2c04b4bb71c7215c42e6a3f7b90011a6bd8b375fadde860693f0fd076b8.jpg
+++ b/力学书籍/FASTKinematics/auto/images/a5ccd2c04b4bb71c7215c42e6a3f7b90011a6bd8b375fadde860693f0fd076b8.jpg
--- a/力学书籍/FASTKinematics/auto/images/aea47d364c3db886f4957a49550f0ee56aef18eef7d0840dede2a171aa848949.jpg
+++ b/力学书籍/FASTKinematics/auto/images/aea47d364c3db886f4957a49550f0ee56aef18eef7d0840dede2a171aa848949.jpg
--- a/力学书籍/FASTKinematics/auto/images/afa45461db2a2b95476a24dfe5f2800411c600e2ef02e1410f267618deae6c23.jpg
+++ b/力学书籍/FASTKinematics/auto/images/afa45461db2a2b95476a24dfe5f2800411c600e2ef02e1410f267618deae6c23.jpg
--- a/力学书籍/FASTKinematics/auto/images/b425e4c4b9ef2bc7d01a78777fb711557ad6ceeba0f84bf46ae380326bc9c813.jpg
+++ b/力学书籍/FASTKinematics/auto/images/b425e4c4b9ef2bc7d01a78777fb711557ad6ceeba0f84bf46ae380326bc9c813.jpg
--- a/力学书籍/FASTKinematics/auto/images/b46f6d5b32c278d5fa117b282e8e89c83efcfe951642463192021963be2383f9.jpg
+++ b/力学书籍/FASTKinematics/auto/images/b46f6d5b32c278d5fa117b282e8e89c83efcfe951642463192021963be2383f9.jpg
--- a/力学书籍/FASTKinematics/auto/images/c5881e166cabde96bd49b6631c6b951f97f8b82ca05b9ad9c40af4ae957d03ed.jpg
+++ b/力学书籍/FASTKinematics/auto/images/c5881e166cabde96bd49b6631c6b951f97f8b82ca05b9ad9c40af4ae957d03ed.jpg
--- a/力学书籍/FASTKinematics/auto/images/e31134e876cef5a881092ca70c43cba1fd09da2d674dd9f5d4cac0be697c2b35.jpg
+++ b/力学书籍/FASTKinematics/auto/images/e31134e876cef5a881092ca70c43cba1fd09da2d674dd9f5d4cac0be697c2b35.jpg
--- a/力学书籍/FASTKinematics/auto/images/e3d6a23ac82c553a6109e1bbd076a589d53454144682502bb72c0663d0760822.jpg
+++ b/力学书籍/FASTKinematics/auto/images/e3d6a23ac82c553a6109e1bbd076a589d53454144682502bb72c0663d0760822.jpg
--- a/力学书籍/FASTKinematics/auto/images/e86624538ba5b2865e2e19027cff55cc7241673860f69c31ddc8f7297c122302.jpg
+++ b/力学书籍/FASTKinematics/auto/images/e86624538ba5b2865e2e19027cff55cc7241673860f69c31ddc8f7297c122302.jpg
--- a/力学书籍/FASTKinematics/auto/images/ef4f76a950c8f1ce0bea0dba255bdc0dc45aee29c9c4dd7e27c2905433d614c8.jpg
+++ b/力学书籍/FASTKinematics/auto/images/ef4f76a950c8f1ce0bea0dba255bdc0dc45aee29c9c4dd7e27c2905433d614c8.jpg
--- a/力学书籍/FASTKinematics/auto/images/f0aa977f81135ff9ada5ba5bbff16d69e441e535ce42fa7471ef27b93ae21afb.jpg
+++ b/力学书籍/FASTKinematics/auto/images/f0aa977f81135ff9ada5ba5bbff16d69e441e535ce42fa7471ef27b93ae21afb.jpg
--- a/力学书籍/FASTKinematics/auto/images/f5e77b445b09c605b565937c9931c544ab43aea41659c32cb20fd54ba1f400a3.jpg
+++ b/力学书籍/FASTKinematics/auto/images/f5e77b445b09c605b565937c9931c544ab43aea41659c32cb20fd54ba1f400a3.jpg
--- a/力学书籍/FASTKinematics/auto/images/fa88774625bec40d9835cefc14b9dab265ecad758ce4f53a290b13815e357f12.jpg
+++ b/力学书籍/FASTKinematics/auto/images/fa88774625bec40d9835cefc14b9dab265ecad758ce4f53a290b13815e357f12.jpg
--- a/力学书籍/FASTKinetics/auto/FASTKinetics.md
+++ b/力学书籍/FASTKinetics/auto/FASTKinetics.md
@ -0,0 +1,478 @@
+The following are derivations of the entire equations of motion used in FAST for a 2-bladed turbine configuration.  The various portions of the equations of motion are organized according to their source.  The equations of motion of a 3-bladed turbine are very similar.  
+
+By a direct result of Newton’s laws of motion, Kane’s equations of motion for a simple holonomic system with 22 DOFs can be stated as follows (Kane and Levinson, 1985):  
+
+$$
+F_{r}+F_{r}^{*}=O\quad\left(r=l,2,...,22\right)
+$$  
+
+where, for a set of $\boldsymbol{w}$ rigid bodies characterized by reference frame $N_{i}$ and center of mass point $X_{i}$ :  
+
+the generalized active forces are:  
+
+$$
+\begin{array}{r l}&{F_{r}=\displaystyle\sum_{i=I}^{w}\varepsilon_{\nu}x_{i}\cdot F^{X_{i}}+{}^{E}\omega_{r}^{N_{i}}\cdot M^{N_{i}}\quad\left(r=I,2,\ldots,22\right)}\\ &{F_{r}^{\ast}=\displaystyle\sum_{i=I}^{w}\varepsilon_{\nu}x_{i}\cdot\left(-m^{N_{i}}\,^{E}a^{X_{i}}\right)+{}^{E}\omega_{r}^{N_{i}}\cdot\left(-\displaystyle^{E}\dot{H}^{N_{i}}\right)\quad\left(r=I,2,\ldots,22\right)}\end{array}
+$$  
+
+and the generalized inertia forces are:  
+
+where it is assumed that for each rigid body $N_{i}$ , the active forces are applied at the center of mass point $X_{i}$ .  The time derivative of the angular momentum of rigid body $N_{i}$ about its center of mass $X_{i}$ in the inertial frame can be found as follows:  
+
+$$
+^{\varepsilon}\dot{H}^{N_{i}}=\left\{\!\!\begin{array}{c}{{\left(\dot{H}^{N_{i}}\right)^{\prime}\!\!+^{E}\pmb{\omega}^{N_{i}}\!\times\!\!^{E}H^{N_{i}}}}\\ {{o r}}\\ {{\left.\overline{{{\bar{I}}}}^{N_{i}}\!\cdot\!\!^{E}\pmb{\alpha}^{N_{i}}\!+\!\!^{E}\pmb{\omega}^{N_{i}}\!\times\!\!\overline{{{\bar{I}}}}^{N_{i}}.^{E}\pmb{\omega}^{N_{i}}\!\right.}}\end{array}\!\!\right.
+$$  
+
+For the wind turbine modeled in FAST, the mass of the platform, tower, yaw bearing, nacelle, structure that furls with the rotor, hub, blades, generator, and tail contribute to the total generalized inertia forces as follows:  
+
+$$
+\left.\!\!\begin{array}{l}{{\left.\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+$$  
+
+Generalized active forces are the result of forces applied directly on the wind turbine system, forces that ensure constraint relationships between the various rigid bodies, and internal forces within flexible members.  Forces applied directly on the wind turbine system include aerodynamic forces acting on the blades, tower, and tail fin; hydrostatic, hydrodynamic, mooring and/or foundation elasticity and damping forces, including added mass effects, acting on the platform; gravitational forces acting on the platform, tower, yaw bearing, nacelle, structure that furls with the rotor, hub, blades, tip brakes, and tail; generator torque; HSS brake; and gearbox friction forces.  Forces that enforce constraint relationships between the various rigid bodies include springs and dampers for yaw, rotor-furl, teeter, and tail-furl (the simple workless constraint forces, for example frictionless pins or rigid connections, do not contribute to the generalized active forces).  Internal forces within flexible members include elasticity and damping in the tower, blades, and drivetrain.  Thus,  
+
+$$
+\begin{array}{r}{F_{r}\big|_{A e r o r}+F_{r}\big|_{A e r o b l}+F_{r}\big|_{A e r o b l}+F_{r}\big|_{A e r o a l}+F_{r}\big|_{H o r e a l}+F_{r}\big|_{G r o w X}+F_{r}\big|_{G r o w X}+F_{r}\big|_{G r o w Y}+F_{r}\big|_{G r o w Y}+F_{r}\big|_{G r o w Y}}\\ {+F_{r}\big|_{S p r o i g Y a w}+F_{r}\big|_{D a m p Y a w}+F_{r}\big|_{S p r o i n g F}+F_{r}\big|_{D a m p R F}+F_{r}\big|_{S p r o i n g T e e t}+F_{r}\big|_{D a m p T e e t}+F_{r}\big|_{S p r o i n g T e}+F_{r}\big|_{S p r o i n g T e}+F_{r}\big|_{H o r e a l}}\\ {+F_{r}\big|_{E l a s t i c T}+F_{r}\big|_{D a m p T}+F_{r}\big|_{E l a s t i c B l}+F_{r}\big|_{D a m p B l}+F_{r}\big|_{E l a s t i c B2}+F_{r}\big|_{D a m p B2}+F_{r}\big|_{E l a s t i c D r i e v}+F_{r}\big|_{D a m p B2}}\end{array}
+$$  
+
+Kane’s equations of motion can be written in matrix form as follows:  
+
+$$
+[C(q,t)]\{\dot{q}\}+\{f(\dot{q},q,t)\}=\{\theta\}\qquad\qquad\mathrm{or,}\qquad[C(q,t)]\{\dot{q}\}=\{-\:f(\dot{q},q,t)\}
+$$  
+
+Platform:  
+
+The rigid lump mass of the platform brings about generalized inertia forces and generalized active forces associated with platform weight, hydrodynamics and hydrostatics, and mooring line and/or foundation elasticity and damping, including added mass effects.  
+
+Fr $\cdot\!\!\left|_{\boldsymbol{X}}={\}^{E}\nu_{r}^{\boldsymbol{Y}}\cdot\left(-m^{\chi\ E}a^{\boldsymbol{Y}}\right)+{}^{E}\omega_{r}^{\chi}\cdot\left(-{}^{E}\dot{\boldsymbol{H}}^{\chi}\right)\quad(r=I,2,\ldots,22)\qquad\qquad\mathrm{where}\qquad\qquad m^{\chi}$ $m^{X}=P t f m M a s s$  
+
+Thus, $\begin{array}{l}{{F_{r}^{*}\Big|_{{\cal X}}={^{E}}_{\psi_{r}}^{Y}\cdot\left(-m^{X}{^{E}}_{a}{^{Y}}\right)+{^{E}}_{\omega_{r}^{X}}\cdot\left(-\overline{{\overline{{{\cal I}}}}}{^{X}}\cdot{^{E}}_{a}{^{X}}-{^{E}}_{\omega}{^{X}}\times\overline{{\overline{{{\cal I}}}}}{^{X}}\cdot{^{E}}_{\omega}{^{X}}\right)\;\;\left(r=I,2,\ldots,22\right)}}\\ {{\mathrm{where}}}&{{\overline{{\overline{{\cal I}}}}{^{X}}=P t f m R l n e r a_{I}a_{I}+P t f m Y l n e r a_{2}a_{2}+P t f m P l n e r a_{3}a_{3}}}\end{array}$  
+
+Or,   
+$F_{r}^{*}\Big|_{X}=^{E}\psi_{r}^{Y}\cdot\left(-m^{X}\left\{\left(\sum_{i=I}^{6}\varepsilon_{\nu_{i}^{Y}}\ddot{q}_{i}\right)+\left[\sum_{i=I}^{6}\frac{d}{d t}\Big(^{\varepsilon}\nu_{i}^{Y}\Big)\dot{q}_{i}\right]\right\}\right)+^{E}\omega_{r}^{X}\cdot\left[-\overline{{I}}^{X}\cdot\left(\sum_{i=I}^{6}\varepsilon_{\omega_{i}^{X}}\ddot{q}_{i}\right)-^{E}\omega_{r}^{X}\ddot{q}_{i}\right]$ ω $\left(r=l,2,\ldots,\theta\right)$   
+Thus,   
+$\begin{array}{l}{\displaystyle\left[{\cal C}\left(q,t\right)\right]_{\scriptstyle\cal X}\left(R o w,C o l\right)=m^{\,\chi\,\,E}\,\nu_{R o w}^{\scriptscriptstyle\cal Y}\cdot{}^{\varepsilon}\nu_{c o l}^{\scriptscriptstyle\cal Y}+{}^{\varepsilon}\omega_{R o w}^{\scriptscriptstyle\cal X}\cdot\overline{{{\overline{{\mathbf{J}}}}}}\,{}^{\varepsilon}\cdot{}^{\varepsilon}\omega_{c o l}^{\scriptscriptstyle\cal X}\quad\left(R o w,C o l=l,2,\ldots,C o l\right)}\\ {\displaystyle\left\{-f\!\left(\dot{q},q,t\right)\right\}\right|_{\scriptscriptstyle\cal X}\left(R o w\right)\!=\!-m^{\,\chi\,\,E}\,\nu_{R o w}^{\scriptscriptstyle\cal Y}\cdot\left[\displaystyle\sum_{i=j}^{\delta}\frac{d}{d t}\!\left({}^{\varepsilon}\nu_{i}^{\scriptscriptstyle\cal Y}\right)\dot{q}_{i}\right]\!\!-\!{}^{\varepsilon}\omega_{R o w}^{\scriptscriptstyle\cal X}\cdot\left({}^{\varepsilon}\omega^{\scriptscriptstyle\cal X}\times\overline{{{\overline{{I}}}}}\,{}^{x}\cdot{}^{\varepsilon}\omega^{\scriptscriptstyle X}\right)\quad(\mathrm{~a~n~d~})}\end{array}$ 5 Row=1,2,,6)   
+$F_{r}\Big|_{G r a v X}=^{E}\nu_{r}^{Y}\cdot\left(-m^{X}g z_{2}\right)\quad\left(r=3,4,...,6\right)\qquad\qquad\mathrm{where}\qquad\qquad g=G r a v i t y$   
+Thus,   
+$\begin{array}{l}{\left[C\left(q,t\right)\right]\Bigr|_{G r a v X}=O}\\ {\left.\left\{-f\left(\dot{q},q,t\right)\right\}\Bigr|_{G r a v X}\left(R o w\right)\!=\!-m^{X}g^{\phantom{X}}\nu_{R o w}^{Y}\cdot z_{2}\quad\left(R o w=3,\mathcal{I},...,\delta\right)}\end{array}$  
+
+$$
+\left.F_{r}\right|_{H y d r o X}={^{E}\nu_{r}^{Y}}\cdot F_{H y d r o}^{Y}+{^{E}\omega_{r}^{X}}\cdot M_{H y d r o}^{X@Y}\quad\left(r=l,2,...,22\right)
+$$  
+
+$F_{H y d r o}^{Z}$ $M_{H y d r o}^{X@Z}$  
+
+$F_{H y d r o}^{Y}=F_{H y d r o}^{Z}$ and $M_{H y d r o}^{X\overline{{{(a)}}}}=M_{H y d r o}^{X\overline{{{(a)}}}Z}+r^{Y Z}\times F_{H y d r o}^{Z}=M_{H y d r o}^{X\overline{{{(a)}}}Z}-r^{Z Y}\times F_{H y d r o}^{Z}$ since $r^{\scriptscriptstyle T Z}=-r^{Z Y}$  
+
+But since ${}^{E}{\pmb{\nu}}_{r}^{Y}={}^{E}{\pmb{\nu}}_{r}^{Z}+{}^{E}{\pmb{\omega}}_{r}^{X}\times{\pmb{r}}^{Z Y}$ , this generalized active force can be expanded to:  
+
+$$
+\left.F_{r}\right|_{H\triangleright d r o X}=\left(^{E}\nu_{r}^{Z}+^{E}\omega_{r}^{X}\times r^{Z Y}\right)\cdot F_{H\triangleright d r o}^{Z}+^{E}\omega_{r}^{X}\cdot\left(M_{H\triangleright d r o}^{X\ G Z}-r^{Z Y}\times F_{H\triangleright d r o}^{Z}\right)\quad(r=I,2,\dots,22
+$$  
+
+Now applying the cyclic permutation law of the scalar triple product, the generalized active force simplifies to:  
+
+$$
+F_{r}|_{H_{y d r o X}}=^{E}\nu_{r}^{Z}\cdot F_{H_{y d r o}}^{Z}+^{E}\omega_{r}^{X}\cdot M_{H y d r o}^{X@Z}\quad\left(r=I,2,...,6\right)
+$$  
+
+But,  
+
+$$
+F_{H y d r o}^{Z}=\left(\sum_{j=I}^{6}F_{H y d r o_{j}}^{Z}\ddot{q}_{i}\right)+F_{H y d r o_{t}}^{Z}
+$$  
+
+$$
+M_{H y d r o}^{X\ @Z}=\left(\sum_{j=I}^{6}M_{H y d r o_{j}}^{X\ @Z}\ddot{q}_{i}\right)+M_{H y d r o_{t}}^{X\ @Z}
+$$  
+
+where,  
+
+$$
+\begin{array}{l l l l l}{\displaystyle{\mathbf{\Theta}_{H y d r o_{j}}^{Z}=-\left(\sum_{i=l}^{3}a_{i j}\mathbf{\Theta}^{E}\boldsymbol{\nu}_{i}^{Z}\right)}}&{\left(j=l,2,...,\theta\right)}&{\quad\mathrm{~and~}\quad\quad}&{\displaystyle{\mathbf{\Theta}_{H y d r o_{j}}^{X\equiv Z}=-\left(\sum_{i=l}^{6}a_{i j}\mathbf{\Theta}^{E}\boldsymbol{\omega}_{i}^{X}\right)}}\end{array}
+$$$$
+M_{H y d r o_{j}}^{X\ @Z}=-\!\left(\sum_{i=4}^{6}{a_{i j}}^{E}\omega_{i}^{X}\right)\;\;\left(j=l,2,...,6\right)
+$$  
+
+with $a_{i j}$ $\left(i,j=l,2,\ldots,\delta\right)$ being the added mass coefficients (or equivalently, $[a]$ being the added mass matrix), $F_{H\!y d r o_{j}}^{Z}\quad\left(j=l,2,\ldots,\right)$ and $\begin{array}{r l}{M_{H_{y}d r o_{j}}^{X(\underline{{\omega}}Z}}&{{}\left(j=l,2,...,6\right)}\end{array}$ being the partial hydrodynamic added mass forces and moments, and $F_{H y d r o_{t}}^{Z}$ and $M_{H y d r o_{t}}^{X@Z}$ being the contributions to $F_{H y d r o}^{Z}$ and $M_{H y d r o}^{X@Z}$ that don’t depend on platform accelerations.  
+
+Thus,  
+
+6 6 Fr HydroX ∑ FZ Hydrojqi + FZ Hydrot Eω ∑ MX@Z Hydrojqi +MHXy@drZo (r=1,2,,6) j =1 j =1  
+
+and $\begin{array}{r l}&{\left[C\left(q,t\right)\right]\Bigr|_{H\triangleright d r o X}\left(R o w,C o l\right)=\left[a\right]\left(R o w,C o l\right)=-\frac{E}{\nu}\nu_{R o w}^{Z}\cdot F_{H y d r o c_{d}}^{Z}-\frac{E}{\nu}\omega_{R o w}^{X}\cdot M_{H y d r o c_{d}}^{X\omega Z}}\\ &{\left\{-f\left(\dot{q},q,t\right)\right\}\Bigr|_{H\triangleright d r o X}\left(R o w\right)=\frac{E}{\nu}\nu_{R o w}^{Z}\cdot F_{H y d r o_{t}}^{Z}+\frac{E}{\nu}\omega_{R o w}^{X}\cdot M_{H y d r o_{t}}^{X\omega Z}\quad\left(R o w=I,2,\ldots,\delta\right)}\end{array}$ =1,2,,6)  
+
+# Tower:  
+
+The distributed properties of the tower bring about generalized inertia forces and generalized active forces associated with tower elasticity, tower damping, tower aerodynamics, and tower weight.  Note that I eliminated the tower mass tuners, since it is redundant to have both mass and stiffness tuners when trying to tune tower frequencies (to tune the frequencies for individual modes, all that is needed is to tune the mass or the stiffness for the individual modes, but not both).  Note also that I eliminated the tower stiffness tuner’s effects on the gravitational destiffening loads.  It is also beneficial to eliminate the tower mass tuners because the tower mass density is needed to compute the tower base loads and thus these tuners affect the tower base loads directly—this makes the form of the tower base load equations considerably more complex and considerably less intuitive. Since the tower elastic stiffness does not directly influence the tower base loads in a fundamental way, the retention of the tower stiffness tuners is much more favorable than the retention of the tower mass tuners (recall that only one set of tuners needs to be retained in order to permit the user to match natural frequencies).  The elimination of the tower stiffness tuner’s effects on the gravitational destiffening was done for the same reason (i.e., the gravity loads directly affect the tower base loads, and thus, tower stiffness tuners make the form of the tower base load equations considerable more complex and considerably less intuitive).  The fact that the gravitational destiffening of the tower is small compared to the overall stiffness of the tower is another reason this elimination of stiffness tuning effects should not be of significant concern.  
+
+![](images/e8b2ca880b8147c9b8b8bc1ebe78e9edd638f1dbb0fa508ed80d0999d727379c.jpg)  
+
+where $k_{\ i j}^{\,\prime T F A}$ and $k_{\mathbf{\Lambda}_{i j}}^{\mathbf{\Lambda}^{T S S}}$ are the generalized stiffnesses of the tower in the fore-aft and side-to-side directions respectively when gravitational destiffening effects are not included as follows:  
+
+k'iTjFA= $\begin{array}{r l}{\sqrt{F A S t T u n r left(i\right)F A S t T u n r\left(j\right)}\overset{T w r F l e c L}{\underset{o}{\int}}E I^{T F A}\left(h\right)\frac{d^{2}\phi_{i}^{T F A}\left(h\right)}{d h^{2}}\frac{d^{2}\phi_{j}^{T F A}\left(h\right)}{d h^{2}}d h}&{\left(i,j=I,2\right)\left(\frac{d^{2}\phi_{i}^{T F A}\left(h\right)}{d h^{2}}\right)}\\ {E I^{T F A}\left(h\right)=A d j F A S t\cdot T w F A S t i f\left(h\right)}&{}\end{array}$ )d2φiTFA2( h)d2φjTFA2( h)dh (i, j=1,2) (which is symmetric)  
+
+$$
+\begin{array}{r l}&{=\sqrt{S S S t T u n r\left(i\right)S S S t T u n r\left(j\right)}\overset{T w r F l e s L}{\underset{\theta}{\int}}E I^{T s s}\left(h\right)\frac{d^{2}\phi_{i}^{T s s}\left(h\right)}{d h^{2}}\frac{d^{2}\phi_{j}^{T s s}\left(h\right)}{d h^{2}}d h\quad\left(i,j=I,2\right)}\\ &{\in\quad\quad\quad E I^{T s s}\left(h\right)=A d j S S t\cdot T w S S S t i f\left(h\right)}\end{array}
+$$  
+
+The coefficient in front of the integral in these generalized stiffnesses represents the individual modal stiffness tuning, which allows the user to vary the stiffness of the tower between the individual modes to permit better matching of tower frequencies.  To be precise, the tuner coefficient only really makes sense when working with a generalized stiffness of a single mode (i.e., $k_{\;\;I I}^{\;\prime T F A},\;k_{\;\;22}^{\;\prime T F A},\;k_{\;\;I I}^{\;\prime T S S}$ , or $k\,_{\,22}^{\prime T S S}$ ), in which case the coefficient for mode $i$ is simply $F A S t T u n r(i)$ or SSStTunr $(i)$ .  However, since the cross-correlation elements of the generalized stiffness matrix will, in general, not vanish, the coefficient in the form above permits the tuning to apply to these terms in a consistent fashion.  
+
+![](images/0b65fb6e4f6782f33fff2b4fc56e3d349a6d994d86f02cd0eed472ec83faca1c.jpg)  
+
+where $\zeta_{i}^{T F A}$ and $\zeta_{i}^{T S S}$ represent  the  structural  damping  ratio  of  the  tower  for  the $i^{\mathrm{th}}$ mode  in  the  fore-aft  and  side-to-side  directions, $T w r F A D m p(i)/l O O$ and $T w r S S D m p(i)/l O O$ respectively.  Also, $f_{\ i}^{\,\prime T F A}$ and $f_{\ i}^{\prime T S S}$ represent the natural frequency of the tower for the $i^{\mathrm{th}}$ mode in the fore-aft and side-to-side directions respectively without tower-top mass or gravitational destiffening effects.  That is,  
+
+$$
+f_{\;\;i}^{\;\prime^{T F A}}=\frac{I}{2\pi}\sqrt{\frac{k_{\;i i}^{\prime^{T F A}}}{m_{\;i i}^{\prime^{T F A}}}}\;\;\;\;\;\mathrm{and}\;\;\;\;\;\;\;f_{\;i}^{\;\prime^{T S S}}=\frac{I}{2\pi}\sqrt{\frac{k_{\;i i}^{\prime T S S}}{m_{\;i i}^{\prime T S S}}}
+$$  
+
+where $m_{\;i i}^{\;\prime T F A}$ and $m_{\ i i}^{\prime T S S}$ represent the generalized mass of the tower for the $i^{\mathrm{th}}$ mode in the fore-aft and side-to-side directions respectively without tower-top mass effects as follows:  
+
+$$
+m_{\ i j}^{\prime^{T F A}}=\int_{0}^{T w r F l e x L}\mu^{T}\left(h\right)\phi_{i}^{T F A}\left(h\right)\phi_{j}^{T F A}\left(h\right)d h\quad\left(i,j=I,2\right)
+$$  
+
+and  
+
+$$
+m_{\it i j}^{\prime T S S}=\int_{0}^{T w r F l e x L}\mu^{T}\left(h\right)\phi_{i}^{T S S}\left(h\right)\phi_{j}^{T S S}\left(h\right)d h\quad\left(i,j=l,2\right)
+$$  
+
+$$
+\left[C(q,t)\right]\Bigr|_{E l a s t i c T}=O
+$$  
+
+#  
+
+Thus,   
+
+
+<html><body><table><tr><td>ElasticT</td><td></td><td>pue</td><td>DampT</td><td></td></tr><tr><td rowspan="5">-f(a,q,t)}</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td>TFA1TFA</td><td></td></tr><tr><td>9TFA1 9TFA2 TSS</td><td>π πf TSS ,TSS</td><td>TFA TFA 11 qTFA2 TFA TFA</td></tr><tr><td rowspan="10">ElasticT</td><td>1TSS1 1TSS2 .......</td><td>{-f(g,q,t)}</td><td>TSS ,TSS</td></tr><tr><td>YTFA1 4TFA2</td><td></td><td>71 TSS TSS TSS2 π</td></tr><tr><td>TSS TSS 4TSS1 YTSS2 22 ·.......................................................</td><td></td><td>....... TFA TFA TFA TFA</td></tr><tr><td></td><td>πfITFA</td><td>qTFA1 22 TFA2</td></tr><tr><td></td><td>21 qTSS1 DampT πfITSS</td><td>TSS TSS TSS 22 TSS2 πfITSS</td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td>...................................</td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td colspan="2"></td><td colspan="2"></td></tr></table></body></html>  
+
+$$
+F_{r}\Big|_{G r a v T}=\int_{0}^{\mathrm{\tiny~\itzurr}/c s L}\varepsilon_{\nu_{r}}^{{\cal T}}\left(h\right)\cdot\left[-\mu^{{\cal T}}\left(h\right)g z_{2}\right]d h+{^{E}\nu_{r}^{o}}\cdot\left(-Y a w B r M a s s\cdot g z_{2}\right)\quad\left(r=3,4,\ldots,4\right)\,.
+$$  
+
+{−f (q, $q,t)\Bigr\}|_{G r a v T}\left(R o w\right)=-\int_{0}^{T w r F|e x d}\mu^{T}\left(h\right)g^{\cal{E}}\nu_{R o w}^{T}\left(h\right)\cdot z_{2}d h-Y a w B r M a s s\cdot g^{\cal{E}}\nu_{R o w}^{o}\cdot z_{2}\quad\left(R o w\right)\times\ r e^{-i\{\frac{1}{2}\}w_{R o w}^{T}\},$ =3,4,,10)  
+
+# TwrFlexL $F_{r}\Big|_{A e r o T}=\int_{0}^{\mathbf{\pi}^{\prime}\cdot\mathbf{\mu}^{\mathrm{max}}}\Big[\,E_{\pmb{\nu}_{r}}^{\pmb{r}}\left(h\right)\cdot F_{A e r o T}^{T}\left(h\right)+\,^{E}\omega_{r}^{\pmb{r}}\left(h\right)\cdot M_{A e r o T}^{F}\left(h\right)\Big]d h\quad\left({r=I,2,\ldots,I\theta}\right)$  
+
+where $F_{A e r o T}^{T}\left(h\right)$ and $M_{A e r o T}^{F}\left(h\right)$ are aerodynamic forces and moments applied to point $\mathrm{T}$ on the tower respectively expressed per unit height. Thus,   
+C (q,t) =0  
+
+{−f (q,q,t $\left.\right)\right\vert_{A e r o T}\left(R o w\right)=\prod_{\theta}^{I W T\ i e x t}\left[\stackrel{E}{\sim}\nu_{R o w}^{T}\left(h\right)\cdot\stackrel{F}{\cdot}\nu_{A e r o T}^{T}\left(h\right)+\stackrel{E}{\cdot}\omega_{R o w}^{F}\left(h\right)\cdot M_{A e r o T}^{F}\left(h\right)\right]d h\quad\left(R o w=I,2\cdot\nu_{R o w}\right).$ ,,10)  
+
+![](images/2785e1e11903d0b60025e83287a636f56201f41f4f0d5fe8d19b39d3a4dd7eb8.jpg)  
+Yaw: The yaw spring and yaw damper bring about yaw moments.  
+
+Nacelle: The rigid lump mass of the nacelle brings about generalized inertia forces and generalized active forces associated with nacelle weight.  
+
+Fr N $=^{E}\nu_{r}^{U}\cdot\left(-m^{N\ E}a^{U}\right)+^{E}\omega_{r}^{N}\cdot\left(-^{E}\dot{H}^{N}\right)\quad(r=I,2,\ldots,22)\qquad\qquad\mathrm{where}\qquad\qquad m^{N}=E\cdot$ $m^{N}=N a c M a s s$  
+
+Thus, $\begin{array}{r l}&{F_{r}^{*}\Big|_{N}=^{\varepsilon}\pmb{\nu}_{r}^{U}\cdot\left(-m^{N\varepsilon}\pmb{a}^{U}\right)+^{\varepsilon}\pmb{\omega}_{r}^{N}\cdot\left(-\overline{{\overline{{I}}}}^{N}\cdot^{\varepsilon}\pmb{a}^{N}-^{E}\pmb{\omega}^{N}\times\overline{{\overline{{I}}}}^{N}\cdot^{\varepsilon}\pmb{\omega}^{N}\right)\quad\left(r=I,2,\ldots,22\right)}\\ &{\mathrm{where}\;\;\;\;\;\;\;\;\;\;\;\;\;\overline{{\overline{{I}}}}^{N}=\left[N a c Y I n e r-m^{N}\left(N a c C M x n^{2}+N a c C M y n^{2}\right)\right]\!d_{2}d_{2}}\end{array}$  
+
+Or, r $\left.^{*}\right|_{N}=\left.^{E}\nu_{r}^{U}\cdot\left(-m^{N}\left\{\left(\sum_{i=l}^{l l}E_{\nu_{i}^{U}}^{U}\ddot{q}_{i}\right)+\left[\sum_{i=d}^{l l}\frac{d}{d t}\!\left(^{E}\nu_{i}^{U}\right)\dot{q}_{i}\right]\right\}\right)+^{E}\omega_{r}^{N}\cdot\left(-\overline{{\overline{{I}}}}^{N}\cdot\left\{\left(\sum_{i=d}^{l l}E_{\omega_{i}^{N}}^{\omega_{i}}\ddot{q}_{i}\right)+\left[\sum_{i=d}^{l l}\frac{d}{d t}\!\left(^{E}\nu_{i}^{U}\right)\dot{q}_{i}\right]\right\}\right)\right|_{N},$ ∑ddt(EωiN)qi−EωN×IN⋅EωN $\left(r=l,2,...,I I\right)$   
+Thus,   
+C (q,t)N(Row,Col)=mNEvRUow⋅EvCUol+EωRNow⋅IN⋅EωCNol (Row,Col=1,2,,11)   
+{−f (q,q,t)} (Row)= −mNEvRUow⋅ ∑ddt(EviU)qi−EωRNow⋅IN⋅∑ddt(EωiN)qi+EωN×IN⋅Eω (Row=1,2,,11)   
+$F_{r}|_{G r a v N}={^{E}\nu_{r}^{U}}\cdot\left(-m{^{N}}g z_{2}\right)\;\;\;\left(r=3,4,...,l l\right)$   
+Thus,   
+$\begin{array}{l}{\left[C\!\left(q,t\right)\right]_{G r a v N}=O}\\ {\left.\left\{-f\!\left(\dot{q},q,t\right)\right\}\right|_{G r a v N}\left(R o w\right)\!=\!-m^{N}g^{\phantom{\frac{N}{L}}}\nu_{R o w}^{U}\cdot z_{2}\quad\left(R o w=3,4,\ldots,I I\right)}\end{array}$  
+
+Rotor-Furl: The rotor-furl springs (linear and stops) and rotor-furl dampers (Coulomb, linear, and stops) bring about rotor-furl moments.  
+
+![](images/e7c96cd1dc15d16c81ae9efd6a454fac0104b89a411f385ed3f5df7a46499a98.jpg)  
+
+Thus,  
+
+# C (q,t) =0 SpringRF  
+
+<html><body><table><tr><td>{-f(q,q,t))</td> rowspan="14"</td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td>-RFrlSpr·qRFrl -IF</td></tr><tr><td>[4Rn > RFrlUSSP, RFrlUSSpr(Rr - RFrlUSSP),0] SpringRF < RFrlDSSP,RFrlDSSpr (qRFr - RFrlDSSP),0</td></tr><tr><td>IF qRFrl</td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr></table></body></html>  
+
+and  
+
+# C (q,t) =0 DampRF  
+
+<html><body><table><tr><td></td> rowspan="14"></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td>.... ........</td></tr><tr><td></td><td></td><td>{-f(g,q,t)} DampRF</td></tr><tr><td>qRFrl</td><td></td><td></td></tr><tr><td colspan="2"><> 0, RFrlCDmp · SIGN(RFrl ),0 - IF[qrFn > RFrlUSDP, RFrlUSDmp ar,0]</td><td></td></tr><tr><td colspan="2">-IF[qRFrl < RFrlDSDP, RFrIDSDmp · rrr,0]</td><td></td></tr><tr><td colspan="2"></td><td></td></tr><tr><td colspan="2"></td><td></td></tr><tr><td colspan="2"></td><td></td></tr><tr><td colspan="2"></td><td></td></tr><tr><td colspan="2"></td><td></td></tr><tr><td colspan="2"></td><td></td></tr><tr><td colspan="2"></td><td></td></tr><tr><td colspan="2"></td><td></td></tr><tr><td colspan="2"></td><td></td></tr><tr><td colspan="2"></td><td></td></tr></table></body></html>  
+
+Structure That Furls with the Rotor (Not Including Rotor):  
+
+The rigid lump mass of the structure that furls with the rotor (not including the rotor) brings about generalized inertia forces and generalized active forces associated with the structure’s weight.  
+
+Fr $\cdot\bigg|_{R}=^{\varepsilon}\nu_{r}^{D}\cdot\left(-m^{R\ E}a^{D}\right)+^{E}\omega_{r}^{R}\cdot\left(-^{\varepsilon}\dot{H}^{R}\right)\quad\left(r=l,2,\dots,22\right)\quad\qquad\qquad\mathrm{where}\qquad\qquad m^{R}\bigg|_{R}^{}$ $m^{R}=R F r l M a s s$  
+
+Thus,  
+
+![](images/57385a94f60492a3dbfbfe875d9397cab105829ef7f2af76d4d87c91a11477e7.jpg)  
+
+$$
+\sum_{i=l}^{l2}\varepsilon_{\nu_{i}^{D}}\ddot{q}_{i}\left.\right)+\left[\sum_{i=l}^{l2}\frac{d}{d t}\Big(^{E}\nu_{i}^{D}\Big)\dot{q}_{i}\right]\Biggr]\dot{\ y}+^{E}\omega_{r}^{R}\cdot\left(-\overline{{{I}}}^{R}\cdot\left\{\left(\sum_{i=l}^{l2}\varepsilon_{\omega_{i}^{R}}\ddot{q}_{i}\right)+\left[\sum_{i=l}^{l2}\frac{d}{d t}\Big(^{E}\omega_{i}^{R}\Big)\dot{q}_{i}\right]\right\}-^{E}\omega_{\sigma_{r}}^{R}\right).
+$$  
+
+$$
+\begin{array}{l}{{\left.\begin{array}{l}{{\left.\begin{array}{l}{{\boldsymbol{\Sigma}\left(q,t\right)}}\end{array}\right|}\right|_{R}\left(R o w,C o l\right)=m^{R}\,^{E}\nu_{R o w}^{D}\cdot^{\nu}\nu_{C o l}^{D}+^{\;E}\omega_{R o w}^{R}\cdot\overline{{{\overline{{I}}}}}^{R}\cdot^{E}\omega_{C o l}^{R}\quad\left(R o w,C o l=I,2,\ldots,I\right)}}\\ {{\left.\cdot f\left(\dot{q},q,t\right)\right\rangle}\right|_{R}\left(R o w\right)\!=\!-m^{R}\,^{E}\nu_{R o w}^{D}\cdot\left[\displaystyle\sum_{i=d}^{l^{2}}\frac{d}{d t}\!\left(^{E}\nu_{i}^{D}\right)\dot{q}_{i}\right]\!-^{\;E}\omega_{R o w}^{R}\cdot\left\{\overline{{{\overline{{I}}}}}^{R}\cdot\left[\displaystyle\sum_{i=7}^{l^{2}}\frac{d}{d t}\!\left(^{E}\omega_{i}^{R}\right)\dot{q}_{i}\right]+}}\end{array}}}\end{array}
+$$  
+
+EωR×IR⋅Eω (Row=1,2,,12)  
+
+$F_{r}|_{G r a v R}={^{E}\nu_{r}^{D}}\cdot\left(-m{^{R}}g z_{2}\right)\;\;\;\left(r=3,4,...,l2\right)$ Thus, $\begin{array}{l}{\left.\left[C\left(q,t\right)\right]\right|_{G r a v R}=O}\\ {\left.\left\{-f\left(\dot{q},q,t\right)\right\}\right|_{G r a v R}\left(R o w\right)\!=\!-m^{R}g^{\phantom{\alpha}E}\nu_{R o w}^{D}\cdot z_{2}\quad\left(R o w=3,4,\ldots,I2\right)}\end{array}$  
+
+Teeter: The teeter springs (soft and hard stop) and teeter dampers (Coulomb and soft stop) bring about teeter moments.  
+
+![](images/b2da3bafdc6945f3f350aaa8c6ec474b49d58328893becb2ef44ad07e034ab2f.jpg)  
+
+# Thus, [C(q,t)]SpringTeet  
+
+<html><body><table><tr><td></td> rowspan="14"></td><td></td></tr><tr><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td>SpringTeet</td></tr><tr><td></td><td>{-f(g,q,t)}</td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td>IF >TeetHStP,TeetHSSp SIGN(qree )(lqree|- TeetHStP),0</td></tr><tr><td>>TeetSStP,TeetSSSp - SIGN(ree )(areeI- TeetSStP),0</td><td></td></tr></table></body></html>  
+
+# and $\left[C(q,t)\right]_{D a m p T e e t}=O$  
+
+<html><body><table><tr><td rowspan="14">{-f(g,q,t)}</td><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td>DampTeet</td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td>IF Tee <> 0,TeetCDmp · SIGN(ree ),0</td></tr><tr><td>IF > TeetDmpP,TeetDmp· iTeet, IaTee</td></tr></table></body></html>  
+
+Hub: The rigid lump mass of the hub brings about generalized inertia forces and generalized active forces associated with hub weight.  
+
+Fr\*H $={}^{E}\nu_{r}^{C}\cdot\left(-m^{H}{}^{E}a^{C}\right)+{}^{E}\omega_{r}^{H}\cdot\left(-{}^{E}\dot{H}^{H}\right)\quad(r=I,2,\ldots,22)\qquad\qquad\mathrm{where}\qquad\qquad m^{H}=1\mathrm{,}$ $m^{H}=H u b M a s s$  
+
+Thus, Fr\* =EvrC⋅(− $\begin{array}{r l}&{\boldsymbol{n}^{H\;E}\boldsymbol{a}^{c}\Big)+{}^{E}\boldsymbol{\omega}_{r}^{H}\cdot\left(-\overline{{\overline{{I}}}}^{H}\cdot\boldsymbol{\varepsilon}_{\alpha}^{H}-{}^{E}\boldsymbol{\omega}^{H}\times\overline{{\overline{{I}}}}^{H}\cdot\boldsymbol{\varepsilon}_{\omega}^{H}\right)\quad(r=I,2,...,22)}\\ &{\quad\overline{{\mathrm{\Omega}}}^{\overline{{\boldsymbol{r}}}N}=\left[\frac{H u b I n e r-\operatorname{\langle{I\rangle}}\big(U n d S l i n g-H u b C M\big)^{2}}{c o s^{2}\big(D e l t a3\big)}\right]\boldsymbol{g}_{I}\boldsymbol{g}_{I}+\left[\frac{H u b I n e r-m^{H}\big(U n d S l i n g-H u b I\big)}{c o s^{2}\big(D e l t a3\big)}\right]\boldsymbol{g}_{I}\boldsymbol{g}_{\alpha}}\end{array}$ where I bCM) g2 g2  
+
+since it is assumed that the hub is essentially a uniform rod directed along the $g_{3}$ axis and passing through the hub center of mass location (point C).  
+
+Note that if: $\left[\frac{H u b I n e r-m^{H}\left(U n d S l i n g-H u b C M\right)^{2}}{c o s^{2}\left(D e l t a3\right)}\right]<0$ , then there must be an error in the input file.  
+
+$\begin{array}{l l}{{\displaystyle\sum_{i=I}^{I d}\varepsilon_{\nu_{i}^{c}}\ddot{q}_{i}\left.\right)+{}^{E}\nu_{T e e t}^{c}\ddot{q}_{T e e t}+\left[\displaystyle\sum_{i=J}^{I d}\frac{d}{d t}\!\left({}^{\varepsilon}\nu_{i}^{c}\right)\dot{q}_{i}\right]+\frac{d}{d t}\!\left({}^{E}\nu_{T e e t}^{c}\right)\dot{q}_{T e e t}\left[\right)}}\\ {{\displaystyle\frac{\pi}{\Gamma}{\cal M}\cdot\left\{\left(\displaystyle\sum_{i=J}^{I d}{}^{E}\varepsilon_{\omega_{i}^{H}}\ddot{q}_{i}\right)+{}^{E}\omega_{T e e t}^{H}\ddot{q}_{T e e t}+\left[\displaystyle\sum_{i=7}^{I d}\frac{d}{d t}\!\left({}^{E}\omega_{i}^{H}\right)\dot{q}_{i}\right]+\frac{d}{d t}\!\left({}^{E}\omega_{T e e t}^{H}\right)\dot{q}_{T e t}\right\}-{}^{E}\omega^{H}\times\overline{{{\cal I}}}^{H}\cdot{\cal E}_{\omega}({}^{E})}}\end{array}$ (r=1,2,,14;Teet)  
+
+H  
+
+Thus,   
+C (q,t) $\begin{array}{r l}&{\left.\big|\right]_{H}\left(R o w,C o l\right)=m^{H\;E}\nu_{R o w}^{C}\cdot^{\;}\nu_{C o l}^{C}+^{.}\omega_{R o w}^{H\;}\cdot\overline{{\overline{{I}}}}^{H}\cdot^{.}\omega_{C o l}^{H}\quad\left(R o w,C o l=I,2,\ldots,I\mathcal{I};22\right)}\\ &{\;}\\ &{\left.q,t\right)\right\}\Bigr|_{H}\left(R o w\right)=-m^{H\;E}\nu_{R o w}^{C}\cdot\left\{\left[\displaystyle\sum_{i=q}^{\mathcal{H}}\displaystyle\frac{d}{d t}\Big(^{.}\nu_{\nu}^{C}\Big)\dot{q}_{i}\right]+\displaystyle\frac{d}{d t}\Big(^{\;}\nu_{\nu_{T e f}^{C}}^{C}\Big)\dot{q}_{T e e t}\right\}}\\ &{\qquad\qquad\qquad-^{\;E}\omega_{R o w}^{H\;}\cdot\left(\overline{{\overline{{I}}}}^{H}\cdot\left\{\displaystyle\sum_{i=q}^{\mathcal{H}}\displaystyle\frac{d}{d t}\Big(^{\;}\omega_{i}^{H}\Big)\dot{q}_{i}\right]+\displaystyle\frac{d}{d t}\Big(^{\;E}\omega_{r e e t}^{H\;}\Big)\dot{q}_{T e e t}\right\}+^{.}\omega^{H}\times\overline{{\overline{{I}}}}^{H}\cdot^{.}}\end{array}$   
+{−f (q, (Row=1,2,,14;2 ω  
+
+$\left.F_{r}\right|_{G r a v H}={^{E}\nu_{r}^{c}}\cdot\left(-m{^{H}}g z_{2}\right)\;\;\;\left(r=3,4,...,I4;T e e t\right)$ Thus, $\begin{array}{l}{{\left[C(q,t)\right]_{G r a v H}=O}}\\ {{\left.\left\{-f\bigl(\dot{q},q,t\bigr)\right\}\right|_{G r a v H}\left(R o w\right)\!=\!-m^{H}g^{\phantom{-}E}\nu_{R o w}^{C}\cdot z_{2}\quad\left(R o w\!=\!3,\mathcal{A},...,I\mathcal{I};22\right)}}\end{array}$  
+
+Blade 1:  
+
+The distributed properties of blade 1 bring about generalized inertia forces and generalized active forces associated with blade elasticity, blade damping, blade weight, and blade aerodynamics.  
+
+![](images/01182863bf924cb79d8a671dd8b07d9c1abe4eb2a38fe98fe5a43765e0690e04.jpg)  
+
+<html><body><table><tr><td>Q BITipE BldFlexL -m</td><td>Ev (r))ai v(r))ai ST (r)) VTeet GTeet dt i=16 18 > BldFlexL)ai > BldFlexL)ai ST BldFlexL) qTeet r =1,2,...,14;16,17,18;Teet =16 2 < BldFlexL )q BldFlexL )qi dt i=16 di BldFlexL qTeet Teet</td></tr></table></body></html>  
+
+Thus,  
+
+![](images/006427ca562878757dfd216aaa53ccf90c4de737d6a2edf30f7fe8b3113dd15a.jpg)  
+
+![](images/5b16b7be773e3aa88a5ea789f4397569fe7963fc771f2e98fadcd04790818bc9.jpg)  
+
+where $k_{\ i j}^{\,\prime B I F}$ and $\boldsymbol{k\,}_{I I}^{\prime B I E}$ are the generalized stiffnesses of blade 1 in the local flap and local edge directions respectively when centrifugal-stiffening effects are not included as follows:  
+
+k' ${}_{i j}^{.B I F}=\sqrt{F l S t T u n r^{B I}\left(i\right)F l S t T u n r^{B I}\left(j\right)}^{B I F l e x L}\int_{0}^{B I F}E I^{B I F}\left(r\right)\frac{d^{2}\phi_{i}^{B I F}\left(r\right)}{d r^{2}}\frac{d^{2}\phi_{j}^{B I F}\left(r\right)}{d r^{2}}d r^{\prime}\quad(i,j\neq2),$ where $E I^{B I F}\left(r\right)=A d j F l S t^{B I}\cdot F l p S t f f^{B I}\left(r\right)$  
+
+and  
+
+where $E I^{B I E}\left(r\right)=A d j E d S t^{B I}\cdot E d g S t f f^{B I}\left(r\right)$  
+
+$$
+k_{\;\;I I}^{\;^{\prime B I E}}=\int_{0}^{B l d F l e x L}E I^{B I E}\left(r\right)\left[\frac{d^{2}\phi_{I}^{B I E}\left(r\right)}{d r^{2}}\right]^{2}d r
+$$  
+
+Similarly, when using the Rayleigh damping technique where the damping is assumed proportional to the stiffness, then  
+
+$$
+\boldsymbol{F}_{r}\Big\rvert_{D a m p B I}=\left\{\begin{array}{l l}{-\frac{\zeta_{I}^{B I F}k_{I I}^{B I F}}{\pi f^{\prime,B I F}}\boldsymbol{\dot{q}}_{B I F I}-\frac{\zeta_{2}^{B I F}k_{I I}^{B I F}}{\pi f^{\prime,B I F}}\boldsymbol{\dot{q}}_{B I F2}}&{f o r}&{r=B I F I}\\ {-\frac{\zeta_{I}^{B I E}k_{I I}^{B I E}}{\pi f^{\prime,B I E}}\boldsymbol{\dot{q}}_{B I E I}}&{f o r}&{r=B I E I}\\ {-\frac{\zeta_{I}^{B I F}k_{I I}^{B I F}}{\pi f^{\prime,B I F}}\boldsymbol{\dot{q}}_{B I F I}-\frac{\zeta_{2}^{B I F}k_{I2}^{B I F}}{\pi f^{\prime,B I F}}\boldsymbol{\dot{q}}_{B I F2}}&{f o r}&{r=B I F2}\\ {0}&{o t h e r w i s e}\end{array}\right.
+$$  
+
+where $\zeta_{i}^{B I F}$ and $\zeta_{i}^{B I E}$ represent the structural damping ratio of blade 1 for the $i^{\mathrm{th}}$ mode in the local flap and edge directions, $B l d F l D m p^{B I}(i)/l O0$ and $B l d E d D m p^{B l}\left(i\right)/l O O$ respectively.  Also, ${f^{\prime}}_{i}^{B I F}$ and $\boldsymbol{f}_{\ i}^{\prime B I E}$ represent the natural frequency of blade 1 for the $i^{\mathrm{th}}$ mode in the local flap and edge directions respectively without centrifugal-stiffening effects.  That is,  
+
+$$
+f_{\;i}^{\prime B I F}=\frac{I}{2\pi}\sqrt{\frac{k_{\;i i}^{\prime B I F}}{m_{\;i i}^{\prime B I F}}}\quad\mathrm{and}\qquad f_{\;i}^{\prime B I E}=\frac{I}{2\pi}\sqrt{\frac{k_{\;i i}^{\prime B I E}}{m_{\;i i}^{\prime B I E}}}
+$$  
+
+where $m_{\ i i}^{\prime B I F}$ and $m_{\ i i}^{\prime B I E}$ represent the generalized mass of blade 1 for the $i^{\mathrm{th}}$ mode in the local flap and edge directions respectively without centrifugal-stiffening and tip mass effects as follows:  
+
+$$
+m_{\ i j}^{\prime B I F}=\int_{\ o}^{\ B I d F l e x L}\mu^{B I}\left(r\right)\phi_{i}^{B I F}\left(r\right)\phi_{j}^{B I F}\left(r\right)d r\quad\left(i,j=I,2\right)
+$$  
+
+and  
+
+$$
+m_{~I I}^{~\!B I E}=\int_{0}^{\!~\!B I d F l e x L}\mu^{B I}\left(r\right)\!\left[\phi_{I}^{B I E}\left(r\right)\right]^{2}d r
+$$  
+
+# Thus, [C(q,t)]E =0 lasticB1  
+
+and $\left[C(q,t)\right]_{D a m p B I}=O$  
+
+<html><body><table><tr><td colspan="3">lasticB</td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td>.....................................................................</td><td></td></tr><tr><td>{-f(g,q,t)} ElasticB1</td><td>.........................................................</td><td>++................</td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td>{-f(a,q,t)}</td><td>DampB1</td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td>yBIF B1F B1F BIF</td></tr><tr><td></td><td></td><td>11 12 qB1F2 BIF 1B1F1 B1F</td></tr><tr><td></td><td>qB1F1 12 B1F2 IB1</td><td>πf πf YBIE B1E</td></tr><tr><td></td><td>9BIE1</td><td>11 B1E1 B1E</td></tr><tr><td>B11 k 21</td><td>B1F B1F1 K B1F2. 22</td><td>πf ................ ...............….</td></tr><tr><td></td><td></td><td>yB1F1 yBIF BIF 2 K 22 1B1F2</td></tr><tr><td></td><td></td><td>BIF B1F1 πf πfIBIF ...........................</td></tr><tr><td></td><td></td><td>.............................. .....................</td></tr><tr><td colspan="2"></td><td>..................................</td></tr><tr><td colspan="2"></td><td></td></tr></table></body></html>  
+
+![](images/53e0033471267f7088b7241519ec842e1c94731ae66e9f17bc5e117c185390a8.jpg)  
+
+${\cal F}_{r}\Big|_{A e r o B I}=\int_{0}^{B H F e x I}\Big[\,{\cal E}\,_{\nu}^{S I}\left(r\right)\cdot{\cal F}_{A e r o B I}^{S I}\left(r\right)+\,^{E}\omega_{r}^{M I}\left(r\right)\cdot{\cal M}_{A e r o B I}^{M I}\left(r\right)\Big]d r+\,^{E}\nu_{r}^{S I}\left(B l d F l e x L\right)\cdot$ $^{t I}_{e r o B I}\left(r\right)\right]d r+^{E}\nu_{r}^{S I}\left(B l d F l e x L\right)\cdot F_{r i p r a g B I}^{S I}\left(B l d F l e x L\right)\quad(r=I,2,\ldots,I\mathcal{I};I\delta,I7,I\delta;T e e t)$  
+
+where $F_{A e r o B I}^{S I}(r)$ and $M_{A e r o B I}^{M I}\left(r\right)$ are aerodynamic forces and pitching moments applied to point S1 on blade 1 respectively expressed per unit span. Note that $M_{A e r o B I}^{M I}\left(r\right)$ can include effects of both direct aerodynamic pitching moments (i.e., $\mathrm{Cm})$ ) and aerodynamic pitching moments caused by an aerodynamic offset (i.e., moments due to aerodynamic lift and drag forces acting at a distance away from the center of mass of the blade element along the aerodynamic chord).   
+Thu  
+
+![](images/128ab85f26d83e90bb74daf0f45cca8fdb8d9558720db62c0b117c68efd0d047.jpg)  
+
+Blade 2:  
+
+Just like blade 1, the distributed properties of blade 2 bring about generalized inertia forces and generalized active forces associated with blade elasticity, blade damping, blade weight, and blade aerodynamics.  The equations for $F_{r}^{*}\Big|_{_{B2}},\;F_{r}\big|_{E l a s t i c B2}\,,\;F_{r}\big|_{D a m p B2},\;F_{r}\big|_{G r a v B2}$ , and $F_{r}|_{A e r o B2}$ are similar to those of blade 1.  
+
+# Drivetrain:  
+
+The inertia of the drivetrain brings about generalized inertia forces and the load in the generator, high-speed shaft brake, gearbox (friction forces resulting from nonzero GBoxEff ) and the flexibility of the low speed shaft bring about generalized active forces.  Note that all of these equations assume that the rotor is spinning about the positive $c_{I}$ axis (they assume that the rotor can’t be forced to rotate in the opposite direction).  This model works for any gearbox arrangement (including no gearbox, single stage, or multi-stage) as long as the generator rotates about the shaft axis (it may not be skewed relative to the shaft, even though it may rotate in the opposite direction of the low-speed shaft due to the gearbox stages).  If there is no gearbox, simply set $G B R a t i o\,=\,G B o x E f f=G e n D i r=1$ (GB R $e\nu e r s e={\mathrm{False}}$ ).  
+
+The mechanical torque within the generator is applied to the high speed shaft and equally and oppositely to the structure that furls with the rotor as follows:  
+
+$$
+\left.F_{r}\right|_{G e n}=\left(^{E}\omega_{r}^{G}-{^{E}\!\omega_{r}^{R}}\right)\cdot M_{G e n}^{G}\quad\left(r=l,2,...,22\right)
+$$  
+
+Thus,  
+
+$$
+\left.F_{r}\right|_{G e n}=\left\{\!\!\!\begin{array}{l l}{{^{E}\pmb{\omega}_{G e d z}^{G}\cdot M_{G e n}^{G}}}&{{f o r~~r=G e A z}}\\ {{0}}&{{o t h e r w i s e}}\end{array}\ \right.
+$$  
+
+$$
+\boldsymbol{M_{G e n}^{G}}=-G e n D i\boldsymbol{r}\cdot\boldsymbol{T^{G e n}}\left(G B R a t i o\cdot\dot{q}_{G e d z},t\right)\boldsymbol{c}_{I}
+$$  
+
+Note that a positive $T^{G e n}$ represents a load (positive power extracted) whereas a negative $T^{G e n}$ represents a motoring-up situation (negative power extracted, or power input).  Thus,  
+
+$$
+F_{r}\Big|_{G e n}=\left\{\begin{array}{l l}{\left(G e n D i r\cdot G B R a t i o\cdot c_{I}\right)\cdot\left[-G e n D i r\cdot T^{G e n}\left(G B R a t i o\cdot\dot{q}_{G e d z},t\right)c_{I}\right]}&{f o r\;\;\;r=G e n D i r}\\ {0}&{o t h e r w i s e}\end{array}\right.
+$$  
+
+Or since $G e n D i r^{2}=I$ ,  
+
+# Thus, $\left[C(q,t)\right]_{G e n}=0$  
+
+<html><body><table><tr><td rowspan="14">{-f(g,q,t)}</td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td>Gen</td></tr><tr><td>GBRatio·T Gen GBRatio·qGeAz'</td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr></table></body></html>  
+
+Similarly, the mechanical torque applied to the high-speed shaft from the high-speed shaft brake is applied equally and oppositely to the structure that furls with the rotor.  Thus,  
+
+$F_{r}\big|_{B r a k e}=\left\{\!\!\!\begin{array}{l l}{{^{E}\pmb{\omega}_{G e A z}^{G}\cdot M_{B r a k e}^{G}}}&{{f o r\;\;\;r=G e A z}}\\ {{0}}&{{o t h e r w i s e}}\end{array}\!\!\right.$ where $M_{B r a k e}^{G}=-G e n D i r\cdot T^{B r a k e}\left(t\right)c_{I}\qquad\mathrm{~and~where}\qquad T^{B r a k e}\left(t\right)=H S S B r k T\left(t\right)$  
+
+which is assumed to be positive in value always.  Thus,  
+
+$$
+F_{r}\bigr|_{B r a k e}=\left\{\begin{array}{l l}{-G B R a t i o\cdot T^{B r a k e}\left(t\right)}&{f o r\;\;\;r=G e A z}\\ {O}&{o t h e r w i s e}\end{array}\right.
+$$  
+
+# Thus,  
+
+$$
+\left[C(q,t)\right]\Bigr|_{B r a k e}={\cal O}
+$$  
+
+<html><body><table><tr><td rowspan="14">(-f(4,q,t) Brake</td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td>GBRatio·TBrake</td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr></table></body></html>  
+
+If the translational inertia of the drivetrain is assumed to be incorporated into that of the structure that furls with the rotor, then the high-speed shaft generator inertia generalized force is as follows:  
+
+Fr\*G= ${^{E}\omega_{r}^{G}}\cdot\left(-\overline{{{\overline{{{I}}}}}}^{G}\cdot{^{E}\alpha^{G}}-{^{E}\omega^{G}}\times\overline{{{\overline{{{I}}}}}}{^{G}}\cdot{^{E}\omega^{G}}\right)\quad(r=I,2,\dots,22)\qquad\mathrm{~where~}\qquad\qquad\overline{{{\overline{{{I}}}}}}^{G}=G e r$ $\overline{{\overline{{I}}}}^{G}=G e n I n e r c_{I}c_{I}$  
+
+or,  
+
+$$
+F_{r}^{*}\Big|_{G}=^{E}\omega_{r}^{G}\cdot\left\{-\overline{{\overline{{I}}}}^{G}\cdot\left\{\left(\sum_{i=\ell}^{l3}\varepsilon_{\theta_{i}^{G}}\ddot{q}_{i}\right)+\left[\sum_{i=\ell}^{l3}\frac{d}{d t}\Big(^{\varepsilon}\omega_{i}^{G}\Big)\dot{q}_{i}\right]\right\}-^{E}\omega^{G}\times\overline{{\overline{{I}}}}^{G}\cdot^{E}\omega^{G}\right\}\quad(r=l,2),
+$$  
+
+However, since $c_{I}\cdot\frac{d}{d t}\Big(\{^{E}\omega_{G e A z}^{G}\Big)\propto c_{I}\cdot\Big(^{E}\omega^{R}\times c_{I}\Big)={^{E}\!\omega^{R}}\cdot\Big(c_{I}\times c_{I}\Big)=0$ (the first $c_{I}$ coming from $\overline{{\overline{{I}}}}^{\,\!\overline{{\sigma}}}$ ), this simplifies as follows:  
+
+$$
+F_{r}^{*}\Big|_{G}=^{E}\omega_{r}^{G}\cdot\left\{-\overline{{\overline{{I}}}}^{G}\cdot\left\{\left(\sum_{i=4}^{l3}\varepsilon_{\theta_{i}^{G}}\ddot{q}_{i}\right)+\left[\sum_{i=7}^{l2}\frac{d}{d t}\Big(^{E}\omega_{i}^{G}\Big)\dot{q}_{i}\right]\right\}-^{E}\omega^{G}\times\overline{{\overline{{I}}}}^{G}\cdot^{E}\omega^{G}\right\}\quad(r=l,2),
+$$  
+
+$$
+F_{r}^{*}\Bigr|_{G}=\left\{\begin{array}{l l}{\displaystyle-\varepsilon_{\omega_{r}}^{\varepsilon}\cdot\overline{{I}}^{\varepsilon}\cdot\left\{\left(\sum_{i=d}^{l/3}\varepsilon_{\omega_{i}^{G}}^{\alpha_{i}}\ddot{q}_{i}\right)+\left[\sum_{i=r}^{l/2}\frac{d}{d t}\big(^{\varepsilon}\omega_{i}^{R}\big)\dot{q}_{i}\right]\right\}-\varepsilon_{\omega_{r}^{R}}^{\varepsilon}\cdot\left({^{\varepsilon}\omega^{R}}\times\overline{{I}}^{G}\cdot{^{\varepsilon}}\omega^{G}\right)}\\ {\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left[\sum_{i=d}^{l/2}\varepsilon_{\omega_{i}^{R}}\ddot{q}_{i}\right]+\left[\sum_{i=l}^{l/2}\frac{d}{d t}\big(^{\varepsilon}\omega_{i}^{R}\big)\dot{q}_{i}\right]\right\}\cdot c_{l}-G e n I n}\\ {\displaystyle-G e n D i r\cdot G B R a t i o c_{l}\cdot\left({^{\varepsilon}\omega^{\varepsilon}}\times\overline{{I}}^{G}\cdot{^{\varepsilon}}\omega^{G}\right)}\\ {\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\omega_{r}^{G}\times\overline{{I}}^{G}\cdot{^{\varepsilon}}\omega^{G}\right)}\\ {\displaystyle\theta}\end{array}\right.
+$$  
+
+However since $c_{\iota}\cdot\left({\,^{E}}{\omega^{G}}\times c_{I}\,\right)={^{E}}{\omega^{G}}\cdot\left(c_{I}\times c_{I}\right)=0$ (the first $c_{I}$ coming from $\overline{{\overline{{I}}}}^{\omega}$ ), this simplifies again as follows:  
+
+![](images/642199797d87d58d786d3a5ee6c8303176b9d0f672825a85f3b35618b3df225b.jpg)  
+
+The terms associated with DOFs 4,5,…,12 represent the fact that the rate of change of angular momentum of the generator can be considered as an additional torque on the structure that furls with the rotor (i.e., in addition to the torques on the structure transmitted directly from the low-speed shaft).  
+
+![](images/20054a9c031e903de79d1d748ec640b3585cfb9c2f27b07cddb92860bea33e91.jpg)  
+
+#  
+
+Thus,   
+
+
+<html><body><table><tr><td colspan="3">ElasticDrive</td><td colspan="3">C(q,t) DampDrive</td></tr><tr><td colspan="3">{-f(q,q,t)}] ElasticDrive</td><td colspan="3">andlikewise (-f(4,q,t)</td></tr><tr><td colspan="3"></td><td></td><td></td></tr><tr><td colspan="3"></td><td></td><td>…....................................................</td></tr><tr><td colspan="3"></td><td></td></tr><tr><td colspan="3">··................................................</td><td></td></tr><tr><td colspan="3"></td><td></td></tr><tr><td colspan="3"></td><td></td></tr><tr><td colspan="3">...............................................</td><td>..............................</td></tr><tr><td colspan="3">+..............................................</td><td></td></tr><tr><td colspan="3"></td><td></td></tr><tr><td colspan="3">..............................................</td><td>IDampDrive</td></tr><tr><td colspan="3">-DTTorSpr · qDrTr</td><td>-DTTorDmp·qDrTr</td></tr><tr><td colspan="3">.................................................</td><td>.....................................................</td></tr><tr><td colspan="3">··.............................................·..</td><td>.·.................................................·.</td></tr><tr><td colspan="3">+ +...........................</td><td>....................................................</td></tr><tr><td colspan="3"></td><td></td></tr><tr><td colspan="3"></td><td></td></tr><tr><td colspan="3"></td><td></td></tr></table></body></html>  
+
+Similar to the generator and HSS brake, the mechanical friction torque applied to the high speed shaft is applied equally and oppositely to the structure that furls with the rotor.  Thus,  
+
+F $=\int_{0}^{\varepsilon}\!\omega_{G e i z}^{G}\cdot M_{\mathit{c B F r i c}}^{G}\quad\mathit{f o r}\quad r=G e d z\qquad\qquad\mathrm{where}\qquad\qquad M_{\mathit{G B F r i c}}^{G}=-\frac{T^{G B F r i c}\left(\ddot{q},\dot{q},\dot{q}\right)}{G B R a t i o\cdot G e^{G B F r i c}}\,,$ GBFric  
+
+where, from a free-body diagram of the high and low-speed shafts, it is easily seen that the friction torque applied on the LSS upon the gearbox entrance, $T^{G B F r i c}\left(\ddot{q},\dot{q},q,t\right)$ , is always positive in value and equal to:  
+
+$$
+,t\bigr)\mathop{=}^{\sim}\biggl[G e n l n e r\cdot G B R a t i o^{2}\cdot\ddot{q}_{G e d z}+G e n D i r\cdot G e n l n e r\cdot G B R a t i o^{E}a^{R}\cdot c_{I}\biggr]\cdot\biggl[\frac{I}{G B o x E/f^{S I G}}\Bigl(G B R a t i o\cdot\dot{q}_{G e d z},t\Bigr)+G B R a t i o\cdot T^{B r a k e}\left(t\right)\biggr]^{2}.
+$$  
+
+Thus,   
+$\left.F_{r}\right|_{G B F r i c}=\left\{\begin{array}{l l}{-T^{G B F r i c}\left(\ddot{q},\dot{q},q,t\right)}&{f o r~~r=G e A z}\\ {0}&{o t h e r w i s e}\end{array}\right.$   
+or, $\cdot\!\!\!\mid_{G B F r i c}=\left\{\!\!\!\left[\!\!\begin{array}{c}{{G e n I n e r\cdot G B R a t i o^{\prime}\cdot\ddot{q}_{G e d z}+G e n D i r\cdot G e n I n e r\cdot G B R a t i o^{\varepsilon}a^{R}\cdot c_{I}}}\\ {{+G B R a t i o\cdot T^{G e n}\left(G B R a t i o\cdot\dot{q}_{G e d z},t\right)+G B R a t i o\cdot T^{B r a k e}\left(t\right)}}\end{array}\!\!\right]\!:\left[\!\!\frac{1}{G B o x}\!\!\!\right]\!:=\!\!\!\left[\!\!\begin{array}{c}{{\Psi_{0}}}\\ {{G e n I}}\\ {{\Psi_{1}}}\end{array}\!\!\!\right]\!\!:=\!\!\!\left[\!\!\begin{array}{c}{{\Psi_{0}}}\\ {{G e n I}}\\ {{\Psi_{2}}}\end{array}\!\!\!\right]\!\!.$ 1   
+F GBoxEffSIGN (LSShftTq) for r=GeAz otherwise  
+
+<html><body><table><tr><td>or, F GBFric</td><td>12 12 M E R wqi GenIner·GBRatio GenDir·GenIner·GBRatio E R 9 + + GeAz b dt i=4 i=7 GBRatio·T Gen GBRatio·q GBRatio·T Brake 十 GeAz 十</td><td>for r= GeAz SIGN(LSShftTq) GBoxEff otherwise</td></tr></table></body></html>  
+
+Thus,  
+
+<html><body><table><tr><td rowspan="11">C(q,t)] GBFric {-f(g,q,t)) GBFric</td><td>Row</td><td>for GBoxEff SIGN(LSShftTq)</td><td rowspan="2">(Row = 13,Col =4,5,...,13 otherwise</td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td colspan="5"></td></tr><tr><td colspan="5"></td></tr><tr><td colspan="5"></td></tr><tr><td colspan="5"></td></tr><tr><td colspan="5"></td></tr><tr><td colspan="5"></td></tr><tr><td colspan="5"></td></tr><tr><td></td><td>(E)a + GBRatio·TGen</td><td colspan="2">" (GBRatio · IGeaz ,t)+ GBRatio ·T</td><td>GBoxEff fSIGN(LSShftTq)</td></tr><tr><td colspan="5"></td></tr><tr><td colspan="5"></td></tr><tr><td colspan="5"></td></tr><tr><td colspan="5"></td></tr><tr><td colspan="5"></td></tr><tr><td colspan="5"></td></tr><tr><td colspan="5"></td></tr></table></body></html>  
+
+It is noted that the gearbox friction generalized active force effects both the mass matrix and the forcing function.  Its effect on the mass matrix can be thought of as an apparent mass in the system (i.e., an active friction force which is a function of the generator acceleration).  The gearbox friction causes the mass matrix to become unsymmetric.  Note that the equation for DOF GeAz is an implicit equation (since the gearbox friction is a function of DOF GeAz), which has no closed-form solution.  To avoid the complications resulting from this implicit characterization, the value of the LSShftTq  from the previous time step is used in the SIGN() function in place of the value of the LSShftTq  in the current time step.  This may be done since it may be assumed that LSShftTq  will not change much between time steps if the time step is considered small enough.  Note that gearbox friction is the only term in the equations of motion that cause the mass matrix to be unsymmetrical.  The mass matrix will only be unsymmetric if GBoxEff  is not $100\%$ —if GBoxEff  is $100\%$ , then the mass matrix is completely symmetric.  
+
+Tail-Furl: The tail-furl springs (linear and stops) and tail-furl dampers (Coulomb, linear, and stops) bring about tail-furl moments.  
+
+![](images/204043d4393282b60c23ffd1a705bb7bcc2a5b8932d1c97208bbbaaa7853497b.jpg)  
+
+# Thus, C (q,t) =0 SpringTF  
+
+<html><body><table><tr><td>{-f(g,q,t)}</td> rowspan="14"</td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td>SpringTF</td></tr><tr><td>-TFrlSpr·qrFrl -IF [4rFr > TFrlUSSP,TFrlUSSpr(qr - TFrlUSSP),0</td></tr><tr><td>< TFrIDSSP,TFrIDSSpr (a - TFrIDSSP),0</td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr></table></body></html>  
+
+# C (q,t)D =0 ampTF  
+
+<html><body><table><tr><td rowspan="14">{-f(g,q,t)}</td><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td>DampTF</td></tr><tr><td></td></tr><tr><td>-IF[qrFrl <TFrIDSDP,TFrIDSDmp ·rrl,0]</td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr><tr><td></td></tr></table></body></html>  
+
+Tail:  
+
+The rigid lump masses of the tail boom and tail fin bring about generalized inertia forces and generalized active forces associated with the structure weight.  Additionally, the tail fin brings about generalized active forces associated with tail fin aerodynamics.  
+
+Fr\*A=EvrI⋅(−mBEaI)+ ${}\quad^{\!\!E}\omega_{r}^{A}\cdot\left(-^{E}\dot{H}^{A}\right)+{}^{E}\nu_{r}^{J}\cdot\left(-m^{F\,E}a^{J}\right)\quad\left(r=I,2,\ldots,22\right)\quad\qquad\mathrm{where}\qquad\qquad m^{B}=B o o m$ $m^{B}=B o o m M a s s\qquad\mathrm{and}\qquad m^{F}=T F i n M a s s$   
+Thus,   
+Fr\* =EvrI⋅(−mBEaI)+EωrA⋅(−IA⋅EαA−EωA×IA⋅EωA)+EvrJ⋅(−mFEaJ) (r=1,2,,22)  
+
+<html><body><table><tr><td colspan="2">A = where (BoomCMxn - TFrlPntxn)²{1-cos²(TFrlSkew)cos²(TFrlTilt)</td><td>TFrlIner-m W tfa tfa or</td></tr><tr><td rowspan="7">A =TFrllner -BoomMass</td><td colspan="2">+(BoomCMzn-TFrlPntzn)cos²(TFrlTilt tfa tfa</td></tr><tr><td colspan="2"></td></tr><tr><td colspan="2">+(BoomCMyn -TFrlPntyn)|1- sin²(TFrlSkew)cos²(TFrlTilt) (BoomCMxn-TFrlPntxn)(BoomCMzn-TFrlPntzn)cos(TFrlSkew)cos(TFrlTilt)sin(TFrlTilt</td></tr><tr><td colspan="2">-2</td></tr><tr><td colspan="2">+(BoomCMxn - TFrlPntxn)(BoomCMyn - TFrlPntyn)cos(TFrlSkew)sin(TFrlSkew)cos² (TFrlTilt)</td></tr><tr><td colspan="2">+(BoomCMzn - TFrlPntzn)(BoomCMyn-TFrlPntyn)sin(TFrlSkew)cos(TFrlTilt)sin(TFrlTilt)</td></tr></table></body></html>  
+
+$$
+\begin{array}{l}{{\displaystyle{1^{B}\left\{\left(\sum_{i=I}^{I I}\varepsilon_{\nu_{i}^{I}}\ddot{q}_{i}\right)+{}^{E}{\nu}_{T F I}^{I}\ddot{q}_{T F r I}+\left[\sum_{i=d}^{I I}\frac{d}{d t}\big({}^{E}{\nu}_{\nu_{i}^{I}}\big)\dot{q}_{i}\right]+\frac{d}{d t}\big({}^{E}{\nu}_{T F I}^{I}\big)\dot{q}_{T F r I}\big\}}}\\ {{\displaystyle{1:\left(-\overline{{{\bar{I}}}}^{A}\cdot\left\{\left(\sum_{i=d}^{I I}\varepsilon_{\nu_{i}^{J}}\ddot{q}_{i}\right)+{}^{E}{\omega}_{T F I}^{A}\ddot{q}_{T F I}+\left[\sum_{i=j}^{I I}\frac{d}{d t}\big({}^{E}{\omega}_{i}^{A}\big)\dot{q}_{i}\right]+\frac{d}{d t}\big({}^{E}{\omega}_{T F I I}^{A}\big)\dot{q}_{T F I}\right\}-{}^{E}{\omega}^{A}\times\overline{{{\bar{I}}}}^{B}\right.}}\\ {{\displaystyle{\cdot\left(-m^{F}\left\{\left(\sum_{i=I}^{I I}\varepsilon_{\nu_{i}^{J}}\ddot{q}_{i}\right)+{}^{E}{\nu}_{T F I}^{J}\ddot{q}_{T F I}+\left[\sum_{i=d}^{I I}\frac{d}{d t}\big({}^{E}{\nu}_{\nu_{i}^{J}}\big)\dot{q}_{i}\right]+\frac{d}{d t}\big({}^{E}{\nu}_{T F I}^{J}\big)\dot{q}_{T F I}\right\}}}\end{array}
+$$  
+
+A⋅EωA (r=1,2,,11;TFrl)  
+
+Thus,  
+
+$\begin{array}{l}{{\left[C\left(q,t\right)\right]_{A}\left(R o w,C o l\right)=m^{B}\,^{E}\nu_{R o w}^{I}\cdot^{\varepsilon}\nu_{{c_{o}}\!\!\!\nu}^{I}+^{E}\omega_{R o w}^{4}\cdot\overline{{{\overline{{I}}}}}\cdot^{A}\cdot^{E}\omega_{C o l}^{A}+m^{F}\,^{E}\nu_{{t o w}}^{J}\cdot^{\varepsilon}\nu_{{c_{o}}\!\!\!\nu}^{J}\quad(R o W)}}\\ {{\left\{-f\left(\dot{q},q,t\right)\right\}_{A}\left[R o w\right]=-m^{B}\,^{E}\nu_{{t o w}}^{I}\cdot\left\{\left[\displaystyle\sum_{i=d}^{I I}\frac{d}{d t}\Big(^{\varepsilon}\nu_{{i}}^{I}\Big)\dot{q}_{i}\right]\!+\!\frac{d}{d t}\Big(^{\varepsilon}\nu_{{r_{I F I}}\!\!\nu_{I}}^{I}\Big)\dot{q}_{{T r i}}\!\right\}}}\\ {{\mathrm{~}\qquad\qquad-\ ^{E}\omega_{R o w}^{A}\cdot\left(\overline{{{\overline{{I}}}}}\cdot\left\{\left[\displaystyle\sum_{i=7}^{I I}\frac{d}{d t}\Big(^{\varepsilon}\omega_{i}^{4}\Big)\dot{q}_{i}\right]\!+\!\frac{d}{d t}\Big(^{\varepsilon}\omega_{T F I}^{4}\Big)\dot{q}_{T r I}\right\}+^{E}\omega_{R o w}^{A}\right)}}\\ {{\mathrm{~}\qquad\qquad\qquad-m^{F}\,^{E}\nu_{R o w}^{J}\cdot\left\{\left[\displaystyle\sum_{i=d}^{I I}\frac{d}{d t}\Big(^{\varepsilon}\nu_{{i}}^{J}\Big)\dot{q}_{i}\right]\!+\!\frac{d}{d x^{i}}\Big(^{E}\nu_{{t r E I}}^{J}\Big)\dot{q}_{{T r I}}\right\}}}\end{array}$ w,Col=1,2,,11;15) A×IA⋅EωA (r=1,2,,11;15) dt  
+
+$$
+F_{r}\big|_{G r a v e d}={^{E}\nu}_{r}^{I}\cdot\left(-m{^{B}g z}_{2}\right)+{^{E}\nu}_{r}^{J}\cdot\left(-m{^{F}g z}_{2}\right)\;\;\left(r=3,4,\ldots,I I;T F r l\right)
+$$  
+
+Thus,   
+$\begin{array}{r l}&{\left\lfloor C\left(q,t\right)\right\rfloor\rfloor_{G r a v d}=O}\\ &{\left\{-f\left(\dot{q},q,t\right)\right\}\Bigl|_{G r a v d}\left(R o w\right)\!=\!-m^{B}g^{\,E}\nu_{R o w}^{I}\cdot z_{2}-m^{F}g^{\,E}\nu_{R o w}^{J}\cdot z_{2}\quad\left(R o w=3,4,\ldots,I I;I\xi\right)}\end{array}$   
+$F_{r}\big|_{A e r o A}={^{E}\nu_{r}^{K}}\cdot F_{A e r o A}^{K}+{^{E}\omega_{r}^{A}}\cdot M_{A e r o A}^{A}\quad\left(r=I,2,...,I I;T F r l\right)$   
+Thus,   
+C (q,t) =0   
+$\begin{array}{r l}{\left.\left\{-f\left(\dot{q},q,t\right)\right\}\right|_{A e r o d}\left(R o w\right)=}&{{}^{E}\nu_{R o w}^{K}\cdot F_{A e r o d}^{K}+\frac{E}{}\omega_{R o w}^{A}\cdot M_{A e r o d}^{A}\quad\left(R o w=I,2,\ldots,I I;I5\right)}\end{array}$  
+
+Overall:  
+
+Combining the results from the previous sections it is seen that the various portions of the equations of motion are related to the various forces as follows (NOTE: $B=B I+B2$ ):  
+
+![](images/3a87020067682bbc60b4cbb4a2760d97dae81172155f29057db82323aae48177.jpg)  
+
+<html><body><table><tr><td>{-f(g,q,t)} SpringTF + DampTF + A+ GravA+ AeroA B1+ElasticB1+ DampB1+GravB1+ AeroB1 B1+ElasticB1+ DampB1+GravB1+ AeroB1 B1+ElasticB1+DampB1+GravB1+AeroB1 B2+ElasticB2+DampB2+GravB2+AeroB2 B2+ElasticB2+DampB2+GravB2+AeroB2 B2+ElasticB2+DampB2+GravB2+ AeroB2 SpringTeet + DampTeet + H + GravH + B+ GravB+ AeroB</td><td>X+ HydroX+T+ AeroT+N+ R+ H+ B+ AeroB+ A+ AeroA ..................... X + HydroX +T+ AeroT + N+ R+H+ B+ AeroB+ A+ AeroA X + GravX + HydroX + T + GravT + AeroT + N + GravN + R+ GravR+ H + GravH + B+ GravB+ AeroB+ A+ GravA + AeroA X + GravX + HydroX + T + GravT + AeroT + N+ GravN + R+ GravR+ H + GravH + B+ GravB+ AeroB+G+ A+ GravA+ AeroA X + GravX + HydroX + T + GravT + AeroT + N + GravN + R+ GravR + H + GravH + B+ GravB+ AeroB + G+ A+ GravA+ AeroA X + GravX + HydroX + T + GravT + AeroT + N + GravN + R+ GravR + H + GravH + B+ GravB+ AeroB+ G+ A+ GravA+ AeroA T + ElasticT + DampT + GravT + AeroT + N+ GravN + R+ GravR+ H+ GravH + B+ GravB+ AeroB+ G+ A+ GravA+ AeroA T + ElasticT + DampT + GravT + AeroT + N+ GravN + R+ GravR+ H + GravH + B+ GravB+ AeroB+G+ A+ GravA+ AeroA ·........ T+ ElasticT + DampT + GravT + AeroT + N+ GravN + R+ GravR+ H + GravH + B+ GravB+ AeroB + G+ A+ GravA+ AeroA T + ElasticT + DampT + GravT + AeroT + N+ GravN + R+ GravR+ H + GravH + B+ GravB+ AeroB+ G+ A+ GravA+ AeroA SpringYaw+ DampYaw+ N + GravN + R+ GravR+ H + GravH + B+ GravB+ AeroB + G+ A+ GravA+ AeroA SpringRF + DampRF + R+ GravR+ H + GravH + B+ GravB+ AeroB +G H + GravH + B+ GravB+ AeroB+ Gen + Brake+ G+ GBFric H + GravH + B+ GravB+ AeroB+ ElasticDrive+ DampDrive ................................................................</td></tr></table></body></html>  
--- a/力学书籍/FASTKinetics/auto/FASTKinetics_origin.pdf
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--- a/力学书籍/FASTKinetics/auto/images/204043d4393282b60c23ffd1a705bb7bcc2a5b8932d1c97208bbbaaa7853497b.jpg
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--- a/力学书籍/FASTKinetics/auto/images/3a87020067682bbc60b4cbb4a2760d97dae81172155f29057db82323aae48177.jpg
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--- a/力学书籍/FASTLoads/auto/FASTLoads.md
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--- a/Show More
+++ b/Show More