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19
.obsidian/plugins/copilot/data.json vendored
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@ -138,6 +138,23 @@
],
"stream": true,
"enableCors": true
},
{
"name": "gemini-2.5-flash",
"provider": "google",
"enabled": true,
"isBuiltIn": false,
"baseUrl": "http://60.205.246.14:8000",
"apiKey": "gyz",
"isEmbeddingModel": false,
"capabilities": [
"reasoning",
"vision",
"websearch"
],
"stream": true,
"enableCors": true,
"displayName": "gemini-2.5-flash"
}
],
"activeEmbeddingModels": [
@ -268,7 +285,7 @@
"name": "Translate to Chinese",
"prompt": "<instruction>Translate the text below into Chinese:\n 1. Preserve the meaning and tone\n 2. Maintain appropriate cultural context\n 3. Keep formatting and structure\n \n </instruction>\n\n<text>{copilot-selection}</text>\n<restrictions>\n1. Blade翻译为叶片，flapwise翻译为挥舞，edgewise翻译为摆振，pitch angle翻译成变桨角度，twist angle翻译为扭角，rotor翻译为风轮，turbine、wind turbine翻译为机组、风电机组，span翻译为展向，deflection翻译为变形，mode翻译为模态，normal mode翻译为简正模态，jacket 翻译为导管架，superelement翻译为超单元，shaft翻译为主轴，azimuth、azimuth angle翻译为方位角，neutral axes 翻译为中性轴\n2. Return only the translated text.\n</restrictions>",
"showInContextMenu": true,
"modelKey": "gemma3:12b|ollama"
"modelKey": "gemini-2.5-flash|google"
},
{
"name": "Summarize",

34
书籍/力学书籍/力学/Flexible Multibody Dynamics (O. A. Bauchau) (Z-Library)/auto/Flexible Multibody Dynamics (O. A. Bauchau) (Z-Library).md
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@ -3328,13 +3328,13 @@ The principle of impulse and momentum involves two sets of new quantities. First
#### Principle of linear impulse and momentum
Figure 3.13 shows a particle of mass $m$ in motion with respect to an inertial frame $\mathcal{F}^{I}=[\mathbf{O},\mathcal{T}=(\bar{\iota}_{1},\bar{\iota}_{2},\bar{\iota}_{3})]$ . The inertial velocity vector of the particle is denoted $\underline{v}$ . The linear momentum vector of a particle is defined as the product of its mass by its inertial velocity vector
图 3.13 示出了质量为 $m$ 的一个粒子相对于惯性系 $\mathcal{F}^{I}=[\mathbf{O},\mathcal{T}=(\bar{\iota}_{1},\bar{\iota}_{2},\bar{\iota}_{3})]$ 的运动。该粒子的惯性速度矢量记作 $\underline{v}$ 。粒子的线性动量矢量定义为其质量与惯性速度矢量的乘积。
$$
\underline{{p}}=m\underline{{v}}.
$$
Taking a time derivative of the linear momentum vector yields $\dot{\underline{{p}}}=m\underline{{a}}.$ Comparing this result with Newton’s second law, eq. (3.4), leads to
对线性动量矢量求时间导数得到 $\dot{\underline{{p}}}=m\underline{{a}}.$ 将此结果与牛顿第二定律（式3.4）进行比较，可得
$$
\underline{{F}}=\underline{{\dot{p}}}.
$$
@ -3343,6 +3343,9 @@ This result implies that the time derivative of the linear momentum vector of a
It is interesting to integrate the above equation in time, between an initial and a final time, denoted $t_{i}$ and $t_{f}$ , respectively. These two instants are chosen arbitrarily, but $t_{i}<t_{f}$ ,
这个结果表明，一个粒子的线性动量矢量的时间导数等于所施加的外部力的总和。显然，这个结果是牛顿第二定律的一个直接推论。
有趣的是，在初始时间和最终时间 $t_{i}$ 和 $t_{f}$ 之间对上述方程进行时间积分。这两个时刻是任意选择的，但 $t_{i}<t_{f}$ 。
![](275f7027bde59c6599909c316acf60741a076c7a58af679d9a6b45c99f3a7cb4.jpg)
Fig. 3.13. Linear and angular momenta vectors of a particle.
@ -3351,14 +3354,21 @@ $$
$$
The term on the left-hand side is called the linear impulse of the externally applied forces, and has units of mass times velocity, or N·s. Equation (3.41) expresses the principle of linear impulse and momentum for a particle.
左侧的项被称为外部施加力的线性冲量，其单位是质量乘以速度，或 N·s。公式 (3.41) 表达了线性冲量和动量原理，适用于一个粒子。
Principle 3 (Principle of linear impulse and momentum for a particle) The linear impulse of the externally applied forces equals the change in linear momentum.
**Principle 3 (Principle of linear impulse and momentum for a particle)** The linear impulse of the externally applied forces equals the change in linear momentum.
In the absence of external forces, this principle implies $\underline{{p}}(t_{f})\,=\,\underline{{p}}(t_{i})$ , i.e., the linear momentum remains constant at all times, since $t_{i}$ and $t_{f}$ are instants chosen arbitrarily. In other words, the linear momentum vector of a particle remains a constant when the externally applied forces vanish.
# Principle of angular impulse and momentum
**第三条原则（单个粒子的线性冲量与动量原理）** 外力作用下的线性冲量等于线性动量的变化。
在没有外力作用的情况下，该原理意味着 $\underline{{p}}(t_{f})\,=\,\underline{{p}}(t_{i})$ ，即线性动量在任何时刻保持不变，因为 $t_{i}$ 和 $t_{f}$ 是任意选择的时刻。换句话说，当外力作用消失时，单个粒子的线性动量矢量保持不变。
#### Principle of angular impulse and momentum
Next, the moment of the particle’s linear momentum vector is computed with respect to point O. This quantity if more often called the angular momentum vector of the particle, $\underline{{h}}_{O}$ , where the subscript, $(\cdot)_{O}$ indicates that the angular momentum is computed with respect to point O. As illustrated in fig. 3.13, the moment of the linear momentum vector is expressed as the cross product of the particle’s inertial position vector, $\underline{{r}}$ , by its linear momentum vector, $m v_{\!}$ , to find
接下来，计算粒子线动量矢量相对于O点的矩。该量通常被称为粒子的角动量矢量$\underline{{h}}_{O}$，其中下标$(\cdot)_{O}$表示角动量是相对于O点计算的。如图3.13所示，线动量矢量的矩表示为粒子的惯性位置矢量$\underline{{r}}$与其线动量矢量$m v_{\!}$的叉积，以求得
Next, the moment of the particle’s linear momentum vector is computed with respect to point O. This quantity if more often called the angular momentum vector of the particle, $\underline{{h}}_{O}$ , where the subscript, $(\cdot)_{O}$ indicates that the angular momentum is computed with respect to point O. As illustrated in fig. 3.13, the moment of the linear momentum vector is expressed as the cross product of the particle’s inertial position vector, $\underline{{r}}$ , by its linear momentum vector, $m v_{\!}$ , to find
$$
\begin{array}{r}{\underline{{h}}_{O}=\widetilde{r}\;m\underline{{v}}.}\end{array}
@ -3368,6 +3378,9 @@ Taking a time derivative of the angular momentum vector yields $\underline{{\dot
The moment of Newton’s second law computed with respect to th e origin of the inertial frame implies $\tilde{r}\underline{{F}}=\tilde{r}m\underline{{a}}$ . Comparing these two results then leads to $\widetilde{r}\underline{{F}}=$ $\underline{{i}}_{O}$ , where the left-han d side t erm can be interpreted as the moment of the exte r nally applied forces evaluated with respect to point $\mathbf{o}$ , denoted $\underline{{M_{O}}}$ . In summary,
对角动量矢量求时间导数得到 $\underline{{\dot{h}}}_{O}=\dot{\tilde{r}}m\underline{{v}}+$ rma。惯性位置矢量的时间导数 $\dot{r}$ 等于惯性速度矢量 $v$，如式 (3.1) 所示；因此，$\dot{\underline{{h}}}_{O}=\widetilde{v}m\underline{{v}}+\widetilde{r}\overline{{m}}\underline{{a}}$。最后，由于 $\widetilde{v}m\underline{{v}}=\underline{{0}}$，角动量矢量的时间导数简化为 $\dot{h_{O}}=\widetilde{r}m\underline{{a}}$。
相对于惯性系原点计算的牛顿第二定律的力矩意味着 $\tilde{r}\underline{{F}}=\tilde{r}m\underline{{a}}$。比较这两个结果，得出 $\widetilde{r}\underline{{F}}=$ $\dot{\underline{{h}}}_{O}$，其中左侧项可以解释为相对于点 $\mathbf{o}$ 计算的外部施加力的力矩，记作 $\underline{{M_{O}}}$。总之，
$$
\underline{{M_{O}}}=\underline{{\dot{h}_{O}}}
$$
@ -3376,16 +3389,25 @@ This result implies that the time derivative of the angular momentum vector of $
As in the case of the linear momentum, the above equation can be integrated in time between two arbitrary instants to yield
此结果意味着，相对于惯性点计算的粒子角动量矢量的时间导数等于相对于同一点计算的外部施加力矩之和。同样，此结果是牛顿第二定律的直接推论。
与线动量的情况一样，上述方程可以在任意两个时刻之间进行时间积分，从而得到
$$
\int_{t_{i}}^{t_{f}}\underline{{M_{O}}}(t)\;\mathrm{d}t=\int_{t_{i}}^{t_{f}}\dot{\underline{{h}}}_{O}\;\mathrm{d}t=\underline{{h}}_{O}(t_{f})-\underline{{h}}_{O}(t_{i}).
$$
The term on the left-hand side is called the angular impulse of the externally applied forces, and has units of $\mathbf{N}{\cdot}\mathbf{m}{\cdot}\mathbf{s}$ . Equation (3.44) expresses the principle of angular impulse and momentum for a particle.
Principle 4 (Principle of angular impulse and momentum for a particle) The angular impulse of the externally applied forces equals the change in angular momentum when both angular impulse and momentum are computed with respect to the same inertial point.
**Principle 4 (Principle of angular impulse and momentum for a particle)** The angular impulse of the externally applied forces equals the change in angular momentum when both angular impulse and momentum are computed with respect to the same inertial point.
In the absence of external moments, this principle implies $\begin{array}{r}{\underline{{h}}_{O}(t_{f})\,=\,\underline{{h}}_{O}(t_{i}).}\end{array}$ , i.e., the angular momentum remains constant at all times. In other words, the angular momentum vector of a particle remains a constant when the externally applied moments vanish.
左侧项称为外施力的角冲量，单位为 $\mathbf{N}{\cdot}\mathbf{m}{\cdot}\mathbf{s}$ 。方程 (3.44) 表达了质点的角冲量和角动量原理。
**原理 4 (质点的角冲量和角动量原理)** 外施力的角冲量等于角动量的变化量，当角冲量和角动量都相对于同一惯性点计算时。
在没有外部力矩的情况下，该原理意味着 $\begin{array}{r}{\underline{{h}}_{O}(t_{f})\,=\,\underline{{h}}_{O}(t_{i}).}\end{array}$ ，即角动量始终保持不变。换句话说，当外施力矩消失时，质点的角动量矢量保持不变。
# Example 3.2. Particle in a pinned tube
Figure 3.14 depicts a particle of mass $m$ connected to inertial point A by means of a spring of stiffness $k$ and dashpot of constant $c$ . At the initial time, the particle is located at $\theta\:=\:0$ , $\phi\,=\,\pi/2$ , and $r~=~r_{0}$ , which corresponds to the un-stretched configuration of the spring; $r$ , $\phi$ , and $\theta$ form a spherical coordinate system, see section 2.7.2. The initial velocity vector of the particle is $\underline{{v}}_{0}$ . Derive the equations of motion of the system.