# States 'States'is a collective term for all time-dependent properties of a simulation model. State Types Working State Initial State State Vector Degrees of Freedom (DOF) The aim of a time domain solution is to find the states' development in time, but they also play a vital role in frequency domain simulations. Together with the model geometry and parameters, which do both not change in time, the model's behavior is consistently described when the states and their change in time is known. Usually, the term 'states' means the Modeling Element states, which are single quantities provided by Modeling Elements and handled by the Equations of Motion. These are described in State Types. In contrast, there is also the term 'model state' or 'state of the model', which comprises the whole of all Modeling Element state values. See, in particular, Working State and Initial State. '状态' 是指模拟模型的所有随时间变化的属性的统称。 状态类型 工作状态 初始状态 状态向量 自由度 (DOF) 时域求解的目标是找到状态随时间的发展,但它们在频域模拟中也起着至关重要的作用。 结合模型几何形状和参数(它们不随时间变化),当已知状态及其随时间的变化时,可以一致地描述模型的行为。 通常,“状态”一词指的是建模元素的“状态”,这些是建模元素提供的单个量,并由运动方程处理。 这些内容在“状态类型”中进行描述。 相反,还有“模型状态”或“模型的状态”一词,它包含了所有建模元素状态值的总和。 尤其参见“工作状态”和“初始状态”。 ## State Types In Simpack there are different types of Modeling Element states.All states are handled by the respective Modeling Elements mentioned in the descriptions below. Joint (including Connections treated as Joints by the multibody formalism) $s_{\mathrm{jnt}}$ (position), $\dot{s}_{\mathrm{jnt}}$ (velocity). These are one of the most common types and describe by means of Joints (and Connections) the kinematical behavior of the Bodies. They represent the generalized coordinates of the Bodies' movements. Due to requirements from the solution process, they consist of two independent values, the so-called first-order states, one for the position and one for the velocity in the respective direction. Joint states are subject to the integration over time in time domain solutions. For more information, see Joint States. The Connection states are not actually states, but their values are mapped to Joint states, which are then solved by the Solver. For elements that support referencing states, most element types allow you to assign either Connection or Joint states. 在Simpack中,存在不同类型的建模元素状态。所有状态均由以下描述中提到的相应建模元素来处理。 关节 (包括多体形式学中被视为关节的连接) $s_{\mathrm{jnt}}$ (位置), $\dot{s}_{\mathrm{jnt}}$ (速度)。 这是一种最常见的状态类型,通过关节(和连接)描述了物体的运动学行为。它们代表了物体运动的广义坐标。由于求解过程的要求,它们由两个独立的数值组成,即所谓的“一阶状态”,一个用于位置,一个用于各自方向上的速度。关节状态会随着时间在时域求解中进行积分。更多信息,请参见关节状态。连接状态实际上不是状态,但它们的值会被映射到关节状态,然后由求解器进行求解。对于支持引用状态的元素,大多数元素类型允许您分配连接状态或关节状态。 Flexible Body states $s_{\mathrm{flx}}$ (position), $\dot{s}_{\mathrm{flx}}$ (velocity) describe the deformation behavior of Flexible Bodies using the modal coordinates or the nodal representation, depending on the Flexible Body type. They also consist of two independent first-order states, one for position and one for velocity, and they are also subject to the time domain integration. See also Modal Approach. 柔性体状态 $s_{\mathrm{flx}}$ (位置), $\dot{s}_{\mathrm{flx}}$ (速度) 用于描述柔性体的变形行为,采用模态坐标或节点表示,具体取决于柔性体的类型。它们也包含两个独立的低阶状态,一个用于位置,一个用于速度,并且也需要进行时域积分。另见模态方法。 Dynamic states $s_{\mathrm{dyn}}$ are used by some Force Elements and Control Elements and describe internal dynamic behavior of these elements. A dynamic state requires only one first-order state. They are also subject to the time domain integration. See States. 动态状态 $s_{\mathrm{dyn}}$ 被某些力元素和控制元素使用,用于描述这些元素的内部动态行为。一个动态状态只需要一个一阶状态变量。它们也受到时域积分的影响。详情请参见“状态”。 Algebraic states $S_{alg}$ are again used by some Force Elements and Control Elements but also by some Markers. The latter describe internal non-dynamic behavior of the Markers, for example contact point positions, and are called 'on position level'. The former describe dynamic behavior, e.g. force values, thus they are called 'on acceleration level'. They are usually handled by the integrator and subject to the integrator's tolerance management but they are not integrated. Some Modeling Elements also have an option to handle algebraic states internally. If they are handled by the integrator then it must be able to solve differential-algebraic equations (DAEs). See also States. 代数状态 $S_{alg}$ 一些力元件和控制元件,以及一些标记(Marker)也会用到代数状态。 后者描述标记的内部非动态行为,例如接触点位置,被称为“位置层级”(on position level)。前者描述动态行为,例如力值,因此被称为“加速度层级”(on acceleration level)。它们通常由积分器处理,并受积分器的容差管理,但本身不参与积分。 一些建模元件也提供处理代数状态的选项。如果由积分器处理,则积分器必须能够求解微分代数方程组(DAEs)。另见 状态。 Constraint and Connection Forces and Torques $s_{\lambda}$ , or Constraint(including Connections treated as Constraints by the multibody formalism) states, are a special kind of algebraic states. They contain the forces and torques invoked by Constraints (and Connections). The integrator must be able to solve differential-algebraic equations (DAEs) to handle Constraint states. 约束与连接力与力矩 $s_{\lambda}$, 或约束(包括通过多体形式学处理为约束的连接)状态,是一种特殊的代数状态。它们包含由约束(和连接)所产生的力和力矩。积分器必须能够求解微分代数方程组 (DAEs) 以处理约束状态。 Root states $s_{\mathrm{root}}$ are used by some Force Elements and Control Elements. They contain the switching status if, for example, a Force Element switches between different force laws. The switching process and the root states are handled by the Time Integration solver. However, not all integrators support this. See Root Functions, Root Functions and Root Functions. 根状态 $s_{\mathrm{root}}$ 被某些力元素和控制元素使用。它们包含切换状态,例如,当力元素在不同的力法则之间切换时。时间积分求解器负责处理切换过程和根状态。然而,并非所有积分器都支持此功能。请参阅根函数、根函数和根函数。 Descriptive states $s_{\mathrm{desc}}$ store intermediate values in Force Elements and Control Elements. They are used to make the intermediate results properly available in the Measurements solver and for the continuation runs (see State Initialization). The descriptive state values are not interpolated between the solver steps (see Output Point Sampling for the model states interpolation); the value of the preceding step is held constant until the end of the current step. This type of state is the only type which is not handled by the solver but by the Modeling Elements themselves. See States. See also the User Routines Descriptive States. More information can be found in the sections describing the Solvers, and also the State Sets. Within the solvers, all state values are handled in coherent SI units. The values entered in the model (user interface, SubVars) are converted before being passed to the solvers. See Unit Handling for more information. 描述状态 $s_{\mathrm{desc}}$ 存储在力元素和控制元素中的中间值。它们用于在测量求解器中和后续运行中(参见状态初始化)正确地提供中间结果。描述状态值不进行求解器步长之间的插值(参见模型状态插值的输出点采样);前一步的值保持恒定直到当前步骤结束。这种类型的状态是唯一由建模元素而非求解器本身处理的状态。参见状态。另见用户例程描述状态。 更多信息请参见描述求解器和状态集的章节。 在求解器内部,所有状态值均以一致的 SI 单位处理。模型(用户界面、子变量)中输入的值在传递给求解器之前会被转换。有关更多信息,请参见单位处理。 ## Working State 'Working state' means the model state that arises from the 'current' values of the single Modeling Element states. The user sees the model in its working state in the 3D Page. The working state can be modified by, for example, · Editing the 'State' values of Modeling Elements that possess states, see State Types ·Applying State Sets to the model · Importing State (.spckst) files, see Exporting and Importing States, into the model ·Resetting states to zero,see Resetting States to Zero The working state is saved with the model and reappears when a model is loaded again. The user may also store the working state in State Sets within the model or export it to State (.spckst) files, see Exporting and Importing States. See also Initial State. “工作状态”指的是基于建模元素状态的“当前”值所产生的模型状态。用户可以在三维页面中看到模型的“工作状态”。工作状态可以通过以下方式进行修改,例如: · 编辑具有状态的建模元素的“状态”值,请参见状态类型; · 将状态集应用于模型; · 导入状态(.spckst)文件,请参见导出和导入状态,到模型中; · 将状态重置为零,请参见将状态重置为零。 工作状态会与模型一起保存,并在模型重新加载时再次出现。 用户还可以将工作状态存储在模型中的状态集中,或将其导出到状态(.spckst)文件,请参见导出和导入状态。 另请参见初始状态。 ## Initial State The initial state is the model state the Solvers begin their actual work with. Online solvers (see On- and Offline Solving) always start from the Working State. Offline solvers normally start from the model state given by the Simpack model file, which is the working state at the time the model was saved. The user may, alternatively, have the solvers start from a given State (.spckst) file, see simpack-slv (Simpack Solver). Additionally, both online and offline solvers may automatically initialize or update some states at each solver start. See Solver Initialization for more information about the initialization and how to avoid it when concatenating solver runs (General: Concatenating Solver Runs). The initial state is the model state after this initialization. 初始状态是求解器开始实际工作时的模型状态。在线求解器(参见在线和离线求解)始终从工作状态开始。离线求解器通常从Simpack模型文件中给定的模型状态开始,该状态是模型保存时的工作状态。用户可以选择,让求解器从给定的状态文件(.spckst文件,参见 simpack-slv(Simpack求解器))开始。 此外,在线和离线求解器在每次求解器启动时都可能自动初始化或更新某些状态。有关初始化以及在连接求解器运行(常规:连接求解器运行)时如何避免初始化的更多信息,请参见求解器初始化。初始状态是经过此初始化后的模型状态。 ## State Vector The first five state types listed in State Types above are collected as subvectors in the so-called 'state vector' This vector is the main input to the equations of motion and is passed to the Time Integration solver, see General Nonlinear. Linear (frequency domain) solvers use a slightly modified state vector, see Linearized. 上面“state types”中列出的前五个state types被收集为所谓的“状态向量”中的子向量。 $$ \mathbf{x}(t) = \begin{pmatrix} \mathbf{S}_{\text{jnt}} \\ \mathbf{S}_{\text{flx}} \\ \mathbf{S}_{\text{dyn}} \\ \dot{\mathbf{S}}_{\text{jnt}} \\ \dot{\mathbf{S}}_{\text{flx}} \\ \mathbf{S}_{\lambda} \\ \mathbf{S}_{\text{alg}} \end{pmatrix} $$ 这个向量是运动方程的主要输入,并传递给时间积分求解器,参见“通用非线性”。线性(频域)求解器使用略微修改过的状态向量,参见“线性化”。 ## Degrees of Freedom (DOF) Another frequently used term is 'degree of freedom', or 'DoF'. There is a well-defined relationship between the degrees of freedom and the states: On the one hand, each single Joint, Flexible Body and dynamic state provides one 'degree of freedom' (DoF) to the model. Thus one DOF corresponds to one or two first-order states (see State Types above). On the other hand, each constraint state removes one degree of freedom by constraining a movement in a particular direction. Thus the resulting number of degrees of freedom of a simulation model $n_{\mathrm{model}}$ can be determined by the equation 另一个常用的术语是“自由度”,或“DoF”。自由度和状态之间存在明确的关系:一方面,每个单独的关节、柔性体和动态状态都会为模型提供一个“自由度”(DoF)。因此,一个DoF对应一个或两个一阶状态(见上方的状态类型)。另一方面,每个约束状态都会通过限制特定方向的运动来减少一个自由度。因此,模拟模型的自由度数 $n_{\mathrm{model}}$ 可以通过以下公式确定。 $$ n_{\mathrm{DOF,model}}=n_{s_{\mathrm{jnt}}}+n_{s_{\mathrm{flx}}}+n_{s_{\mathrm{dyn}}}-n_{s_{\lambda}} $$ with $n_{s_{\mathrm{jnt}}}$ the number of Joint states $s_{\mathrm{jnt}},\,n_{s_{\mathrm{flx}}}$ the number of Flexible Body states $s_{\mathrm{flx}}$ $n_{s_{\mathrm{dyn}}}$ the number of Force Element and Control Element dynamic states $s_{\mathrm{dyn}}$ and $n_{s_{\lambda}}$ the number of Constraint Forces and Torques $s_{\lambda}$ 其中,$n_{s_{\mathrm{jnt}}}$ 表示关节状态 $s_{\mathrm{jnt}}$ 的数量,$n_{s_{\mathrm{flx}}}$ 表示柔体状态 $s_{\mathrm{flx}}$ 的数量,$n_{s_{\mathrm{dyn}}}$ 表示力元素和控制元素动态状态 $s_{\mathrm{dyn}}$ 的数量,而 $n_{s_{\lambda}}$ 表示约束力矩 $s_{\lambda}$ 的数量。 The number of equations to be solved by the integrator in a time domain simulation is 积分器在时域仿真中需要求解的方程数量是: $$ n_{\mathrm{Eq,model}}=2\left(n_{s_{\mathrm{jnt}}}+n_{s_{\mathrm{flx}}}\right)+n_{s_{\mathrm{dyn}}}+n_{s_{\mathrm{alg}}}+n_{s_{\lambda}} $$ where $n_{s_{\mathrm{alg}}}$ is the number of algebraic states $s_{\mathrm{alg}}$ . All state equations are always solved, although some degrees of freedom may be removed by Constraints or Connections. The Simpack solvers convert the Modeling Elements and model structure into a set of nonlinear ordinary differential equations (ODE). Models containing Constraints or Connections in closed kinematic loops, and/or algebraic states require an additional set of algebraic equations. The complete set of equations is then called differential-algebraic equations (DAE). See State Types for an explanation of the different state types. 其中,$n_{s_{\mathrm{alg}}}$ 表示代数状态 $s_{\mathrm{alg}}$ 的数量。所有状态方程始终会被求解,尽管一些自由度可能会因约束 (Constraints) 或连接 (Connections) 被移除。 Simpack 求解器将建模元素和模型结构转换为一组非线性常微分方程 (ODE)。包含闭合运动学回路中的约束或连接,以及/或代数状态的模型,需要额外的一组代数方程。 这样,完整的方程组被称为微分代数方程 (DAE)。有关不同状态类型的解释,请参阅“状态类型”部分。 # Equations of Motion ## General Nonlinear The Simpack solvers convert the Modeling Elements and model structure into a set of nonlinear ordinary differential equations (ODE). Models containing Constraints or Connections in closed kinematic loops, and/or algebraic states require an additional set of algebraic equations. The complete set of equations is then called differential-algebraic equations (DAE). See State Types for an explanation of the different state types. Simpack求解器会将建模元素和模型结构转换为一组非线性常微分方程 (ODE)。包含约束或连接形成闭合运动学回路,以及/或代数状态的模型,需要额外的一组代数方程。 这样,完整的方程组被称为微分代数方程 (DAE)。有关不同状态类型的解释,请参阅“状态类型”部分。 ### System of Equations The nonlinear DAE set in its most general, explicit form is as follows: $$ \begin{array}{r l}{\dot{\mathbf{p}}=\mathbf{T}\left(\mathbf{p}\right)\mathbf{v}} &{}(kinematic)&{}\\{\mathbf{M}\left(\mathbf{p}\right)\dot{\mathbf{v}}=\mathbf{f}\left(\mathbf{p},\mathbf{v},\mathbf{c},\mathbf{s},t,\mathbf{u},\lambda\right)-\mathbf{G}^{T}\left(\mathbf{p},\mathbf{c},\mathbf{s},t,\mathbf{u}\right)\lambda}&{}(momentum)\\ {\dot{\mathbf{c}}=\mathbf{f}_{c}\left(\mathbf{p},\mathbf{v},\mathbf{c},\mathbf{s},t,\mathbf{u},\lambda\right)} &{}(dynstates)&{}\\ {0=\mathbf{g}\left(\mathbf{p},\mathbf{c},\mathbf{s},t,\mathbf{u}\right)}&{}(cnsts)&{}\\ {0=\mathbf{b}\left(\mathbf{p},\mathbf{v},\mathbf{c},\mathbf{s},t,\mathbf{u},\lambda\right)}&{}(algebraic)&{}\end{array} $$ where - p are the position states of Joints (including Connections treated as Joints by the multibody formalsim), and Flexible Bodies ( $\mathbf{\delta}_{s_{\mathrm{jnt}}}$ and $\scriptstyle{s_{\mathrm{flx}}}$ ) - $\mathbf{T}$ is the transformation matrix for angles - $\mathbf{v}$ are the Joint, Connection, and Flexible Body states on velocity level ( $\dot{s}_{\mathrm{jnt}}$ and $\dot{s}_{\mathrm{flx}}$ - M is the mass matrix - f are the force and torque equations of Force Elements - c are the dynamic states of Force Elements and Control Elements $(s_{\mathrm{dyn}})$ - s are the algebraic states $(s_{\mathrm{alg}})$ on position and acceleration level - t the model time - u the values of u-Vector Elements and the excitations from driven Joints and Connections on position and velocity level - $\lambda$ the Constraint forces and torques ( ${s_{\lambda}},$ only applies if the model contains Constraints or the model has a closed kinematic loop with Connections) - G the Jacobian matrix of the constraint conditions, G(p,c,s,t,u) = - $\mathbf{f}_{c}$ the dynamic state equations of Force Elements and Control Elements - g the algebraic constraint conditions related to Constraints (only applies if the model contains them) - b the algebraic constraint conditions related to algebraic states (only applies if the model contains them) No difference is made between independent and dependent Joint and Flexible Body states (see State Dependencies and Constraint Redundancy). In Simpack, the mass matrix $\mathbf{M}\left(\mathbf{p}\right)$ is shifted to the right-hand side of equation (momentum), where it appears accordingly as $\mathbf{M}^{-1}\left(\mathbf{p}\right)$ , if the explit formalism is used, see Explicit and Residual Formalism. 独立和从属联合柔性体状态之间没有区别(参见状态依赖性和约束冗余)。在Simpack中,质量矩阵 $\mathbf{M}\left(\mathbf{p}\right)$ 被移到方程(动量)的右侧,并相应地出现为 $\mathbf{M}^{-1}\left(\mathbf{p}\right)$,如果使用显式形式,请参见显式与残余形式。 In this formulation, the state vector is $$ \mathbf{x}\left(t\right)=\left(\begin{array}{l}{\mathbf{p}}\\ {\mathbf{c}}\\ {\mathbf{v}}\\ {\lambda}\\ {\mathbf{s}}\end{array}\right) $$ which directly corresponds to the symbol naming in State Vector. Equation (kinematic) is the so-called kinematic differential equation and reduces the system from second to first order because common time integration solvers used in multibody simulation are available for first-order systems of equations only (nonlinear state-space representation). Equation (momentum) is the well-known principle of (linear) momentum according to Newton and Euler, with the Constraint forces and torques brought in as Lagrange multipliers. Equation (dynstates) covers the first-order dynamic states of Force or Control Elements. The algebraic equation (cnsts) describes the constraint conditions used by Constraints. The second algebraic equation (algebraic) describes the conditions used for algebraic states of Force Elements and ControlElements. 运动方程(运动学)是所谓的运动学微分方程,它将系统降阶为一阶,因为多体动力学模拟中常用的时间积分求解器仅适用于一阶方程组(非线性状态空间表示)。 动量方程(动量)是根据牛顿和欧拉的(线性)动量守恒原理,并将约束力矩引入作为拉格朗日乘子而得到的。 (dynstates) 方程涵盖了力或控制元件的一阶动态状态。代数方程 (cnsts) 描述了约束所使用的约束条件。第二个代数方程 (algebraic) 描述了用于力元件和控制元件代数状态的条件。 ### Preconditions for Solvability This system of equations is solvable if the following conditions are fulfilled: 1. Positive definite mass matrix: The mass matrix M must be symmetric and positive definite. Thus, Bodies with zero or even negative mass or main inertia moments are not allowed. 2. Force law continuity: All force laws $\mathbf{f}\left(\mathbf{p},\mathbf{v},\mathbf{c},\mathbf{s},t,\mathbf{u},\lambda\right)$ used in Force Elements must be continuous in all their independents. This is a tough requirement and certain effort is needed to fulfil it even in strongly nonlinear cases like contact loss and regain and stick-slip friction. Simpack's integrators (see Available Integrators) may, depending on the tolerance settings (Tolerances Tab), just skip small discontinuities. However, it is also possible that they significantly reduce the stepsize, and the calculation time increases drastically. If the discontinuity is too large and/or very fine tolerances are used then the integrator may even abort with a fatal error. Simpack's Modeling Elements and the Time Integration solver provide special functionality to smooth or circumvent discontinuous force laws. Some Force Elements and Control Elements provide root functions, causing the integrator to automatically stop and restart the integration when a discontinuity is detected, see Root Functions, Root Functions and Root Functions. 3. Unique constraint conditions: The Jacobian of the constraint conditions $\mathbf{G}$ must have full rank. This means in particular that constraining conditions must neither be redundant nor conflicting, see State Dependencies and Constraint Redundancy. Redundant constraining conditions are ignored (by default) automatically by the solver. Conflicts cause an error as the constraining conditions cannot be fulfilled and need to be resolved by the user. 4. Initial constraint conditions: The initial values of p, v, C, S, $\mathbf{g}$ and $\lambda$ must be consistent. This is ensured by the Assemble System solver. 5. Constraint conditions differentiability: The constraint conditions $\mathbf{g}\left(\mathbf{p},\mathbf{c},\mathbf{s},t,\mathbf{u}\right)$ must be twice continuously differentiable. This is normally ensured by the Constraints themselves. 6. Correctly defined algebraic states: The algebraic state residuals in b must depend on their associated algebraic states $s_{\mathrm{alg}}$ . Simpack's library Modeling Elements ensure this automatically, but in User Routines and Control Element 175: Algebraic State Feedback the user is responsible for defining this relationship properly. 本方程组求解可行,若满足以下条件: 7. 正定质量矩阵:质量矩阵 M 必须对称且正定。因此,不允许具有零甚至负质量或主惯性矩的刚体。 8. 作用力规律连续性:作用于 Force Element 中使用的所有作用力规律 $\mathbf{f}\left(\mathbf{p},\mathbf{v},\mathbf{c},\mathbf{s},t,\mathbf{u},\lambda\right)$ 必须在其所有自变量上连续。这是一个苛刻的要求,即使在接触损失/恢复和黏滑摩擦等强非线性情况下,也需要付出一定的努力才能满足。Simpack 的积分器(参见 Available Integrators)可能会根据容差设置(Tolerances Tab)直接跳过小的间断。然而,它们也可能显著减小步长,导致计算时间大幅增加。如果间断过大且/或使用了非常精细的容差,积分器甚至可能因致命错误而中止。Simpack 的 Modeling Elements 和时间积分求解器提供了特殊功能,用于平滑或规避不连续的作用力规律。某些 Force Element 和 Control Element 提供了根函数,导致积分器在检测到间断时自动停止和重新开始积分,参见 Root Functions, Root Functions and Root Functions。 9. 唯一约束条件:约束条件 Jacobian $\mathbf{G}$ 必须具有满秩。这意味着约束条件既不能冗余,也不能相互冲突,参见 State Dependencies and Constraint Redundancy。求解器会默认自动忽略冗余的约束条件。冲突会导致错误,因为约束条件无法满足,需要用户解决。 10. 初始约束条件:p, v, C, S, $\mathbf{g}$ 和 $\lambda$ 的初始值必须一致。Assemble System 求解器负责确保这一点。 11. 约束条件可微性:约束条件 $\mathbf{g}\left(\mathbf{p},\mathbf{c},\mathbf{s},t,\mathbf{u}\right)$ 必须满足二阶连续可微性。这通常由 Constraints 本身保证。 12. 正确定义的代数状态:代数状态残差 b 中的各项必须依赖于其相关的代数状态 $s_{\mathrm{alg}}$ 。Simpack 的 Modeling Elements 库会自动确保这一点,但在 User Routines 和 Control Element 175: Algebraic State Feedback 中,用户负责正确定义这种关系。 ### Explicit and Residual Formalism Most integrators use the explicit form of equation (kinematic) and the follwing equations. Some integrators, however, can solve the equations of motion also in an implicit form, i.e. they try to ensure that the difference of left-hand and right-hand side (e.g. $\dot{\mathbf{p}}-\mathbf{T}\left(\mathbf{p}\right)\mathbf{v})$ becomes zero. This method is also called 'residual formalism' and is often significantly faster than the explicit formalism. See also Explicit and Residual Formalism for more information. 大多数积分器使用方程(运动学)的显式形式以及以下方程。然而,有些积分器也可以用隐式形式求解运动方程,即他们试图确保左侧和右侧之间的差值(例如,$\dot{\mathbf{p}}-\mathbf{T}\left(\mathbf{p}\right)\mathbf{v}$) 变为零。这种方法也被称为“残差形式法”,通常比显式形式法快得多。有关更多信息,请参阅显式和残差形式法。 ## Linearized The linear solvers Eigenvalues, Linear System Analysis and State-Space Matrices Export need a linearized version of the equations of motion stated in General Nonlinear. See Linearization for information on how the linearization is performed. The state equation in the state-space representation of the (now linear and time invariant, 'LTI') system describes the eigen behavior of the system and the influence of excitations: 线性求解器 Eigenvalues、线性系统分析和状态空间矩阵导出需要将通用非线性中陈述的运动方程线性化。有关线性化过程的信息,请参阅线性化部分。 状态空间表示法中(现在是线性且时不变,即“LTI”)系统的状态方程,描述了系统的特征行为以及激励的影响: $$ {\dot{\mathbf{x}}}\left(t\right)=\mathbf{A}\mathbf{x}\left(t\right)+\mathbf{B}\mathbf{u}\left(t\right) $$ where - $\mathbf{x}$ the state vector, containing the positions and velocities of the independent Joint and Flexible Body states (see State Dependencies and Constraint Redundancy) as well the Force or Control Element dynamic states, see State Vector - $t$ the model time - A the system matrix ·B the input matrix - u the input vector, defined via u-Vector Elements The output equation describes the outputs of the system, which the user may have arbitrarily defined via y-Outputs: $$ \mathbf{y}\left(t\right)=\mathbf{C}\mathbf{x}\left(t\right)+\mathbf{D}\mathbf{u}\left(t\right) $$ with the additional variables $\mathbf{y}$ the output vector, containing an arbitrary set of quantities to be measured, defined via y-Outputs $\mathbf{C}$ theoutputmatrix ·Dthefeedthroughmatrix All four matrices A, $\scriptstyle\mathbf{B},$ C and $\mathbf{D}$ are constant. The Eigenvalues solver only needs the system matrix A, which describes the system's eigenbehavior. The Linear System Analysis solver requires all four matrices. The State-Space Matrices Export solver exports the matrices, along with the state, input and output vector in the linearization state, for use with externalsoftware. # Flexible Bodies ## Approach in Finite Element Software ### Introduction to Finite Element Analysis The Finite Element Method (FEM) is one of the numerical approaches developed over the last decades for modeling the dynamic behavior of the deformable bodies/components. The method consists of three stages: ·Preprocessing: Once the geometry of interest has been generated using a Computer Aided Design (CAD) software, the resulting computation domain is subdivided (meshed) into elements of finite dimensions (known as finite elements) connected by nodes, and the appropriate material properties, boundary conditions and loads are applied. · Solution: The equations for the applied nodal forces are defined in terms of the unknown nodal displacements. From this, the equations of equilibrium of each element are assembled in a matrix form (i.e. stiffness $\cdot$ displacement $=$ load). · Postprocessing: The results (i.e. displacement, element stresses and strains fields, natural frequencies) are obtained. Although the FEM approach can be easily aplied ina range of geometrically complex problems, a large number of fite elements and nodes have to be considered (up to six degrees of freedom per node) in order to sufficiently describe the dynamic behavior of a multibody system.Often the FEMis based on models with more than one million degreesof freedomwhichimplieshighcomputationalcostshusit isuualiited inapplicationswithcomparativelymallnumeof iloadcaseseigenmodesoritis applied to acertain component of a system. The major challenge remains on describing the dynamic of a complete mechanical system. The coupling of multibody and finite element simulations has proven to be a crucial approach for this purpose. Though, in order to enable the coupling procedure during the modeling workflow, a method for reducing the size of the finite element model (degrees of freedom) is necessary (see Basics of Craigh Bampton Reduction). 有限元法 (FEM) 是过去几十年里为模拟变形体/构件的动态行为而开发的一种数值方法。该方法包括三个阶段: · 预处理:一旦使用计算机辅助设计 (CAD) 软件生成了感兴趣的几何体,则将计算域划分为具有有限尺寸的单元(称为有限元),这些单元通过节点连接,并施加适当的材料属性、边界条件和载荷。 · 求解:针对施加的节点力,定义未知节点位移的方程。由此,每个单元的平衡方程以矩阵形式组装起来(即:刚度 $\cdot$ 位移 $=$ 载荷)。 · 后处理:获得结果(例如:位移、单元应力和应变场、固有频率)。 尽管有限元法可以轻松应用于各种几何形状复杂的难题,但为了充分描述多体系统的动态行为,需要考虑大量的有限元和节点(每个节点可达六个自由度)。 往往有限元法基于具有超过一百万自由度的模型,这暗示着高昂的计算成本,因此通常仅限于具有相对较少载荷工况、特征模态或应用于系统某个特定构件的应用中。 描述整个机械系统的动态仍然是主要挑战。多体动力学与有限元模拟的耦合已被证明是实现这一目标的关键方法。然而,为了在建模工作流程中启用耦合过程,需要一种减少有限元模型尺寸(自由度)的方法(参见 Craig-Bampton 降阶方法的基础)。 ### Basics of Craigh Bampton Reduction The'Craig-Bampton' method [Craigh1968a] is a method of reducing the size (degrees of freedom) of the FE model in order to integrate the flexible model (known as superelement) into the MBS software.The dynamic analysis is based on fundamental frequencies and the corresponding mode shapes are better performed considering a few numbers of degrees of freedom. The modes (constraint modes/constrained normal modes) of the superelement need to be specified by the user in the finite element code. The 'Craig-Bampton method’ is based on the projection of large finite element models into small matrices that contain mass stiffness and mode shape data of the structure representing the low frequency response modes. For yielding the mode shape information, the modes are expressed in physical coordinates at the interface nodes in addition to a set of modes expressed in modal or generalized coordinates. The mode shapes corresponding to higher frequency responses, when transformed to modal coordinates, can be truncated without loss of information.The resulting matrices can be easilyused for the coupling with themultibody systems. Here, a study case is utilized for the demonstration of the introduced concepts.Figure 1 shows a multibody system of a steam mechanism with a flexible connection rod. In this example,the connection rod,reproduced infinite element code is coupled with the multibody system at the interface nodes, shown in Figure 1. In Simpack at least the interface nodes must be accessible since Simpack interacts with the FE superelement on these nodes. 克雷格-班普顿还原法基础 “克雷格-班普顿”法 [Craigh1968a] 是一种减少有限元模型尺寸(自由度)的方法,以便将柔性模型(称为超单元)集成到运动学软件中。基于基本频率的动力学分析,考虑少数自由度能更好地表现模态。超单元的模态(约束模态/约束正交模态)需要在有限元代码中由用户指定。 “克雷格-班普顿”法基于将大型有限元模型投影到小矩阵的过程,这些矩阵包含结构的质量、刚度和模态信息,从而代表低频响应模态。为了得到模态信息,模态不仅以界面节点处的物理坐标表示,还以模态或广义坐标表示。对应于较高频率响应的模态,在变换到模态坐标后,可以进行截断而不损失信息。得到的矩阵可以轻松用于与多体系统的耦合。 在此,利用一个案例研究来演示所介绍的概念。图1显示了一个蒸汽机理的多体系统,具有柔性连杆。在这个例子中,连杆的有限元代码与多体系统在界面节点处耦合,如图1所示。在Simpack中,至少需要访问界面节点,因为Simpack在这些节点上与有限元超单元进行交互。 Figure 1. Multibody system of a steam engine. ![](68c7e83f4bfce9fe445f7021c0b0d125ca71b389e5770bacbd736551ad23bfe5.jpg) ![](6a4b62c5d10bf1defe1f42fc1a83d91d1730c0b643e172d055a271fc3ceb5399.jpg) interface node interface node The modes that can be generated, are shown in the following sections. ### Constraint Modes The constraint modes are used to account static deformations when coupling the Body at nodes to other components through Joints, Constraints and Force Elements. These modes are the static shape assumed by the Body when setting one degree of freedom to unity while setting all other degrees of freedom at the interface nodes fixed. Consequently,the number of resulting modes is equal to the number of degrees of freedom set at the interface nodes. Accordingly, a constraint mode is computed by the finite elementcodefor eachretaineddegreeof freedom. In Figure 2 are shown the mode shapes obtained when setting one degree of freedom to unity at the interface nodes (retained nodes) while all others are held fixed. These constraint modes represent vertical bending. Setting one degree of freedom in the left hand side node (translation in 之 or rotation in a) yields the corresponding shapes coloured inblue. 约束模式 约束模式用于在将实体在节点处通过铰接件、约束和力元件与其他组件连接时,考虑静态变形。这些模式是实体在将一个自由度设置为 unity,同时将接口节点处的所有其他自由度固定时所假设的静态形状。因此,得到的模式数量等于接口节点处设置的自由度数量。相应地,对于每个保留的自由度,有限元代码会计算出一个约束模式。 图2显示了当一个自由度在接口节点(保留节点)处设置为 unity,而其他自由度固定时所获得的模式形状。这些约束模式代表垂直弯曲。将左侧节点的一个自由度设置为 unity(或旋转),会得到相应的蓝色显示的形状。 Figure 2. Constraint modes representing vertical bending. ![](0e83b7864233b0a48b093caf26501a18a5eec7ea2401e31decbec9e8efd18f4d.jpg) Note: The total static deformation due to forces and constraints at the interface nodes of the superelement is the linear combination of the constraint modes.The deformations at the retained nodes are the linear factors. The linear combination of constraint modes yields a statically correct solution, if concentrated loads act at the interface nodes. 注意:由于超单元界面节点处的力和约束引起的总体静变形,是约束模态的线性组合。保留节点的变形是线性因子。如果集中载荷作用于界面节点,则约束模态的线性组合可以得到一个静态正确的解。 # Constrained Normal Modes In order to obtain a better aproximated solution inthe higher frequency range, the constrained normal modes can be computed. Constrained normal modes are the eigenmodes when allinterface degrees of freedom are held fixed representing the natural vibration of the Body. The finite element code computes these modes by performing a modal analysis whilst all degrees of freedom are constrained at the interface nodes. The more constrained normal modes are considered, the better deformation results are obtained in Simpack. In Figure 3 are shown representative mode shapes obtained when setting zero degrees of freedom at the interface nodes. Figure3.Totalstaticdeformationofthesuperelement. ![](00ba76b2803a159600731ca15035d3e0b4b5ef77653a136c5b4a52e5fb584dd9.jpg) # Reduced Finite Element Model The total deformation of the super element is the linear combination of constraint modes and constrained normal modes. The coordinates of the super elements are the retained degrees of fredom (deformation of interface nodes) and the generalised coordinates of the constrained normal modes. The total deformation can be written as follows: Figure4.TotalDeformation. ![](fd3d662c843a5663ea73a44b221ef9b47c9fb8cda5e1020fd1ecfc08f695d48d.jpg) deformation z at left node deformation alpha at left node deformation z at right node deformation alpha at right node generalized coordinate 1 generalized coordinate 2 generalized coordinate 3 Therecoverymatrix $\mathbf{T}$ collects all the mode shapes. The expanded solution of the reduced model deformation can be written as follows: $$ \mathbf{u}={\left[\mathbf{t}_{1}\ \mathbf{t}_{2}\ \mathbf{t}_{3}\ \mathbf{t}_{4}\ \mathbf{t}_{5}\ \mathbf{t}_{6}\ \mathbf{t}_{7}\ \right]}\left\{\begin{array}{l}{{\mathbf{w}}_{1}}\\ {\beta_{1}}\\ {{\mathbf{w}}_{2}}\\ {\beta_{2}}\\ {{\mathbf{q}}_{1}}\\ {{\mathbf{q}}_{2}}\\ {{\mathbf{q}}_{3}}\end{array}\right\}} $$ $$ \mathbf{u}=\mathbf{T}\mathbf{u}_{\mathbf{S}\mathbf{E}} $$ where, $\mathbf{T}$ is the recovery matrix. use is the vector of the coordinates of the reduced finite element model The equations of motionfor thefinite element model before thereduction canbe written as: $$ \mathbf{M}_{\mathrm{FEM}}\ddot{\mathbf{u}}+\mathbf{D}_{\mathrm{FEM}}\dot{\mathbf{u}}+\mathbf{K}_{\mathrm{FEM}}\mathbf{u}=\mathbf{p}_{\mathrm{FEM}} $$ where, $\mathbf{u}$ is the deformation vector which contains the nodal displacements. i is the second derivative value with respect to time. The matrices and vectors (mass matrix $\mathbf{M}_{\mathbf{FEM}},$ dampingmatrix $\mathbf{D_{FEM}},$ stiffnes matrix ${\bf K}_{{\bf F E M}},$ load vectors PprEM) of the finite element model are reduced using the recovery matrix $\mathbf{T}$ as shown below to yield the reduced finite element model: $$ \begin{array}{r l}&{\mathbf{M}_{\mathbf{S}\mathbf{E}}=\mathbf{T}^{\mathbf{T}}\mathbf{M}_{\mathbf{F}\mathbf{E}\mathbf{M}}\mathbf{T}}\\ &{\mathbf{D}_{\mathbf{S}\mathbf{E}}=\mathbf{T}^{\mathbf{T}}\mathbf{D}_{\mathbf{F}\mathbf{E}\mathbf{M}}\mathbf{T}}\\ &{\mathbf{K}_{\mathbf{S}\mathbf{E}}=\mathbf{T}^{\mathbf{T}}\mathbf{K}_{\mathbf{F}\mathbf{E}\mathbf{M}}\mathbf{T}}\\ &{\mathbf{p}_{\mathbf{S}\mathbf{E}}=\mathbf{T}^{\mathbf{T}}\mathbf{p}_{\mathbf{F}\mathbf{E}\mathbf{M}}}\end{array} $$ where, $\mathbf{T}^{\mathbf{T}}$ isthetransposematrixof $\mathbf{T}$ The equations of motion for the reduced finite element modelcan be written as: $$ \mathbf{M}_{\mathbf{SE}}\mathbf{i}_{S E}+\mathbf{D}_{\mathbf{SE}}\mathbf{i}_{S E}+\mathbf{K}_{\mathbf{SE}}\mathbf{u}_{S E}=\mathbf{p}_{\mathbf{SE}} $$ The superelement matrices $(\mathbf{M}_{\mathbf{S}\mathbf{E}},\,\mathbf{D}_{\mathbf{S}\mathbf{E}},\,\mathbf{K}_{\mathbf{S}\mathbf{E}})$ andvectors $(\bf{p_{S E}})$ of the reduced finite element model are used for the modal Approach in Simpack. This section briefly describes by means of equations, how Simpack approaches the behavior of Flexible Bodies that are integrated into multibody systems. The Flexible Bodies are connected at the interface points by Force Elements, Constraints and Joints with the multibody system. When loads act in these connections the Flexible Bodies undergo deformations, which is represented by mode shapes. These mode shapes are either eigenmodes or modes that account for the local deformation at the interface points. The next sections will explain the equations of motion that are generated by Simpack. # Kinematic Description of Flexible Bodies in Simpack The position of a point of the Flexible Body (kinematic description) is input for setting up the equations of motion using the principle of virtual work as will be explained in Equations of Motion. For this, before describing the equations of motion in Equations of Motion, a brief description is given here by using an example of a multibody system, on how this position is represented. Figure Figure 1 shows the multibody system of a steam mechanism with a flexible connection rod. Figure 1. Multibody system of a steam engine. ![](0fc6dd95654afe820159fd2c87b21b622aff537cfca9a33553520e1d4770c5c0.jpg) The position of a point of the Flexible Body (flexible connection rod): $$ \mathbf{r}\left(\mathbf{c},t\right)=\mathbf{A}\left(\mathbf{t}+\mathbf{c}+\mathbf{u}\left(\mathbf{c},t\right)\right) $$ is represented by ${\mathbf{u}},\,{\mathbf{c}},\,{\mathbf{t}},$ A as shown in Figure 1 Wwhere, $\mathbf{u}$ is the vector of deformation which depends on the location c and the time t. c is the vector of rigid body configuration with respect to the Body Reference Frame t is the vector of translation of the Body Reference Frame with respect to Body Reference Frame A is the transformation matrix of the Body Reference Frame with respect to the inertia system. The displacement vector u depends on the location c and the time t for transient load cases. Assumption: Small, linear rigid body deformation is imposed on large rigid body motion. # Modal Approach Modeshapes $\Phi_{\mathbf{i}}\left(\mathbf{c}\right)$ are eigenmodes and/or interface modes and are formed by the superelemet matrices and vectors of the finite element reduced model described inReduced FiniteElementModel. Eigenmodes The eigenmodes account for the global deformation and/or free oscillations of the structure. $$ \left(\mathbf{K}_{\mathbf{SE}}-\omega^{2}\mathbf{M}_{\mathbf{SE}}\right)\Phi_{\mathbf{i}}\left(c\right)=0 $$ Interface modes The interface modes represent the deformation due to loads acting in Force Elements, Constraints, Joints that are connected to the Body, i.e. for the 'frequency responce mode' FRM it can be written: $$ \left(\mathbf{K}_{\mathbf{SE}}-{\Omega_{0}}^{2}\mathbf{M}_{\mathbf{SE}}\right)\Phi_{\mathbf{i}}\left(c\right)=\mathbf{p}_{S E} $$ In Equation 1, small elastic displacements will be taken into account, expressed by linear combinations of mode shapes $\Phi_{\mathbf{i}}\left(c\right)$ weightedwithtimedependent modalcoordinates $s_{\mathrm{flx},i}\left(t\right)$ , as shown in Equation 2 (modal approach): $$ \begin{array}{r}{\mathbf{u}\left(\mathbf{c},\mathbf{t}\right)=\sum\Phi_{\mathbf{i}}\left(c\right)s_{\mathrm{flx},i}\left(t\right)}\end{array} $$ where $\Phi_{\mathrm{i}}\left(c\right)$ the mode shapes (eigenmodes, interface modes) $s_{\mathrm{flx},i}\left(t\right)$ the modal coordinates For selecting the mode shapes in the user interface, see Linear flexible Bodies: Modes Tab. # Equations of Motion Inserting the kinematic description (position of a point of the Flexible Body $\mathbf{r},$ (see Kinematic Description of Flexible Bodies in Simpack) into the principle of virtual work, yields the equation of motion fora Flexible Body of a multibody system in which allterms of the principle of virtual work can be expressed as functions of the acceleration ${\bf a},$ the angular velocity $\omega$ and angular accelerating $\dot{\omega}$ and finally the modal coordinates $\dot{\mathbf{\rho}}_{s\mathrm{flx},i}\left(t\right)$ Particularly, from the mode shapes $\Phi_{\mathrm{i}}\left(c\right)$ in Equation 3 the strain vector $\varepsilon$ and stress vector $\sigma$ are computed (Equation 4). The time dependent quantities (i.e. themodal coordinates $s_{\mathrm{flx},i}$ (t), the transformation matrix A) are extracted, yielding the invariants, and from the invariants the coefficients (i.e. coeefficients of the mass matrix, the corresponding coefficients of the centrifugal forces) of the equation of motion are obtained. Position: $$ \begin{array}{r}{\int_{V}\rho\delta\mathbf{r}^{I T}\ddot{\mathbf{r}}^{I}\,\mathrm{d}V+\int_{V}\delta\boldsymbol{\varepsilon}^{T}\boldsymbol{\sigma}\,\mathrm{d}V=\int_{V}\delta\mathbf{r}^{I T}\mathbf{p}\,\mathrm{d}V}\end{array} $$ The resulting equation of motions is separated into quantities that only depend on the position (invariants over volume integral) and quantities that depend on the time. The invariants are computed by Simpack, and the computation is done automatically during the preprocessing steps (see Automatic Set-up/Update of the Equations ofMotion). ![](4d3e943d23099ddf90d36a0675769fc4a95e96e226d78ab2e883e5155195261c.jpg) # Boundary Conditions for Mode Computation The Joint of the Body determines the boundary condition that is to be considered when computing the eigenmodes and the interface modes (see Modal Approach). Directions constrained by the Joint of the Flexible Body are automatically constrained by Simpack in the mode computation steps (see Automatic Set-up/Update of the Equations of Motion). In Figure 2 are shown the boundary conditions applied on the multibody system of a steam engine. ![](a0650bb8410aaba225017baaab88257732d018de40d3881700a8625f76984599.jpg) Boundary Condition Requirements: · For six degree of freedom Joints · Any Marker is allowed as the Joint To Marker. ● For other Joints, all of the following conditions must be met: · The Joint To Marker must be connected to and coincident with a single node. Please be aware that not all options of Flexible type: in the 'Marker Properties' dialog can be used. · If the Joint does not have a degree of freedom in a certain direction then the node in the FE model must have this degree of freedom (this is a requirement for the correct calculation of the Joint forces). · The Marker Flexible type: must be At node, Interpolation (Auto), Rigid Link (User) or Rigid Link (Auto). ·The boundary condition must be independent with respect to time. The Joint types that can be used must not have exactly two rotation axes. The Joint types that cannot be used are, for example,the'universal' Joints (12: Universal al-be, 13: Universal be-ga, 14: Universal al-ga), and the Joint 25: User Defined with 2 rotational degrees of freedom. The free and locked directions are defined in the Joint From Marker. If two free rotations are defined in the Joint From Marker, no two rotations can be found that are permanently free in the Joint To Marker (that is, the boundary condition is time-dependent). ![](eeb715b43d5c68f587c2f6d7451be89694fe99899c2fc6485fdd7e40df49aaa6.jpg) ·When working with Connections, there are no restrictions. Note: If the Joint conditions above are fulfilled, the formalism will treat the connection as a Joint. In other cases, the formalism treats the Connection as a 6 degree-of-freedom Joint with a Constraint blocking the respective motions. To check how the formalism treats the Connections, run a Test Call. Any Connections that cannot be treated as Joint are provided in the Test Call output. # Automatic Set-up/Update of the Equations of Motion The automatic set-up/update of the equations of motion during the preprocessing steps are shown in this section. A schematic of the automated process is illustrated in Figure 3 Figure 3. Automatic Set-up/Update of the Equations of Motion. ![](8579aa567ba0d1bd1796bf5eef8d7e205d4b532bb306afee43d5f082f01cc007.jpg) The user does not intervene in the above described automated process. Though, when working with large models, the user should go through these steps in numerical order in order to reduce the time required for seting-up the model: 2.Modification of the Joint type of the Body (see Joints) 3. Modification of the frequency range (see Linear flexible Bodies: Modes Tab) 4. Addition/deletion of a Force Element (see Force Elements), Constraint (see Constraints) or Joint pointing to another Body 5. Definition/modification of structural damping (see 'Damping type' option in Linear flexible Bodies: Modes Tab) and 'Damping matrix' option in Linear flexibleBodies:OptionsTab. This section describes the modeling assumptions when using Linear SIMBEAM Bodies. Shape Functions and Stress Resultants Kinematic Assumptions Input Element Matrices Influence of the Shear Center Offsets Damping Rotary Inertia All SIMBEAM element formulations have 3 translational and 3 rotational degrees of freedom per node. Hence the element matrices have 12 degrees of freedomperelement. # Shape Functions and Stress Resultants This section describes the local element frame of SIMBEAM element formulations. The element's axis coincides with the $\times$ -axisoftheelementreference frame. The element y-axis points into the 'width-direction' of a basic Cross Section (for example; 1: Circle, 4: Rectangle), whereas the element ${\sf z}$ axispoints into the according 'height-direction'. All SIMBEAM element formulations representing linear flexible behavior are using the set of shape functions shown in the followingtable:
DirectionShapeFunctionDegreesoffreedompernode
Longitudinal deformationLinearLagrange polynomialTranslationaldeformationintheelementx-direction
TorsionLinearLagrange polynomialRotationaldeformationaboutelementx-direction
BendingCubicHermitepolynomialTranslationaldeformationinelementy/z-directionandrotationaldeformationabouttheelementz/y direction
Hence, the stress resultants have the following form along the element axis:
StressresultantFormalongtheelementaxis
LongitudinalforceConstant(discontinuousatnodes)
TorsionalmomentConstant(discontinuousatnodes)
Bending momentLinear (COsteady)
Bending/ShearforceConstant(discontinuousatnodes)
# Kinematic Assumptions Linear SIMBEAM element formulations use either the Euler-Bernoulli Hypothesis or the Timoshenko hypothesis. In SIMBEAM's linear elements, warping is not yet taken into account, that is, the cross-sections are supposed to be 'rigid' during the deformation so that the deformation is only represented by the beam axis. In addition, the kinematic behavior, as described in the following table, is associated to the Cross Section .
Cross SectionComment
4: Rectangle, 5: Rectangle, hollowLongitudinalandbendingdeformationuncoupled,torsionandbendinguncoupled,thetwobendingdirectionsareuncoupled
1: Circle, 2: Circle, hollowLongitudinalandbendingdeformationuncoupled,torsionandbendinguncoupled,the twobendingdirectionsareuncoupled.
6:Ellipse, 7: Ellipse, hollowLongitudinal and bending deformation uncoupled, torsion and bending uncoupled, the two bending directions are uncoupled.
10: General Basic
11: General AdvancedTakes into account twist bend coupling and coupling of bending and longitudinal deformation. 11: General Advanced Cross Sections can be used for modeling arbitrarily shaped cross-sections. In addition, depending on the cross-section's shape, the two bending directions may be coupled. Thelattereffectisdescribedbytheareadeviationmomentofinertia,seeTable1.
12:Matrix InputNotyetavailablefor linearSIMBEAMelementformulations.CompositeBeamtechnologywith in-plane andout-planewarping.Fullypopulated 6x6 prismaticbeam sections.The warping is always free,that is,also thisformulationhas 12 degrees of freedom per element.
# Input Table 1 shows the data that is used to the setup the element formulations representing linear elastic behavior: Table 1. Input Data
InputtoElement MatricesSymbol used in formulaeComment
Young'sModulusEItisusedtomodelisotropiclinearmaterial.
ShearModulusGItisusedtoisotropiclinearmaterial.
Cross-sectionareaABendingstiffnesswhenmultipliedwithYoung'smodulus.Inputtomassmatrixwhenmultipliedwithdensity.
Areamomentof inertiaabout elementx-axisIcInputtoelementmassmatrix.
Areamomentof inertiaabout elementv-axisIyBendingstiffnesswhenmultipliedwithYoung'smodulus.Inputtomassmatrixwhenmultipliedwithdensity.
The shear factors are given by:
Area moment ofBending stiffness when multiplied with Young's modulus. Input to mass matrix when multiplied with density.
inertia about elementz-axisI2
Torsional stiffness constantItYieldstorsionalstiffnesswhenmultipliedwithshearmodulus.
Area deviation moment of inertiaIyzFor 11:General Advanced Cross Section only.
Static moment about element y- axisnSFor 11:General Advanced Cross Section only.
Static moment aboutelement z- axisSFor11:GeneralAdvancedCrossSectiononly.
Shear center offset in element y- directionycsFor 11: General Advanced Cross Section only.
Shearcenteroffset in element z- directionZCSFor 11:General Advanced Cross Section only.
Center of gravity offset in element y- directionyCGFor 11: General Advanced Cross Section only.
Center ofgravity offsetinelementz- directionZCGFor 11:General Advanced Cross Section only.
Shear factor in elementy-directionVyThe dimensionless shear factor describes the amount of shearing in a range from 0 to 1. With a shear factor of 0, the SIMBEAM elements to which the Cross Section is assigned generate the maximum of shear deformation, whereas no shear deformation occurs if the shear factor is 1.
Shear factor in elementz-directionSIMBEAM elements to which the Cross Section is assigned generate the maximum of shear deformation, whereas no shear deformation occurs if the shear factor is 1.
$$ \begin{array}{r}{\psi_{z}=\frac{1}{1+12\frac{E l_{Y}}{A_{s z}}}}\end{array} $$ $$ \psi_{y}=\frac{1}{1+12\frac{E l_{z}}{A_{s y}}} $$ where the shear area $A_{s}$ is defined by: $$ \bar{\tau}A_{S}=\int\tau d\!\!\!/A $$ with the averaged (that is, constant) shear stress $\bar{\tau}$ # Element Matrices The element length is given by $l$ . The rest of the symbols are explained in Table 1. Stiffness Longitudinal Deformation Stiffness $$ K_{l}=\frac{E A}l\left(\begin{array}{c c}{{1}}&{{-1}}\\ {{-1}}&{{1}}\end{array}\right) $$ Mass $$ M_{l}=\rho A l{\binom{1/3}{1/6}} $$ Torsional Deformation Stiffness $$ K_{t}=\frac{G l_{t}}{l}\left(\begin{array}{c c}{{1}}&{{-1}}\\ {{-1}}&{{1}}\end{array}\right) $$ Mass $$ M_{t}=I_{x}l{\left(\begin{array}{l l}{1/3}&{1/6}\\ {1/6}&{1/3}\end{array}\right)} $$ Bending Stiffness for bending in $Z$ -direction $$ K_{b z}=\frac{E I_{y}}{l^{3}}\left(\begin{array}{r r r r}{{12\psi}}&{{-12\psi}}&{{-6l\psi}}&{{-6l\psi}}\\ {{-12\psi}}&{{12\psi}}&{{6l\psi}}&{{6l\psi}}\\ {{-6l\psi}}&{{6l\psi}}&{{l^{2}\left(1+3\psi\right)}}&{{l^{2}\left(-1+3\psi\right)}}\\ {{-6l\psi}}&{{6l\psi}}&{{l^{2}\left(-1+3\psi\right)}}&{{l^{2}\left(1+3\psi\right)}}\end{array}\right) $$ Mass for bending in $Z$ -direction $$ M_{b z}=\frac{\rho A l}{840}\left(\begin{array}{c c c c}{{4\left(70+7\psi+\psi^{2}\right)}}&{{4\left(35-7\psi-\psi^{2}\right)}}&{{-l\left(35+7\psi+2\psi^{2}\right)}}&{{l\left(35-7\psi-2\psi^{2}\right)}}\\ {{4\left(35-7\psi-\psi^{2}\right)}}&{{4\left(70+7\psi+\psi^{2}\right)}}&{{-l\left(35-7\psi-2\psi^{2}\right)}}&{{l\left(35+7\psi+2\psi^{2}\right)}}\\ {{-l\left(35+7\psi+2\psi^{2}\right)}}&{{-l\left(35-7\psi-2\psi^{2}\right)}}&{{l^{2}\left(7+\psi^{2}\right)}}&{{-l^{2}\left(7-\psi^{2}\right)}}\\ {{l\left(35-7\psi-2\psi^{2}\right)}}&{{l\left(35+7\psi+2\psi^{2}\right)}}&{{l^{2}\left(7-\psi^{2}\right)}}&{{l^{2}\left(7+\psi^{2}\right)}}\end{array}\right) $$ In the above listed bending matrices translational deformation is described by the rows 1 and $^{2,}$ and rotational deformation is described by the rows 3 and 4. Setting $A_{s}$ to 1 in the above bending matrices yields the Euler-Bernoulli element formulation. For the Timoshenko beam formulation, the shear factor is $0<\psi<1$ ![](c607aaeb08919b0c086d08d7973925b6818bf9976713f13833fe0c4ee2d2aad6.jpg) # Influence of the Shear Center Offsets With the 11: General Advanced Cross Section, you can take into account coupling terms between torsion and bending Figure 1 shows the coordinate system of the cross-section with origin $o$ and the axes $_y$ and $_z$ . The beam axis is perpendicular to the drawing plane and located at the origin $o$ . The shear forces $F_{y}$ and $F_{Z}$ are the resultants of the shear stress in y- and ${_Z}$ -direction. The shear center has the offset ycs and zcs with respect to the origin o.The 11: General Advanced Cross Section takes the influence of the shear center into account as an additional torsional moment $_T$ that follows from the equilibrium $T=F_{y}z_{C S}-F_{z}y_{C S}$ $_T$ is imposed on the torsional moment. Figure 1.Schematicof the shearcenter definition ![](56b367d7d545b5b005cb520b498fe9b83070f56aed29c84737308fc5f34af8ba.jpg) Taking this into account in the derivation of the beam stiffness matrix yields the shear center influence as shown in the following table: Table 2. Shear center influence (expressions that couple bending in z-direction with torsion)
Torsionatnode1Torsionatnode2
Translationalbendingdeformationinz-directionatnode112Ew(Iyycs+Iy²cs) [312E(Iyycs+Iyz²cs) [3
Translationalbendingdeformationinz-directionatnode212E(Iyycs+Iyz²cs) [312E(Iyycs+Iyz²cs) [3
Rotationalbendingdeformationabouty-directionatnode16E(IyyCs+Iyz²Cs) [26E(Iyycs+IyzCs) [2
Rotationalbendingdeformationabouty-directionatnode26E(Iyycs+Iyz²Cs) [26E(Iyycs+Iyz²Cs) [2
Bending in y-direction is modeled accordingly. Figure 2 shows the element reference system, the position of node 1, and the position of node 2. Figure2.Beamcoordinate system and nodes ![](19e66878f84764fd2ade8b93c458627da4d33651293d9189393e340dafdae64a.jpg) # Damping Linear SIMBEAM element formulations assume viscous damping directly in the modal space, therefore, no damping matrices are used. The modal damping matrixissupposedtobediagonal. # Rotary Inertia In the SIMBEAM mass matrix formulation, the moments of inertia are taken into account $\begin{array}{r}{\theta_{y}=\int z^{2}d\boldsymbol{m},}\end{array}$ which is often not the case in standard beam element formulations. The above formula describes the rotary inertia about the element y-axis and the coordinate $\scriptstyle\mathbf{Z}$ is also defined in the element reference frame. The following example of a shaft with a circular cross-section shows the effects of the rotary inertia terms: Natural frequencies in $\mathbf{1}/\mathbf{s},$ ratiolength/diameter ${\mathrel{\sim}}5$ ,kinematics:Euler-Bernoulli
BendingmodeSIMBEAMwithoutrotaryinertiaSIMBEAMwithrotaryinertiaANsYS(volumeelements)Referencefrom Gasch1989a
1115110539791163
23173265520983208
Naturalfrequenciesin1/s,ratiolength/diameter=5,kinematics:Euler-Bernoulli
BendingmodeSIMBEAMwithoutrotaryinertiaSIMBEAMwithrotaryinertiaANsYS(volumeelements)ReferencefromGasch1989a
1288276274290
2793711696802
As expected, the influence of the rotary inertia terms increases with the ratio diameter/length and the eigenfrequency. The rotary inertia terms shift the eigenfrequencies toward a finite element representation. You cannot deactivate the rotary inertia terms.