From c53698c3425c55b6fcd3e26e672216b6f757bc83 Mon Sep 17 00:00:00 2001 From: gyz Date: Tue, 22 Apr 2025 17:43:12 +0800 Subject: [PATCH] vault backup: 2025-04-22 17:43:12 --- ...ics of Multibody Systems (Shabana A.A.) (Z-Library).md | 13 +++++++------ 1 file changed, 7 insertions(+), 6 deletions(-) diff --git a/力学书籍/力学/Dynamics of Multibody Systems (Shabana A.A.) (Z-Library)/auto/Dynamics of Multibody Systems (Shabana A.A.) (Z-Library).md b/力学书籍/力学/Dynamics of Multibody Systems (Shabana A.A.) (Z-Library)/auto/Dynamics of Multibody Systems (Shabana A.A.) (Z-Library).md index f94bee1..fa75954 100644 --- a/力学书籍/力学/Dynamics of Multibody Systems (Shabana A.A.) (Z-Library)/auto/Dynamics of Multibody Systems (Shabana A.A.) (Z-Library).md +++ b/力学书籍/力学/Dynamics of Multibody Systems (Shabana A.A.) (Z-Library)/auto/Dynamics of Multibody Systems (Shabana A.A.) (Z-Library).md @@ -6445,7 +6445,7 @@ where $k_{1},k_{2}$ , and $k_{3}$ are constants. Determine whether or not these # 5 FLOATING FRAME OF REFERENCE FORMULATION In this chapter, approximation methods are used to formulate a finite set of dynamic equations of motion of multibody systems that contain interconnected deformable bodies. As shown in Chapter 3, the dynamic equations of motion of the rigid bodies in the multibody system can be defined in terms of the mass of the body, the inertia tensor, and the generalized forces acting on the body. On the other hand, the dynamic formulation of the system equations of motion of linear structural systems requires the definition of the system mass and stiffness matrices as well as the vector of generalized forces. In this chapter, the dynamic formulation of the equations of motion of deformable bodies that undergo large translational and rotational displacements are developed using the floating frame of reference formulation. It will be shown that the equations of motion of such systems can be written in terms of **a set of inertia shape integrals** in addition to the mass of the body, the inertia tensor, and the generalized forces that appear in the dynamic formulation of rigid body system equations of motion and the mass and stiffness matrices and the vector of generalized forces that appear in the dynamic equations of linear structural systems. These inertia shape integrals that depend on the assumed displacement field appear in the nonlinear terms that represent the inertia coupling between the reference motion and the elastic deformation of the body. It will be also shown that the deformable body inertia tensor depends on the elastic deformation of the body, and accordingly it is an implicit function of time. -本章将使用近似方法，推导包含互连变形体的多体系统的有限集动态运动方程。正如第三章所示，刚体多体系统的动态运动方程可以根据物体的质量、惯性张量以及作用在物体上的广义力来定义。另一方面，线性结构系统运动方程的动态公式需要定义系统质量矩阵和刚度矩阵，以及广义力向量。在本章中，我们将使用浮动参考系公式，推导经历大位移和旋转的变形体运动方程。将证明，此类系统的运动方程可以表示为一组惯性形状积分，除了物体的质量、惯性张量以及出现在刚体系统运动方程动态公式化中的广义力，以及出现在线性结构系统动态方程中的质量和刚度矩阵和广义力向量。这些取决于假设位移场的惯性形状积分出现在代表参考运动与物体弹性变形之间惯性耦合的非线性项中。还将证明，变形体的惯性张量取决于物体的弹性变形，因此它随时间呈隐式函数变化。 +本章将使用近似方法，推导包含互连变形体的多体系统的有限集动态运动方程。正如第三章所示，刚体多体系统的动态运动方程可以根据物体的质量、惯性张量以及作用在物体上的广义力来定义。另一方面，线性结构系统运动方程的动态公式需要定义系统质量矩阵和刚度矩阵，以及广义力向量。在本章中，我们将使用浮动参考系公式，推导经历大位移和旋转的变形体运动方程。将证明，**此类系统的运动方程可以表示为一组惯性形状积分**，除了物体的质量、惯性张量以及出现在刚体系统运动方程动态公式化中的广义力，以及出现在线性结构系统动态方程中的质量和刚度矩阵和广义力向量。这些取决于假设位移场的惯性形状积分出现在代表参考运动与物体弹性变形之间惯性耦合的非线性项中。还将证明，变形体的惯性张量取决于物体的弹性变形，因此它随时间呈隐式函数变化。 In the floating frame of reference formulation presented in this chapter, the configuration of each deformable body in the multibody system is identified by using two sets of coordinates: reference and elastic coordinates. Reference coordinates define the location and orientation of a selected body reference. Elastic coordinates, on the other hand, describe the body deformation with respect to the body reference. In order to avoid the computational difficulties associated with infinite-dimensional spaces, these coordinates are introduced by using classical approximation techniques such as Rayleigh–Ritz methods. The global position of an arbitrary point on the deformable body is thus defined by using a coupled set of reference and elastic coordinates. The kinetic energy of the deformable body is then developed and the inertia coupling between the reference motion and the elastic deformation is identified. The kinetic energy as well as the virtual work of the forces acting on the body are written in terms of the coupled sets of reference and elastic coordinates. Mechanical joints in the multibody system are formulated by using a set of nonlinear algebraic constraint equations that depend on the reference and elastic coordinates and possibly on time. These algebraic constraint equations can be used to identify a set of independent coordinates (system degrees of freedom) by using the generalized coordinate partitioning of the constraint Jacobian matrix, or can be adjoined to the system differential equations of motion by using the vector of Lagrange multipliers. 在本书这一章所介绍的浮动参考系表述中，多体系统中每个可变形体的状态配置由两组坐标来确定：参考坐标和弹性坐标。参考坐标定义了选定体参考系的位姿；而弹性坐标则描述了相对于该体参考系的体变形。为了避免与无限维空间相关的计算困难，这些坐标是通过使用诸如瑞利–里兹方法等经典近似技术引入的。因此，可变形体上任意一点的全局位置由一组耦合的参考坐标和弹性坐标来定义。随后，推导了可变形体的动能，并识别了参考运动和弹性变形之间的惯性耦合。作用于该体的力的动能以及虚功，都以耦合的参考坐标和弹性坐标来表示。多体系统中的机械连接，则通过一组依赖于参考坐标和弹性坐标（以及可能依赖于时间）的非线性代数约束方程来构建。这些代数约束方程可以通过约束雅可比矩阵的广义坐标划分来识别一组独立的坐标（系统自由度），或者可以通过拉格朗日乘子向量附加到系统的运动微分方程中。 @@ -6453,7 +6453,7 @@ In the floating frame of reference formulation presented in this chapter, the co # 5.1 KINEMATIC DESCRIPTION Multibody systems in general include two collections of bodies. One collection consists of bulky and compact solids that can be treated as rigid bodies, while the other collection includes typical structural components such as rods, beams, plates, and shells. As pointed out in previous chapters, rigid bodies have a finite number of degrees of freedom; for instance, a rigid body in space has six degrees of freedom that describe the location and orientation of the body with respect to the fixed frame of reference. On the other hand, structural components such as beams, plates, and shells have an infinite number of degrees of freedom that describe the displacement of each point on the component. As was shown in the preceding chapter, the behavior of such components is governed by a set of simultaneous partial differential equations. Using the separation of variables, the solution of these equations, if possible, leads to representation of the displacement field in terms of infinite series that can be written in the following form: -多体系统通常包含两类构件。一类是由粗大、紧凑的实体构成，这些实体可以被视为刚体；另一类则包括典型的结构元件，如杆、梁、板和壳体。正如前几章所指出的，刚体具有有限的自由度；例如，在空间中的刚体有六个自由度，用于描述其相对于固定参考系的位姿。另一方面，梁、板和壳体等结构元件具有无限的自由度，用于描述元件上每个点的位移。正如前一章所示，这类构件的行为由一组联立偏微分方程支配。利用变量分离法，如果可行，这些方程的解可以导向位移场的表示，其形式为无穷级数，可以写成如下形式： +多体系统通常包含两类构件。一类是由粗大、紧凑的实体构成，这些实体可以被视为刚体；另一类则包括典型的结构元件，如杆、梁、板和壳体。正如前几章所指出的，刚体具有有限的自由度；例如，在空间中的刚体有六个自由度，用于描述其相对于固定参考系的位姿。另一方面，**梁、板和壳体等结构元件具有无限的自由度，用于描述元件上每个点的位移**。正如前一章所示，这类构件的行为由一组联立偏微分方程支配。利用变量分离法，如果可行，这些方程的解可以导向位移场的表示，其形式为无穷级数，可以写成如下形式： $$ @@ -6463,8 +6463,8 @@ $$ where $\bar{u}_{f1},\,\bar{u}_{f2}$ , and $\bar{u}_{f3}$ are the components of the displacement of an arbitrary point that has coordinates $(x_{1},x_{2},x_{3})$ in the undeformed state. The vector of displacement $\bar{\mathbf{u}}_{f}=[\bar{u}_{f1}~\bar{u}_{f2}~\bar{u}_{f3}]^{\mathrm{T}}$ is space- and time-dependent. The coefficients $a_{k},\,b_{k}$ , and $c_{k}$ are assumed to depend only on time. These coefficients are called the coordinates, and the functions $f_{k},\,g_{k}$ , and $h_{k}$ are called the base functions. Each of the functions $f_{k},\,g_{k}$ , and $h_{k}$ must be admissible; that is, the function has to satisfy the kinematic constraints imposed on the boundary of the deformable body. It is also required that the infinite series of Eq. 1 converge to the limit functions $\bar{u}_{f1},\,\bar{u}_{f2}$ , and $\bar{u}_{f3}$ and that these limit functions give an accurate representation to the deformed shape. 其中，$\bar{u}_{f1},\,\bar{u}_{f2}$ , 和 $\bar{u}_{f3}$ 分别是具有坐标 $(x_{1},x_{2},x_{3})$ 的任意点在未变形状态下的位移分量。位移矢量 $\bar{\mathbf{u}}_{f}=[\bar{u}_{f1}~\bar{u}_{f2}~\bar{u}_{f3}]^{\mathrm{T}}$ 是空间和时间相关的。系数 $a_{k},\,b_{k}$ , 和 $c_{k}$ 被假定仅随时间变化。这些系数被称为坐标，而函数 $f_{k},\,g_{k}$ , 和 $h_{k}$ 被称为基函数。每个函数 $f_{k},\,g_{k}$ , 和 $h_{k}$ 都必须是可容许的；也就是说，该函数必须满足施加在变形体边界上的运动约束。此外，还需要无限级数（如公式1所示）收敛于极限函数 $\bar{u}_{f1},\,\bar{u}_{f2}$ , 和 $\bar{u}_{f3}$，并且这些极限函数能够准确地表示变形后的形状。 -Rayleigh–Ritz Approximation A simple example of Eq. 1 is the displacement representation that arises when one writes the partial differential equation of a vibrating beam and uses the separation of variables technique to solve this equation. In this particular case, the base functions are the eigenfunctions and the coordinates that are infinite in dimension are the time-dependent modal coordinates. Because of the computational difficulties encountered in dealing with infinite-dimensional spaces, classical approximation methods such as the Rayleigh–Ritz method and the Galerkin method are employed wherein the displacement of each point is expressed in terms of a finite number of coordinates. In this case the series of Eq. 1 are truncated, and this leads to -瑞利-里兹逼近法 +**Rayleigh–Ritz Approximation** A simple example of Eq. 1 is the displacement representation that arises when one writes the partial differential equation of a vibrating beam and **uses the separation of variables technique** to solve this equation. In this particular case, the base functions are the eigenfunctions and the coordinates that are infinite in dimension are the time-dependent modal coordinates. Because of the computational difficulties encountered in dealing with infinite-dimensional spaces, classical approximation methods such as the Rayleigh–Ritz method and the Galerkin method are employed wherein the displacement of each point is expressed in terms of a finite number of coordinates. In this case the series of Eq. 1 are truncated, and this leads to +**瑞利-里兹逼近法** 公式1的一个简单例子是振动梁的偏微分方程求解过程中出现的位移表示。通过变量分离法求解该方程时，基函数是特征函数，而无限维坐标则是随时间变化的模态坐标。由于处理无限维空间会遇到计算困难，因此采用诸如瑞利-里兹法和伽辽金法等经典逼近方法，其中每个点的位移用有限数量的坐标来表达。在这种情况下，公式1的级数被截断，这导致 @@ -6475,10 +6475,11 @@ $$ The functions $\bar{u}_{f1},\,\bar{u}_{f2}$ , and $\bar{u}_{f3}$ represent, in this case, partial sums of the series of Eq. 1. For the approximation of Eq. 2 to be valid, the sequences of partial sums of Eq. 2 must converge to the limit functions of Eq. 1. In other words, we require the sequences of partial sums to be Cauchy sequences. A sequence of functions $(s_{1},s_{2},\ldots)$ is said to be a Cauchy sequence if, given a small number $\varepsilon>0$ , there exists a natural number $M(\varepsilon)$ such that if $n$ and $m$ are two arbitrary natural numbers that are greater than or equal to $M(\varepsilon)$ and $m>n$ , we have $|s_{m}-s_{n}|\,<\,\varepsilon$ . By assuming that the sequences of partial sums of the series in Eq. 1 are Cauchy sequences, and provided $l,m$ , and $n$ of Eq. 2 are relatively large, we are guaranteed that the approximation of Eq. 2 is acceptable. 函数 $\bar{u}_{f1},\,\bar{u}_{f2}$ , 和 $\bar{u}_{f3}$ 在本例中分别代表公式 1 中的级数的偏和。为了使公式 2 的近似有效，公式 2 的偏和序列必须收敛到公式 1 的极限函数。换句话说，我们要求偏和序列是柯西序列。一个函数序列 $(s_{1},s_{2},\ldots)$ 被称为柯西序列，如果给定一个任意小的数 $\varepsilon>0$ ，存在一个自然数 $M(\varepsilon)$，使得当 $n$ 和 $m$ 是两个任意自然数，且都大于或等于 $M(\varepsilon)$ 且 $m>n$ 时，有 $|s_{m}-s_{n}|\,<\,\varepsilon$ 。通过假设公式 1 中级数的偏和序列是柯西序列，并且在 $l,m$ 和 $n$ 在公式 2 中足够大的前提下，我们保证公式 2 的近似是可接受的。 -Equation 2 implies also that approximations of the limit functions $\bar{u}_{f1},\ \bar{u}_{f2}$ , and $\bar{u}_{f3}$ can be obtained as linear combinations of the base functions $f_{k},\;g_{k}$ , and $h_{k}$ , respectively. This property, in addition to the fact that the sequences of partial sums of the series of Eq. 1 are Cauchy sequences, is called completeness; that is, completeness is achieved if the exact displacements, and their derivatives, can be matched arbitrarily closely if enough coordinates appear in the assumed displacement field. The assumed displacement field is, in general, either exact or stiff. This is mainly because the structure is permitted to deform only into the shapes described by the assumed displacement field. 公式 2 也暗示，极限函数 $\bar{u}_{f1}$、$\bar{u}_{f2}$ 和 $\bar{u}_{f3}$ 的近似值可以分别作为基函数 $f_{k}$、$g_{k}$ 和 $h_{k}$ 的线性组合得到。 这一性质，以及公式 1 的级数偏和序列是柯西序列这一事实，合称为完备性；也就是说，如果足够多的坐标出现在假设的位移场中，能够任意接近地匹配精确位移及其导数，则实现了完备性。 假设的位移场通常要么是精确的，要么是刚性的。 这主要是因为结构仅被允许变形为假设的位移场所描述的形状。 +Equation 2 implies also that approximations of the limit functions $\bar{u}_{f1},\ \bar{u}_{f2}$ , and $\bar{u}_{f3}$ can be obtained as linear combinations of the base functions $f_{k},\;g_{k}$ , and $h_{k}$ , respectively. This property, in addition to the fact that the sequences of partial sums of the series of Eq. 1 are Cauchy sequences, is called completeness; that is, completeness is achieved if the exact displacements, and their derivatives, can be matched arbitrarily closely if enough coordinates appear in the assumed displacement field. The assumed displacement field is, in general, either exact or stiff. This is mainly because the structure is permitted to deform only into the shapes described by the assumed displacement field. +公式 2 也暗示，极限函数 $\bar{u}_{f1}$、$\bar{u}_{f2}$ 和 $\bar{u}_{f3}$ 的近似值可以分别作为基函数 $f_{k}$、$g_{k}$ 和 $h_{k}$ 的线性组合得到。 这一性质，以及公式 1 的级数偏和序列是柯西序列这一事实，合称为完备性；也就是说，如果足够多的坐标出现在假设的位移场中，能够任意接近地匹配精确位移及其导数，则实现了完备性。 假设的位移场通常要么是精确的，要么是刚性的。 这主要是因为结构仅被允许变形为假设的位移场所描述的形状。 Floating Frame of Reference In the development presented in the subsequent sections, we assume that the displacement field of Eq. 2 describes the deformation of the body with respect to a selected body reference as shown in Fig. 5.1. The motion of the body is then defined as the motion of its reference plus the motion of the material points on the body with respect to its reference. If the assumed displacement field contains rigid body modes, a set of reference conditions has to be imposed to define a unique displacement field with respect to the selected body reference. This subject is discussed in more detail in the following chapter where a finite element floating frame of reference formulation is presented. -浮动参考系 +**浮动参考系** 在后续章节所述的发展过程中，我们假设公式2中的位移场描述了相对于所选的刚体参考系（如图5.1所示）的物体变形。然后，物体的运动被定义为其参考系的运动加上物体上各质点相对于其参考系的运动。如果假设的位移场包含刚体运动模式，则必须施加一组参考条件，以定义相对于所选刚体参考系的唯一位移场。本主题将在下一章中更详细地讨论，届时将介绍有限元浮动参考系公式。