Stability analysis of constraints in flexible multibody systems dynamics

Automated algorithms for the dynamic analysis and simulation of constrained multibody systems assume that the constraint equations are linearly independent. During the motion, when the system is at a singular configuration, the constraint Jacobian matrix possesses less than full rank and hence it results in singularities. This occurs when the direction of a constraint coincides with the direction of the lost degree of freedom. In this paper the constraint equations for deformable bodies are modified for use in the neighborhood of the singular configuration to yield the system inertia matrix which is nonsingular and also to take the actual generalized constraint forces into account. The procedures developed are applicable to both the augmented approach and the coordinate reduction methods. For the modeling of the constrained flexible multibody systems, a general recursive formulation is developed using Kane's equations, finite element method and modal analysis techniques. The system may contain revolute, prismatic, spherical or other types of joints, as well as geometrical nonlinearities; the rotary inertia is also automatically included. Simulation of a two-link flexible manipulator is presented at a singular configuration to demonstrate the utility of the method.     

A recursive method for the dynamic analysis of mechanical systems in spatial motion

Summary. In the present study, a recursive method for generating the equations of motion of mechanical systems that undergo spatial motion is presented. The method uses the force and moment equations to generate the rigid body equations of motion in terms of the Cartesian coordinates of a dynamically equivalent constrained system of particles, without introducing any rotational coordinates and the corresponding rotation matrices. For the open loop case, the equations of motion are generated recursively along the serial chains. Closed loop systems are transformed to open loop systems by cutting suitable kinematic joints and introducing cut-joint constraints. The method is simple and suitable for computer implementation. An example is chosen to demonstrate the generality and simplicity of the developed formulation.     

A matrix formulation for the dynamic analysis of spatial mechanisms using point coordinates and velocity transformation

Summary. This paper presents a matrix formulation for the dynamic analysis of spatial mechanisms with common types of kinematic joints. The formulation is derived in two steps. Initially an equivalent constrained system of particles that replaces the rigid bodies is constructed and used to define the configuration of the mechanical system. This results in a simple and straightforward procedure for generating the equations of motion in terms of the rectangular Cartesian coordinates of the particles without introducing any rotational coordinates. The equations of motion are then derived in terms of relative joint coordinates through the use of a velocity transformation matrix. The velocity transformation matrix relates the relative joint velocities to the Cartesian velocities. For the open loop case, this process automatically eliminates all of the non-working constraint forces and leads to an efficient integration of the equations of motion. For the closed loop case, suitable joints should be cut and few cut-joints constraint equations should be included for each closed loop. An example is used to demonstrate the generality and efficiency of the proposed method.     

Hamiltonian direct differentiation and adjoint approaches for multibody system sensitivity analysis

The design of multibody systems involves high fidelity and reliable techniques and formulations that should help the analyst to make reasonable decisions. Given that constrained equations of motion for the simplest of multibody systems are highly nonlinear, determining the sensitivity terms is a computationally intensive and complex process that requires the application of special procedures. In this article, two novel Hamiltonian-based approaches are presented for efficient sensitivity analysis of general multibody systems. The developed direct differentiation and the adjoint methods are based on constrained Hamilton's canonical equations of motion. This formulation provides solutions, which are more stable as compared to the results of direct integration of equations of motion expressed in terms of accelerations due to a reduced differential index of the underlying system of differential-algebraic equations and explicit constraint imposition at the velocity level.The proposed Hamiltonian based methods are both capable of calculating the sensitivity derivatives and keeping the growth of constraint violation errors at a reasonable rate. The Hamiltonian-based procedures derived herein appear to be good alternatives to existing methods for sensitivity analysis of general multibody systems.     

On the eigenvalues of a uniform cantilever beam carrying any number of spring–damper–mass systems

The eigenvalues of a uniform cantilever beam carrying any number of spring–damper–mass systems with arbitrary magnitudes and locations were determined by means of the analytical‐and‐numerical‐combined method (ANCM). First of all, each spring–damper–mass system was replaced by a massless effective spring with spring constant k_eff, which is the main point that the ANCM is available for the present problem. Next, the equation of motion for the ‘constrained’ beam (with spring–damper–mass systems attached) was derived by using the natural frequencies and normal mode shapes of the ‘unconstrained’ beam (without carrying any attachments) incorporated with the expansion theorem. Finally, the equation of motion for the ‘constrained’ beam in ‘complex form’ is separated into the real and the imaginary parts. From either part, a set of simultaneous equations were obtained. Since the simultaneous equations are in ‘real form’, the eigenvalues of the ‘constrained’ beam were determined with the conventional numerical methods. To confirm the reliability of the presented theory, all the numerical results obtained from the ANCM were compared with the corresponding ones obtained from the conventional finite element method (FEM) and good agreement was achieved. Because the order of the property matrices for the equation of motion derived by using the ANCM is much lower than that by using the conventional FEM, the storing memory and the CPU time required by the ANCM are much less than those required by the FEM. Besides, the solution of the equation of motion derived from the ANCM can always be obtained with the general personal computers, but that from the FEM can sometimes be obtained only with the computers of workstations or main frames when the total degrees of freedom exceeding a certain limit. Copyright © 1999 John Wiley & Sons, Ltd.     

On the construction of a geometric constraint frame by integral manifold theory for constrained manipulators

Abstract

For industrial constrained manipulators, the interactive motion and force dynamics between the end effector and the constrained environment occur at both the normal space and the tangent space to the constraint manifold. In order to characterize such interactive behaviors, it is advantageous to develop a specific coordinate frame to facilitate the formulation of such constrained dynamics. Most previous approaches solved the problem from the algebraic viewpoint. This paper, on the other hand, develops a systematic method to construct a geometric constraint frame from the perspective of integral manifold theory. A constraint frame, distinct from previous orthogonal moving frames, behaves as a partially orthogonal coordinate system and is endowed with intrinsic geometric implications which are beneficial to the dynamic analysis and the controller design. The algebraic and geometric points of view concerning the constrained systems are compared and discussed. An illustrative example for constrained manipulators will be demonstrated.     

Constrained dynamics of geometrically exact beams

 Geometrically exact beams are regarded from the outset as constrained mechanical systems. This viewpoint facilitates the discretization in space and time of the underlying continuous beam formulation without using rotational variables. The present semi-discrete beam equations assume the form of differential–algebraic equations which are discretized in time. The resulting energy–momentum scheme satisfies the algebraic constraint equations on both configuration and momentum level. Dedicated to the memory of Prof. Mike Crisfield, for his cheerfulness and cooperation as a colleague and friend over many years.     

非对称粘滞阻尼系统的约束部件模态综合

本文对具非对称粘滞阻尼的振动系统提出一种约束部件模态综合法。首先消除状态空间部件运动方程内不独立的界面位移与界面速度，然后以截断的约束部件复模态、位移约束模态及速度约束模态作为里兹基，减缩总体方程。文末以算例说明了方法的有效性。     

Iterative procedures for improving penalty function solutions of algebraic systems

The standard implementation of the penalty function approach for the treatment of general constraint conditions in discrete systems of equations often leads to computational difficulties as the penalty weights are increased to meet constraint satisfaction tolerances. A family of iterative procedures that converges to the constrained solution for fixed weights is presented. For a discrete mechanical system, these procedures can be physically interpreted as an equilibrium iteration resulting from the appearance of corrective force patterns at the nodes of ‘constraint members’ of constant stiffness. Three forms of the iteration algorithm are studied in detail. Convergence conditions are established and the computational error propagation behaviour of the three forms is analysed. The conclusions are verified by numerical experiments on a model problem. Finally, practical guidelines concerning the implementation of the corrective process in large-scale finite element codes are offered.     

Stochastic response determination of nonlinear structural systems with singular diffusion matrices: A Wiener path integral variational formulation with constraints

The Wiener path integral (WPI) approximate semi-analytical technique for determining the joint response probability density function (PDF) of stochastically excited nonlinear oscillators is generalized herein to account for systems with singular diffusion matrices. Indicative examples include (but are not limited to) systems with only some of their degrees-of-freedom excited, hysteresis modeling via additional auxiliary state equations, and energy harvesters with coupled electro-mechanical equations. In general, the governing equations of motion of the above systems can be cast as a set of underdetermined stochastic differential equations coupled with a set of deterministic ordinary differential equations. The latter, which can be of arbitrary form, are construed herein as constraints on the motion of the system driven by the stochastic excitation. Next, employing a semi-classical approximation treatment for the WPI yields a deterministic constrained variational problem to be solved numerically for determining the most probable path; and thus, for evaluating the system joint response PDF in a computationally efficient manner. This is done in conjunction with a Rayleigh-Ritz approach coupled with appropriate optimization algorithms. Several numerical examples pertaining to both linear and nonlinear constraint equations are considered, including various multi-degree-of-freedom systems, a linear oscillator under earthquake excitation and a nonlinear oscillator exhibiting hysteresis following the Bouc–Wen formalism. Comparisons with relevant Monte Carlo simulation data demonstrate a relatively high degree of accuracy.     

特殊约束条件下多跨梁静定超静定特征的共生现象

工程中存在的一些特殊约束形式，如约束绳系、液压支承系统会引出一些新的力学问题，本文揭示的反力强制分配效应及双特殊约束系统约束多跨梁（或结构）静定超静定特征的共生现象是引人关注的两个主要力学问题。     

A direct violation correction method in numerical simulation of constrained multibody systems

 A new direct violation correction method for constrained multibody systems is presented. It can correct the value of state variables of the systems directly so as to satisfy the constraint equations of motion. During the integration of the dynamic equations of constrained multibody systems, this method can efficiently control the violations of constraint equations within any given accuracy at each time-step. Compared to conventional indirect methods, especially Baumgarte's Constraint Violation Stabilization Method, this method has clear physical meaning, less calculation and obvious correction effect. Besides, this method has minor effect on the form of the dynamic equations of systems, so it is stable and highly accurate. A numerical example is provided to demonstrate the effectiveness of this method. Received: 17 December 1999     

谐变力作用功能梯度旋转圆板强非线性主共振

功能梯度材料作为一种新型材料,具有良好的力学性能,近年来被广泛关注和应用。该文针对金属-陶瓷功能梯度圆板,考虑周边夹支边界约束条件,选取多项式形式的振型函数,利用伽辽金法,推得旋转运动状态和热效应作用下系统的纵横耦合非线性振动方程,求得由旋转及密度差引起的静挠度项。用改进多尺度法求解方程,得到强非线性系统的频幅响应方程和解析解。通过算例,给出功能梯度圆板的幅频曲线、幅值-激励力曲线、幅值-温度曲线,分析了不同物理量对结构共振幅值的影响规律,并且比较了解析解和数值解,两者结果较为吻合。     

An effective solver for absolute variable formulation of multibody dynamics

This paper presents an effective and general method for converting the equations of motion of multibody systems expressed in terms of absolute variables and Lagrange multipliers into a convenient set of equations in a canonical form (constraint reaction-free and minimal-order equations). The method is applicable to open-loop and closed-loop multibody systems, and to systems subject to holonomic and/or nonholonomic constraints. Being aware of the system configuration space is a metric space, the Gram-Schmidt ortogonalization process is adopted to generate a genuine orthonormal basis of the tangent (null, free) subspace with respect to the constrained subspace. The minimal-order equations of motion expressed in terms of the corresponding tangent speeds have the virtue of being obtained directly in a resolved form, i.e. the related mass matrix is the identity matrix. It is also proved that, in the case of absolute variable formulation, the orthonormal basis is constant, which leads to additional simplifications in the motion equations and fits them perfectly for numerical formulation and integration. Other useful peculiarities of the orthonormal basis method are shown, too. A simple example is provided to illustrate the convertion steps.The research leading to this paper was supported in part by the State Committee for Scientific Research, Grant No. 3 0955 91 01     

Dynamic Performance of Stacked Packaging Units

Stacked packaging units is the main form of distribution packaging of products. Its dynamic performance is not fully understood. This paper investigated the influence of the constraint, input vibration, location and contact nonlinearity on the dynamic performance of three layers stacked packaging units. The dynamic contact force between surfaces and acceleration response of products were obtained. In sine sweep vibrations, the constraint to stacked packaging units has an obvious influence on the dynamic characteristics. The acceleration response of product is associated with the vibration mode. The force amplification factor is in general between 1.5 and 2, but it can close to 3 on top layer in the case of no fixed. In random vibrations, non‐Gaussian data of dynamic contact force appear when Gaussian data of input vibration pass through the stacked packaging units, resulting a Weibull distribution of force level‐crossing. The force level‐crossing diagram becomes more abrupt with the decreasing of input vibration level, smoother from the top contact surface to middle and bottom ones, and moves right and becomes smoother with the constraint strengthen. In the case of lower input level, Gaussian distribution of force level‐crossing appears. The force power spectral density (PSD) between bottom box and table is much larger than that between boxes, which is significantly influenced by the first resonance frequency. However, the acceleration PSD of product is significantly influenced by both the first and second resonance frequencies, and controlled by the vibration mode. It depends much on the value of input acceleration PSD around the resonance frequencies.     

Evolution equation of moving defects: dislocations and inclusions

Evolution equations, or equations of motion, of moving defects are the balance of the “driving forces”, in the presence of external loading. The “driving forces” are defined as the configurational forces on the basis of Noether’s theorem, which governs the invariance of the variation of the Lagrangean of the mechanical system under infinitesimal transformations. For infinitesimal translations, the ensuing dynamic J integral equals the change in the Lagrangean if and only if the linear momentum is preserved. Dislocations and inclusions are “defects” that possess self-stresses, and the total driving force for these defects consists only of two terms, one expressing the “ self-force” due to the self-stresses, and the other the effect of the external loading on the change of configuration (Peach–Koehler force). For a spherically expanding (including inertia effects) Eshelby (constrained) inclusion with dilatational eigenstrain (or transformation strain) in general subsonic motion, the dynamic J integral, which equals the energy-release rate, was calculated. By a limiting process as the radius tends to infinity, the driving force (energy-release rate) of a moving half-space plane inclusion boundary was obtained which is the rate of the mechanical work required to create an incremental region of eigenstrain (or transformation strain) of a dynamic phase boundary. The total driving force (due to external loading and due to self-forces) must be equal to zero, in the absence of dissipation, and the evolution equation for a plane boundary with eigenstrain is presented. The equation applied to many strips of eigenstrain provides a system to solve for the position/ evolution of strips of eigenstrain.     

Unified Relaxation Spectrum for Polymers of Multiple Scales Based upon a Fuzzy Constraint Method and Non-Equilibrium Statistical Thermodynamics

The molecular relaxation mechanisms of polymers withmulti-scale units of motion in glassy, rubbery and melt states areproposed based upon a fuzzy constraint method and non-equilibriumstatistical thermodynamics. The entanglement effects due to cohesiveforce and steric hindrance are expressed quantitatively in terms of amembership function. The micro-Brownian motion of a polymer chain isgoverned by the Langevin equation, which accounts for viscous force,nonuniform tension, entanglement constraint force and random force.Perturbation solutions have been established for different time and sizescales. The solutions account for the effects of both intramolecular andintermolecular interactions in the relaxation process. The unifiedrelaxation spectrum over many orders of time scale is a naturalconsequence of macromolecular structure, which satisfies thetime-temperature equivalence in the form of the Arrhenius equation atlow and high temperatures and in the form of the WLF equation near theglass transition temperature. The barrier model, the normal mode theory,the retraction and reptation theories can be taken as special casescorresponding to different scales.     

Magnetophoretic velocity modulation mass analysis of a single microparticle in an atmosphere

A new principle of the magnetophoretic velocity modulation mass analysis of microparticles, which can determine simultaneously the mass and magnetic susceptibility of a single microparticle, has been proposed, and the measurement system was constructed by applying a magnetophoretic force on a falling microparticle through a magnetic field gradient in an atmosphere. A polystyrene microparticle as a test particle adsorbed on a glass plate was selectively knocked off by a pulsed Nd:YAG laser impact into a narrow gap of pole pieces of permanent magnets having a magnetic field gradient with a maximum intensity of 850 T2 m(-1). The falling particle was irradiated by a He-Ne laser, and the scattered light was detected through a slit array mask as a function of time. A bundle of spiked signals of scattered light intensity was analyzed to obtain velocities, which gave acceleration and deceleration of the falling particle. On the basis of the equation of motion under the magnetic field gradient, the mass and magnetic susceptibility of the test particle were reasonably determined.     

激光干涉法进行正弦力校准研究

采用绝对法进行正弦力校准,动态力由安装在振动台上的质量块产生,力值根据力的定义直接溯源到质量、加速度和时间。为了减少测量不确定度,质量块上的加速度分布必须进行准确测量。文中介绍了采用激光干涉法进行动态力校准的原理和信号处理方法,同时介绍了与德国PTB加速度实验室进行加速度测量的比对情况。结果表明,采用所介绍的正弦力校准方法可以提高校准的准确度。     

Stochastic rotational response of a parametric pendulum coupled with an SDOF system

In this paper, parametric excitation of a lumped mass pendulum is investigated when having its pivot point vertically excited. Such a system is described by a sinus type nonlinear Mathieu equation and attention is drawn to the rotational motion of the mass. A single-degree-of-freedom (SDOF) system is used as a base for the pendulum, floating on ocean waves and being externally excited by them thus falling into the subclass of autoparametric systems. Due to the random nature of waves, a narrow-band stochastic process is used to model the environmental excitation. The effect of dry friction is considered and the interaction between the pendulum's rotational response and the base is studied.