An efficient computational method for dynamic stress analysis of flexible multibody systems

This paper presents an efficient computational method of dynamic stress history calculation for a general three-dimensional flexible body by combining flexible multibody dynamic simulation and quasi-static finite element analysis (FEA). In the dynamic simulation of flexible multibody systems, flexible components can undergo nonsteady gross motion and small elastic deformation that is described with respect to the body reference frame by using the assumed mode method. D'Alembert inertia loads from the gross body motion and the elastic deformation are expressed as a combination of space-dependent and time-dependent terms that are obtained from the dynamic simulation. D'Alembert inertia loads that are associated with each unit value of the time-dependent terms are then distributed to all finite element nodes in order to compute a corresponding stress influence coefficient through quasi-static structural analyses. Total dynamic stresses due to D'Alembert inertia loads are obtained by multiplying actual magnitude of time-dependent terms with the associated stress influence coefficients. By the proposed method, it is shown that, for a general three-dimensional component, the required number of FEAs can be significantly reduced.     

A method for improving dynamic solutions in flexible multibody dynamics

This paper presents a method for improving dynamic solutions that are obtained from the dynamic simulation of flexible multibody systems. The mode-acceleration concept in linear structural dynamics is utilized in the proposed method for improving accuracy in the postprocessing stage. A theoretical explanation is made on why the proposed method improves the dynamic solutions in the context of the mode-acceleration method. A mode-acceleration equation for each flexible body is defined and the load term in the right hand side of the equation is represented as a combination of space-dependent and time-dependent terms so that efficient computation of dynamic solutions can be achieved. The load term is obtained from dynamic simulation of a flexible multibody system and a finite element method is used to compute dynamic solutions by quasi-static analyses. Numerical examples show the effectiveness of the proposed method.     

Comment on spanning trees for multibody dynamic simulation

The results of [1] are generalized to accept an arbitrary search method for the construction of a spanning tree of the mechanical system graph.     

Stability evaluation and system identification of flexible multibody systems

This paper is concerned with the linearized stability analysis and system identification of flexible multibody systems. Two closely related stability analysis approaches are summarized. Next, these approaches are extended to provide robust system identification procedures that combine least squares techniques and Kalman filters. The singular value decomposition, a numerically stable mathematical tool, is used to improve the robustness of the algorithm. The proposed algorithm identifies a minimum order plant based on input-output data, and is applicable to both experimental measurements or numerically computed responses. The proposed approaches are computationally inexpensive and consist of purely post processing steps that can be used with any multi-physics computational multibody tool or with experimental data. Commemorative Contribution.     

Dynamics simulation of multibody chains on a transputer system

The paper describes the implementation on a transputer system of a novel parallel algorithm for dynamics simulation of a multibody chain. The algorithm is formulated at a level of parallelism which is natural for the problem but is essentially unavailable to other simulation dynamics algorithms. The experimental results demonstrate that one can improve efficiency of computation by exploiting this level of parallelism. However, analysis of the performance shows that the serial component of the resulting parallel algorithm grows to be a large fraction of the total parallel execution time and therefore limits the speedup that can be achieved with this approach.     

Geometric stiffening of flexible link system with large overall motion

In the conventional hybrid-coordinate formulation, the Cartesian deformation variables are employed with a linear Cauchy strain measure. It has been found that such modeling method fails to capture the motion-induced stiffness terms and provides erroneous dynamic results in case of high rotating speed. In this paper, geometric stiffening of flexible link system is investigated. Using a non-Cartesian deformation variable, the equations of motion of each link, which include the stiffening terms, are obtained based on the virtual power principle, and forward recursive formulation is employed to derive the equations of flexible link system. Relative generalized coordinates are employed to derive the equations of motion of the link system. Numerical examples are presented to investigate the stiffening effect on large overall motion as well as deformation of the flexible link system and to testify the accuracy and efficiency of the formulation.     

Parameter identification of damping models in multibody dynamic simulation of mechanical systems

To reduce vibration and noise, a damping mechanism is often required in mechanical systems. Many types of dampers are currently used. In this paper, several typical damping models, i.e., structural damping, frictional damping, and viscoelastic damping, are illustrated, and their parameters are identified for multibody dynamic simulation. Linear damping, widely adopted for structural damping, is applied to beam deflection. Quadratic damping including air resistance is applied to plate deflection. To model stick phenomenon in mechanical dampers, a STV (stick-transition velocity) model was first introduced. To identify parameters, an optimization process is applied to the damping parameters. A new MSTV (modified stick-transition velocity) model is proposed for a friction damper. A modified Kelvin–Voight model is suggested for a rubber bushing model used in vehicle dynamics, and its parameters are identified. A modified Bouc–Wen model is also proposed; it includes the hysteretic behavior of an elastomer, and optimized results with parameter identification are compared to test results.     

A regularized approach for frictional impact dynamics of flexible multi-link manipulator arms considering the dynamic stiffening effect

The present study offers a regularized approach for multi-link flexible manipulator arms with frictional impacts. The complex risks of global dynamics simulation, which involve nonlinear frictional impact, stick–slip, and foreshortening deformation, as well as multi-scale numerical problems, were implemented. The system is described as an assembly of \(n\) flexible links connected by \(n\) rotary joints. The stretching, bending, and the torsional deformations of the flexible links were considered in addition to the flexibility and mass of the joint. The introduction of a contact force potential energy approach transformed the non-differentiable functions of the normal and tangential frictions into differentiable ones, thereby generating Lagrange equations for the general recursive formulation of the systems. A numerical simulation for the double pendulum and spatial manipulator arms collision with targets was generated, thereby allowing the calculation of the frequent switching between the stick/sliding and forward/backward sliding. Several normal contact and friction models were adopted, and their corresponding results were analyzed. The generated ordinary differential equations of the proposed smoothed algorithm were solved using explicit solvers to verify any improvements in the global computational efficiency of the frictional collision dynamics for the flexible manipulator arms.     

A nonlinear two-node superelement for use in flexible multibody systems

A nonlinear two-node superelement is proposed for the modeling of flexible complex-shaped links for use in multibody simulations. Assuming that the elastic deformations with respect to a corotational reference frame remain small, substructuring methods may be used to obtain reduced mass and stiffness matrices from a linear finite element model. These matrices are used in the derivation of potential and kinetic energy expressions of the nonlinear two-node superelement. By evaluating Lagrange’s equations, expressions for the internal and external forces acting on the superelement can be obtained. The inertia forces of the superelement are derived in terms of absolute nodal velocities and accelerations, which greatly simplifies the dynamic formulation. Three examples are included. The first two examples are used to validate the method by comparing the results with those obtained from nonlinear beam element solutions. We consider a benchmark simulation of the spin-up motion of a flexible beam with uniform cross-section and a similar simulation in which the beam is simultaneously excited in the out-of-plane direction. Results from both examples show good agreement with simulation results obtained using nonlinear finite beam elements. In a third example, the method is applied to an unbalanced rotating shaft, illustrating the potential of the proposed methodology for a more complex geometry.     

A two-step approach for model reduction in flexible multibody dynamics

In this work, a two-step approach for model reduction in flexible multibody dynamics is proposed. This technique is a combination of the Krylov-subspace method and a Gramian matrix based reduction approach that is particularly suited if a small reduced-order model of a system charged with many force-inputs has to be generated. The proposed methodology can be implemented efficiently using sparse matrix techniques and is therefore applicable to large-scale systems too. By a numerical example, it is demonstrated that the suggested two-step approach has very good approximation capabilities in the time as well as in the frequency domain and can help to reduce the computation time of a numerical simulation significantly.     

Use of penalty formulations in dynamic simulation and analysis of redundantly constrained multibody systems

The determination of particular reaction forces in the analysis of redundantly constrained multibody systems requires the consideration of the stiffness distribution in the system. This can be achieved by modeling the components of the mechanical system as flexible bodies. An alternative to this, which we will discuss in this paper, is the use of penalty factors already present in augmented Lagrangian formulations as a way of introducing the structural properties of the physical system into the model. Natural coordinates and the kinematic constraints required to ensure rigid body behavior are particularly convenient for this. In this paper, scaled penalty factors in an index-3 augmented Lagrangian formulation are employed, together with modeling in natural coordinates, to represent the structural properties of redundantly constrained multibody systems. Forward dynamic simulations for two examples are used to illustrate the material. Results showed that scaled penalty factors can be used as a simple and efficient way to accurately determine the constraint forces in the presence of redundant constraints.     

Topology optimization based on level set for a flexible multibody system modeled via ANCF

A topology optimization methodology is proposed for the flexible multibody system undergoing both large overall motion and large deformation. The system of concern is modeled via the absolute nodal coordinate formulation. The equivalent static load method is employed to transform the topology optimization of the nonlinear dynamic response of the system into a static one, and evaluated to adapt to the absolute nodal coordinate formulation by splitting the elastic deformations of the flexible components from the overall motions of those components. During the static topology optimization, the material interface is implicitly described as the zero level set of a higher-dimensional scalar function. Then, the semi-implicit level set method with the additive operator splitting algorithm is employed to solve the corresponding Hamilton-Jacobi partial differential equation. In addition, the expert evaluation method of weights based on the grey theory is utilized to define the objective function, and a modified augmented Lagrange multiplier method is proposed to treat the inequality volume constraint so as to avoid the oscillation and drift of the volume. Finally, two numerical examples are provided to validate the proposed methodology.     

An improved dynamic modeling of a multibody system with spherical joints

A dynamic modeling of multibody systems having spherical joints is reported in this work. In general, three intersecting orthogonal revolute joints are substituted for a spherical joint with vanishing lengths of intermediate links between the revolute joints. This procedure increases sizes of associated matrices in the equations of motion, thus increasing computational burden of an algorithm used for dynamic simulation and control. In the proposed methodology, Euler parameters, which are typically used for representation of a rigid-body orientation in three-dimensional Cartesian space, are employed to represent the orientation of a spherical joint that connects a link to its previous one providing three-degree-of-freedom motion capability. For the dynamic modeling, the concept of the Decoupled Natural Orthogonal Complement (DeNOC) matrices is utilized. It is shown in this work that the representation of spherical joints motion using Euler parameters avoids the unnecessary introduction of the intermediate links, thereby no increase in the sizes of the associated matrices with the dynamic equations of motion. To confirm the efficiency of the proposed representation, it is illustrated with the dynamic modeling of a spatial four-bar Revolute-Spherical–Spherical-Revolute (RSSR) mechanism, where the CPU time of the dynamic modeling based on proposed methodology is compared with that based on the revolute joints substitution. Finally, it is explained how a complex suspension and steering linkage can be modeled using the proposed concept of Euler parameters to represent a spherical joint.     

Frame loads accuracy assessment of semianalytical multibody dynamic simulation methods of a recreational vehicle

The design of a vehicle frame is largely dependent on the loads applied on the suspension and heavy parts mounting points. These loads can either be estimated through full analytical multibody dynamic simulations, or from semi-analytical simulations in which tire and road sub-models are not included and external vehicle loads, recorded during field testing, are used as inputs to the wheel hubs. Several semi-analytical methods exist, with various modeling architectures, yet, it is unclear how one method over another improves frame loads prediction accuracy.

This study shows that a semi-analytical method that constrains the vehicle frame center of gravity movement along a recorded trajectory, using a control algorithm, leads to an accuracy within 1% for predicting frame loads, when compared to reference loads from a full analytical model. The control algorithm computes six degrees of freedom forces and moments applied at the vehicle center of gravity to closely follow the recorded vehicle trajectory. It is also shown that modeling the flexibility of the suspension arms and controlling wheel hub angular velocity both contribute in improving frame loads accuracy, while an acquisition frequency of 200 Hz appears to be sufficient to capture load dynamics for several maneuvers. Knowledge of these loads helps engineers perform appropriate dimensioning of vehicle structural components therefore ensuring their reliability under various driving conditions.

     

Static form-finding analysis of a railway catenary using a dynamic equilibrium method based on flexible multibody system formulation with absolute nodal coordinates and controls

This paper proposes a dynamic equilibrium method for finding the initial equilibrium configuration of a railway catenary. In the proposed method, the catenary composed of flexible wires is modeled using two-node cable elements with absolute nodal coordinates based on a flexible multibody system formulation. Dynamic conditions that characterize the initial equilibrium configuration of the catenary are given and addressed as control processes in the form-finding procedure. The key feature of the proposed method is that the catenary configuration is continually evolved by dynamic simulation until characterization conditions are attained and an equivalent configuration of the centenary at static equilibrium is thus computed. It is validated using two examples and applied to the form-finding analysis of a two dimensional simple railway catenary. The obtained results are analyzed and discussed. It is general and can be applied to catenaries with complex configurations.     

Scaling problems in dynamic system simulation

Dynamic system simulation using analog or hybrid computers requires scaling of the variables. This problem arises also in digital simulation when fixed-point arithmetic is used. Prediction of variables extrema is the most difficult problem in scaling. Authors developed methods of estimation of variables based on direct Lyapunov method. General solution for the linear time-invariant systems was obtained. Extension of this solution for some classes of non-linear systems common in control engineering is also shown.     

Analysis of dynamic strains in tibia during human locomotion based on flexible multibody approach integrated with magnetic resonance imaging technique

Bone is known to adapt to the prevalent strain environment while the variation in strains, e.g., due to mechanical loading, modulates bone remodeling, and modeling. Dynamic strains rather than static strains provide the primary stimulus of bone functional adaptation. The finite element method can be generally used for estimating bone strains, but it may be limited to the static analysis of bone strains since the dynamic analysis requires expensive computation. Direct in vivo strain measurement, in turn, is an invasive procedure, limited to certain superficial bone sites, and requires surgical implementation of strain gauges and thus involves risks (e.g., infection). Therefore, to overcome difficulties associated with the finite element method and the in vivo strain measurements, the flexible multibody simulation approach has been recently introduced as a feasible method to estimate dynamic bone strains during physical activity. The purpose of the present study is to further strengthen the idea of using the flexible multibody approach for the analysis of dynamic bone strains. Besides discussing the background theory, magnetic resonance imaging is integrated into the flexible multibody approach framework so that the actual bone geometry could be better accounted for and the accuracy of prediction improved.     

Discussion on the optimization problem formulation of flexible components in multibody systems

This paper is dedicated to the structural optimization of flexible components in mechanical systems modeled as multibody systems. While most of the structural optimization developments have been conducted under (quasi-)static loadings or vibration design criteria, the proposed approach aims at considering as precisely as possible the effects of dynamic loading under service conditions. Solving this problem is quite challenging and naive implementations may lead to inaccurate and unstable results. To elaborate a robust and reliable approach, the optimization problem formulation is investigated because it turns out that it is a critical point. Different optimization algorithms are also tested. To explain the efficiency of the various solution approaches, the complex nature of the design space is analyzed. Numerical applications considering the optimization of a two-arm robot subject to a trajectory tracking constraint and the optimization of a slider-crank mechanism with a cyclic dynamic loading are presented to illustrate the different concepts.