Showing papers in "Mechanics of Structures and Machines in 1997"

On the Design of Compliant Mechanisms Using Topology Optimization

Abstract: This paper presents a method for optimal design of compliant mechanism topologies. The method is based on continuum-type topology optimization techniques and finds the optimal compliant mechanism topology within a given design domain and a given position and direction of input and output forces. By constraining the allowed displacement at the input port, it is possible to control the maximum stress level in the compliant mechanism. The ability of the design method to find a mechanism with complex output behavior is demonstrated by several examples. Some of the optimal mechanism topologies have been manufactured, both in macroscale (hand-size) made in Nylon, and in microscale (<.5mm)) made of micromachined glass.

A Topology-Based Approach for Exploiting Sparsity in Multibody Dynamics in Cartesian Formulation*

Abstract: In this paper, a new method for efficiently computing accelerations and La-grange multipliers in the equations of multibody dynamics is presented. These quantities are the solution of a system of linear equations that has a coefficient matrix with the special structure of an optimization matrix. This matrix is likely to have a large number of zero entries, according to the connectivity among bodies of the mechanical system. This method takes advantage of both the special structure and the sparsity of the coefficient matrix. Simple manipulations bring the original problem of solving a system of n + m equations in n + m unknowns to an equivalent problem in which a positive definite system of dimension m × m has to be solved for the Lagrange multipliers. Accelerations are then efficiently determined. For certain mechanical system models, the bandwidth of the m ×m positive definite matrix can be reduced significantly by appropriately numbering the joints connecting bodies of the model. Numerical expe...

A State-Space-Based Implicit Integration Algorithm for Differential-Algebraic Equations of Multibody Dynamics*

Abstract: An implicit numerical integration algorithm based on generalized coordinate partitioning is presented for the numerical solution of differential-algebraic equations of motion arising in multibody dynamics. The algorithm employs implicit numerical integration formulas to express independent generalized coordinates and their first time derivative as functions of independent accelerations at discrete integration times. The latter are determined as the solution of discretized equations obtained from state-space, second-order ordinary differential equations in the independent coordinates. All dependent variables in the formulation, including Lagrange multipliers, are determined by satisfying the full system of kinematic and kinetic equations of motion. The algorithm is illustrated using the implicit trapezoidal rule to integrate the constrained equations of motion for three stiff mechanical systems with different generalized coordinate dimensions. Results show that the algorithm is robust and has the ...

An efficient three-dimensional tire model for vehicle dynamics simulations

Abstract: For vehicle dynamics simulations, tire models are lacking that can accurately predict spindle loads resulting from durability road events, such as curb impact or chuckhole impact, and for off-road scenarios. In this paper, an efficient three-dimensional tire model is presented for use in simulating such events. The model is based upon a precomputed table that characterizes the force-deflection behavior of the tire sidewall, which is coupled to standard shell elements to model the tire tread. Comparisons are given between computed and experimental results for several different loading cases that demonstrate the effectiveness of the proposed model

A Topology-Based Approach to Exploiting Sparsity in Multibody Dynamics: Joint Formulation*

Abstract: In this paper, advantage is taken of the problem structure in multibody dynamics simulation when the mechanical system is modeled using a minimal set of generalized coordinates. It is shown that the inertia matrix associated with any open- or closed-loop mechanism is positive definite by finding a simple mathematical expression for the quadratic form expressing the kinetic energy in an associated state space. Based on this result, an algorithm that efficiently solves for second time derivatives of the generalized coordinates is presented. Significant speed-ups accrue due to both the no fill-in factorization of the composite inertia matrix technique and the degree of parallelism attainable with the new algorithm.

On Computational Issues in Large Deformation Analysis of Rubber Bushings

Abstract: Accurate bushing analysis requires a locking free finite element formulation, an appropriate selection of the strain energy density function, and an adequate use of bulk modulus to assure numerical stability and accuracy. In this paper, the pressure projection finite element method is employed. The method projects displacement-calculated pressure onto a lower order pressure field, based on the Babuska-Brezzi condition, to avoid volumetric locking and pressure oscillation. Mooney-Rivlin and Cubic strain energy density functions are used to study the material effect on the predicted rubber behavior in tension-compression and shear deformation modes, and the need to use a higher order strain energy density function for bushing analysis is identified. The effect of bulk modulus on bonded rubber behavior in bushings with respect to bushing shape factor is studied, and the minimum allowable bulk modulus to impose incompressibility in bushing analysis is characterized. The load-deflection response of an...

Elastodynamics of Multibody Systems Using Generalized Inertial Coordinates and Structural Damping

Abstract: A new formulation to describe the elastodynamics of flexible multibody systems using efficient generalized inertial coordinates is presented and discussed in this paper. The finite element method is initially applied to the continuum mechanics equations of the system components, leading to equations of motion for flexible bodies in which the linear elastodynamics is effectively coupled with the body gross motion by a time variant mass matrix. However, the coefficients of the mass matrix must be derived for each panicular type of finite element used in the description of the flexible body. Applying a lumped mass formulation and referring nodal displacements to the inertial frame, rather than to the body-fixed coordinate frame, yields a constant diagonal mass matrix for a flexible body. Coupling between the large rigid body motion and the small elastic deformations is still preserved. Kinematic constraints are introduced in the multibody system equations, using the new coordinates. Efficiencies and...

A Flexible, Reconfigurable, Transient Multi-cylinder Diesel Engine Simulation for System Dynamics Studies*

Abstract: A new generation, transient, multi-cylinder, turbocharged diesel engine simulation is developed for predictions of dynamic response and performance of engine powertrain systems, for assessment of alternative system configurations, and for integration studies in conjunction with the rest of the components of ground vehicles. The simulation is based on a comprehensive single-cylinder engine model with built-in physical submodels and transient capability to ensure high fidelity predictions. The single-cylinder model has been converted into a module within the flexible and reconfigurabie MATLAB-SIMULINK environment to readily accommodate design changes, submodel upgrades, and interfacing with other vehicle models. It is shown that transient, multi-cylinder simulation of an arbitrarily-selected number of cylinders that are configured to form the engine can be readily accomplished. Illustrative studies are conducted to demonstrate the capability of the simulation to perform system dynamic studies and p...

Perturbation Method for Probabilistic Dynamic Finite Element Analysis of a Rectangular Plate

Abstract: In this study, the statistical dynamic response of rectangular plates with stochastic mass density and Young's modulus of elasticity is investigated, using the perturbation technique in conjunction with the finite element method. The proposed method produces the expectation, variance, and autocorrelation function of the deflection, strain, and stress that are useful in estimating structural safety and reliability. Some statistical responses obtained by the perturbation method are checked by the Monte Carlo Simulation. Also, the statistical dynamic responses of a rectangular plate are evaluated for parameters that control the shape of the power spectra of the stochastic field.

Shape Design Sensitivity and Optimization Using P-Version Design Modeling and Finite Element Analysis*

Abstract: A methodology that supports shape design modeling, structural shape design sensitivity analysis (DSA), and optimization of elastic solids, using p-version finite element analysis (FEA), is developed. The p-version FEA approach is attractive for shape design optimization, due to its high accuracy of analysis results, even with coarse mesh; ease of creation of design and finite element models; consistency between design and finite element models; insensitivity to finite element mesh distortion and aspect ratio; and tolerance for large shape changes during design optimization iterations. A continuum-based DSA method that is applicable to both planar and spatial structures is developed with an established p-version FEA code, STRESS CHECK. A shape design parameterization method that uses the geometric entities of STRESS CHECK is proposed and implemented. In STRESS CHECK, the shape design model is identical to the finite element model, so finite element modeling errors are eliminated. Design optimizati...

A variational approach for the design of spatial four-bar mechanism

Abstract: An analytical formulation for computing kinematic sensitivity of the spatial four-bar mechanism is presented. The experimental code developed is used to compute an assembled configuration for the mechanism that accounts for the effect of a design variation. The mechanism is modeled using graph theory, in which a body is defined as a node and a kinematic joint is defined as an edge. The spherical joint is cut to convert the model into a tree structure by cutting an edge and introducing constraints. The effect of variation in mechanism design using concepts of virtual displacement and rotation is introduced. The variation of the spherical constraint is computed, maintaining joint-attachment vectors and orientation matrices as variables. A system of equations that has a greater number of design variables than equations is then solved using the modified Moore-Penrose pseudo inverse. A recursive formulation is introduced, which can be used to obtain the state variation of a body in terms of the state ...

Sensitivity Analysis and Optimal Design of Elastic Hinges and Supports in Beam and Frame Structures

Abstract: This paper considers the problem of optimal design of elastic beam-like structures with unspecified location and stiffness of hinges and supports. Sensitivity analysis for an arbitrary behavioral functional is carried out, using the direct and adjoint approaches. Relevant optimality conditions are discussed for the assumed objective functional. Simple examples illustrate the theory.

Optimal Trajectories of Open-Chain Mechanical Systems: An Explicit Optimality Equation with a Multiple Shooting Solution*

Abstract: This paper addresses the problem of finding optimal trajectories of multi-degree-of-freedom open-chain systems, while minimizing integral cost function-als. The contributions of this paper are (i) explicit optimality equations in terms of inertia matrix and potential energy of the system, (ii) compatibility conditions on the terminal conditions of the system that are consistent with the cost functional, (iii) a robust computational procedure for solving the two-point boundary value problem using the multiple shooting method, and (iv) integrated software for dynamic optimization and simulation of open-chain systems.

Optimal Design of a Hybrid Electric Powertrain System

Abstract: Optimal design of an electric hybrid powertrain system using a decomposition-based approach is presented. In this approach, a general system design problem is first formulated without specifying objectives. The mathematical model is analyzed using partitioning techniques, and an optimal design problem that can be readily decomposed and solved using an appropriate coordination strategy is derived. Basic concepts for hybrid powertrains in automotive applications and a mathematical design model are introduced. Different ways of synthesizing a hierarchically decomposed optimization problem statement are described, and one such problem is solved using a sensitivity-based coordination strategy.

Some Non-linear Elastic Solutions for Masonry Solids*

Abstract: This paper deals with equilibrium problems for solids made of elastic materials of bounded tensile strength and for which exact solutions are achieved. A constitutive equation is adopted and its main properties with regard to uniqueness of the solution to boundary problems are also analyzed. Four distinct equilibrium problems are then considered. The first three are characterized by specific symmetry conditions—polar, spherical, and cylindrical, respectively.

Postbifurcation equilibrium paths modified by nonlinear configuration-dependent conservative loading using nonsmooth analysis

Abstract: This paper discusses the effect of deformation-sensitive loading devices because the nature of loading is generally not perfectly dead, being independent of the deflections that occur. This paper presents the effect of nonlinear variable load. Postbifurcation equilibrium paths and structural tangent stiffness are modified on the basis of a polygonal approximation and nonsmooth analysis. The effects of dead and variable loads are compared. Configuration-dependent loading devices can be characterized by some load-deflection functions, much like the nature of material behavior can be characterized by stress-strain functions. The effect of a deformation-sensitive load is similar to that of the material. Consequently, in the stability analysis of structures, a configuration-dependent loading program can be handled like material behavior. Thus, in the tangent stiffness of the structure, much like the tangent modulus of the material, the tangent modulus of the load appears. Previous research has shown t...

An Analytical Solution of Equivalent Stress for Structure Fatigue Life Prediction under Broad Band Random Loading

Abstract: An analytical solution of an equivalent stress-range calculation, based on the power-spectral density of stress in critical points of structures, and a statistical theory for the peak distribution of a stationary Gaussian random process are presented in this paper for fatigue life assessment under broad-band random loading. This model has more advantages than similar existing models.

On Developing New Statistical Energy Methods for the Analysis of Vibration Transmission in Complex Vehicle Structures

Abstract: This paper presents an overview of research being conducted on vibration transmission in complex vehicle structures. Complex structures are assemblages of coupled component systems. Random parameter uncertainties in such structures can dramatically affect the high- to mid-frequency dynamics, mandating a statistical analysis. New statistical energy methods for predicting vibration transmission are explored by relaxing assumptions which limit the applications of current high-frequency techniques. Particular attention is paid to those assumptions that restrict the analysis to higher frequency ranges. A parameter-based approach that provides a more faithful representation of the structure's forced response is presented. For an example system, numerical simulations and analytical approximations based on this approach are shown to provide improved estimates of the average power transmission. This improvement is most significant in the mid-frequency range

Influence of Lateral Restraint on the Stability of Stiffened Plates Under Uniform Compression

Abstract: This paper investigates the influence of lateral restraint on the buckling behavior of stiffened plates under uniform compression. As a first stage, a numerical procedure is presented for stability analysis of stiffened plates. The structure is idealized as assembled plate and beam elements, rigidity connected at their junctions. Strain energy components for plate and stiffener elements are then derived in terms of out- and in-plane displacement functions. The Sequential Quadratic Programming (SQPr technique is then used to determine the critical buckling load for given plate/stiffener geometric properties. Results are presented, showing the transition from the overall mode to the local mode by increasing stiffener depth for various concentric and eccentric stiffening configurations. Finally, the influence of lateral restraint on buckling behavior of multi-stiffened plates is illustrated.

A System-Level Component Mode Approach for Flexible Multibody Dynamic Simulation*

Abstract: Computationally efficient, high fidelity flexible multibody dynamic simulation requires that a flexible body model be of low dimension, yet account for effects of interest. Model reduction methods are reviewed, and a new type of component: mode, called a system-level component mode, is extracted from results of analysis of the entire system. A system-level component mode is decomposed into two parts: deformation of the component and rigid-body motion. It is shown that the latter degrades simulation accuracy. Extraction of rigid-body modes from the system-level component modes is carried out by maximizing strain energy for unit displacement norm. Dependency, or near dependency, in the resulting set of deformation modes is identified and removed using a method based on singular value decomposition. Numerical examples are presented to demonstrate the effectiveness of the approach.

Dynamic Stability of Spinning Beams with an Unsymmetrical Cross-Section and Distinct Boundary Conditions Subjected to Time-Dependent Spin Speed*

Abstract: The equations of motion of a spinning beam with a rectangular cross-section are formulated using the Euler beam theory and the assumed mode method. The spin speed consists of steady-state and time-dependent portions. The resulting equations of motion are not in standard Mathieu-Hill's equation form, due to the time-dependent coefficient of the gyroscopic term. These equations of motion are then reduced to a set of first-order differential equations with time-dependent coefficients. The regions of instability due to parametric excitations are determined using the multiple scale method. Numerical results are presented for a spinning beam subjected to combinations of end conditions in the two orthogonal planes of transverse vibration. Widths of the unstable regions are found to decrease as the aspect ratio of the rectangular cross-section approaches unity for spinning beams with an identical set of end conditions in both transverse vibration planes.These regions vanish when the aspect ratio becomes ...

Linearization of the dynamics of closed-chain mechanical systems

Abstract: This paper presents an analytical-numerical method for linearizing the equations of motion of mechanical systems with closed chains The algorithm developed here linearizes basic recursive kinematic relationships and then applies the chain rule to the derivation of the equations of motion under the framework of recursive formulation This method can be incorporated into the formulation of recursive equations of motion for general multibody dynamic systems to handle large-scale systems The method is directly applicable to system Jacobian matrix computation Since the proposed algorithm uses no numerical differentiation, its accuracy is comparable to a symbolic, closed-form linearization Moreover, without needing repetition computation in search of proper perturbation quantity, this method is computationally more efficient than the finite difference method

Variable Fidelity Differential-Algebraic Equation Model Correlation

Abstract: A formulation for selecting operator and control inputs to a high fidelity dynamics model, governed by differential-algebraic equations, is presented to minimize deviation in its response relative to that of a lower fidelity model that is also governed by differential-algebraic equations of motion. An adjoint variable method for computing sensitivity of the error measure defined is derived and implemented in a nonlinear programming formulation that is suitable for iterative minimization of the error functional. A numerical example using a multibody mechanism is presented to demonstrate effectiveness of the method and provide insights into means for effectively formulating problems of model correlation and strategies for their solution