A mathematical model for a rotary harmonograph

A shape design sensitivity analysis and optimization are proposed for the ine nitesimal elasto ‐plasticity with a frictional contact condition. Rate-independent plasticity is considered with a return mapping algorithm and a von Mises yield criterion. The contact condition is formulated using the penalty method and the modie ed coulomb friction law. A continuum-based shape design sensitivity formulation is developed for structural and frictional contact variational equations. The direct differentiation method is used to compute the displacement sensitivity, and the sensitivities of various performance measures are computed from the displacement sensitivity. Path dependency of the sensitivity equation due to the constitutive relation and friction is discussed. It is shown that no iteration is required to solve the sensitivity equation. Response analysis and the proposed sensitivity formulation are implemented using the mesh-free method where the mesh distortion problem can be resolved. Numerical examples show accurate results of the proposed method compared to the e nite difference method. Dife culties in the sensitivity formulation for the e nite deformation problem are discussed. I. Introduction B ECAUSEoftherecentdevelopmentofcomputationalmechanics, it is now possible to analyze practical examples of complicated structural problems. Many design engineers, who are not satise ed with response analysis alone, have keen interests in the methodology of the design. For more than two decades, signie cant research efforthasbeen focused on the rate of response with respect to the changes in structural shape under shape design sensitivity analysis (DSA). Analysis of the design sensitivity information is the most important and costly procedure in the automated optimum design process. It supplies useful quantitative information to the design engineer about the direction of the desired design change. In a classical linear problem, DSA research has proved the differentiability of the solution of the response analysis using the linear operator theory and has derived specie c sensitivity expressions for variousproblems. 1 Aresultworthy of attention in linear DSAis that theoriginalresponseandthesensitivityoftheresponsebelongtothe samekinematicallyadmissiblespaceandhavethesameregularities. Owing to the development of the response analysis capability, engineers have directed their interest to the nonlinear problems that are dealt with efe ciently. Because many design application problems are accompanied by plastic deformation, the design sensitivity of nonlinear problems has been actively developed, and many research results are reported. In the procedure of nonlinear response analysis, the projection, called a return mapping, of the elastic trial stress is carried out to satisfy the variational inequality (VI)through an iteration in the stress space. 2 The DSA, on the other hand, computes the rate of change of the projected response in the tangential direction of the constraint set without iteration. Note that the sensitivity analysis is linear and can be computed without iteration even thoughtheresponseanalysisisnonlinear. 3 Unlikethenonlinearelastic problem, the sensitivity equation of the plastic problem requires sensitivity of the stress and internal evolution variables at the previous time. The sensitivity equation is solved at each time without iteration, and the sensitivity of the stress and evolution variables are

An implicit Runge-Kutta numerical integration algorithm based on generalized coordinate partitioning is presented for solution of the differential-algebraic equations of motion and design sensitivity analysis for rigid multibody systems. The direct differentiation method for dynamic sensitivity analysis is employed to obtain differential-algebraic sensitivity equations. The implicit integration formula expresses design sensitivity of independent generalized coordinates and their first-time derivative as functions of sensitivities of independent accelerations at discrete integration times. The same integration Jacobian is used to determine generalized independent accelerations and their sensitivities, by iteratively solving the equations obtained from state-space, second-order ordinary differential equations in independent coordinates. Design sensitivities of dependent variables in the formulation, including design sensitivities of Lagrange multipliers, are recovered by satisfying the full design sensitivity system of kinematic and kinetic equations of motion. Design sensitivity analysis results are verified using finite differences, and a comparison with an explicit numerical integration algorithm is performed to validate the efficiency of the proposed algorithm for design sensitivity analysis of stiff vehicle models.

Recursive formulas for design sensitivity analysis of mechanical systems

This paper together with the following two papers present emerging technologies for development of design sensitivity analysis (DSA) and optimization that can be used for concurrent engineering. A summary of recently developed unified continuum DSA methods for linear and nonlinear structural systems is presented. Design sensitivities of static and dynamic responses of elastic solids and built-up structures with respect to material property, sizing, shape, and configuration design variables are considered. For DSA of acoustic response, acousto-elastic systems are treated. For DSA of nonlinear structural systems, both geometric and material nonlinearities are considered. The adjoint variable and direct differentiation methods are used to derive explicit design sensitivity expressions that can be evaluated numerically using analysis results from established finite element analysis (FEA) codes. It is demonstrated that the continuum based DSA method allows design sensitivity computations to be carried out using established FEA codes with respect to geometric design parameters that are employed in computer-aided design (CAD) tools, so that industry standard tools can be exploited in concurrent engineering. A DSA and optimization (DSO) tool with a visually driven user interface is developed to allow design engineers to easily create geometric, design, and analysis models; define performance measures; perform DSA; and carry out a four-step interactive design process that includes visual display of design sensitivity, what-if study, trade-offs, and interactive design optimization.

Configuration design sensitivity analysis of dynamics for constrained mechanical systems

An improved upper bound analysis for study of the void closure behavior in the plane strain extrusion

Vortex extrusion (VE) as a severe plastic deformation method that imposes high values of strain with simultaneous torsion and extrusion of a workpiece was studied in this paper. Response surface methodology was used for investigation the relationship between reduction in area (RA), twist angle (φ), length of twist zone (L) and position of control points in Beziers’ formulation (C1) as input parameters, and efficient twisting of the material and punch load (steady state load) as output parameters using streamline proposed 6-indent VE die in different frictional conditions. Accuracy of the finite element analysis (FEA) results was evaluated with experimental data before of using FEA to obtain the responses. Analysis of variance was used to evaluate the effect of the parameters and their interaction on the responses and to develop a mathematical model for the responses. Results showed that, beside the effects of φ, RA, L on twisting of the materials in both frictionless and frictional conditions, interaction of φ and RA (φ × RA) is a significant parameter in the frictional condition. However, φ, RA and interaction of RA with RA (RA × RA) were considered as the significant parameters that affect the load of the process in both frictional conditions. Finally the proposed mathematical model for load of the process was evaluated with experimental data in similar conditions. It was concluded that the results of the modeling and experimental data are in a good agreement.

The product of high complex profile, high strength, high productivity and excellent material properties with infinite length can be produced by Continuous Extrusion (CE) process. The numerical simulation of Aluminum (AA 1100) feedstock material at different wheel velocities, product diameter, feedstock temperature, die temperature and friction condition has been carried out using 3D simulation tool Design Environment for Forming (DEFORM-3D) in this paper. The development of mathematical model is carried out to investigate the influence of wheel velocity, extrusion ratio, feedstock temperature, die temperature and friction conditions on total load required for the deformation and extrusion of feedstock material through Response Surface Methodology (RSM). The statistical significance of mathematical model is verified through analysis of variance (ANOVA). The most optimum value of extrusion load has been found to be 136.4[Formula: see text]kN through iterative process of Genetic Algorithm (GA) using Artificial Neural Network (ANN). The optimized value of input process variables for minimum value of extrusion load obtained has been found to be 13 Revolutions per Minute (RPM) as wheel velocity, 5[Formula: see text]mm as product diameter, 0.95 as friction condition, 650[Formula: see text]C as feedstock temperature and 550[Formula: see text]C as die temperature. This paper with proposed methodology will be helpful for industries working in the area of CE in terms of minimizing energy consumption during production process of bus bars, tubes, wires, cables, sheets, plates, strips, etc.

Causes of chatter in a hot strip mill: Observations, qualitative analyses and mathematical modelling

This article proposes a frictional resistance description approach in sheet metal forming and the objective is to characterize the friction coefficient value under a wide range of friction conditions without performing time-consuming experiments. To describe the friction condition in sheet metal forming simulations, the friction coefficient should be quantified using friction models. Realistic friction models must account for the influence of surface roughness and surface topography on the lubricant flow and dry friction conditions. Due to considerable amount of factors that affect the friction coefficient value, building the analytical friction model for specified process conditions is too demanding. Thus, mathematical models that describe the friction behaviour using multiple regression analysis (MRA) and artificial neural networks (ANN) are utilized. The regression analysis was performed using the user subroutine in the MATLAB, while the ANN model was built in STATISTICA Neural Networks. As input variables for regression model and training of multilayer perceptron (MLP) the results of strip drawing friction test were utilized.

For configuration design of non-linear dynamic structures, such as crashworthiness design, a continuum-based configuration design sensitivity analysis (DSA) and optimization methods are developed. The same transient dynamic analysis method used in Part I of this paper is employed here. The elastic–plastic material and finite strain and rotation effect are considered. The first-order variations of the energy forms, load form and kinematic and structural responses with respect to configuration design variables are derived. For the configuration design, both the shape and orientation variations contribute to the first-order variation of the equations of motion. The angular design velocity field associated with Euler angles is used to represent the orientation variation to obtain accurate design sensitivity results. Like Part I of the paper, the updated Lagrangian formulation and direct differentiation method are used for DSA for the path-dependent problem. For the updated Lagrangian formulation, the design velocity field that defines the mapping between the initial and perturbed designs should also be updated at each configuration. Numerical implementation of configuration DSA and optimization is carried out for fixed time steps using DYNA3D and the modified feasible direction (MFD) method. It is observed that the proposed DSA method yields better sensitivity results than the finite difference method for the highly non-linear problem. Design optimization is carried out using the design sensitivity information.Copyright © 2000 John Wiley & Sons, Ltd.

A continuum-based design sensitivity analysis (DSA) method is presented for configuration design of nonlinear structural systems using Mindlin plate and Tim-oshenko beam theories. Both displacement and critical load performance measures are considered. Configuration design variables are characterized by shape and orientation changes of structural components. The material derivative that is used to develop the continuum-based shape DSA method is extended to account for effects of configuration design variation. The piecewise linear design velocity field, i.e., C0-regular, is used to support configuration design changes for a broad class of built-up structures with beams and plates. To allow use of the C0-design velocity field, mathematical models of beam and plate bending must be second-order partial differential equations, so that only first-order derivatives appear in the integrand of the energy equation and, thus, in the integrand of the configuration design sensitivity expression. Since the Mindlin plate and Timoshenko beam theories use displacement and rotation to describe structural response, mathematical models of beam and plate bending are reduced to second-order partial differential equations. The isoparametric finite element formulations are used for numerical evaluation of continuum design sensitivity expressions.

Based on a bond-based peridynamics theory for dynamic crack propagation problems, this paper presents a design sensitivity analysis and optimization method. Peridynamics has a peculiar advantage over the existing continuum theory in the mathematical modelling of problems where discontinuities arise. For the design optimization of the crack propagation problems, a non-shape design sensitivity is derived using the adjoint variable method. The obtained adjoint sensitivity of displacement and strain energy turns out to be very accurate and efficient compared to the finite different sensitivity. The obtained design sensitivities are futher utilized to optimally control the position of bifurcation point in the design optimization of crack propagation in a plate under tension. A numerical experiment demonstrates that the optimal distribution of material density could delay the position of bifurcation.

Most state-of-the-art multibody systems are multidisciplinary and encompass a wide range of components from various domains such as electrical, mechanical, hydraulic, pneumatic, etc. The design considerations and design parameters of the system can come from any of these domains or from a combination of these domains. In order to perform analytical design sensitivity analysis on a multidisciplinary system (MDS), we first need a uniform modeling approach for this class of systems to obtain a unified mathematical model of the system. Based on this model, we can derive a unified formulation for design sensitivity analysis. In this paper, we present a modeling and design sensitivity formulation for MDS that has been successfully implemented in the MIXEDMODELS (Multidisciplinary Integrated eXtensible Engine for Driving Metamodeling, Optimization and DEsign of Large-scale Systems) platform. MIXEDMODELS is a unified analysis and design tool for MDS that is based on a procedural, symbolic-numeric architecture. This architecture allows any engineer to add components in his/her domain of expertise to the platform in a modular fashion. The symbolic engine in the MIXEDMODELS platform synthesizes the system governing equations as a unified set of non-linear differential-algebraic equations (DAE’s). These equations can then be differentiated with respect to design to obtain an additional set of DAE’s in the sensitivity coefficients of the system state variables with respect to the system’s design variables. This combined set of DAE’s can be solved numerically to obtain the solution for the state variables and state sensitivity coefficients of the system. Finally, knowing the system performance functions, we can calculate the design sensitivity coefficients of these performance functions by using the values of the state variables and state sensitivity coefficients obtained from the DAE’s. In this work we use the direct differentiation approach for sensitivity analysis, as opposed to the adjoint variable approach, for ease in error control and software implementation. The capabilities and performance of the proposed design sensitivity analysis formulation are demonstrated through a numerical example consisting of an AC rectified DC power supply driving a slider crank mechanism. In this case, the performance functions and design variables come from both electrical and mechanical domains. The results obtained were verified by perturbation analysis, and the method was shown to be very accurate and computationally viable.

Resonance phenomena occurs at large vertical pump which is operating to cool down the hot steam using sea water in the power plant. To avoid the resonance, the natural frequency needs to be isolated about 20% from motor operating speed. Yet, excessive vibration occurs especially at low tide. At first, natural frequency of the whole pump system and each part is calculated using ANSYS. As it is revealed in the previous journal papers that only circular pipe part is related to resonance, the FSI technique is applied for free vibration analysis. The natural frequency is reduced to 60% (compared to that) of the frequency measured in air as it is similar to other published results. And the frequency obtained by finite element analysis is almost same to that obtained from modal test. Based on the accurate finite element model and analysis, design change is tried to avoid the resonance by changing the thickness of pipe and base supporting plate. In stead of doing optimization process, design sensitivity is computed and used to find such designs to avoid resonance.