Dynamic Response Optimization Research Articles

Dynamic Response Optimization of Piezoelectrically Excited Thin Resonant Beams

Piezoelectrically excited, resonant, elastic beams find wide use as piezoelectric fans, optical choppers, MEMS sensors, and piezoelectric motors. The devices consist of either one piezoelectric ceramic patch (piezopatch) bonded on one side (asymmetric configuration), or of two oppositely poled patches placed symmetrically on either side of a thin, flexible elastic beam (symmetric configuration). Field equations of the coupled structure governing the coupled longitudinal and bending motions of the resonator are derived using linear constitutive equations, slender beam approximations, and Hamilton’s principle. Analytical solutions are found to the coupled eigenvalue problem. Eigenvalues and eigenfunctions for the short-circuited and open-circuited configurations are predicted analytically and are found to be in excellent agreement with results from three-dimensional finite element simulations. Electromechanical coupling factors (EMCF) are computed using the analytical and finite element model and optimal resonator geometries are identified for maximal EMCF. The EMCF predictions are also compared with experiments for an asymmetrically configured resonator. The analytical solution provides a convenient tool for the optimal design of such devices.

Structural shape optimization using equivalent static loads transformed from dynamic loads

In structural optimization, static loads are generally utilized although real external forces are dynamic. Dynamic loads have been considered only in small-scale problems. Recently, an algorithm for dynamic response optimization using transformation of dynamic loads into equivalent static loads has been proposed. The transformation is conducted to match the displacement fields from dynamic and static analyses. This algorithm can be applied to large-scale problems. However, the application has been limited to size optimization. The present study applies the algorithm to shape optimization. Because the number of degrees of freedom of finite element models is usually very large in shape optimization, it is difficult to conduct dynamic response optimization with conventional methods that directly treat dynamic response in the time domain. The optimization process is carried out by interfacing an optimization system and an analysis system for structural dynamics. Various examples are solved to verify the algorithm. The results are compared to the results from static loads. It is found that the algorithm using static loads transformed from dynamic loads based on displacement is valid for very large-scale shape optimization problems. Copyright © 2005 John Wiley & Sons, Ltd.

Optimal gains for active vibration control by the beam analogy

A numerical approach to determine optimal gains for active optimal vibration control of continuous structures by the finite element method is presented. The approach uses an analogy between the optimality equations for control in the time domain and the governing equations for a set of static beams in the spatial domain. The finite element model of control is generated and analyzed in a fictitious spatial domain. The results are then transferred to the time domain to describe the optimal dynamic response of the system. The analogy (referred to as the beam analogy) allows for an efficient application of the finite element method to solve two-point-boundary-value problems for the finite time control cases. Here it is applied to the infinite time control cases. An algorithm for a direct calculation of optimal gains for closed loop control of time-invariant problems without using Riccati’s equation is discussed in detail.

Interval optimization of dynamic response for uncertain structures with natural frequency constraints

This paper presents an interval optimization method for the dynamic response of uncertain structures with natural frequency constraints. The matrices of structures with interval parameters are given. Combining the interval extension of function with the perturbation theory of dynamic response and eigenvalue analysis, the optimization method for interval dynamic response analysis is derived. The interval optimization problem is transformed into a corresponding deterministic one. Because the mean values and the uncertainties of the interval parameters can be selected as the optimization parameters, more information of the optimization results can be obtained by the present method than that obtained by the deterministic one. The present method is implemented for a truss structure and a frame structure. The numerical results show that the method is effective.

Interval optimization of dynamic response for structures with interval parameters

This paper presents an interval optimization method for the dynamic response of structures with interval parameters. The matrices of structures with interval parameters are given. Combining the interval extension of function with the perturbation theory of dynamic response, the method for interval dynamic response analysis is derived. The interval optimization problem is transformed into a corresponding deterministic one. Because the mean values and the uncertainties of the interval parameters can be elected as the design variables, more information of the optimization results can be obtained by the present method than that obtained by the deterministic one. The present method is implemented for a truss structure and a frame structure. The numerical results show that the method is effective.

Modeling, analysis and optimization of a scissors linkage seat suspension

Under-the-seat scissors linkage suspensions have been in use for the off-road and construction vehicle market for many years. The off-road environment requires that the suspension be rugged while providing comfort to a seat occupant over a rough terrain. Since different operators can use the equipment at different times, the seat suspension must be adjustable for both the height and weight of the seat occupant. This study utilizes static and dynamic mathematical modeling to develop an optimal geometric and damping configuration for a recently developed scissors seat suspension. The proposed design allows for the suspension to be adjusted in order to change the seat height without substantially affecting its load-deflection characteristics. Since the height adjust travel is included within the scissors linkage stroke, the best configuration of internal components to provide this feature is not intuitive. Therefore, a suspension model was optimized until a satisfactory configuration with essentially constant load-deflection characteristics across the range of height and weight adjustments was achieved. The dynamic response optimization using a 4-DOF human surrogate-suspension model shows that for the given vibrational input, a non-linear damping function that increases slightly as it moves away from the static equilibrium position transmitted the least acceleration to the operator.

동하중으로부터 변환된 등가정하중을 통한 구조물의 형상최적설계

In structural optimization, static loads are generally utilized although real external forces are dynamic. Dynamic loads have been considered in only small-scale problems. Recently, an algorithm for dynamic response optimization using transformation of dynamic loads into equivalent static loads has been proposed. The transformation is conducted to match the displacement fields from dynamic and static analyses. The algorithm can be applied to large-scale problems. However, the application has been limited to size optimization. The present study applies the algorithm to shape optimization. Because the number of degrees of freedom of finite element models is usually very large in shape optimization, it is difficult to conduct dynamic response optimization with the conventional methods that directly threat dynamic response in the time domain. The optimization process is carried out via interfacing an optimization system and an analysis system for structural dynamics. Various examples are solved to verify the algorithm. The results are compared to the results from static loads. It is found that the algorithm using static loads transformed from dynamic loads based on displacement is valid even for very large-scale problems such as shape optimization.

Dynamic response optimization design for engineering structures based on reliability

In many practical structures, physical parameters of material and applied loads have random property. To optimize this kind of structures, an optimum mathematical model was built. This model has reliability constraints on dynamic stress and displacement and upper & lower limits of the design variables. The numerical characteristic of dynamic response and sensitivity of dynamic response based on probability of structure were deduced respectively. By equivalent disposing, the reliability constraints were changed into conventional forms. The SUMT method was used in the optimization process. Two examples illustrate the correctness and practicability of the optimum model and solving approach.

Optimization of Elevated Concrete Foundations for Vibrating Machines

The objective of this study is to describe a problem formulation and an optimization procedure for the design of elevated reinforced concrete foundations for vibrating machines. Special emphasis is placed on structures composed of footings, beams, and columns. The dimensions of the structure and its reinforcement are the design variables for the optimization problem. The objective function consists of costs of the concrete, the steel, the form, and the propping form. Constraints related to material and soil failures, as well as geometrical limits and human comfort are imposed. A new failure surface for columns and beams subjected to biaxial bending and axial loads is defined and used in the formulation. The main motivation of using the new failure surface is to save a large amount of computational effort in the solution of this dynamic response optimization problem. The problem of minimizing the structural cost while satisfying the operating and safety requirements is solved using an augmented Lagrangian method. A large number of constraints (time-dependent and time-independent) is treated without any difficulty in the method. The numerical methods used in the solution process are described. Optimal solutions for an example problem are obtained and discussed.

Optimal Dynamic Response Control of Elastically Tailored Nonuniform Thin-Walled Adaptive Beams

A dual approach based on structural tailoring and adaptive materials technology aimed at controlling the dynamic response of tapered thin-walled beams exposed to external pressure pulses is presented. Whereas structural tailoring uses the directionality properties of advanced composite materials, the adaptive materials technology exploits the actuating and sensing capabilities of piezoelectric material systems bonded or embedded into the host structure. The structure is modeled as a doubly tapered composite thin-walled beam incorporating a number of nonclassical features such as transverse shear, warping inhibition, anisotropy of constituent materials, and rotatory inertias. The cases of piezoactuators spread over the entire span of the structure or in the form of a patch are considered, and issues related to the influence of patch location and size on the control efficiency are discussed. Other issues related to the implications of the inclusion/discard in the quadratic performance index of time-dependent external excitations are also addressed. The displayed numerical results provide a comprehensive picture of the synergistic implications of the application of both techniques, namely, the tailoring and optimal control on vibration response of nonuniform thin-walled beam structures exposed to external time-dependent excitation.

Optimum design of stochastically excited non‐linear dynamic systems without geometric constraints

AbstractThis paper presents an efficient procedure for min–max dynamic response optimization of stochastically excited non‐linear systems with multiple time‐delayed inputs. This procedure employs a stochastic linearization technique to overcome system non‐linearity and an auto‐covariance analysis technique to represent the original stochastic mechanical model in a suitable form for optimization. Special attention is given to the sensitivity analysis, due to the complex nature of the problem. Therefore, exact expressions are obtained in a simple form for the evaluation of the required gradients, which greatly improve the stability and efficiency of the optimization algorithm. The numerical results and performance are presented by means of solving two min–max dynamic response optimization problems. Copyright © 2002 John Wiley & Sons, Ltd.

Composite Nonsmooth Optimization Using Approximate Generalized Gradient Vectors

A new approximation method is presented for directly minimizing a composite nonsmooth function that is locally Lipschitzian. This method approximates only the generalized gradient vector, enabling us to use directly well-developed smooth optimization algorithms for solving composite nonsmooth optimization problems. This generalized gradient vector is approximated on each design variable coordinate by using only the active components of the subgradient vectors; then, its usability is validated numerically by the Pareto optimum concept. In order to show the performance of the proposed method, we solve four academic composite nonsmooth optimization problems and two dynamic response optimization problems with multicriteria. Specifically, the optimization results of the two dynamic response optimization problems are compared with those obtained by three typical multicriteria optimization strategies such as the weighting method, distance method, and min-max method, which introduces an artificial design variable in order to replace the max-value cost function with additional inequality constraints. The comparisons show that the proposed approximation method gives more accurate and efficient results than the other methods.

An efficient dynamic response optimization using the design sensitivities approximated within the estimate confidence radius

In order to reduce the expensive CPU time for design sensitivity analysis in dynamic response optimization, this study introduces the design sensitivities approximated within estimated confidence radius in dynamic response optimization with ALM method. The confidence radius is estimated by the linear approximation with Hessian of quasi-Newton formula and qualifies the approximate gradient to be validly used during optimization process. In this study, if the design changes between consecutive iterations are within the estimated confidence radius, then the approximate gradients are accepted. Otherwise, the exact gradients are used such as analytical or finite differenced gradients. This hybrid design sensitivity analysis method is embedded in an in-house ALM based dynamic response optimizer, which solves three typical dynamic response optimization problems and one practical design problem for a tracked vehicle suspension system. The optimization results are compared with those of the conventional method that uses only exact gradients throughout optimization process. These comparisons show that the hybrid method is more efficient than the conventional method. Especially, in the tracked vehicle suspension system design, the proposed method yields 14 percent reduction of the total CPU time and the number of analyses than the conventional method, while giving similar optimum values.

Efficient approximation method for constructing quadratic response surface model

For a large scaled optimization based on response surface methods, an efficient quadratic approximation method is presented in the context of the trust region model management strategy. If the number of design variables inn, the proposed method requires only 2n+1 design points for one approximation, which are a center point and two additional axial points within a systematically adjusted trust region. These design points are used to uniquely determine the main effect terms such as the linear and quadratic regression coefficients. A quasi-Newton formula then uses these linear and quadratic coefficients to progressively update the two-factor interaction effect terms as the sequential approximate optimization progresses. In order to show the numerical performance of the proposed method, a typical unconstrained optimization problem and two dynamic response optimization problems with multiple objective are solved. Finally, their optimization results compared with those of the central composite designs (CCD) or the over-determined D-optimality criterion show that the proposed method gives more efficient results than others.

A new penalty parameter update rule in the augmented lagrange multiplier method for dynamic response optimization

Based on the value of the Lagrange multiplier and the degree of constraint activeness, a new update rule is proposed for penalty parameters of the ALM method. The theoretical exposition of this suggested update rule is presented by using the algorithmic interpretation and the geometric interpretation of the augmented Lagrangian. This interpretation shows that the penalty parameters can effect the performance of the ALM method. Also, it offers a lower limit on the penalty parameters that makes the augmented Lagrangian to be bounded. This lower limit forms the backbone of the proposed update rule. To investigate the numerical performance of the update rule, it is embedded in our ALM based dynamic response optimizer, and the optimizer is applied to solve six typical dynamic response optimization problems. Our optimization results are compared with those obtained by employing three conventional update rules used in the literature, which shows that the suggested update rule is more efficient and more stable than the conventional ones.

Optimal Dynamic Response Control of Adaptive Thin-Walled Cantilevers Carrying Heavy Stores and Exposed to Blast Pulses

The optimal dynamic response control of adaptive thin-walled beam cantilevers carrying externally mounted stores and exposed to blast pulses is investigated. The structure under consideration is modeled as a thin-walled beam of closed cross-section contour, encompassing a number of non-classical features such as transverse shear, secondary warping, and anisotropy of the constituent materials. The structure can carry along its span heavy concentrated masses and/or tip stores. The vibration control is achieved via the use of actuating and sensing capabilities provided by piezoelectric devices which are bonded or embedded into the host structure. The cases of piezoactuators spread over the entire span of the structure, or in the form of a patch are considered, and issues related to the influence of patch location and size upon the vibration control efficiency are highlighted. Other issues related to the minimization of the required input power and of that generated by the inclusion/discard in the quadratic performance index of time-dependent external excitations are also addressed.

Optimal Dynamic Response Control of Adaptive Thin-Walled Cantilevers Carrying Heavy Stores and Exposed to Blast Pulses

Optimal Dynamic Response Control of Adaptive Thin-Walled Cantilevers Carrying Heavy Stores and Exposed to Blast Pulses

Buckling design optimization of complex built-up structures with shape and size variables

The design optimization of buckling behaviour is studied for complex built-up structures composed of various kinds of elements and implemented within JIFEX95, a general-purpose software for finite element analysis and design optimization. The direct and adjoint methods of sensitivity analysis for critical buckling loads are presented with detailed computational procedures. Particularly, the variations of prebuckling stresses and external loads have been accounted for. The design model and solution methods presented in this paper are available for both shape and size optimization, and buckling optimization can also be combined with static, frequency and dynamic response optimization. The numerical examples show the applications of the buckling optimization method and the effectiveness of the methods and the program of this paper.

Design Optimization of Engine Room of Ship with Dynamic Response Using Genetic Algorithms

In this paper, the dynamic response optimization of engine room structure of ship is carried out by using the Genetic Algorithms (GAs). The static stresses and dynamic response to the external excitation are included into the constraint conditions. To save the computing storage, the floating point representation of solution in GAs is used. The operators of arithmetical crossover, boundary mutation and non-uniform mutation are introduced to improve the convergence of the GA. The effects of them are investigated. Finite element model for the static and dynamic analyses is employed in developing the Genetic Algorithm for the optimization of structural system. Modal superposition principle is used to calculate the response of the structure to the propeller excitation. The optimum results for the numerical example are presented and it shows that the GA for the dynamic response optimization is applicable.

Fuzzy Optimum Design of Engine Room of Ship with Dynamic Response Using Genetic Algorithms

In this paper, the theory of fuzzy set is introduced to the optimization of engine room structure considering the dynamic response constraints. Two ways of solving the fuzzy optimization are investigated. Firstly, the constraints of static stresses and dynamic response to the external excitation are handled by the fuzzy set theory. By this approach, a family of optimal designs under different degrees of satisfaction with constraints for the structure can be obtained. Secondly, the objective is also considered to be a fuzzy variable with its own membership function. Through maximizing the degree of membership to the fuzzy decision, a crisp solution can be provided to designers. A Genetic Algorithm (GA) is applied to do the optimization. The optimum results for the numerical example are presented and it is shown that the proposed method for the dynamic response optimization is applicable.