Sensitivity Analysis of Nonlinear Transient Response Based on Perturbation in the Complex Domain

Sensitivity analysis of transient response plays a crucial role in structural dynamics optimization. In this paper, the sensitivity analysis method for calculating the first-order derivatives of the nonlinear transient response using the real-imaginary perturbations (RIP) in the complex domain is proposed. Independent design parameters are synchronously perturbed using real and imaginary perturbations, respectively. The complex variable finite element method is employed to compute the complex transient response of the perturbed model. The nonlinear transient response sensitivities with respect to each parameter are obtained by separately extracting the real and imaginary responses. The computational accuracy and efficiency of the proposed method are demonstrated by employing a nonlinear multidegree-of-freedom system and a cantilever beam with nonlinear elastic supports. Results show that the response sensitivity with respect to multiple design parameters is obtained accurately by using the proposed method, the computational efficiency of which is increased compared with the complex variable method (CVM). The stability of the transient response sensitivity is significantly affected by the parameter perturbations in the real and imaginary parts. The RIP-imaginary-based response sensitivity is feasible even for an extremely small perturbation in the imaginary part; the RIP-real-based response sensitivity is stable only within a limited real perturbation range.

Geometrically Nonlinear Transient Response of Laminated Plates with Flexible Supports

Laminated plates are loading-bearing components that are generally connected to flexible pads and exhibit complicated mechanical responses. To investigate the geometrically nonlinear transient responses of a laminated plate with flexible pad supports, a varied constraint reaction model and a systematic numerical procedure are presented in this paper. The flexible pad supports of the plate were treated as viscoelastic boundary conditions, wherein the strip-type pad per unit length was modeled as a cantilever beam. The nonlinear Kelvin–Voigt model was developed to simulate the nonlinear viscoelastic behaviors of the flexible pads. The dynamically varied constraint reactions generated by the viscoelastic supports, which depend upon the displacement and velocity of the nodes along the plate edge, were determined by the deflection and slope equations of the beam theory used, and they were applied on the plate edges by using the nonlinear load functions. Thus, the dynamical responses of the laminated plate with viscoelastic supports were obtained. Numerical results show that the present method can effectively treat the geometrically nonlinear transient response of the laminated plate with viscoelastic supports, and it is essential to consider the effects of non-ideal boundary conditions in the nonlinear transient analysis.

Geometrically Nonlinear Transient Response of Laminated Plates with Nonlinear Elastic Restraints

To investigate the dynamic behavior of laminated plates with nonlinear elastic restraints, a varied constraint force model and a systematic numerical procedure are presented in this work. Several kinds of typical relationships of force-displacement for spring are established to simulate the nonlinear elastic restraints. In addition, considering the restraining moments of flexible pads, the pads are modeled by translational and rotational springs. The displacement- dependent constraint forces are added to the right-hand side of equations of motion and treated as additional applied loads. These loads can be explicitly defined, via an independent set of nonlinear load functions. The time histories of transverse displacements at typical points of the laminated plate are obtained through the transient analysis. Numerical examples show that the present method can effectively treat the geometrically nonlinear transient response of plates with nonlinear elastic restraints.

Geometrically Nonlinear Transient Response of Laminated Plates With Nonlinear Viscoelastic Restraints

In this work, geometrically nonlinear transient response of laminated plates with nonlinear viscoelastic restraints is investigated by a numerical method, in the computational platform - MSC.Nastran. The variable constraining forces as functions of dynamic displacements and velocities are added to the right-hand side of equation of motion as additional applied loads. These loads are represented by an independent set of functions that satisfy the various constraint conditions for specific cases. The nonlinear equations are solved by using the load increments scheme in conjunction with Newton-Raphson iteration. The time history of transverse displacement at a typical point is given through a series of transient analysis. Then the comparisons of the responses for different parameters of the Kelvin-Voigt model and boundary conditions are made. The numerical results show that the present method is validated to be effective for treating geometrically nonlinear transient problems with nonlinear viscoelastic restraints.