Search

Highlights • The first and second derivatives for the transient response of non-viscously damped systems are calculated. • The direct differentiate method is adopted. • The sensitivities are derived by using a modified precise integration method. • The computational considerations of the methods are investigated and compared. • Two numerical examples are given to show the performances of the method. Abstract Calculation of the first and second derivatives of transient response with respect to design variables is a prerequisite when gradient-based methods are adopted for optimization design in time-domain. In this paper, a design sensitivity analysis (DSA) method for calculating the first and second derivatives of the transient response for non-viscously damped systems is developed. The assumed damping forces depend on the past history of motion via convolution integrals over some kernel functions. The direct differentiate method (DDM) is selected to derive the DSA method. By introducing a generalized damping model in expression of fraction formula, the equations of motion of the non-viscously damped system are transformed into a state-space form without the convolution integral terms. Then, the first and second derivatives of the transient response are formulated based on a modified precise integration method using the DDM. The numerical stability, accuracy and implementation effort of the DDM are discussed. Two numerical examples are comparatively demonstrated using the DDM, the discretize-then-differentiate adjoint variable method (AVM) and the differentiate-then-discretize AVM. The results indicate that, by considering all the computational considerations, the proposed DDM is more suitable than the other two methods for the sensitivity analysis of transient response for non-viscously damped systems. Besides, it is also the only existing method to capture the second-order derivatives of transient response for the non-viscously damped systems. [ABSTRACT FROM AUTHOR]

In this paper, design sensitivity analysis methods for the transient response of non-viscously damped systems are considered. The damping forces of the non-viscously damped systems depend on the past history of motion via convolution integrals over some suitable kernel functions. The adjoint variable method (AVM) is adopted to develop the design sensitivity analysis. Two numerical solution schemes, namely the discretize-then-differentiate method and the differentiate-then-discretize method, are introduced to complete the AVM for the sensitivity analysis of non-viscously damped systems. The discretize-then-differentiate AVM discretizes the equations of motion based on the Newmark- β implicit integration method first and then differentiates the discrete equations. On the contrary, the differentiate-then-discretize AVM firstly differentiates the equations of motion after transforming it into a state-space form, and then discretizes the equations based on a modified precise integration method (PIM). The numerical accuracy, efficiency, consistency and implementation effort are discussed and compared. Two numerical examples are presented to show the effectiveness of both methods. The results indicate that, by considering both computational accuracy and efficiency, the PIM based differentiate-then-discretize method is more suitable than the Newmark- β based discretize-then-differentiate method for the sensitivity analysis of transient responses for non-viscously damped systems. [ABSTRACT FROM AUTHOR]

Sophisticated engineering systems are usually assembled by subcomponents with significantly different levels of energy dissipation. Therefore, these damping systems often contain multiple damping models and lead to great difficulties in analyzing. This paper aims at developing a time integration method for structural systems with multiple damping models. The dynamical system is first represented by a generally damped model. Based on this, a new extended state-space method for the damped system is derived. A modified precise integration method with Gauss-Legendre quadrature is then proposed. The numerical stability and accuracy of the proposed integration method are discussed in detail. It is verified that the method is conditionally stable and has inherent algorithmic damping, period error and amplitude decay. Numerical examples are provided to assess the performance of the proposed method compared with other methods. It is demonstrated that the method is more accurate than other methods with rather good efficiency and the stable condition is easy to be satisfied in practice. [ABSTRACT FROM AUTHOR]

• The first and second derivatives for PSD function subject to stochastic excitations are derived. • The proposed methods are applicable to non-classically damped systems. • The highly efficient Pseudo-Excitation-Method (PEM) is adopted. • Two correction methods are developed without the need to calculate the eigen-sensitivities. • Computational accuracy is greatly improved by considering truncated high-order complex modes. Calculating the first and second derivatives of Power Spectrum Density (PSD) function with respect to various design variables is a prerequisite for random responses when gradient-based algorithms are adopted. This paper presented two numerical methods to capture the sensitivity and Hessian matrix of the PSD function for non-classically damped systems subject to stationary stochastic excitations. The direct differentiate method (DDM) is adopted to develop the design sensitivity analysis (DSA). By using Pseudo-Excitation Method (PEM), the governing equations of the non-classically damped system subject to stationary stochastic excitations are transformed into a corresponding deterministic harmonic response problem. Then, the first and second derivatives of the PSD function are given in detail using the DDM. Two numerical methods, namely PEM-modal displacement method (MDM) and PEM-hybrid expansion method (HEM), are proposed to compute the sensitivity and Hessian matrix of the PSD function. The computational accuracy and efficiency of both methods are discussed and compared theoretically and numerically by two illustrations. The results indicate that both methods are valid for the DSA of the PSD function for non-classically damped systems and the PEM-HEM is more suitable than the PEM-MDM with all computational considerations. [ABSTRACT FROM AUTHOR]

The mode superposition method (MSM), which is used for dynamic response calculations of nonviscously damped systems, requires state–space eigensolutions, which incur a high computational cost. This study proposed an alternative symmetric formulation-based time integration method for analyzing structural dynamic systems. An anelastic displacement field (ADF) model was introduced to capture damping mechanisms in nonviscous systems. The differential equation describing the ADF model was transformed into an augmented coupled system via the introduction of dissipative variables. Two symmetric state formulations based on coupled augmented systems were derived for nonviscous systems. Consequently, alternative asymmetric state formulations were derived for nonviscous systems involving the ADF model. Thereafter, a new method was developed based on these state–space formulations by assuming linear approximations. A numerical implementation of the new method for dynamic systems was demonstrated. Three benchmark examples of discrete and continuous systems were considered to evaluate the asymmetric formulation and the computational efficiency, accuracy, stability, and time-step-size sensitivity of the new method. Further, the response of the new method was validated using MSM and precise time integration (PTI) methods. An asymmetric state formulation was also validated against MSM and direct frequency response methods. The results revealed that the new method accurately matched the MSM and PTI methods at various time steps. The results also showed that the new method was more efficient than the MSM and PTI methods, rendering it suitable for solving large-scale structural dynamics problems. • Mode superposition method (MSM) requires significant computation for the dynamic response calculations of non-viscously damped systems. • New method is proposed for the response calculations of large-scale dynamic systems. • Symmetric and asymmetric formulations based on the ADF model are derived for non-viscous systems. • New method gives precise results that approach accurate solutions at different time-step sizes. • New method has a much higher computational efficiency than the MSM and PTI methods. [ABSTRACT FROM AUTHOR]