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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]

Large and complex engineering systems are usually assembled by subcomponents with different energy dissipation levels. Therefore, these systems often contain multiple damping models, which may lead to great difficulties in analyzing efficiently. In this paper, an efficient modal projection-based reduction method, which accounts for transient dynamic responses of structural system with multiple damping models, is proposed in the framework of a modified precise integration method. Two robust modal reduction bases, namely multi-model method (MM) and modal strain energy by first-order correction method (MSEC), are introduced to reduce the order of the original system. Based on the reduced system and a general damping model (GDM), a reduced state-space formalism for the structural system with multiple damping models is developed. Finally, the transient dynamic responses are derived using a modified precise integration method on the reduced stage. The numerical stability, accuracy and complexity are discussed. Two numerical examples are illustrated to assess the performances of the computational accuracy and efficiency. The results indicate that the proposed method is more efficient than other methods and most suitable for large-scale problems with rather good accuracy. [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]

Engineering structures are often assembled by subsections with various levels of energy dissipation, which lead to great difficulties in analyzing. In this paper, a direct time integration method is proposed to solve the dynamic responses of structural systems involving multiple nonviscous damping models. Based on a general non-viscously damped model, an extended state-space scheme for the damped system is derived. Then, an implicit time integration method using linear approximations is developed built on the extended state-space formalism. The stability and efficiency of the method are discussed by theoretical and numerical analyses. The method is illustrated by two examples. [ABSTRACT FROM AUTHOR]