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This paper investigates the optimal distribution of damping material in vibrating structures subject to harmonic excitations by using topology optimization method. Therein, the design objective is to minimize the structural vibration level at specified positions by distributing a given amount of damping material. An artificial damping material model that has a similar form as in the SIMP approach is suggested and the relative densities of the damping material are taken as design variables. The vibration equation of the structure has a non-proportional damping matrix. A system reduction procedure is first performed by using the eigenmodes of the undamped system. The complex mode superposition method in the state space, which can deal with the non-proportional damping, is then employed to calculate the steady-state response of the vibrating structure. In this context, an adjoint variable scheme for the response sensitivity analysis is developed. Numerical examples are presented for illustrating validity and efficiency of this approach. Impacts of the excitation frequency as well as the damping coefficients on topology optimization results are also discussed.

PurposeThe main objectives of this paper are to develop a novel perturbation method (PM) to solve the complex-orthogonal eigenvalue problem and further propose an exact complex mode superposition method (CMSM) for the non-proportionally rate-independent damped systems.Design/methodology/approachA novel PM is developed to solve the eigenvalue problem. The PM reduced the N-order generalized complex eigenvalue problem into a set of n algebraic equations by the perturbation theory. The convergence and accuracy of the PM are demonstrated by several numerical examples. Further, an exact CMSM is presented. The influences of the imaginary part response of the modal coordinate and the off-diagonal elements of the damping matrix as well as the modal truncation on the solution by CMSM are discussed to illustrate the effectiveness of the developed CMSM.FindingsThe eigenvalues obtained by PM would converge to the exact ones with the increase of the modal numbers. For seismic response, the influence of the imaginary part solutions of the modal coordinate would increase with the increase of the coupling factor. The contribution of higher modes to acceleration response is greater than that to the displacement. The cumulative mode contribution coefficient of acceleration is developed to estimate the numbers of the complex modes for the acceleration seismic response by the CMSM.Originality/value1. An eigenvalue perturbation method for a rate-independent damped system is proposed. 2. PM is carried out by the real mode and accomplishes the reduction of the matrix. 3. CMSM is established for rate-independent damped systems. 4. CMSM considers the effect of imaginary part solutions of the modal coordinate. 5. Modal truncation index is developed to estimate the complex mode number for CMSM.

In this study, we explore the application of the mode superposition method for obtaining the dynamic response of a barrel to shot pressure excitation. This method is used to effectively analyze the vibrational behaviour of the barrel when subjected to the transient forces generated during firing. By decomposing the barrel’s response into a series of mode shapes, the complex dynamic interaction between shot pressure and the barrel structure is simplified, allowing for a more efficient and accurate computational process. The study provides a detailed examination of the influence of the dynamic response on bullet dispersion. Experimental validation is carried out through a series of tests on a representative barrel, with the results demonstrating a high correlation between the predicted and observed dynamic responses. This approach not only enhances understanding of barrel dynamics under shot pressure but also offers a valuable tool for the design and optimizing of ammunition, with the goal of improving performance while mitigating adverse vibrational effects. The findings suggest that the mode-superposition method is a robust and reliable technique for dynamic analysis in ballistics engineering.

A new preconditioned modified conjugate gradient algorithm based on improved gradient operator and preconditioned technology is proposed for moving force identification of bridge structure in this paper. First, the moving load identification problem is converted into the problem of solving large-scale linear equations by the time-domain deconvolution technology and modal superposition method. Then the large-scale linear equations problem is transformed into easily solved equivalent problem by preprocessing. Subsequently, it is transformed into an unconstrained linear optimization problem by constructing the corresponding objective function. Finally, the problem is solved by the proposed conjugate gradient method. The innovation of the proposed method lies in two aspects. First, the proposed conjugate gradient method is proved by mathematical theory. Second, before constructing the objective function, the preconditioned technique is utilized to simplify the original problem. A series of numerical simulations are carried out to verify the stability and effectiveness of the proposed approach under 21 kinds of noise levels and 6 different sensor configurations, and its performances are compared with several conjugate gradient methods. The results show that the proposed method can reduce the iteration number, and also ensure the load identification accuracy, which indicates that the proposed method can improve the speed of identification and effectively reduce the cost. Meanwhile, the identification situation of different load components is studied by the frequency spectrum analysis method. It is found that the proposed method is a stable and a reliable identification method for static and low-frequency components, which provides a new idea for dynamic weighing of low-frequency loads on bridges.

With the wide use of compressors in various fields, the vibration problem of pipeline systems caused by airflow pulsation also occurs frequently. Through the analysis of the vibration causes of the reciprocating compressor pipeline system, it can be seen that the structural resonance caused by the airflow pulsation is the main cause of the pipeline vibration. Based on the actual pipeline and equipment data of the gas storage gathering station, this paper uses the modal superposition method combined with Abaqus finite element simulation to study the pipeline vibration characteristics caused by airflow pulsation, establishes a full-scale pipeline-equipment finite element model, and simulates the pipeline vibration process under each mode. Through the modal analysis, harmonic response analysis, and stress safety assessment of the pipeline, it is calculated that the natural frequency of each order is much larger than the resonance area of the excitation frequency of the compressor pipeline. The deviation between the calculated results and the measured data is within 10%. The research results provide a theoretical basis for the constrained optimization design of the compressor pipeline.

This paper considers a vibration control problem for a multi-span beam bridge under pier base vibrating excitation by using nonlinear quasi-zero stiffness (QZS) vibration isolators. Three linear springs are needed to construct a nonlinear vibration isolator with quasi-zero stiffness. The vibration of the multi-span beam bridge under control and without control is governed by partial differential equation and several ordinary differential equations which are derived from Galerkin method. Modal superposition method with numerical modes of the structure and an iterative method are combined to predict the vibration response of the structure under pier base excitation. The influence of the quasi-zero stiffness vibration isolators on isolation of multimodal vibration of beam bridge is studied. The absolute motion transmissibility is proposed to evaluate the performance of the quasi-zero stiffness vibration isolator and is compared with an equivalent linear viscoelastic vibration isolator. The results demonstrate the effectiveness of the these two potential control method as well as a good control performance in suppressing vibration for high frequencies. But at low frequencies, only the quasi-zero stiffness vibration isolator can reduce the vibration amplitude of the beam bridge around the resonance frequency region. The effects of each control parameter on the absolute motion transmissibility of steady-state behaviors are investigated for a better isolation performance.

Based on geometrically nonlinear FEM, combined with stochastic imperfection mode, the stochastic imperfection mode superposition method was proposed for domes. Using the method the least favorable distribution of imperfection can be found and the corresponding critical buckling load of the structure can be calculated. Moreover the process of the structure with the load increment can be tracked. Finally the example demonstrated effectiveness of the method in finding the least favorable distribution of imperfection and the critical buckling load.

Identifying the characteristic coordinates or modes of nonlinear dynamical systems is critical for understanding, analysis, and reduced-order modeling of the underlying complex dynamics. While normal modal transformation exactly characterizes any linear systems, there exists no such a general mathematical framework for nonlinear dynamical systems. Nonlinear normal modes (NNMs) are natural generalization of the normal modal transformation for nonlinear systems; however, existing research for identifying NNMs has relied on theoretical derivation or numerical computation from the closed-form equation of the system, which is usually unknown. In this work, we present a new data-driven framework based on physics-integrated deep learning for nonlinear modal identification of unknown nonlinear dynamical systems from the system response data only. Leveraging the universal modeling capacity and learning flexibility of deep neural networks, we first represent the forward and inverse nonlinear modal transformations through the physically interpretable deep encoder–decoder architecture, generalizing the modal superposition to nonlinear dynamics. Furthermore, to guarantee correct nonlinear modal identification, the proposed deep learning architecture integrates prior physics knowledge of the defined NNMs by embedding a unique dynamics-coder with physics-based constraints, including generalized modal properties, dynamics evolution, and future-state prediction. We test the proposed method by a series of study on the conservative and non-conservative Duffing systems with cubic nonlinearity and observe that the proposed data-driven framework is able to identify NNMs with invariant manifolds, energy-dependent nonlinear modal spectrum, and future-state prediction for unknown nonlinear dynamical systems from response data only; these identification results are found consistent with those from theoretically derived or numerically computed from closed-form equations. We also discuss its implementations and limitations for nonlinear modal identification of dynamical systems.

This paper proposes a dynamic model for the first time in order to investigate nonlinear time-varying dynamic behavior of a drivetrain including parallel axis gears (such as spur and helical gears) and intersecting axis gears (such as spiral bevel gears). Flexibilities of shafts and bearings are included in the dynamic model by the use of finite element modeling. Finite element models of shafts are coupled with each other by the mesh models of gear pairs including backlash nonlinearity and fluctuating mesh stiffness. A system of nonlinear algebraic equations is established from the resulting nonlinear differential equations of motion by utilizing multi-harmonic harmonic balance method (HBM) in conjunction with continuous-time Fourier transform (CFT). Since the number of nonlinear equations is large, potential convergence problems are avoided by utilizing continuous-time Fourier transform, in contrast with gear dynamics studies that use discrete Fourier transform (DFT). Solutions obtained by utilizing CFT and DFT are compared, and the advantages of utilizing CFT are shown. Fourier coefficients are calculated by utilizing analytical integration rather than numerical integration for a further improvement in computational time. A new solution method, modal superposition method, is introduced for the first time to study nonlinear dynamics of drivetrains with multiple gear meshes, which is impractical if traditional solution methods are used due to the increased number of nonlinear equations. Using modal superposition method, the number of nonlinear equations becomes proportional to the number of modes employed which is significantly less than the number of degrees of freedom associated with nonlinearities, especially as the number of gear meshes in the drivetrain increases. Consequently, the proposed method decreases the computational effort drastically in the forced response analysis of multi-mesh, multi-stage gear systems and also makes it possible to model gear shafts by using finite element method. The resulting system of nonlinear algebraic equations is solved by utilizing Newton’s method with arc-length continuation. Solutions obtained by HBM are validated by the solutions obtained by direct numerical integration. Several parametric studies are carried out in order to investigate the effects of design parameters on the dynamics of the drivetrain. It is observed that nonlinear modeling of helical gear pairs is necessary if they are coupled with spur or spiral bevel gear pairs.

In this study, analytical solutions are presented for the flexural–torsional coupled vibration response analyses of thin-walled beams under bidirectional moving random loads. Based on classical Euler–Bernoulli and Vlasov beam theories, the governing dynamic equations considering the influence of additional torque have been established. The modal superposition method, the Laplace transform, and the Duhamel's integral technique have been employed to obtain the average value and standard deviation of beam displacements in vertical, lateral, and torsional directions. For the validation of the proposed formulations, the results obtained in this paper are compared with the results acquired by the Newmark-β method and the Monte Carlo method. Comparisons of the results prove the accuracy of the suggested formulations. Through the parametric analysis, it is confirmed that the position where the average value reaches its maximum is related to the load velocity. But the maximum standard deviation always occurs at the end of the beam, which decreases with the growth of velocity and the drop in span. When the velocity does not exceed 30 m/s, the displacement response is mainly controlled by low-order modes.