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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.

For convenience, a beam without any attachments is called “bare” beam, and the beam carrying concentrated elements (CEs) is called “loaded” beam. One of predominant differences between an unconstrained beam (UB) and a constrained beam (CB) is that the free oscillations of an UB consist of the “rigid-body” motions and “elastic” vibrations, while those of a CB consist of the “elastic” vibrations only. In practice, most beams are the CBs, thus, the literature regarding “rigid-body” motions are rare. However, an elastically supported beam is a kind of UB so that its natural frequencies and mode shapes for both the “rigid-body” motions and “elastic” vibrations are required by the engineers. For the last reason, with the effects of rigid-body (heave and pitch) motions considered, this paper presents a modified mode-superposition method (MMSM) to study the topic being not yet to appear in the existing literature: free vibration characteristics of the free–free (F–F) loaded double-tapered beams (DTBs). In which, based on the normal mode shapes for both “rigid-body” motions and “elastic” vibrations of the F–F bare DTB, the partial differential equation of motion for the F–F loaded DTB was transformed into the matrix equation so that the free vibration characteristics of the F–F loaded DTB can be easily obtained. Numerical examples reveal that the MMSM can provide not only the heave and pitch frequencies for the “rigid-body” motions but also the satisfactory natural frequencies and mode shapes for the “elastic” vibrations of the F–F loaded DTB. However, this is not true for the conventional mode-superposition method (CMSM). Good agreements between the numerical results obtained from the MMSM and the corresponding ones obtained from the finite element method (FEM) confirm the reliability of the presented theories and developed computer programs for this paper.

An industrial building is a non-classically damped system due to the different damping properties of the primary structure and equipment. The objective of this paper is to quantify the range of applicability of the real model superposition approximation method to the seismic response calculation of industrial buildings. The analysis using lumped mass-and-shear spring models indicates that for the equipment-to-structure frequency ratios γf > 1.1 or γf < 0.9, the non-classical damping effect is limited, and the real mode superposition approximation method provides accurate estimates. For 0.9 < γf < 1.1, the system may have a pair of closely spaced frequency modes, and the non-zero off-diagonal damping terms have a non-negligible effect on the damping ratios and mode shape vectors of these modes. For 0.9 < γf < 1.1 and the equipment-to-structure mass ratios γm < 0.07, the real mode superposition approximation method results in large errors, while the approximation method can provide an accurate estimation for 0.9 < γf <1.1 and γm > 0.07. Furthermore, extensive parametric analyses are conducted, where both steel structures and reinforced concrete structures with equipment with various damping ratios are considered. Finally, the finite element analysis of a five-story industrial building is adopted to validate the proposed range of applicability.

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.

Propeller, as one of the main excitations, causes the vibration of the ship, which seriously affects the life and living comfort of the ship. Therefore, the structure of the marine excitation source region, which acts as a direct action area for the propeller excitation, is essential to the improvement of vibration. Based on the modal superposition method, the modal vibration analysis and vibration transmission law are worked out to conduct a study on the vibration characteristics of the marine excitation source region. The first-order vertical mode shapes of the plate frame model, the plate model, and the lattice model are 35.48 Hz, 42.42 Hz, and 131.42 Hz, respectively. In the vertical direction, the transfer function slightly decreases from the bottom to the 01 deck. In the longitudinal direction, the transfer function visibly decreases from the excitation source to the bow. In the transverse direction, the transfer function significantly decreases from the mid-longitudinal profile to the side shell. An optimization scheme of the marine excitation source region is proposed to reduce vibration response by increasing the number of the bottom longitudinal stiffeners. This paper gives an ideal for optimizing structural vibration, which is of guiding significance for the structural design of ships and the improvement of living comfort.

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 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.

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.