Mode Superposition Method Research Articles

Dynamic performance of sleeper-damping track with elastic side-supporting pad

ABSTRACT Vibration and noise induced by urban rail transit has been a growing public concern worldwide. A new sleeper-damping track with elastic side-supporting pad (SDTESSP) has been proposed to reduce the environmental vibration caused by metro trains. To evaluate the dynamic performance of SDTESSP system, a refined 3D vehicle-SDTESSP coupled dynamics model was developed in this work. Firstly, the SDTESSP model was established based on the mode superposition method, and the mode shape vectors, and natural frequencies are obtained through a finite element method, and the finite element model of the SDTESSP was verified by the mechanical property and impact tests. Then, the influences of rail irregularities, vehicle running speed, and structural parameters of elastic side-supporting pad (ESSP) on the dynamic performances of the vehicle and SDTESSP are evaluated in detail. Finally, under the same track stiffness, the dynamic responses of the vehicle and track system caused by the conventional elastic under-supporting pad (USP) and the new side-supporting pad (SSP) are compared. The results show that the proposed dynamics model can avoid the difficulty in casting dynamics equation and enables detailed evaluations for the dynamic performance of the SDTESSP system. Furthermore, the optimal thickness and taper range of the ESSP are presented by comprehensively considering all dynamic indexes, and the damping performance of the sleeper-damping track using SSP is better than that using USP.

Research on One-to-Two Internal Resonance of Sling and Beam of Suspension Sling–Beam System

An approach is presented to investigate the 1:2 internal resonance of the sling and beam of a suspension sling–beam system. The beam was taken as the geometrically linear Euler beam, and the sling was considered to be geometrically nonlinear. The dynamic equilibrium equation of the structures was derived using the modal superposition method, the D’Alembert principle and the Hamilton principle. The nonlinear dynamic equilibrium equations of free vibration and forced oscillation were solved by the multiple-scales method. We derived the first approximation solutions for the single-modal motion of the system. Numerical examples are provided to verify the correctness of formula derivation and obtain the amplitude–time response of free vibration, the primary resonance response, the amplitude–frequency response, and the amplitude–force response of forced oscillation. According to the analysis, it is evident that the combination system exhibits robust nonlinear coupling properties due to the presence of internal resonance, which are useful for engineering design.

Refined nonlinear fractional derivative model of vehicle-track coupling dynamics

The coupled vehicle-track system (CVTS) dynamics have been extensively investigated for decades. However, the calculation accuracy of prevailing vehicle-track coupling models needs to be improved in the high frequency range due to the inappropriate model simplification and neglect of material nonlinearity. In this study, we propose a refined numerical model of the CVTS that considers the nonlinear properties of the railpads and primary suspension using the fraction derivative Zener model. Furthermore, we more realistically simulate the wheelset, rail and railpad configuration with the elastic axle, solid finite element and surface-support models, respectively, and improve the computation efficiency by employing the mode superposition method. The results demonstrate that the refined CVTS model is more accurate than the classical model in simulating vehicle-track coupling dynamics above 2 kHz. In particular, there are significant differences in the dynamic response of the elastic wheelset model compared to the rigid model over a broad frequency range, with an 11% difference in the bogie acceleration response at the first dominant frequency. When the railpads are modeled using the surface-support model, the rail acceleration differences exceed 41% near 1 kHz and 44% near 2650 Hz, compared to the point-support model. Additionally, the rail response at various locations across the rail cross section can be calculated using the finite element method in this refined model. Overall. the proposed CVTS model provides high accuracy and efficiency for random vibration analysis, especially in the high frequency domain.

Semi-analytical method for liquid sloshing in the rigid super-elliptical tank with the ring baffle

Tanks with super-elliptical cross-sections can improve space utilization. The effect of the submerged ring baffle on liquid sloshing in the tank is studied. The semi-analytical method is developed to solve the three-dimensional (3D) sloshing problem in the baffled non-axisymmetric tank, using linear potential theory by dividing the liquid domain into sub-domains. The formal solutions are obtained analytically for the 3D non-axisymmetric sub-domains with arbitrary cross-sections, allowing for the determination of natural frequencies and modes by substituting the formal solutions into the linearized free surface condition, the boundary conditions on the baffle and tank wall and the continuity conditions at the interfaces of the interconnecting sub-domains. The dynamic response equation is derived using the mode superposition method with the damping term for swaying excitation. The obtained solution is validated by comparing it with a numerical model, demonstrating excellent accuracy and efficiency. Parametric studies are performed to determine the effect of the baffle on liquid sloshing, indicating that the baffle can considerably reduce the liquid sloshing when considering an optimal baffle position and width. This technique can be extended to investigate the sloshing reduction in vertical containers with arbitrary cross-sections.

Flow-Induced Vibration Hybrid Modeling Method and Dynamic Characteristics of U-Section Rubber Outer Windshield System of High-Speed Trains

The flow-induced vibration characteristic of the U-section rubber outer windshield structure of high-speed train is the key factor to limit its high-speed movement. Accurate and effective flow-induced vibration analysis of windshield structures is an important topic. In this paper, a hybrid modeling method for the analysis of flow-induced vibration of windshield structure is innovatively proposed for the U-section rubber windshield system of high-speed train. The method uses the external aerodynamic load obtained by aerodynamic simulation as the input condition of the flow-induced vibration model, and maps the aerodynamic load to the structural dynamics model characterized by the modal test data of the windshield structure. The flow-induced vibration model is established by means of modal superposition method and the time-domain response is effectively integrated by Runge Kutta method with variable step size. The results show that this method can effectively simulate the flow induced vibration of the wind baffle structure, and the real-time relationship between the aerodynamic load and the modal characteristics of the structure and the response of displacement and velocity can be obtained. On this basis, the comprehensive dynamic performance of the windshield system of high-speed trains at 400 km/h under external aerodynamic load is studied, that is, the force, displacement and velocity variation rules of the flexible structure are examined. It is determined that the displacement and velocity response curve of the measuring point near the lower side of the U-section rubber outer windshield is significantly higher than that of other parts. Moreover, the contribution of the first mode to the dynamic response of the structure is very obvious. This method provides an efficient calculation method for analyzing the flow-induced vibration characteristics of complex flexible structures.

Experimental investigation on vortex/wake-induced force of double unequal-diameter flexible cylinders in tandem

An experimental investigation on the vortex/wake-induced force of double unequal-diameter cylinders in tandem was performed. In the experiment, the upstream-to-downstream cylinder diameter ratio was 1.569, and the gap spacing ratio between the cylinder models ranged from 3 to 8. Under uniform flow with Reynolds number up to 11 328, the strains caused by the vortex/wake-induced vibrations were measured for the double flexible cylinders. Based on the measured strains, the displacements were reconstructed with the mode superposition method. Then, the vortex/wake-induced forces were obtained inversely. Subsequently, the time-varying excitation and added mass coefficients were identified by the forgetting factor least squares method. The results show that the wake-induced force coefficient and excitation coefficient of the downstream cylinder are smaller than those of the single cylinder, while the added mass coefficient becomes larger. In addition, the downstream cylinder under frequency capture can absorb a higher proportion of energy from the wake to excite larger vibration amplitudes, especially in the in-line direction.

Study on the Transient Forced Vibration Characteristics of a Radial Turbine Impeller

This work focuses on transient forced vibration response characteristics of a radial turbine passing through the resonant frequency at different speeds. The transient structure analysis based on the modal superposition method is used to calculate the vibration response of the impeller, and the frequency sweep tests are carried out to verify the calculation results. The experimental results are in good agreement with the simulation results. The impeller response curve becomes steeper as the frequency sweep time increases; however, the elastic potential energy of the blade is dissipated more by the structural damping at 13000 Hz. The response waveform of frequency sweep time 10s is close to the situation of 60s. When the frequency sweep takes the 60s, the excitation energy is input to the impeller, and the damping dissipation of the system reaches a state close to equilibrium.

Vibrational Power Flow Analysis for the Sandwich Cylindrical Shell Structure with a Metal–Rubber Core in the Thermal Environment

This research is designed to investigate the vibrational characteristics of a sandwich cylindrical shell structure with an elastic-porous metal–rubber core in the thermal environment by the power flow method. The finite element models of the homogeneous and sandwich cylindrical shell structures are established for harmonic response analysis completed by the mode superposition method. Further, the structural power flow is calculated and visualized based on the results of the finite element analysis. A comparison is made with the results for a plate available in the literature validating the effectiveness of the power flow visualization method. The effects of some key factors, such as the frequency, the metal–rubber core, the length-to-radius ratio, and the temperature on the power flow of the sandwich cylindrical shell structure, are analyzed using power flow cloud pictures and vector diagrams. The results reveal that the frequency affects the distribution of power flow, and the metal–rubber core can dissipate energy, whereas the length-to-radius ratio and temperature do not have a significant influence on the power flow of the sandwich cylindrical shell structure with a metal–rubber core (SCSMR). This work should be valuable for the noise and vibration control application of the SCS-MR.

A semi analytic method for the analysis of the symmetric hydroelastic response of a container ship under slamming and green water loads

This paper presents a semi analytic time domain method for the analysis of the symmetric hydroelastic response of a container ship subject to slamming and green water loads. An Impulse Response Function (IRF) is adopted for the calculation of radiation, diffraction and wave excitation forces. Local hydrodynamic forces associated with green water on decks and slamming loads are respectively modelled by the Buchner’s Dam Break Model and a Generalised Wagner Model. The structural responses are captured by Euler-Bernoulli beam theory and solved by the modal superposition method. The Duhamel Integral technique is used to evaluate the dynamic response. A parametric study demonstrates how external forces may affect the global wave induced vertical bending moments and shear forces. Numerical simulations are compared against a hybrid method that combines computational fluid dynamics, boundary element and finite element methods for low to medium frequency induced dynamic response. It is concluded that the proposed semi analytic methodology is fast and accurate and may be useful at concept ship design stage.

Dynamic analysis of segmental linings of shield tunnels using a state space method and its application in physical test interpretation

Based on the state space method and D'Alembert's principle, a new analytical method is established to study the dynamic response of circular segmental linings of shield tunnels. The tunnel lining segments are considered to be curved Euler–Bernoulli beams, the joints connecting the segments are modeled by a series of distributed springs, and the lining–surrounding soil interaction is represented by distributed reaction springs with viscous dampers. With the state space method, the governing equations can be concisely expressed in matrix form, and a separate variable method in conjunction with a frequency scanning strategy is employed to obtain the natural frequencies and corresponding modal shapes of the free vibration of the linings. The orthogonality of the vibration modes of the linings is demonstrated using a symplectic inner product, and the response of the linings under arbitrary dynamic loads is solved using the modal superposition method. Numerical and model experimental examples are presented to verify the efficiency and robustness of the proposed method. We prove that the proposed method can solve the dynamic response of segmental linings by considering the effect of joint and asymmetric loading conditions with high efficiency and accuracy.

Wind load identification of lattice towers using multi-source heterogeneous monitoring data

Wind load is one of the governing design loads for lattice tower structures. However, such information is difficult to be measured directly, and the traditional indirect identification methods highly depend on the accuracy of finite element model and are limited to fluctuating wind loads. To solve this problem, a novel wind load identification method from limited multi-source heterogeneous monitoring data is proposed. First, the complex model of a lattice tower is simplified into a reduced-order model based on the similarity principle of dynamic characteristics. And then, limited strain and acceleration data are fused to reconstruct wind-induced responses and modal parameters based on Kalman filter algorithm and mode-superposition method. Finally, the total wind loads, including both mean and fluctuating components, are obtained from identified modal parameters and modal responses. Numerical simulation of a lattice tower verified that the proposed method can reconstruct wind load and base reaction forces with high accuracy, and the anti-noise analysis shows that the proposed method has strong noise robustness. Moreover, the field test of a transmission tower during Typhoon Mangkhut was further carried out to validate the proposed method.

A Study on Shear Frame Structure by Modal Superposition Method

A Study on Shear Frame Structure by Modal Superposition Method

Sparse accelerometer-aided computer vision technology for the accurate full-field displacement estimation of beam-type bridge structures

Computer vision-based sensors are considered effective means to monitor full-field displacements; nevertheless, several limitations prevent their applications in full-scale structures. First, the accuracy of computer vision-based sensors is affected by measured distances. Second, due to the limited frame rate of cameras, computer vision-based sensors cannot record high-frequency responses. To effectively address these issues, a novel full-field displacement estimation approach based on the data fusion of camera-based measurements and limited accelerations was proposed. Full-field pseudo-static displacements were extracted from camera-based measurements, and full-field dynamic displacements were expanded by accelerations at sparse locations based on the modal superposition method. Subsequently, high-fidelity full-field displacements were obtained by data fusion. The proposed approach was numerically and experimentally verified. Further, the full-field displacements of an experimental-scale beam model and a full-scale bridge were estimated by a consumer-grade camera and several accelerometers.

Implementation of Timoshenko curved beam into train-track-bridge dynamics modelling

Curved bridge has been widely applied in railway lines due to its strong terrain adaptability, a more refined model that could capture curved bridge vibrations in the train-track-bridge (TTB) dynamic interaction issues is urgently required. In this paper, an improved semi-analytical approach for the vibrations of a Timoshenko curved beam is newly proposed based on the Chebyshev-tau method, which considers the rotary inertia and shear deformation, and is firstly implemented into the train-track-curved bridge coupled dynamics modelling (TTCBCD). First, by utilizing Galerkin method to discretize the partial differential equation of the curved beam and employing modal superposition method to decouple its ordinary differential equation, the forced vibration equations of the curved beam subject to three-dimensional moving loads are derived. Then, comprehensive comparisons of the eigenvalue analyses and dynamic performance of a pinned-pinned curved beam with published literature and finite element method demonstrate the accuracy and reliability of the current scheme. Moreover, the natural frequencies of curved beams with variable boundary conditions (BC) are investigated to illustrate its strong versatility by supplementing BC formulas. Finally, the established model is further implemented into the dynamics analysis of the TTCBCD featured by the nonlinear wheel-rail contact, the spatial flexibility of the track structure and the track slab-curved bridge interactions. The numerical analyses indicate that the bridge modelling approach exhibits remarkable distinction between the conventional method (idealized as a straight beam) and the proposed method, and some new insights in the applicable ranges of the proposed model with variable bridge lengths are pointed out. This work could be beneficial to a more accurately evaluation of the curve effects induced by train-track-bridge dynamic interactions.

A reduced-plate model transmission method for fast dynamic analysis of vehicle–pavement interaction

A reduced-plate model transmission method is developed in this study to accurately and efficiently analyze the vehicle–pavement interaction dynamics over long travel distances. A wide pavement is reduced to many small areas (referred to as the reduced-plate models) along the vehicle-moving trajectory, and each reduced-plate model accounts for the vibration of the plate structure caused by the moving vehicle in a small period. All reduced-plate models are assumed as stationary in the global coordinate system, and the vehicle moves from one reduced-plate model to another. The dynamic equations of the reduced-plate model are derived using the modal superposition method (MSM) and Lagrange's equations, and the calculation results of one reduced-plate model are considered as the initial conditions of the next reduced-plate model. Because only the reduced-plate model that is located inside the vehicle needs to be calculated in different periods, the number of degrees of freedom of the model is significantly reduced, and its calculation efficiency is increased. In addition, the present model is simple and better reflects the real situation of the vehicle–pavement interaction. After validating the present model, it is used in the vehicle–pavement interaction dynamic analysis during a long-distance travel. The calculation results obtained from the reduced-plate model transmission method are in good agreement with the MSM results for an entire plate, and the calculation time of the reduced-plate model transmission method is only one-third of that using the MSM for the entire plate. The calculation results further show that road roughness has a significant influence on the vehicle–pavement interaction dynamics in different situations. The Young's modulus and thickness of the base course layer also have a leading influence on pavement dynamic responses in the vehicle-pavement interaction system, where the variation of these parameters can lead to excessive responses of the pavement structure.

Combining Reduced-Order Stick Model with Full-Order Finite Element Model for Efficient Analysis of Self-Elevating Units

Reduced-order stick models are frequently employed to obtain dynamic amplification factors of self-elevating units (SEU), while the full-order finite element (FE) models are used for quasi-static analyses. This paper develops an efficient framework to create structural digital twins for SEUs by combining both stick models and full-order FE models. A stick model and a detailed FE model of an in-house developed generic SEU are established, respectively, following the standard industry guideline. Dynamic analyses are performed for the stick model based on the modal superposition method. Assuming that the stick model contains the key dynamic characteristics of the full-order FE model, the modal participation factors are multiplied by the corresponding mode shapes of the full FE model to derive the global dynamic responses of the entire SEU. The derived nodal displacements are imposed on the full-order model to obtain the member stresses. The global responses and member stresses are benchmarked with the results from a direct full-order dynamic FE analysis for various environmental conditions. The presented framework is found to significantly increase the efficiency of the simulation while retaining a similar accuracy, and it forms a critical step for the ongoing development of digital twins of fixed offshore structures.

Analysis and Experimental Study of Excitation Force Transfer Path of Optical Satellite Refrigeration Module

When the problem of the reaction force under the receiver structure is complicated, the existing force-based transfer path analysis method is not suitable for its analysis. Therefore, an improved reaction force transfer path analysis method is proposed in this paper. In the transfer path analysis method based on the modal superposition method, the penalty function method is introduced, and the path contribution of the receiving end reaction force is evaluated by analyzing the displacement response of the connection between the transfer path and the receiving end. A simulation analysis of a satellite refrigeration module vibration isolation device designed by the research group identified the main factors affecting the vibration isolation efficiency of the vibration isolation ring, and provided guidance for subsequent further optimization. The device was tested and analyzed by building an experimental platform, and the obtained simulation data fit well with the test data. These conclusions further show that the proposed analysis method can more accurately and conveniently analyze the path contribution of the receiving end reaction force under specific working conditions. This method is suitable for analyzing the transfer path of the reaction force at the receiving end in the design stage.

浮体式洋上風車の風と波による応答および構造損傷発生確率評価に関する研究

An evaluation method of structural damage probability of floating offshore wind turbines under wind and wave actions is presented. The structural damage probability of the floating wind turbines is analyzed by using the first order reliability method in the structural reliability evaluation. In the evaluation of the probability, the response analysis results of the floating wind turbine for the wind and wave actions are used. The response for the action by the mean wind speed is analyzed by using the statically equation of equilibrium. On the other hand, the response for the dynamic actions by the variable wind speed and the wave is analyzed by using a mode superposition method and a stationary random vibration theory. Numerical model of the floating wind turbine is discretized by finite elements, whereas interface of the floating wind turbine and the sea water is discretized by boundary elements. Moreover, variation of the damage probability of the floating wind turbine for different wind speed is discussed by using the proposed method.

Transmission loss and directivity of sound transmitted through a slit on ground.

An analytical model is proposed for sound transmission through a slit on a rigid ground based on the modal superposition method to investigate the transmission loss (TL). A simple formula is derived for estimation of the TL for plane waves with and without the ground, which gives a more precise prediction than existing approaches. It is found that a larger slit height generally decreases the TL, except at the resonant frequencies of the slit. The slit width has little effect on the TL at high frequencies, and the slit depth affects the resonant frequencies significantly even though it has little effect on the overall TL. Compared with the same size slit in the free field, the rigid ground reduces the TL at most frequencies, and that reduction is a constant between 3 and 9 dB in the low frequency range. It is also found that the sound transmitted through the slit is almost omnidirectional at low frequencies, while most of the sound energy at high frequencies falls within the range where the long side of the slit is located. The experimental results demonstrate the validity of the analytical model and the findings in numerical simulations.

Vibration and Sound Radiation of Cylindrical Shell Covered with a Skin Made of Micro Floating Raft Arrays Excited by Turbulence

To reduce the vibration and sound radiation of underwater cylindrical shells, a skin composed of micro floating raft arrays and a compliant wall is proposed in this paper. A vibroacoustic coupling model of a finite cylindrical shell covered with this skin for the case of turbulence excitation is established based on the shell theories of Donnell. The model is solved with the modal superposition method to investigate the effects of the structural parameters of micro floating raft elements on the performance of reducing vibration and sound radiation of the cylindrical shell of this skin. The results indicate that increasing the stiffness ratio, damping ratio, mass ratio, or decreasing the interval between micro floating raft elements can improve the vibration and sound radiation reduction performance of this skin over the frequency range 0~2000 Hz. Moreover, the mean quadratic velocity level and sound radiation power level of the finite cylindrical shell with this skin can be reduced by 12.00 dB and 9.65 dB respectively compared to the finite cylindrical shell with homogeneous viscoelastic coating in the frequency range from 0~2000 Hz, implying a favorable performance of this skin for reducing the vibration and sound radiation of cylindrical shells.