Mode Superposition Method Research Articles

Electromechanical modelling and vibration control analysis of cyclic symmetric structures with a piezoelectric vibration absorber

This study explored the application of piezoelectric vibration absorbers (PVAs) equipped with synthetic circuits for the vibration control of cyclic symmetric structures. A cyclic symmetric electromechanical model (CSEM) integrated with PVAs was introduced to facilitate the design and optimization of the PVAs. This model was formulated analytically based on the circulant matrix theory, and the dynamic response was computed using the complex mode superposition method with explicit consideration of the intrinsic resistance within the PVA. A case study involving a simplified blisk model and an experimental test rig was conducted to validate the proposed CSEM. The validity of the model was confirmed through a comparative analysis of both the natural and dynamic characteristics obtained using the finite element model and experimental results. Furthermore, the impact of synthetic circuit parameters (inductance and negative capacitance values) and the placement of piezoelectric patches on the vibration control performance of PVA were investigated. The results suggested that the optimal inductance coincide with the analytical value, whereas the optimal negative capacitance of the series was slightly greater than the intrinsic capacitance of the patch. Additionally, mounting the piezoelectric patch on the blade root improved the vibration control.

Plane Wave Finite Element Method for the Dynamic Response of the Periodic Pile-Type Composite Ground Under a Harmonic Concentrated Load

In this study, a simplified model for the pile-type composite ground, namely, the periodic pile-type composite ground (PPCG) model is proposed. The PPCG in this study is assumed to be periodic in the horizontal directions and finite in the vertical direction. For investigation of the dynamic response of the PPCG to a harmonic concentrated load, a plane wave finite element (PWFE) method for the PPCG under the load is developed in this study. To develop the method, the concentrated load is decomposed into its wavenumber domain components by the 2D Fourier transform first. The response of the PPCG to a wavenumber domain load component is represented by a set of plane waves, and the variables associated with each plane wave are discretized via the FEM along the vertical direction of the PPCG. The finite element equations for each plane wave of the PPCG are developed via the virtual work principle. To obtain the overall dynamic response of the PPCG to the concentrated load by synthesizing the responses due to all load components effectively, the mode superposition method is used to construct the response of the PPCG to the load component, and the polar coordinate system-based inverse Fourier transform method is developed for the inversion of the 2D Fourier transform with respect to the wavenumber. With the developed method, some numerical results about the dynamic response of the PPCG to the concentrated load are presented.

A Fast Simulation Model of Pantograph–Stitched-Catenary Interaction in Long-Distance Travel

The increasing operation speed of high-speed trains allows the pantograph to continuously interact with the catenary over a long distance in a short time, and many new methods have been developed to efficiently calculate its dynamics. However, the existing methods only consider simple catenary systems, which limits their application in high-speed railway systems. In this work, a reduced pantograph–stitched-catenary interaction model is developed to simulate pantograph–stitched-catenary interactions during long-distance travel. Based on the existing reduced catenary model, the stitched catenary system is first considered, where the stitched wire is simplified into a part of the messenger wire supported by two spring-damping elements. The present model is validated by test results and the EN 50318:2018 standard, and it is subsequently used to study the dynamic performance of the pantograph–stitched-catenary system at an overdesigned speed in Sweden. The results show that the proposed model can be seven times faster than the traditional modal superposition method with the same accuracy in a stitched catenary system, and the existing catenary system cannot be operated at an overdesigned speed without increasing the contact wire tension. The present model gives an efficient solution to pantograph–stitched-catenary interaction problems.

Numerical simulations for the flow-induced vibrations of tandem dual flexible circular cylinders

The three-dimensional large-eddy-simulation coupled with a mode superposition method was applied to numerically simulate the flow-induced vibrations (FIVs) of tandem dual flexible cylinders at Re = 1000 with three different spacing ratios (Sx/D = 2.5, 3.5, and 5, Sx is center-to-center spacing for tandem cylinders, and D is the diameter of the cylinder), corresponding to the reattachment flow, transition flow, and co-shedding flow regimes in stationary tandem cylinders, respectively. The effects of Sx/D on structural vibrations, flow fields, distributions of the surface pressures, and energy properties were investigated to reveal the mechanism for the FIV. Increasing Sx/D weakens the influence of the upstream cylinder on maximum response amplitudes and lock-in region for downstream cylinder. The wake patterns for tandem flexible cylinders are more complex compared to stationary or vibrated rigid tandem cylinders. The shielding effect reduces surface pressure on the downstream cylinder significantly when its vibrations are smaller, leading to a decrease in mean power as well. Furthermore, different mechanisms contribute to amplified FIV in downstream cylinders depending on Sx/D: when Sx/D = 2.5, the upstream vortices collide with the downstream cylinder's front surface and merge with the vortices generated by the downstream cylinder, increasing negative pressure on both front and rear surfaces of the downstream cylinders and promoting FIV; however when Sx/D =3.5 and 5, a binary vortex street forms behind the downstream cylinder without obvious negative pressures on its front surface, the dominant causes of FIV are primarily attributed to interactions among upstream and downstream vortices.

Design of piezoelectric quasi— zero—stiffness metastructures for improved low—frequency vibration isolation

Abstract In this paper, a novel type of piezoelectric quasi–zero stiffness (QZS) metastructure is proposed and the improved vibration isolation performance is investigated. A piezoelectric QZS metastructure is composed of curved beams covered with piezoelectric macro–fiber composite patches, to which digital circuits are connected. Based on the principle of minimum potential energy and the mode superposition method, the constrain on the geometry and material parameters of the piezo–curved beams to achieve QZS is given. According to this design criterion, 1D, 2D and 3D piezoelectric QZS metastructures are designed, and their improved low–frequency vibration isolation characteristics are comprehensively analyzed through theoretical, numerical and experimental studies. It is demonstrated that by optimizing the parameters of the resonant transfer functions implemented in the digital circuits, high–level damping localized near the first resonant peak can be introduced into the curved beams. As a result, the resonant peaks of the metastructures can be reduced without compromising the isolation performance beyond those peaks.

Strain response prediction of offshore wind turbine tower under free vibration

Strain monitoring of critical locations in offshore wind turbine tower is essential for fatigue life prediction and safe operation of wind turbine. This paper theoretically analyses the feasibility of predicting tower strain under free vibration using the modal superposition method. A finite element numerical model of the tower is established, and the displacement mode and strain mode parameters of the tower are extracted. The initial displacement and strain of the tower at cut-out wind speed are analyzed, and the first three modal coordinate of the tower are calculated using test points of displacement or strain separately. Subsequently, the full field strain of the tower is predicted using the modal superposition method. Comparative results demonstrate higher accuracy in modal coordinate values calculated using test points of displacement, and the accuracy and errors of strain prediction using different numbers of test points are compared, emphasizing the crucial role of selecting optimal positions and numbers of test points in improving prediction accuracy.

Seismic performance analysis of wood-steel frame structures

Wood and steel have different physical properties. By combining the two kinds of materials as bonded parallel beams, wood-steel frame structures can be prepared. This work considers the seismic performance of such engineering structures. Based on the Rayleigh damping model of substructures, the non-proportional damping coefficient is used to quantify the structural non-proportional damping characteristic of wood-steel frame structures. A complex mode superposition method is used to calculate seismic responses. Numerical cases showed that compared with the frame structure with upper steel and lower wood, the overall lateral performance is lower and the local lateral performance is higher for the frame structure with upper wood and lower steel. The overall lateral seismic design needs to be improved for the wood-steel frame structure with upper wood and lower steel. The local lateral seismic design needs be improved for the wood-steel frame structure with upper steel and lower wood.

Nonlinear metamaterial enabled aeroelastic vibration reduction of a supersonic cantilever wing plate

The violent vibration of supersonic wings threatens aircraft safety. This paper proposes the strongly nonlinear acoustic metamaterial (NAM) method to mitigate aeroelastic vibration in supersonic wing plates. We employ the cantilever plate to simulate the practical behavior of a wing. An aeroelastic vibration model of the NAM cantilever plate is established based on the mode superposition method and a modified third-order piston theory. The aerodynamic properties are systematically studied using both the timedomain integration and frequency-domain harmonic balance methods. While presenting the flutter and post-flutter behaviors of the NAM wing, we emphasize more on the pre-flutter broadband vibration that is prevalent in aircraft. The results show that the NAM method can reduce the low-frequency and broadband pre-flutter steady vibration by 50%–90%, while the post-flutter vibration is reduced by over 95%, and the critical flutter velocity is also slightly delayed. As clarified, the significant reduction arises from the bandgap, chaotic band, and nonlinear resonances of the NAM plate. The reduction effect is robust across a broad range of parameters, with optimal performance achieved with only 10% attached mass. This work offers a novel approach for reducing aeroelastic vibration in aircraft, and it expands the study of nonlinear acoustic/elastic metamaterials.

A semi-analytical approach for dynamic responses of monopile-supported offshore wind turbines subjected to accidental loads

Offshore wind energy is leading the way in sustainable energy generation. Offshore wind turbines (OWTs) need to be designed to withstand extreme or accidental loads in Ultimate Limit States (ULS) and Accidental Limit States (ALS), whereby high order eigenmodes may become significant. This paper introduces a semi-analytical approach for efficient and reliable analysis of the dynamic responses of monopile-supported OWTs subjected to accidental loads. A Rayleigh-Ritz solution is initially adopted to determine the high-order natural frequencies and eigenmodes of monopile OWTs, explicitly considering tapered towers and soil-pile interactions. Dynamic responses of OWTs are then calculated depending on the interaction schemes between accidental loads and the structures, e.g. soft contact loads (e.g. slamming, wind) and hard contact loads (e.g. ship collisions, ice impact). For soft contact loading conditions, the loads are considered independent of turbine responses, and the transient dynamic response is computed using the classical modal superposition method. While for hard contact loading conditions, the load actions depend on the contact stiffnesses and responses of the interacting bodies. A numerical contact algorithm is thus developed, and numerical iterations are performed to ensure convergence. The proposed approach is applied to the DTU 10 MW monopile-supported OWT subjected to extreme water slamming and ship collisions, respectively as example applications of soft and hard contact scenarios. The results are verified against nonlinear finite element analysis using USFOS and discussed with respect to the turbine natural frequencies and eigenmodes, contact forces and dynamic responses. A parametric analysis is conducted for ship-OWT collisions, exploring different impact scenarios by varying ship sizes, contact stiffnesses, and initial impact velocities. The proposed approach can serve as a promising tool for accidental load response analysis and the design of monopile OWTs.

A frequency domain analysis method for the dynamic response of a submerged floating tunnel under wave action

To investigate the dynamic response of a submerged floating tunnel (SFT) under the action of waves, a modal superposition method in the frequency domain was established, and the amplitudes of the modals were determined using the generalized wave loads and hydrodynamic coefficients of the structural modals. In this model, the SFT was simplified as a constant-cross-section Euler–Bernoulli beam with multiple anchor cables, and its structural modals were resolved using the finite element method (FEM). The generalized wave loads and generalized hydrodynamic coefficients were obtained using the Fourier expansion strategy of the modal function, by which the hydrodynamic problem can be converted into 2D problems with the modified Helmholtz controlling equation and solved using the matched higher-order boundary element method (HOBEM) and eigenfunction expansion. The effects of the diameter of the anchor cables and length of the SFT on the stiffness of the structure and the dynamic response of an SFT with anchor cables under wave action were investigated. Furthermore, the effects of different added mass calculation methods on the structural response were studied. In addition, the wave exciting forces were calculated using potential flow theory or linearized Morison equations according to the wave scale. Finally, the effects of the incident wave frequencies and incident angles were investigated in detail to reveal the combined resonance mechanism.

Rapid assessment of seismic performance of large monopile-supported offshore wind turbines under scour

This study focuses on investigating the seismic performance of a reference 15 MW offshore wind turbine (OWT) supported by a large-diameter monopile, under various scour conditions. Fully coupled nonlinear finite element models considering 3D soil continuum for seismic response analysis of monopile-supported OWTs are very complicated and computationally expensive. Therefore, it is important to have good insight on the dynamic behaviour of monopile-supported OWTs subjected to earthquake excitations before performing high-fidelity fully nonlinear 3D simulations, which will not generally be practical in cases considering various seismic loading and scouring scenarios. The purpose of this study is to provide a fast and efficient tool to understand the interaction between the seismic behaviour of emerging large-capacity OWTs when scour erosion affects the dynamic properties of the structure. The dynamic stiffness method (DSM) is applied for the free vibration analysis of the structure. Modal superposition is used to calculate the seismic responses of the OWT by using the exact mode shapes obtained from the dynamic stiffness formulations. The rotor-nacelle-assembly (RNA) is idealized as a lumped mass at the tower top. The tower and monopile are modelled as Timoshenko beam-columns under axial compressive loading. The soil-structure interaction (SSI) is modelled using the Winkler framework, whereby the soil contribution is assumed as elastic to take advantage of the modal superposition method. Two different approaches are considered for the soil stiffness; a constant profile, and a linearly-varying subgrade modulus along the embedded depth of the pile, and both are appraised in this paper. Firstly, natural frequencies obtained from dynamic stiffness formulations are validated using experimental data from literature for the case of controlled dynamic SSI experiments, and for operating OWTs. Additionally, the natural frequency provided by a technical document for the reference 15 MW OWT is compared to the result from the DSM, where a very good agreement is observed. The seismic response time-histories of the OWT model are obtained under different input earthquake conditions, and subsequently verified by comparison with the results of finite element simulations. The effects of different scour scenarios and foundation modelling approaches on the free vibration and seismic responses are subsequently investigated. A parametric analysis is also performed using different soil properties to observe the interaction between the soil stiffness, scour condition, and seismic response. Finally, the influence of the frequency content of seismic records on the dynamic response of the OWT, specifically the tower top acceleration, the mudline rotation, and the mudline stress, is investigated considering various scour scenarios. The specific numerical case study undertaken for a reference 15 MW OWT, which is supported by a large monopile in very dense sand, shows that the seismic responses are very limited and that the elastic soil behaviour assumption is reasonable. Additionally, considering the computational efficiency, the proposed combination of DSM and modal superposition can be used for rapid seismic performance assessment of large diameter monopile-supported OWTs at the preliminary design stage to gain insight to the global dynamic response of the whole vibrating system.

The Vibration Characteristics Analysis of Fiber-Reinforced Thin-Walled Conical–Cylindrical Composite Shells with Artificial Spring Technique

In this paper, the vibration characteristics (free vibration and forced vibration) of fiber-reinforced thin-walled conical–cylindrical composite shells (FTCCS) are analyzed by combining theory and experiment. Based on the classical shell theory and Kirchhoff–Love assumption, the overall structure of the FTCCS is theoretically modeled. The artificial spring technology is used to simulate the arbitrary boundary conditions at the joint, which is divided into a main spring and a secondary spring. At the same time, the displacement functions of the two sub-structure shells of the FTCCS are established by the orthogonal polynomial method, and the energy function of the FTCCS is constructed according to the continuity condition of the bolt joint. Then, the vibration characteristics of the FTCCS are solved using the Rayleigh–Ritz method and the modal superposition method. Finally, the TC300/epoxy resin-based fiber-reinforced thin-walled conical–cylindrical composite shells are selected as the research object, and the rationality of the mathematical model is verified by pulse, frequency sweeping and resonance excitation tests. The experimental results show that the errors between the natural frequency and the resonance displacement calculated by the theory and the experimental results are 1.75–4.76% and 3.79–9.28%, which confirms the rationality of the proposed model. By changing the length of the shell, the lamination angle and the semi-cone angle of the conical shell, the vibration characteristics parameters under different conditions are calculated to evaluate the influence of three physical parameters on the vibration characteristics of the FTCCS.

Analytical solutions for out-of-plane response of curved beams resting on an elastic foundation under a random moving load

In this study, the vibration behaviour of a horizontally curved beam resting on an elastic foundation and subjected to a random moving force is studied. The governing coupled differential equations for the out-of-plane vibration of curved beams are derived based on the mode superposition method, Duhamel's integral, and the Laplace transformation. Note that the out-of-dynamic responses of curved beams induced by a random moving load are composed of the centred random and mean value. Analytical solutions are proposed for the beam’s vertical and torsional deflections and the bending moment, where the rotary inertia, warping constant, and foundation stiffness are incorporated in these formulations. Because analytical studies on the random vibration of a curved beam resting on an elastic foundation are very limited, the aim of this study is to investigate the effect of relative parameters, namely, high-mode vibrations, the radius of curvature, the moving velocity, and foundation vertical and torsional stiffnesses, on the centred random and mean values of the beam dynamic responses. The results show that high-mode vibrations have a greater impact on the beam responses at higher velocities. To obtain precise results, the first 30 modes are considered in the calculation for the dynamic responses of the beams. The critical velocity is an inherent property of bridges, and it increases as the radius of curvature and foundation stiffness increase. In addition, an increase in the velocity and foundation stiffness results in a reduction in the centred random values for responses.

Investigation on dynamic modelling and nonlinear vibration behaviors of composite structures: A case of cable-beam model

This paper conducts a nonlinear analysis of cable-beam model of cable-stayed bridges by using the exact mode superposition method (EMSM) and the cable-beam dragging method (CBDM), respectively, comparing and exploring their theoretical foundations and practical implications. The EMSM is based on the global mode function of the cable-beam structure for nonlinear analysis, yet it requires more computational resources. The CBDM is based on the cable-beam dragging equations for nonlinear analysis, which can quickly obtain the static equilibrium state and dynamic response of the cable-beam system, but it requires some simplifying assumptions on the cable-beam connection conditions. Research results demonstrate qualitative and quantitative differences between these two methods through parametric analysis on dynamic behaviors, which provide a significant methodological study and a reference for the design and dynamics of composite structures.

Study on the Length of the Effective Vibration Area of the Catenary in a Pantograph–Catenary Interaction System

The effective vibration area includes most of the catenary vibration caused by pantograph–catenary interactions and is the basis of the real-time catenary model for hardware-in-the-loop simulation. However, while the length of the effective vibration area is one of the most important parameters of the real-time catenary model, it has not been fully studied at present. In this paper, the length of the effective vibration area is first investigated. A pantograph–catenary interaction model is developed based on the modal superposition method. After the validation of the model, the vibration energy distribution of the catenary is used to determine the length of the effective vibration area based on the converged total energy. The influence of vehicle velocity and contact wire tension on the vibration energy distribution and length of the effective vibration area is investigated. The obtained appropriate length of effective vibration area is validated by a real-time catenary model and online measurement data of the contact force. The investigation results show that the energy distribution of the catenary can accurately determine the length of effective vibration area, and it increases with increasing vehicle velocity but decreases with increasing contact wire tension. The appropriate length of effective vibration area should be at least 160 m (approximately three spans) in the pantograph–catenary system.

Numerical study on slamming characteristics of an elastic wedge based on a two-way coupled finite volume and modal superposition method

Numerical study on slamming characteristics of an elastic wedge based on a two-way coupled finite volume and modal superposition method

Self-oscillation of cantilevered silicone oil paper sheet system driven by steam

Self-oscillation is to draw energy from a stable external environment to maintain its own continuous motion, making it a highly relevant concept for intelligent materials, soft robots and medical devices. Exploring a simple and durable system driven by steam for self-sustained motion is a tough challenge. In this article, a novel steam-driven self-oscillation cantilevered silicon oil paper is proposed, incorporating a rectangular silicon oil paper sheet, a fixed end and a set of vertically rising steam. Experiments show that the silicone oil paper sheet can engage in continuous periodic self-oscillation in a steady steam. According to the above phenomenon, the dynamic model of the self-oscillation system is established, and the control equation is derived. The semi-analytical form of solution is obtained using the modal superposition method. It shows that the paper sheet has two motion patterns, namely, the static pattern and the self-oscillation pattern. The theoretical predictions are in general agreement with the experimental results. The silicone oil paper sheet maintains the self-oscillation by absorbing thermal energy to offset the damping dissipation. In addition, the critical conditions for self-oscillation and the key system parameters affecting its frequency and amplitude are also examined thoroughly. The oscillation amplitude can be controlled by adjusting the inclined angle, steam temperature factor, damping coefficient and thermal characteristic time, while the frequency mainly depends on the inclined angle and steam temperature factor. The cantilevered silicone oil paper sheet has advantages of simple structure, flexible adjustment, customizable size, and easy assembly. It is expected to have a certain guiding significance in the fields of remote drives, sensors, and micro-robot design.

Rigid-flexible coupling dynamic modelling and dynamic accuracy analysis of planar cam four-bar mechanism with multiple clearance joints

Combination mechanism of conjugate cam and four-bar is a complex type of multi-body system, in which the coupling problems such as cam profile law, flexible deformation of connecting rods, and multiple joint clearance collision make it difficult to model and optimise its dynamics. Just as the beating-up mechanism of high-speed rapier looms for multi-layer weaving is a representative conjugate cam four-bar mechanism. In response to the current development of textile machinery in the direction of high-speed and precision, and in order to improve the work efficiency and operational accuracy of the conjugate cam four-bar beating-up mechanism, the following work is done for the purpose of taking into account the multi-joint clearance and the flexible rod based on the mechanism. Firstly, based on Kane's equation, a multi-body dynamic model with multiple clearances is established combined with clearance collision theory. Secondly, a dynamic model of a rigid-flexible coupling system with multiple revolute clearances is established using finite element method combined with modal synthesis technology. Subsequently, the variables separation method and mode superposition method are used to solve the dynamic equations of the coupled system. Finally, numerical examples are used to analyse the effects of clearance quantity, clearance size, mode truncation order, cam speed, and component material properties on the dynamic response and accuracy of the system. The results show that, in the rotational speed range of 600–800 rpm, the dynamic performance of the system is little affected by the speed; Through comparative analysis of multiple materials, it is found that selecting carbon fibre composite materials has the smallest impact on the motion accuracy of the system. In this paper, the clearance collision model, finite element model and rigid-flexible coupling dynamic model are integrated, which are applied to the cam-linkage combined multi-body mechanism, and the dynamic problems of the actual machine are analysed and solved.

Integrated hydrodynamic-structural analysis of flexible floating structures

This paper presents a comprehensive approach for the linearized frequency-domain analysis of hydrodynamic loading and structural responses on a deformable body. The structural and hydrodynamic analyses are integrated by employing the mode superposition method. Commencing with an eigenvalue analysis on the structural model, the shapes of selected modes, mass, and structural stiffness matrices are extracted as input to the subsequent hydrodynamic analysis, where the global response is solved in the modal space. The resulting hydrodynamic and inertial loads are then transferred to the same finite element model for stress assessment. To exemplify the proposed methodology, a bottom-fixed flexible cylindrical monopile and a long box-shaped barge model are investigated. For the barge model, the sectional loads are obtained from the integral of stresses in the cuts along the structure model, which are found to be consistent with those from the hydrodynamic analysis, demonstrating the consistency of the entire workflow. In particular, this paper introduces a straightforward formulation for evaluating the generalized restoring matrices, eliminating the need for spatial derivatives of the mode shape function and thereby significantly reducing numerical uncertainties in using the flexible modes derived from finite element model for hydrodynamic analysis.

Fast Calculation for Vehicle–Road Coupled Response Based on Cyclic Road Modeling Method

In order to improve the computational efficiency of the vehicle–road coupled response in long-distance driving, this paper proposes a cyclic road modeling method to equivalent an infinitely long road. The road is built using a closed model, and the vehicle–road coupled response is calculated assuming that the vehicle numerically cyclic driving. First of all, the concept of the cyclic road model is introduced, and the modal analysis is performed. Secondly, the principle for determining the optimal length of the cyclic road model is proposed based on the attenuation properties of the road response in time and space. Next, the vibration analysis of the vehicle–road coupled system is conducted, and its equation of motion is constructed using the mode superposition method (MSM). Further, the optimal length of the cyclic road model is derived regarding to road parameters via numerical analysis. At last, the application of this cyclic road model on computing the vehicle–road coupled response in long-distance driving is discussed via numerical simulation. The results indicate that this model has a consistent performance with the widely used models, while significantly improves computational efficiency. The proposed method also provides a new idea for the fast modeling of roads.