Modal Superposition Approach Research Articles

Vibration behaviours of composite conical–cylindrical shells with damping coating: Theory and experiment

This study presents the theoretical and experimental investigations on vibration behaviours of carbon fiber/ resin composite conical-cylindrical shells (or CC shells) with damping coating. First, an analytical model of such a combined shell with coating material considers the effects of base harmonic excitation load under arbitrary boundary conditions is proposed using the first-order shear deformation theory, virtual artificial spring technique, modal superposition approach, the Rayleigh-Ritz method, etc. Subsequently, the solutions of fundamental frequencies, mode shapes, and vibration displacements in the frequency domain are acquired by deriving the corresponding energy expressions and motion equations. Convergence analysis is utilized to determine the virtual spring stiffness value and the appropriate truncation numbers. Both literature and experimental results are conducted to give an adequate validation of the current model, in which the continuous 3D printing technology is adapted to fabricate two CC shell specimens with one of the outside surfaces being sprayed by coating material using an atomization deposition approach. Finally, the effects of the coating thickness ratio, Young's modulus ratio of the coating and the substrate shell, the semi-vertex angle, and the fiber laying angle on the free vibration and forced vibrations of the coated structure are discussed, with some practical recommendations being provided to improve the vibration suppression of such a coated structure.

Virtual sensing for real-time strain field estimation and its verification on a laboratory-scale jacket structure under water waves

This study aims to achieve real-time estimation of the full-field strain distribution in a structure by signals measured from several strain gauges attached to the structure. Our virtual sensing procedure is developed based on finite element formulation and employs the mode superposition approach. To verify the feasibility of the proposed procedure, numerical and experimental tests are conducted on a laboratory-scale offshore jacket structure subjected to water waves. Key aspects addressed in this study include the selection of displacement modes and the division of strain signals. The experiments are performed in an ocean basin, and comprehensive explanations are provided for the jacket prototype design, implementations, experimental setup, and wave loading conditions. The performance of the proposed virtual sensing procedure is thoroughly assessed through various evaluation measures, enhancing the understanding of its capabilities and limitations in practical applications.

Low-frequency vibration reduction of an underwater metamaterial plate excited by a turbulent boundary layer

Flow-induced structural noise is an important component of hydrodynamic noise of underwater structures. Local resonance metamaterials are considered to have excellent performance and enormous potential in the field of low-frequency vibration and noise control. To verify its potential, the paper derived the underwater band gap of a lateral local resonance (LLR) plate through the plane wave expansion (PWE). Then, utilizing the modal superposition approach and Rayleigh integral technique, the vibro-acoustic response of a LLR plate under a turbulent boundary layer (TBL) excitation is obtained. Finite element certification is also conducted through an uncorrelated wall plane wave technique. Parametric study is conducted to analyse the factors which influence the control effects. The result shows that the plate exhibits excellent suppression performance for flow-induced vibration at band gap frequencies. The band gaps and suppression ranges generated by the underwater metamaterial plate, are dramatically narrowed due to the thick fluid load. The paper provides theoretical guidance for the control of flow-induced structural vibration and the application of acoustic metamaterials.

A refined mode superposition method for dynamical responses of an underwater cylindrical shell with substructures

To overcome the difficulties of time-consuming and inefficient in the response calculations of the submerged cylindrical shell with internal substructures, this paper firstly presents a refined mode superposition method. In view of the necessity and difficulties of directly and quickly obtaining the frequencies and modes of structures submerged in fluid (named the wet modes), the wet modes are expanded by the modes in vacuum (dry modes) and solved by the energy method. The added kinetic energy of the fluid is calculated via boundary integration, and the Lagrange equations of the second kind is applied to the fluid-structure coupling equations. Then the wet modes are solved by eigenvalue calculation and the modal mass and stiffness of each order wet mode are obtained. Finally, they are used for establishing a mode superposition approach for response calculations. The accuracy of the present method is verified by ANSYS software. In this method, all the required data are obtained from the structural analysis and the traditional complicated fluid force modeling is no longer required. Thus it has the advantages of high computational efficiency and applicability. Compared to the traditional semi-analytic model, this modeling methodology has broad application potential for vibration problems of complex underwater structures since the structural dry modes can be solved efficiently by commercial software. It also has practical value as a theoretical reference for developing mode-superposition-based calculations for fluid-structure problems using commercial software.

Analysis of dynamic interaction and response in multispan girder bridges subjected to near-fault earthquakes

This study delves into the separation and collision behavior of multi-span bridges with plate rubber bearings, focusing on the influence of seismic excitation period and vertical/horizontal acceleration (V/H) ratio during near-fault earthquakes. Employing the modal superposition approach, a four-span bridge model is scrutinized to understand the dynamic interaction between its components under seismic forces. The main girder, supports, and piers are simplified into beam, spring, and rod elements, respectively, to analyze vertical movements and collisions. Different bridge positions are evaluated for separation using the transient wave function, and the vertical impact force during collisions is assessed via the indirect modal method. It emerges that bridge separation is likely when excitation periods align with the bridge's natural frequency, especially for multi-span structures. These bridges also demonstrate a 25–110% increase in vertical collision forces compared to double-span bridges under various excitation periods, surpassing V/H= 2/3 standard. Theoretical energy-based calculations corroborate these observations.

A fully nonlinear BEM-beam coupled solver for fluid–structure interactions of flexible ships in waves

A fully nonlinear hydroelastic numerical method is proposed for the fluid–structure interactions of elastic ships. The boundary value problem is solved by the three-dimensional boundary element method. To simulate the instantaneous evolution on the free surface, an effective mesh optimization solution consisting of the local coordinate system and an elastic mesh technique has been developed. The mixed Eulerian–Lagrangian method is implemented to update the fully nonlinear free surface conditions in the local coordinate system. The hull structure is simplified as a non-uniform Timoshenko beam, and the modal superposition approach is adopted to analyse the symmetric responses in the head sea. Convergence of the model is evaluated for mesh resolutions and the time step size, and then comprehensive investigations of vertical motions as well as vertical bending moment are carried out for an advancing ship in a range of wave frequencies. The numerical solutions are compared with the experimental data and a satisfactory agreement is demonstrated. The distribution of high-order harmonics and whipping responses are predicted by the amplitude spectrum, in which the water entry and water exit are considered. The high-frequency vertical bending moments are investigated, in which the effects of higher harmonics and whipping responses are addressed.

Low-frequency sound and vibration reduction of a metamaterial plate submerged in water

Sound and vibration reduction of a metamaterial plate submerged in water is discussed theoretically and numerically. The modal superposition approach and Rayleigh integral technique are exploited in terms of the structural displacement and fluid pressure to investigate the low-frequency vibroacoustic characteristics of the underwater metamaterial plate and radiated sound field. Premised on the fluid-solid coupling mechanism, the acoustic band gaps generated by the underwater metamaterial plate to prevent the flexural waves propagation, are dramatically narrowed due to the extra fluid load, resulting in a reduction of the attached resonator's mass ratio over the host plate. Furthermore, from the theoretical formulae, extra sound radiation impedance can be observed to contribute to the plate's mechanical impedance owing to the presence of thick fluid, which greatly influences acoustic performance. As a result, it is concluded that in the low-frequency range, the vibroacoustic characteristic of the metamaterial plate is significantly affected by the ambient water. Results obtained from the theoretical analysis can provide guidelines on the design and optimization of acoustic metamaterials submerged in water to achieve prescribed acoustic properties.

Semianalytical Model for Dynamic Analysis of Missile Vertical Cold Launch

To quickly determine the dynamic characteristics of a missile excited by wind load during a vertical cold launch, a semianalytical model has been established by using the transfer matrix method and modal superposition approach, where the missile is simplified as a segmental Timoshenko beam and the adapter is equivalent to a spring with compressive and flexural stiffness. Then, the semianalytical model is verified by a finite element method and used to analyze the effects of wind load, thrust deviation, and the adapter’s elastic modulus on missile launch. The fact validates that the semianalytical model is effective and feasible, as well as more efficient, as compared with the finite element model, which is beneficial to perform the aforementioned influence analysis. The research proves that wind load has almost no effect on the missile’s vertical displacement but makes an obvious influence on the horizontal displacement and yaw angle. The missile rises more slowly, and the horizontal motion is delayed with less thrust; consequently, the horizontal displacement and yaw angle are increased when the missile reaches a specified altitude. The horizontal displacement and yaw angle are decreased with the adapter’s elastic modulus increased when the adapter’s assembly stress remains constant.

Seismic soil–structure interaction analysis of wind turbine support structures using augmented complex mode superposition response spectrum method

Abstract. In this study, the seismic soil–structure interaction (SSI) of wind turbine support structures is investigated based on the complex mode superposition approach. For accurate and efficient estimation of seismic loadings on wind turbine support structures, an augmented complex mode superposition response spectrum method (RSM) is developed, where the maximum shear force and bending moment of the non-classically damped system are analytically derived. An empirical formula of the modal damping ratios with a threshold value for the allowable damping ratio is also proposed to improve the prediction accuracy of the shear force acting on the footing. Furthermore, additional loadings to consider the contribution of the mass moment of inertia of rotor and nacelle assembly and P−Δ effect to the bending moment on the tower are analytically derived. The proposed formulae are first demonstrated upon a 2 MW wind turbine supported by two different types of foundations. A parametric study is then carried out by changing the tower geometries and soil conditions to propose the threshold value for the allowable damping ratio.

Vibration Energy Harvesting by Means of Piezoelectric Patches: Application to Aircrafts.

Vibration energy harvesters in industrial applications usually take the form of cantilever oscillators covered by a layer of piezoelectric material and exploit the resonance phenomenon to improve the generated power. In many aeronautical applications, the installation of cantilever harvesters is not possible owing to the lack of room and/or safety and durability requirements. In these cases, strain piezoelectric harvesters can be adopted, which directly exploit the strain of a vibrating aeronautic component. In this research, a mathematical model of a vibrating slat is developed with the modal superposition approach and is coupled with the model of a piezo-electric patch directly bonded to the slat. The coupled model makes it possible to calculate the power generated by the strain harvester in the presence of the broad-band excitation typical of the aeronautic environment. The optimal position of the piezoelectric patch along the slat length is discussed in relation with the modes of vibration of the slat. Finally, the performance of the strain piezoelectric harvester is compared with the one of a cantilever harvester tuned to the frequency of the most excited slat mode.

Analysis and Control of Vibrations of a Cartesian Cutting Machine Using an Equivalent Robotic Model

The vibrations of a Cartesian cutting machine caused by the pneumatic tool are studied with a sub-system approach. The cutting head is modeled as an equivalent robot arm which is able to mimic the measured resonances. The Cartesian structure is modeled according to the mode superposition approach. A global analytical model is obtained coupling the aforementioned models, and is solved in MATLAB. The full model is able to predict the variations in the response of the machine to tool excitation that are caused by the motion of the head along the rails of the Cartesian structure. Comparisons with experimental results are made.

Study of the characteristics of train-induced dynamic SIFs of tunnel lining cracks based on the modal superposition approach

Cracks on a tunnel lining are commonly observed during in-site tunnel inspection. The cracks are generated owing to various reasons and can potentially further propagate as they are subjected to repeated dynamic train loads during the entire tunnel service life. The study investigated train-induced dynamic stress intensity factors (SIFs) of tunnel lining cracks by numerical simulations using the modal superposition approach and examined the effects of a wide range of factors. The methodology of the modal superposition approach was adopted and validated to determine dynamic SIFs. Subsequently, the coupled dynamic FE model was established to calculate train-induced SIFs of tunnel lining cracks and radiation damping owing to the infinite surrounding medium was determined. Finally, a parametric analysis was conducted to investigate the effects of crack location, crack depth, wave velocity of the surrounding medium, and material damping on dynamic SIFs of tunnel lining cracks subjected to metro train loads. The results demonstrate that the modal superposition approach can be employed to accurately and effectively calculate the train-induced dynamic SIFs of tunnel lining cracks when radiation damping is considered. The effects of crack location, crack depth, and wave velocity of the surrounding medium are significant, while the effect of material damping is not considerably evident. The maximum dynamic SIF significantly increases with crack depth, while it decreases with the wave velocity of the surrounding medium and material damping ratio. Cracks with greater depth in the poorer medium are more likely to propagate under dynamic train loads.

Frequency-based investigation of a floating wave energy converter system with multiple flaps

In the present paper, a new Floating Wave Energy Converter System (FloWECS), consisting of a floating platform and multiple rotating flaps, is proposed and is preliminary investigated. The numerical analysis is implemented in the frequency domain under the action of regular head and oblique incident waves and it is based on a “dry” mode superposition approach. In this approach, the generalized modes concept is utilized for describing the relative to the platform rotation of the flaps, additionally to the six rigid-body modes. The required mode shapes of the generalized modes are determined through appropriate vector shape functions, which are derived in the present paper. The diffraction/radiation problem is solved using a numerical model based on the conventional boundary integral equation method. The proposed numerical formulation is, initially, validated by comparing results with the numerical ones of other investigators for the case of a single, bottom-hinged, rotating plate. Next, the exciting forces and the response of the “isolated” platform are assessed, while, finally, the hydrodynamic behaviour and the energy absorption of the FloWECS are investigated. The results illustrate, that for both examined incident wave directions, the generalized modes’ response and, thus, the system's power absorption is characterized by the existence of distinctive peaks at two different wave frequency ranges, attributed to resonance effects, as well as to the strong interaction effects between the activated flaps.

Analysis of Vibration and Acoustic Characteristics of a Simply Supported Double-Panel Partition under Thermal Environment

Numerical studies on the vibration and acoustic characteristics of a simply supported double-panel partition under the thermal environment are presented by the modal superposition approach and temperature field theory. Many factors are considered in this theoretical research, including acoustic refraction, dynamic response of the panel under thermal and acoustic load, vibroacoustic coupling characteristic analysis, and the variation of material properties. To access the accuracy and feasibility of the theoretical model, a finite element method is proposed to calculate the natural frequencies and mode shapes. The results show that the vibration and acoustic responses change obviously with the change of thermal stress and material properties. The rise of the graded thermal environment and thermal load decreases the natural frequencies and moves response peaks to the low-frequency range. The first valley of sound transmission loss is well consistent with the mode frequency. Finally, the relation between the average sound insulation and the thickness ratio is analyzed.

An efficient model for vehicle-slab track coupled dynamic analysis considering multiple slab cracks

Abstract This paper presents an efficient model for vehicle-slab track coupled dynamic analysis addressing the challenge problem of multiple slab cracks in high-speed railways. The multi-cracked track slabs are described as free Euler-Bernoulli beams supported on the cement asphalt (CA) mortar. The concentrated cracks are assumed to be in the through-transverse form, which are considered to locally affect the vertical bending stiffness of the beam and treated by applying the uniform flexural stiffness with Dirac’s delta singularities. The introduced damage parameters are expressed by incorporating the rotational springs using the linear elastic fracture mechanics. The vibration equations of slab track subsystem involving multiple slab cracks are finally set up by adopting the modal superposition approach, and are coupled with vehicle subsystem motions through nonlinear wheel-rail contact forces. An explicit integration method is applied to solve the whole coupled system excited by the random track irregularities. The developed model can also be expediently degenerated to the traditional model with intact slabs by setting the damage parameters to zero. Time-frequency analysis is conducted through fully investigating the effect of the crack depth, the crack numbers with different distribution properties and the CA mortar stiffness on the system dynamic responses. Some practical conclusions are drawn and may provide potent information for the maintenance in railway engineering and vibration based damage identification.

Dynamic response of non-classically damped structures via reduced-order complex modal analysis: Two novel truncation measures

The Caughey-O’Kelly condition, which defines a specific property of the damping matrix with respect to the mass and stiffness matrix, is violated in several cases of interest in earthquake engineering (e.g. seismic base isolation, supplemental energy dissipation devices, etc.). In these circumstances, non-classically damped systems are associated with complex-valued natural modes of vibration, and the complex modal superposition approach can be invoked as a generalization of the real-valued classical modal analysis to decouple the equations of motion and to compute the earthquake response. In analogy to the classical modal analysis, measures that quantify the error resulting from truncating up to a certain mode are desirable, especially for structures with hundreds or thousands of degrees of freedom. The concept of modal participating mass ratio, widely used in classical modal analysis, is no longer applicable to non-classically damped systems. The authors introduce here two novel measures related to a generalized modal mass ratio and to a modal dissipation ratio of each mode in the complex modal analysis framework. These two real-valued measures, whose sum is equal to one when all the modes are present, should be simultaneously considered to anticipate the importance of each mode in the complex modal superposition approach. Therefore, they are useful to determine the number of modes to retain in a reduced-order model. These two measures and the novel concepts are illustrated through some simple numerical examples of free vibration response and earthquake response of non-classically damped systems, including structures equipped with fluid viscous dampers and base-isolated buildings.

Minimization of drift force on a floating cylinder by optimizing the flexural rigidity of a concentric annular plate

The deployment of suitable configurations of mutually interacting floating bodies for efficiently controlling their hydrodynamic interactions towards the reduction of the wave drift forces and, thus, of the mooring lines’ loads, has, nowadays, gained a great scientific interest. In this paper, the hydrodynamic behaviour of a floating cylinder and a concentric annular flexible plate is analysed in the frequency domain aiming at the minimization of the drift forces acting on the cylinder by optimizing the flexural rigidity of the plate. The diffraction/radiation problem is solved using a higher-order boundary element method. The analysis is implemented assuming that both floating bodies oscillate freely in heave, while for the plate, flexible modes are, additionally, considered for describing its structural deformations. The required modes shapes are determined in vacuum (“dry” mode superposition approach) through analytical expressions. The flexural rigidity of the plate, D, is optimized at a specific wave number using a real-coded genetic algorithm. Initially, results are compared with numerical results of other investigators for the case of two rigid concentric floating cylinders. Next, extended results are presented, focusing on the effect of D, including its optimum value, on various physical quantities describing the behaviour of both the cylinder and the plate. Contrary to the isolated cylinder, the presence of the plate introduces sharp peaks in the variation pattern of the drift force of the cylinder, bounded at specific wave numbers, where resonance of the seiche mode of water motion in the annular cavity or of specific flexible modes of the plate occurs. However, by reducing D to its optimum value, the cylinder’s drift force obtains practically zero values at the target wave number, due to an efficient improvement of the wave field in the annular cavity around the cylinder. Moreover, a great reduction of the drift force compared to the isolated cylinder is achieved in the subsequent high frequency range.

Nonlinear dynamic behaviour of a cable with multiple intra-span elastic supports

This paper studies the nonlinear vibration of a cable with multiple intra-span elastic supports under primary resonance excitation. The elastic supports are treated as linear springs in the analysis. Lagrange’s equation and modal superposition approach are employed to derive the dynamic equilibrium equations of the cable based on the energy of system. The nonlinear dynamic behaviour of a cable is analysed in this work, and 3:1 internal resonance is investigated, with coupled in-plane and out-of-plane motions discussed as well. A cable with both two edges simply supported subjected to harmonic excitation is studied by using homotopy analysis method. The amplitude-frequency response and the first approximation solutions of the double-modal motion are obtained. Through comparison with four-order Runge-kutta approach, it demonstrates this method is accurate. Finally, numerical analysis is performed to study how the pretensions, the damping coefficients and the linear stiffness coefficients of the elastic supports influence the amplitude-frequency and amplitude-excitation curves. A range of nonlinear dynamic properties of a cable are acquired, especially the internal resonance issue, which requires the cables stay beyond the region of resonance prior to perform structural design.

Nonlinear hybrid modal synthesis based on branch modes for dynamic analysis of assembled structure

Abstract This paper describes a simple and fast numerical procedure to study the steady state responses of assembled structures with nonlinearities along continuous interfaces. The proposed strategy is based on a generalized nonlinear modal superposition approach supplemented by a double modal synthesis strategy. The reduced nonlinear modes are derived by combining a single nonlinear mode method with reduction techniques relying on branch modes. The modal parameters containing essential nonlinear information are determined and then employed to calculate the stationary responses of the nonlinear system subjected to various types of excitation. The advantages of the proposed nonlinear modal synthesis are mainly derived in three ways: (1) computational costs are considerably reduced, when analyzing large assembled systems with weak nonlinearities, through the use of reduced nonlinear modes; (2) based on the interpolation models of nonlinear modal parameters, the nonlinear modes introduced during the first step can be employed to analyze the same system under various external loads without having to reanalyze the entire system; and (3) the nonlinear effects can be investigated from a modal point of view by analyzing these nonlinear modal parameters. The proposed strategy is applied to an assembled system composed of plates and nonlinear rubber interfaces. Simulation results have proven the efficiency of this hybrid nonlinear modal synthesis, and the computation time has also been significantly reduced.

Finite element analysis of tires in rolling contact

AbstractIn this publication an overview on sophisticated modeling approaches for the finite element computation of rolling tires based on an Arbitrary Lagrangian Eulerian (ALE) formulation of the kinematic will be outlined. After a brief introduction into the ALE‐concepts special attention is laid onto the treatment of the tangential contact formulation. As the path‐lines of the material points are not computed directly within the relative kinematics framework, techniques for the treatment of inelastic material properties and the coupled thermo‐mechanical analysis with emphasis to rolling loss analysis will be given. Furthermore a modal superposition approach is suggested for the analysis of the high frequency response of rolling tires excited from the surface roughness of the road. Finally a mesoscopic modeling approach for the transient dynamic analysis of the tread rubber with the rough roads surface is presented. In summary these techniques enable for a broad bandwidth of tire specific behavior in rolling contact. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)