Modal Superposition Theory Research Articles

Differentiated tactile effects of vibration modal superposition considering the time course

Several theoretical studies on modal superposition methods to achieve tactile differentiation have been reported. However, there is still a lack of in-depth analysis and understanding of the regularity of the superposition of vibration modes and the resulting differences in tactile sensation. The touch beam vibrations are no longer in the form of standing waves after the modal superposition. The vibration points on the touch beam will present a periodic motion over time, instead of simply moving harmonically. The vibration response of the touch beam at high frequency, gave subjects a dramatically different tactile perception from that of a single inherent mode. Therefore, this paper reshapes the modal superposition theory of the haptic feedback technique to classify the vibration modes generated by the modal superposition of touch beams showing differences in tactile sensations considering the time course. The analysis and tactile texture prediction theories are improved compared to previous studies. The tactile sensation generated by the vibration patterns of left-right sway were found in the touch beam. Finally, the correctness of the tactile differentiation achieved by the modal superposition method was demonstrated by psychophysical assessment for different finger conditions.

Effect of Support Stiffness Nonlinearity on the Low-Frequency Vibro-Acoustic Characteristics for a Mechanical Equipment—Floating Raft—Underwater Cylindrical Shell Coupled System

The low-frequency vibro-acoustic characteristics of a mechanical equipment—floating raft—cylindrical shell—underwater acoustic field coupled system with nonlinear supports are studied in this paper. Firstly, the state space equations were established by a modal superposition theory for the coupled system, and a modal parameter identification method was deduced and verified for the cylindrical shell—underwater acoustic field coupled subsystem. On this basis, the formulas were derived for transmitted power flow in the coupled system, and the nonlinear stiffness constitutive relation of the vibration isolation supports was expressed by softening and hardening characteristics. Finally, dynamic simulations were carried out by the Runge—Kutta method to analyze the effect of nonlinear stiffness characteristic parameters on the low-frequency vibration modes and vibro-acoustic transfer characteristics in the coupled system. The research shows that a superharmonic phenomenon is common in the steady vibration mode of the coupled system with a nonlinear softening (or hardening) stiffness characteristic under harmonic excitation. The stronger the softening (or hardening) stiffness characteristic is, the more complex the vibration form is, and the smaller (or larger) the low-frequency vibro-acoustic transfer level in resonance regions is.

Multiobjective-clustering-based optimal heterogeneous sensor placement method for thermo-mechanical load identification

The direct measurement of external loads acting on structures remains a challenge in many engineering applications. In this context, mechanical load identification has received considerable attention to inverse them using some response signals. Obviously, sensor placement is fundamental to the success of load identification. Considering temperature effects in the environment, this study investigates an optimal heterogeneous sensor placement framework for multi-case mechanical load identification. Firstly, the temperature field approximation method is developed using the measuring points with the minimum error in RBF interpolation. Based on the modal superposition theory, the formulas of mechanical load identification under static and dynamic cases are then deduced utilizing the responses eliminating thermal-oriented components. With the specific loading position, an index of the modal contribution rate is defined to provide a reasonable modal selection. Further, an optimal placement framework of multi-type response sensors (strain gauges and accelerometers) considering distance constraints is constructed for mechanical load identification under static/dynamic multiple cases, in which the optimization objective integrates the global performance and local evaluation of the interested modal loads. More specifically, an algorithm of collaborative clustering for heterogeneous modal matrices is involved to avoid redundant information and alleviate the computation burden. Eventually, two numerical examples are discussed to demonstrate the validity and feasibility of the developed approach.

Vibration characteristics analysis of composite floating rafts for marine structure based on modal superposition theory

Abstract Composite materials have become a research hotspot in the field of vibration and noise reduction for their high strength, high damping, and other outstanding mechanical properties in recent years. In this paper, the effect of laminated materials on the dynamic performance of floating rafts is investigated based on modal superposition theory using the finite element method. The detailed derivation of the modal superposition theory was made, and taking T700 fiber-reinforced composite material as an example the damping effect of the floating raft structure in three cases was discussed: whether the composite material is laid or not, different layup angles, and different layup positions. The research shows that laying composite materials can improve damping effect of the floating raft and the changes in both the laying angle and the laying position will affect its dynamic performance. Moreover, the damping performance of the structure is inversely correlated with its stiffness within a certain range.

Complex frequency identification using real modal shapes for a structure with proportional damping

AbstractModal identification is based on modal superposition theory. For structures with proportional damping, the structural responses are recognized as real modal shapes multiplied by real modal coordinates. However, it is not true because the actual modal shapes should be complex even though they can be normalized to real ones. If the modal shapes are recognized as real ones, the modal superposition theory cannot reflect the actual modal parameters. This paper proposes an innovative method to identify complex frequencies for a structure with proportional damping using the identified real modal shapes, where the complex modal shapes are considered. The proportional damping is ideal but always assumed in practical engineering. The characteristic of the proposed method is that the complex property of frequencies is considered along with modal identification process. First, the reason why the complex modal shapes are hidden is revealed. When the real modal shapes are obtained, the Hilbert transform of analytical responses is used to estimate complex coordinates. Then, formulas to identify the complex frequencies are derived. The k‐means clustering method is employed. Finally, the proposed method is used in a numerical example and a practical bridge example. The results show that the proposed method can estimate the complex frequencies effectively, and can be used in practical engineering.

SOUND TRANSMISSION LOSS ACROSS A FINITE CLAMPED DOUBLE-LAMINATED COMPOSITE PLATE WITH POROELASTIC MATERIAL

A theoretical study of sound transmission loss across a clamped double-laminated composite plate filled with poroelastic material is formulated. Biot’s theory is employed to describe wave propagation in elastic porous media. The two face composite plates are modeled as classical thin plates. By using the modal superposition theory, a double series solution for the sound transmission loss of the structure is obtained with the help of the Galerkin method. The analytical model is validated against previous experimental results of a single sound wave under normal incidence. The numerical results suggest that the density of poroelastic material, the type of composite materials and the composite plies arrangement have significant effects on the sound transmission loss of considered structure.

SOUND TRANSMISSION LOSS ACROSS A FINITE CLAMPED DOUBLE-LAMINATED COMPOSITE PLATE WITH POROELASTIC MATERIAL

A theoretical study of sound transmission loss across a clamped double-laminated composite plate filled with poroelastic material is formulated. Biot’s theory is employed to describe wave propagation in elastic porous media. The two face composite plates are modeled as classical thin plates. By using the modal superposition theory, a double series solution for the sound transmission loss of the structure is obtained with the help of the Galerkin method. The analytical model is validated against previous experimental results of a single sound wave under normal incidence. The numerical results suggest that the density of poroelastic material, the type of composite materials and the composite plies arrangement have significant effects on the sound transmission loss of considered structure.

An optimal modal coordination strategy based on modal superposition theory to mitigate low frequency oscillation in FCWG penetrated power systems

Full converter-based wind power generation (FCWG, e.g. permanent magnet synchronous generator (PMSG)) becomes prevalent in power electronics dominated multi-machine power system (MMPS). With flexibly modified FCWG oscillation modes (FOMs), FCWG has the potential to actuate conducive dynamic interactions with electromechanical oscillation modes (EOMs) of MMPS. In this paper, a mathematical model of FCWG and MMPS is firstly derived to examine the dynamic interactions. Then a novel modal superposition theory is proposed to classify the modal interactions between FOMs and EOMs in the complex plane for the first time. The modal coupling mechanism is graphically visualized to investigate the dynamic interactions, and the eigenvalue shift index is proposed to quantify the dynamic interaction impact on critical EOM. Based on different manifestos in modal coupling mechanism and eigenvalue shift index, a novel methodology to optimize the dynamic interactions between the FCWG and MMPS is designed within the existing control frame. The optimized dynamic interactions (i.e. modal counteraction) can significantly enhance the LFO stability of MMPS, effectiveness of which is verified by both modal analysis and time domain simulations.

Reduction of Sound Transmission Through Finite Clamped Metamaterial-Based Double-Wall Sandwich Plates with Poroelastic Cores

Finite structures play a more realistic role in applications designed for sound and vibration isolation. Doublepanel structure with poroelastic cores is able to exhibit a superior sound insulation performance in mid-high frequency range, while is relatively inferior to isolate waves at low frequencies. In order to further reduce sound transmission at low frequencies and cater for the actual situation, this paper decides to introduce the metamaterial concept into finite double-wall sandwich plates and presents an analytical model to calculate the sound transmission loss through the metamaterial-based double-panel with fully clamped boundary conditions. The metamaterial-based double-wall sandwich plates are constructed by replacing the bare panel with the metamaterial plate, consisting of a homogeneous plate and periodically attached local resonators. Biot's theory is used to examine the wave propagation in the poroelastic medium. The vibro-acoustic problem of the proposed sandwich plate is solved by employing the modal superposition theory and the Galerkin method. Numerical results show that the sound transmission is significantly reduced at low frequencies. Unique phenomena caused by attached local resonators are explained and the eff ects of resonator inerter, incident angles and damping on the sound insulation properties are also studied.

Analytical modelling of sound transmission loss across finite clamped triple-wall sandwich panels in the presence of external mean flow

• The triple-wall panels have superior sound performance to their double-wall counterparts due to an additional partition. • The finite dimensions and clamped boundary influence the sound transmission of triple-wall sandwich panels. • The external flow enhances sound transmission loss significantly in the low-frequency domain. This paper analytically addresses the sound transmission loss (STL) performance of finite triple-wall sandwich panels lined with poroelastic materials and clamp mounted on an infinite rigid baffle in the presence of an external mean flow. The wave propagation in poroelastic media is addressed using Biot's theory. The coupling methods between the poroelastic core and facing plates are associated with different boundary conditions and various panel configurations. The Galerkin method and the modal superposition theory are used to deal with the clamped boundary conditions and finite extensions as well as to transfer the integral function to a matrix equation. The analytical model is verified by comparing to previous results of infinite triple-wall sandwich panels and finite triple-wall panels without poroleastic materials. The numerical results show that the triple-wall panels have superior STL to their double-wall counterparts due to the existence of an additional partition. The finite dimensions of the triple-wall sandwich panels with clamped boundary strongly influence the STL at low frequencies, and this effect is coupled with that of the external mean flow. The air layer plays a critical role in the sound insulation properties that vary among different panel configurations, and the configurations with two air layers exhibit the optimal overall sound insulation performance.

Effects of external and air gap flows on sound transmission through finite clamped double-panel sandwich structures

In this paper, previous theories on the prediction of sound transmission loss (STL) through a finite clamped double-panel sandwich structure lined with poroelastic material are extended to address the influence of both external and air gap flows. Biot’s theory is employed to describe wave propagation in poroelastic material. The clamped boundary condition along the panel edge introduces the modal superposition theory and a double Fourier series solution for the STL is obtained by employing the weighted residual (Galerkin) method. The analytical model is validated against theoretical results of a single incident wave under normal and oblique incidences. Various parameters such as flow velocity and gap flow orientations are considered. The numerical results suggest that the STL performance can be enhanced dramatically below the fundamental (1,1) resonance frequency with the increasing gap flow Mach numbers. Opposite orientation of air gap flows further improve the STL in the low-frequency range, and reduces the limiting angle of incidence which can trigger total internal reflection across the panel system. Although the air gap flows have less influence than the external flow, their noticeable effects on the STL show the potential in active/passive noise control.

External mean flow influence on sound transmission through finite clamped double-wall sandwich panels

This paper studies the influence of an external mean flow on the sound transmission through finite clamped double-wall sandwich panels lined with poroelastic materials. Biot's theory is employed to describe wave propagation in poroelastic materials and various configurations of coupling the poroelastic layer to the facing plates are considered. The clamped boundary of finite panels are dealt with by the modal superposition theory and the weighted residual (Garlekin) method, leading to a matrix equation solution for the sound transmission loss (STL) through the structure. The theoretical model is validated against existing theories of infinite sandwich panels with and without an external flow. The numerical results of a single incident wave show that the external mean flow has significant effects on the STL which are coupled with the clamped boundary effect dominating in the low-frequency range. The external mean flow also influences considerably the limiting incidence angle of the panel system and the effect of the incidence angle on the STL. However, the influences of the azimuthal angle and the external flow orientation are negligible.

Analytical modelling of sound transmission through finite clamped double-wall sandwich panels lined with poroelastic materials

This paper addresses analytically the vibroacoustic problem of sound transmission across a rectangular double-wall sandwich panel clamp mounted on an infinite rigid baffle and lined with poroelastic materials. The wave propagation in poroelastic media is described by Biot’s theory and the coupling methods between the poroelastic core and the panel determine the various configurations and associated boundary conditions. The modal superposition theory and the weighted residual (Galerkin) method are employed to account for the finite extension with the clamped boundary and to obtain a double-series solution of the problem through a matrix equation. The sound transmission loss (STL) of the structure is calculated for a single incident wave after validating the analytical model against previous theories and experiments. The numerical results show that the finite panel and clamped boundary influence the STL dominantly in the low-frequency range. The poroelastic materials exhibit strong damping effects on the resonances of STL spectra at high frequencies as well as panel vibrations and enhance significantly the sound insulation performance of the sandwich panel.

Identification Method for Moving Loads over Continuous Beam Based on Bending Moment Influence Lines

Vehicle axle loads are modeled as moving loads and bridge is considered as a continuous beam. Based on the modal superposition theory, the model accelerations can be obtained from the accelerations of the bridge at several sections. Then based on the d’Alembertian theory, the inertia force of the bridge can be expressed approximately. Using the bending moment influence lines, the equations about flexural moment and moving force is obtained. Using the formulas, the moving force can be obtained at any time. Examples show that the method has high accuracy in identifying varying time moving force as well as constant moving force. This method has highly efficiency and appropriate to applying in engineering.

A generally applicable 3D truck ride simulation with coupled rigid bodies and finite element models

The aim of this paper is to present a three dimensional ride simulation for a medium weight truck equipped with two axles. The dynamics of both a rigid body and a flexible frame structure are considered in the vehicle model. The road surface irregularities are modelled as stationary Gaussian random excitations taking into account the cross correlation between the vehicle axles and tracks. The free-free modal properties of the frame structure have been calculated using the FEM (finite-element method) considering two different configurations for the vehicle frame. The results from the FE models are incorporated with the general vehicle model to study their effect on the vehicle ride behaviour. The equations of motion of the general vehicle model have been formulated applying the Lagrangian approach and the modal superposition theory. Random responses, including power spectral densities and root mean square values, are calculated and analysed. The results covered 23 response variables including driver, cab and body accelerations and the induced dynamic forces in the cab and main suspension systems and tyres. The results showed that the structure dynamics of the frame strongly affects several response variables. Moreover, it has been found that the present simulation technique is practical, powerful, and economic, and can be commercially used for the assessment of truck dynamics. Conclusions and recommendations for further work are given.

Effect of frame flexibility on the ride vibration of trucks

The aim of the paper is to study the ride vibration behaviour of trucks and in particular the effect of including a flexible frame. The work is a theoretical investigation and concentrates on the ride comfort, cargo protection, structural integrity and vehicle safety. The road surface irregularities are represented as stationary Gaussian random excitations. To analyse the vehicle dynamic responses, the spectral analysis technique has been used and the driver acceleration reponses have been weighted according to the ISO ride comfort techniques. To study the effects of the frame flexibility, the truck frame has been modelled using the FEM (finite element method) and its modal properties have been calculated. An explanation of the modal superposition theory is given and the possibility of incorporating this theory with Lagrange's equations for modelling a truck containing flexible subsystems is discussed and applied. Numerical results are presented for the truck, including power spectral densities and root mean square values of the vehicle dynamic response variables. The results show that there is good agreement with the previous experimental published work and that the modelling technique established is very powerful and economical for the analysis of complex vehicle structures. From comparison of the responses of the rigid and flexible body models it has been found that the frame flexibility strongly affects the accelerations of both the driver and the truck body. Therefore, it is critical that frame flexibility effects are taken into account in the design of primary, cab and engine suspension systems.