Complex eigensolutions of coupled flexural and longitudinal modes in a beam with inclined elastic supports with non-proportional damping

Structure borne vibration and noise in an automobile are often explained by representing the full vehicle as a system of elastically coupled beam structures representing the body, engine cradle and body subframe where the engine is often connected to the chassis via inclined viscoelastic supports. To understand more clearly the interactions between a beam structure and isolators, this article examines the flexural and longitudinal motions in an elastic beam with intentionally inclined mounts (viscoelastic end supports). A new analytical solution is derived for the boundary coupled Euler beam and wave equations resulting in complex eigensolutions. This system is demonstrated to be self-adjoint when the support stiffness matrices are symmetric; thus, the modal analysis is used to decouple the equations of motion and solve for the steady state, damped harmonic response. Experimental validation and computational verifications confirm the validity of the proposed formulation. New and interesting phenomena are presented including coupled rigid motions, modal properties for ideal angled roller boundaries, and relationships between coupling and system modal loss factors. The ideal roller boundary conditions when inclined are seen as a limiting case of coupled longitudinal and flexural motions. In particular, the coupled rigid body motions illustrate the influence of support stiffness coupling on the eigenvalues and eigenfunctions. The relative modal strain energy concept is used to distinguish the contribution of longitudinal and flexural deformation modes. Since the beam is assumed to be undamped, the system damping is derived from the viscoelastic supports. The support damping (for a given loss factor) is shown to be redistributed between the system modes due to the inclined coupling mechanisms. Finally, this article provides valuable insight by highlighting some technical issues a real-life designer faces when balancing modeling assumptions such as rigid or elastic formulations, proportional or non-proportional damping, and coupling terms in multidimensional joint properties.     

Guided wave propagation and mode differentiation in hollow cylinders with viscoelastic coatings

Guided wave propagation theories have been widely explored for about one century. Earlier theories on single-layer elastic hollow cylinders have been very beneficial for practical nondestructive testing on piping and tubing systems. Guided wave flexural (nonaxisymmetric) modes in cylinders can be generated by a partial source loading or any nonaxisymmetric discontinuity. They are especially important for guided wave mode control and defect analysis. Previous investigations on guided wave propagation in multilayered hollow cylindrical structures mostly concentrate on the axisymmetric wave mode characteristics. In this paper, the problem of guided wave propagation in free hollow cylinders with viscoelastic coatings is solved by a semianalytical finite element (SAFE) method. Guided wave dispersion curves and attenuation characteristics for both axisymmetric and flexural modes are presented. Due to the fact that dispersion curve modes obtained from SAFE calculations are difficult to differentiate from each other, a mode sorting method is established to distinguish modes by their orthogonality. Theoretical proof of the orthogonality between guided wave modes in a viscoelastic coated hollow cylinder is provided. Wave structures are also calculated and discussed in view of wave mechanics in multilayered cylindrical structures containing viscoelastic materials.     

Experimental modal substructuring to estimate fixed-base modes from tests on a flexible fixture

Fixed boundary conditions are often difficult if not impossible to simulate experimentally, but they are important to consider in many applications. In principle, modal substructuring or impedance coupling approaches can be used to predict the fixed base modes of a system from tests where the system has some other boundary condition if the motion at the connection point can be measured, but this approach can be highly sensitive to imperfections in the experimental measurements. This work presents two alternatives that reduce the sensitivity to experimental errors, capitalizing on recent works where additional degrees of freedom are used to improve the robustness of substructure uncoupling. The system of interest is tested while mounted on a stiff fixture, where some modes of the fixture inevitably interact with those of the system of interest. The modes of the system–fixture assembly are extracted using a modal test and then a modal substructuring approach is used to apply constraints to eliminate the motion of the fixture. Two types of constraints are proposed, one based on the modes of the fixture and the other on a singular value decomposition of the fixture motion that was observed during the test. Neither approach requires an estimate of the displacements or rotations at the points where the system of interest is connected to the fixture. The methods are validated by applying them to experimental measurements from a simple test system meant to mimic a flexible satellite on a stiff shaker table. A finite element model of the subcomponents was also created and the method is applied to its modes in order to separate the effects of measurement errors and modal truncation. The proposed method produces excellent predictions of the first several modes of the fixed-base structure, so long as modal truncation is minimized. The proposed approach is also applied to experimental measurements from a wind turbine blade mounted in a stiff frame and found to produce reasonable results.     

Localized modes at adsorbed atoms studied by the method of response functions

The problem of localized modes at atoms adsorbed on surfaces of semi-infinite solids is studied as a special case of the theory of linear response functions. The only dynamical properties of the host lattice model which are required are the Laplace transforms of some of the “response functions” of the lattice. Using special cases of both one-dimensional and three-dimensional systems as illustrations, it is shown how the modal frequency and modal vector (that is, all modal properties) may be determined for each localized mode. Some results from one-dimensional and three-dimensional systems are compared, and an interesting similarity is uncovered. The construction of parametrized analytic forms for the response functions is discussed, and it is concluded that, in the present context, such forms must contain only terms of odd order in time, rendering currently-available parametrized forms useless in this context.     

Modified independent modal space control of m.d.o.f. systems

The performance of the independent modal space control (IMSC) algorithm for structural vibration control is examined in this paper. Both the theoretical analysis and numerical simulation show that, for a multi-degree-of-freedom system, the modal control forces may increase the contributions of the vibration of higher modes (uncontrolled modes) to the system response if the IMSC algorithm is used to design a structural control system. Therefore, the responses of the controlled structure may be underestimated if the effects of control forces on the higher modes are not considered in the response analysis. A new control algorithm—modified independent modal space control (MIMSC) algorithm is proposed in this paper for eliminating the effect of modal control force on the uncontrolled modes. Numerical example shows that the structural responses can be effectively reduced when control system design is carried out based on the proposed algorithm. By comparing the simulated results obtained by the IMSC and MIMSC algorithms, it is found that, in order to achieve the same control objective, the proposed algorithm is more effective than IMSC since the modal control forces do not have any effect on the uncontrolled modes. In order to verify the effectiveness of the proposed algorithm, a practical example—active control design of UCLA Math-Science Building is presented and discussed.     

A new method for the determination of damping in cocured composite laminates with embedded viscoelastic layer

A new method for the determination of damping in cocured composite laminates with embedded viscoelastic layer is developed based on mode superposition and modal strain energy method. The calculated damping value is not modal loss factor but a combination of damping from the contributing modes. The dynamic mechanical properties of the viscoelastic material cocured with composites were investigated and were substituted in the present method for calculating the damping in cocured composites. The analytical results were compared with the experimental results by dynamic mechanical thermal analysis (DMTA). The results demonstrate a good agreement between analytical and experimental results. This work provides a means for the study of damping in this structure with different environment temperature and excited frequency.     

Acoustic-structural interaction analysis using the component mode synthesis method

The main objective is to analyse the effects of the imposition of common velocity on the acoustic-structural interface via the Component Mode Synthesis Method (CMS). The original contribution of this analytical study is to show the importance of including kinematic compatibility on the structural-acoustic problem. Some background information about the method is provided as a basis for assisting the understanding of the process. Following this, the formulation of the structural-acoustic problem in terms of ‘components’ is described. The results obtained using CMS are compared to those obtained using both a one dimensional wave approach and standard modal analysis. Finally, conclusions are drawn based on the analysis of the results and the extension to three-dimensional acoustic systems discussed.     

Modal correlation approaches for general second-order systems: Matching mode pairs and an application to Campbell diagrams

Modal correlation is well developed for undamped and proportionally damped vibrating systems. It is less well defined for generally damped linear systems. This paper addresses the fundamental problem of comparing two general second-order linear systems through modal information. It considers precisely the problem of how to achieve matching of modes (mode pairs).There are several possible motivations for modal correlation of which the most important is probably the model updating application. In that application, one set of modes derives from a numerical model and the other from measured data. This paper focuses mainly on a different application—constructing Campbell diagrams for rotating machines. There are two significant differences here: (a) the two sets of modes being compared at any one time are from the same numerical model but for different spin speeds and (b) there is generally a strong distinction between the left and right modes of the system. Without some modal correlation approach, the Campbell diagram is constructed simply as a set of points on the frequency-speed graph. With modal correlation, the eigenvalue problem can be solved at far fewer speeds and the points can be joined meaningfully.A dimensionless (n×n) modal-matching array is produced whose entries indicate which pairs of modes from the first system best correlate with any particular pair of modes from the second system. The presented work is motivated mainly by the application of developing Campbell diagrams for rotating machines by means which are more effective than simply plotting a large set of discrete points. Wider applications of this paper include model updating procedures where mode pairs must be matched initially to ensure convergence towards the exact system.     

Finite element analysis and experimental study on dynamic properties of a composite beam with viscoelastic damping

This paper investigates the frequency dependent viscoelastic dynamics of a multifunctional composite structure from finite element analysis and experimental validation. The frequency-dependent behavior of the stiffness and damping of a viscoelastic material directly affects the system's modal frequencies and damping, and results in complex vibration modes and differences in the relative phase of vibration. A second order three parameter Golla–Hughes–McTavish (GHM) method and a second order three fields Anelastic Displacement Fields (ADF) approach are used to implement the viscoelastic material model, enabling the straightforward development of time domain and frequency domain finite elements, and describing the frequency dependent viscoelastic behavior. Considering the parameter identification a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Agreement between the curve fits using both the GHM and ADF and experiment is within 0.001 percent error. Continuing efforts are addressing the material modulus comparison of the GHM and the ADF model. There may be a theoretical difference between viscoelastic degrees of freedom at nodes and elements, but their numerical results are very close to each other in the specific frequency range of interest. With identified model parameters, numerical simulation is carried out to predict the damping behavior in its first two vibration modes. The experimental testing on the layered composite beam validates the numerical predication. Experimental results also show that elastic modulus measured from dynamic response yields more accurate results than static measurement, such as tensile testing, especially for elastomers.     

A parametric multi-body section model for modal interactions of cable-supported bridges

A parametric section model is formulated to synthetically describe the geometrically nonlinear dynamics of cable-stayed and suspended bridges through a planar elastic multi-body system. The four-degrees-of-freedom model accounts for both the flexo-torsional motion of the bridge deck and for the transversal motion of a pair of hangers or stay cables. After linearization around the pre-stressed static equilibrium configuration, the coupled equations of motion governing the global deck dynamics and the local cable motion are obtained. A multi-parameter perturbation method is employed to solve the modal problem of internally resonant systems. The perturbation-based modal solution furnishes, first, explicit formulae for the parameter combinations which realize the internal resonance conditions and, second, asymptotic approximations of the resonant frequencies and modes. Attention is focused on the triple internal resonance among a global torsional mode of the deck and two local modes of the cables, due to the relevant geometric coupling which maximizes the modal interaction. The asymptotic approximation of the modal solution is found to finely describe the multiple veering phenomenon which involves the three frequency loci under small variation of the most significant mechanical parameters, including terms of structural coupling or disorder. Moreover, the veering amplitude between any two of the three frequency loci can be expressed as an explicit parametric function. Finally, the disorder is recognized as the only parameter governing a complex phenomenon of triple modal hybridization involving all the resonant modes. The entire hybridization process is successfully described by an energy-based localization factor, presented in a new perturbation-based form, valid for internally resonant system.     

The decoupling of damped linear systems in free or forced vibration

The purpose of this paper is to extend classical modal analysis to decouple any viscously damped linear system in non-oscillatory free vibration or in forced vibration. Based upon an exposition of how exponential decay in a system can be regarded as imaginary oscillations, the concept of damped modes of imaginary vibration is introduced. By phase synchronization of these real and physically excitable modes, a time-varying transformation is constructed to decouple non-oscillatory free vibration. When time drifts caused by viscous damping and by external excitation are both accounted for, a time-varying decoupling transformation for forced vibration is derived. The decoupling procedure devised herein reduces to classical modal analysis for systems that are undamped or classically damped. This paper constitutes the second and final part of a solution to the “classical decoupling problem.” Together with an earlier paper, a general methodology that requires only the solution of a quadratic eigenvalue problem is developed to decouple any damped linear system.     

Optical fibre excitation by partially coherent sources

Sources for optical fibre excitation have previously been assumed to be either totally coherent or incoherent. This paper formulates the modal excitation problem for partially coherent sources. The modal excitation coefficients are given in terms of the source complex degree of coherence and numerical results and simple analytical expressions appropriate to multimode step index fibres are presented. The assumption of equal modal power when excitation is by a very incoherent source is examined and the following simple criterion developed: for highly incoherent sources, modes with eigenvaluesU < the reciprocal of the coherence length, measured in units of fibre radius, are approximately equally excited, while the remaining modes carry little power.     

Finite element reduced order model for noise and vibration reduction of double sandwich panels using shunted piezoelectric patches

This work concerns the control of sound transmission through double laminated panels with viscoelastic core using semi-passive piezoelectric shunt technique. More specifically, the system consists of two laminated walls, each one composed of three layers and called sandwich panel with an air cavity in between. The external sandwich panel has a surface-mounted piezoelectric patches. The piezoelectric elements, connected with resonant shunt circuits, are used for the vibration damping of some specific resonance frequencies of the coupled system. Firstly, a finite element formulation of the fully coupled visco-electro-mechanical-acoustic system is presented. This formulation takes into account the frequency dependence of the viscoelastic material. A modal reduction approach is then proposed to solve the problem at a lower cost. In the proposed technique, the coupled system is solved by projecting the mechanical displacement unknown on a truncated basis composed by the first real short-circuit structural normal modes and the pressure unknown on a truncated basis composed by the first acoustic modes with rigid boundaries conditions. The few initial electrical unknowns are kept in the reduced system. A static correction is also introduced in order to take into account the effect of higher modes. Various results are presented in order to validate and illustrate the efficiency of the proposed finite element reduced order formulation.     

A modal coupling procedure to improve residual modal effects based on experimentally generated data

In this paper a modal coupling procedure is developed to improve residual modal effects based on experimentally generated data. While showing that Craig–Chang's proposed approximation function for higher frequency modes does not compare well with the exact function, a new procedure is developed in order to re-express the dynamic residuals. For this purpose, all frequency dependent terms of the Maclaurin series in Craig–Chang's proposed method are replaced by a general compatible function, which includes the neglected higher order terms. However, this function no longer appears in final equation of motion. Results from case studies for the discrete, continuous and experimental models are presented to verify the procedure and to show the capabilities of the proposed method.     

Damped vibration analysis of an elastically connected complex double-string system

The main purpose of the present paper is to consider theoretically damped transverse vibrations of an elastically connected double-string system. This system is treated as two viscoelastic strings with a Kelvin-Voigt viscoelastic layer between them. A theoretical analysis has been made for a simplified model of the system, in which assumed physical parameters make it possible to decouple the governing equations of motion by introducing the principal co-ordinates. Applying the method of separation of variables and the modal expansion method, exact analytical solutions for damped free and forced responses of the system subjected to arbitrarily distributed transverse continuous loads are determined in the case of arbitrary magnitude of linear viscous damping. It is important to note that the solutions obtained are explicitly expressed in terms of parameters characterizing the physical properties of the system under discussion. For the sake of completeness of the analysis, solutions for undamped free and forced vibrations are also formulated.     

Complete modal decomposition for optical waveguides

Virtually all electromagnetic waveguiding structures support a multiplicity of modes. Nevertheless, to date, an experimental method for unique decomposition of the fields in terms of the component eigenmodes has not been realized. The fundamental problem is that all current attempts of modal decomposition do not yield phase information. Here we introduce a noninterferometric approach to achieve modal decomposition of the fields at the output of a general waveguiding structure. The technique utilizes a mapping of the two-dimensional field distribution onto the one-dimensional space of waveguide eigenmodes, together with a phase-retrieval algorithm to extract the amplitudes and phases of all the guided vectorial modes. Experimental validation is provided by using this approach to examine the interactions of 16 modes in a hollow-core photonic-band gap fiber.     

System identification-guided basis selection for reduced-order nonlinear response analysis

Reduced-order nonlinear simulation is often times the only computationally efficient means of calculating the extended time response of large and complex structures under severe dynamic loading. This is because the structure may respond in a geometrically nonlinear manner, making the computational expense of direct numerical integration in physical degrees of freedom prohibitive. As for any type of modal reduction scheme, the quality of the reduced-order solution is dictated by the modal basis selection. The techniques for modal basis selection currently employed for nonlinear simulation are ad hoc and are strongly influenced by the analyst's subjective judgment. This work develops a reliable and rigorous procedure through which an efficient modal basis can be chosen. The method employs proper orthogonal decomposition to identify nonlinear system dynamics, and the modal assurance criterion to relate proper orthogonal modes to the normal modes that are eventually used as the basis functions. The method is successfully applied to the analysis of a planar beam and a shallow arch over a wide range of nonlinear dynamic response regimes. The error associated with the reduced-order simulation is quantified and related to the computational cost.     

Stationary response of combined linear dynamic systems to stationary excitation

The stationary response of a broad class of combined linear systems to stationary random excitation is determined by the normal mode method. The systems are characterized by a viscously damped simple beam (or string, membrane, thin plate or shell, etc.) connected at discrete points to a multiplicity of viscously damped linear oscillators and/or masses. The solution of the free vibration problem by way of Green functions and the deterministic forced vibration problem by modal analysis for both proportional and non-proportional damping is reviewed. The orthogonality relation for the natural modes of vibration is used to derive a unique relationship between the cross-spectral density functions of the applied forces and the cross-spectral density functions of the generalized forces. Finally, the response spectral density functions and the mean square responses of the beam and oscillators are derived in closed form, exact for the proportionally damped system and approximate for the non-proportionally damped system.     

Truncating cylindrical wave modes in two-dimensional multiple scattering

When analyzing the two-dimensional multiple scattering of electromagnetic waves by cylinders, the incident, scattered, and transmitted fields need to be represented by infinite sums of cylindrical wave modes. These infinite sums need to be truncated to a finite limit in order to calculate the scattering matrix. The accuracy of the scattered field representation and the stability of the matrix inversion are both critically dependent on the truncation limit. The parameters involved in the scattering are analyzed to determine their effect on the upper and lower bounds of an appropriate modal truncation.