Damped Beam Research Articles

Nuclearity of Hankel operators for ultradifferentiable control systems

Nuclearity of the Hankel operator is a known sufficient condition for convergence of Lyapunov-balanced truncations. We show how a previous result on nuclearity of Hankel operators of systems with an analytic semigroup can be extended to systems with a semigroup of class D p with p ≥ 1 (the case p = 1 being the analytic case). For semigroups that are generated by a Dunford–Schwartz spectral operator we prove that being of class D p is equivalent to being (Gevrey) ultradifferentiable of order p . We illustrate how for certain partial differential equations our results lead to an easy way of showing nuclearity of the Hankel operator for a wide range of control and observation operators by considering several examples of damped beams.

Optimum Design of Segmented Passive-Constrained Layer Damping Treatment Through Genetic Algorithms

The optimization of geometry and material properties of Passive Constrained Layer Damping (PCLD) treatments has been the subject of many research efforts in recent years. In this study, a genetic algorithm is used to investigate the relationship of the viscoelastic layer thickness, the constraining layer thickness, and the number of cuts initiated in the PCLD treatment as they all relate to optimum damping of beams. Genetic algorithms mimic the biological evolution process where potential solutions compete for survival based on their relative fitness. As a measure of the fitness of the PCLD treatment, a finite element model is used to compute the loss factor for the first mode of vibration with the capability of segmenting the treatment. It is found that, while the loss factor increases asymptotically with the increase in the viscoelastic layer thickness, an optimum constraining layer thickness for each viscoelastic layer thickness exists. The number of cuts in the segmented treatment tends to decrease with increasing thickness of the damping treatment. Furthermore, it was found that the optimum constraining layer thickness decreases with increasing the thickness of the viscoelastic layer. Physical insight is shed on all observations made which makes future optimization of PCLD treatment more focused in terms of criteria and goals.

Enhanced damping of a cantilever beam by axial parametric excitation

A uniform cantilever beam under the effect of a time-periodic axial force is investigated. The beam structure is discretized by a finite-element approach. The linearised equations of motion describing the planar bending vibrations of the beam structure lead to a system with time-periodic stiffness coefficients. The stability of the system is investigated by a numerical method based on Floquet’s theorem and an analytical approach resulting from a first-order perturbation. It is demonstrated that the parametrically excited beam structure exhibits enhanced damping properties, when excited near a specific parametric combination resonance frequency. A certain level of the forcing amplitude has to be exceeded to achieve the damping effect. Upon exceeding this value, the additional artificial damping provided to the beam is significant and works best for suppression of vibrations of the first vibrational mode of the cantilever beam.

Structural Damage Detection Using Energy Flow Models

The presence of a crack in a structure modifies the energy dissipation pattern. As a consequence, damaged structures can present high localized damping. Experimental tests have revealed that crack nucleation and growth increase structural damping which makes this phenomenon useful as a damage locator. This paper examines the energy flow patterns caused by localized damping in rods, beams and plates using the Energy Finite Element Method (EFEM), the Spectral Element Method (SEM) and the Energy Spectral Element Method (ESEM) in order to detect and locate damage. The analyses are performed at high frequencies, where any localized structural change has a strong influence in the structural response. Simulated results for damage detection in rods, beams, and their couplings calculated by each method and using the element loss factor variation to model the damage, are presented and compared. Results for a simple thin plate calculated with EFEM are also discussed.

Evaluation of dynamic properties of extra light weight concrete sandwich beams reinforced with CFRP

Analytical and experimental investigation on dynamic properties of extra lightweight concrete sandwich beams reinforced with various lay ups of carbon reinforced epoxy polymer composites (CFRP) are discussed. The lightweight concrete used in the core of the sandwich beams was made up of extra lightweight aggregate, Lica. The density of concrete was half of that of the ordinary concrete and its compressive strength was about <TEX>$100Kg/cm^2$</TEX>. Two extra lightweight unreinforced (control) beams and six extra lightweight sandwich beams with various lay ups of CFRP were clamped in one end and tested under an impact load. The dimension of the beams without considering any reinforcement was 20 cm <TEX>${\times}$</TEX> 10 cm <TEX>${\times}$</TEX> 1.4 m. These were selected to ensure that the effect of shear during the bending test would be minimized. Three other beams, made up of ordinary concrete reinforced with steel bars, were tested in the same conditions. For measuring the damping capacity of sandwich beams three methods, Logarithmic Decrement Analysis (LDA), Hilbert Transform Analysis (HTA) and Moving Block Analysis (MBA) were applied. The first two methods are in time domain and the last one is in frequency domain. A comparison between the damping capacity of the beams obtained from all three methods, shows that the damping capacity of the extra lightweight concrete decreases by adding the composite reinforced layers to the upper and lower sides of the beams, and becomes most similar to the damping of the ordinary beams. Also the results show that the stiffness of the extra lightweight concrete beams increases by adding the composite reinforced layer to their both sides and become similar to the ordinary beams.

Dynamic analysis of slip damping in clamped layered beams with non-uniform pressure distribution at the interface

Slip damping is a mechanism exploited for dissipating noise and vibration energy in aerodynamic and machine structures. Such slip in layered structures can be simulated by applying pressure to hold the members together at the interface. However, while most analyses of the mechanism assume an environment of uniform pressure at the interface, experiments to date have confirmed that this is rarely the case. There have been recent attempts to relax the restriction of uniform interface pressure to allow for more realistic pressure profiles that are encountered in practice. However, such works have mostly been limited to static loading for which it has been established that the interfacial pressure gradient does play a dominant role in modulating the level of energy dissipation. This paper is an attempt to extend such analyses to account for cases of realistic dynamic loading that drive such structural vibration in the first instance. In particular, it is shown that under dynamic loads, frequency variation more than non-uniformity in the interface pressure can have significant effect on both the energy dissipation and the logarithmic damping decrement associated with the mechanism of slip damping in such layered structures.

Free-Response Simulation via the Proper Orthogonal Decomposition

The proper orthogonal decomposition is applied to the response of linear and nonlinear beam models to construct a reduced-order model for predicting the response to various initial conditions without requiring the equations of motion for the structure. The proper orthogonal decomposition is interpreted as a modal sum, and the structural response to an initial condition is expressed as a weighted summation of proper orthogonal modes and corresponding proper orthogonal coordinate histories. A method for calculating the weights of each mode in the response to an altered initial condition is presented. This method is applied to predict the responses of a linear undamped beam model and a damped beam model with a nonlinear spring to various initial displacement and velocity profiles. The results obtained are compared with those from respective finite element models for each beam.

Dynamic modeling of active constrained layer damping of composite beam under thermal environment

The present work deals with the active vibration control of composite beam under thermal environment using finite element method. The effect of fiber orientation and temperature on natural frequency and damping of the system are investigated for different boundary conditions. The temperature-dependent properties of viscoelastic core are considered. Results are presented for frequencies and loss factor with different core material and core thickness. Characteristics of loss factor and frequency near the buckling temperature are discussed, and the effect of core thickness on buckling temperature is also investigated.

Non-local finite element analysis of damped beams

Non-local viscoelastic beam models are used to analyse the dynamics of beams with different boundary conditions using the finite element method. Unlike local damping models the internal force of the non-local model is obtained as weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure. In the finite element analysis, the interpolating shape functions of the element displacement field are identical to those of standard two-node beam elements. However, for non-local damping, nodes remote from the element do have an effect on the energy expressions, and hence on the damping matrix. The expressions of these direct and cross damping matrices may be obtained explicitly for some common spatial kernel functions and Euler–Bernoulli beam theory. Alternatively numerical integration may be applied to obtain solutions. Examples are given where the eigenvalues are compared to the exact solution for a pinned–pinned beam to demonstrate the convergence of the finite element method. The results for beams with other boundary conditions are used to demonstrate the versatility of the finite element technique.

The measurement of the loss factors of beams and plates with constrained and unconstrained damping layers: A critical assessment

Methods of measuring the loss factors of heavily damped beams and plates damped by uniform layers of visco-elastic damping are reviewed and compared. Deducing loss factors from measured spatial decay rates of free flexural waves in long beams is shown to be valid only for wave motion governed by the fourth-order flexural wave equation, so the method can only be used for unconstrained layer configurations. The usual approximation made in this method limits its accuracy and is examined. The flexural wave equation for three-layered sandwich beams and plates with constrained damping layers is of sixth order, so the spatial decay rate method is inapplicable for deducing loss factors. This is shown by both analysis and computer experiment. Reliable methods of damping measurement on finite damped sandwich beams are then investigated by computer experiment. Frequency-response functions for three-layered sandwich beams are computed for free–free and encastré beams, centrally excited. Loss factors of the fundamental symmetric modes are deduced from this computed response data as though it were experimental data. They are deduced by established methods such as the ‘half-power-point’ method, detailed analysis of Nyquist diagrams and the energy input method. Most of the deduced loss factors agree closely with the loss factors of the theoretical damped normal modes.

Damping analysis of beams covered with multiple PCLD patches

An analytical approach for vibration damping analysis of beams with partial passive constrained layer damping (PCLD) treatments is extended to the case where the structure is covered by multiple patches. The energy-based approach and assumed-mode method are used to derive the equations governing the vibration displacement responses of the system, which is characterized in terms of the identical transverse displacement of all layers, and the longitudinal displacements of the base beam and the constraining layers. Furthermore, the longitudinal displacements of the base beam and constraining layer are independent of each other, which is in contrast to Kerwin's weak core assumption. The direct frequency response method is employed to determine the modal loss factors of the damped beam in a frequency range of interest. A convergence study is made on the damped beam to determine appropriate numbers of the mode functions for frequency response function (FRF) calculations. Parametric studies are performed with the established analytical approach and the determined numbers of mode functions to study the effects of dividing a PCLD patch into two pieces in different length ratios and also the spacing between the patches on the damping performance of a simply supported beam.

Multiple doubly periodic solutions of semi-linear damped beam equations

The purpose of this work is to investigate the weakened Ambrosetti–Prodi type multiplicity results for weak doubly periodic solutions of damped beam equations. By using the topological degree theory, the author obtains a result which is similar to the result for damped wave equations in the literature.

Prediction of vibration of rotating damped beams with arbitrary pretwist

A solution procedure for the bending–bending vibration of a rotating damped beam with arbitrary pretwist and an elastically restrained root is derived. The viscous damping is assumed to be proportional to the distributed mass. The general complex system is divided into two subsystems. The physical meanings of the subsystems are studied. The exact complex frequency relations between two viscously damped beams with arbitrary pretwist and elastic root are revealed. The underdamping, critical damping and overdamping systems are analyzed. Moreover, the influence of the parameters on the decay rate, the natural frequencies, the critical damping, and the phenomenon of divergence instability are investigated.

Modal Analysis of Nonviscously Damped Beams

Linear dynamics of Euler–Bernoulli beams with nonviscous nonlocal damping is considered. It is assumed that the damping force at a given point in the beam depends on the past history of velocities at different points via convolution integrals over exponentially decaying kernel functions. Conventional viscous and viscoelastic damping models can be obtained as special cases of this general damping model. The equation of motion of the beam with such a general damping model results in a linear partial integro-differential equation. Exact closed-form equations of the natural frequencies and mode shapes of the beam are derived. Numerical examples are provided to illustrate the new results.

Integral equation formulation and analysis of the dynamic stability of damped beams subjected to subtangential follower forces

This paper presents a mathematical model based on integral equations for numerical investigations of stability analyses of damped beams subjected to subtangential follower forces. A mathematical formulation based on Euler–Bernoulli beam theory is presented for beams with variable cross sections on a viscoelastic foundation and subjected to lateral excitation, conservative and non-conservative axial loads. Using the boundary element method and radial basis functions, the equation of motion is reduced to an algebro-differential system related to internal and boundary unknowns. Generalized formulations for the deflection, the slope, the moment and the shear force are presented. The free vibration of loaded beams is formulated in a compact matrix form and all necessary matrices are explicitly given. The load–frequency dependence is extensively investigated for various parameters of non-conservative loads, of internal and viscous dampings and for various positions of the concentrated foundation. For an undamped beam, a dynamic stability analysis is illustrated numerically based on the coalescence criterion. The flutter load and instability regions with respect to various parameters are identified. The effects of internal and viscous dampings on the critical flutter load are examined separately and relative effects are evaluated. The dynamic responses, before, near and after the flutter are investigated. A simple and quite general methodological approach is presented. Comprehensive numerical tests for flutter analysis are reported and discussed.

Space-dependent damping for reducing vibration of continuous systems

Structural vibration in continuous systems is a common source of problems for many mechanical systems. Much research has considered ways to reduce the vibration in continuous systems. Some research uses passive techniques that are simple to implement, whereas other methods implement active control that can be quite complex. The research presented here investigates the effect space-dependent damping has on the vibration of continuous systems. Specifically, the work at hand will focus on the flexural vibration of beams. In a distributed damping beam system, the damping is variable as a function of position along the length of the beam. Such a system could be physically realized through the use of magneto-rheological or electro-rheological fluids. Along its length, a beam could be divided into several isolated sections containing such a fluid. By controlling the magnetic or electric field across each section of the beam, the damping would vary along the length of the beam. The work presented here optimizes the damping as a function of beam position to reduce the broadband frequency response of the beam. A number of support conditions for the vibrating beam are considered.

Optimal time-delayed open/closed-loop control of a damped beam

Vibration suppression of a structurally damped beam via open and closed-loop controls is investigated. In doing so, minimization of a weighted, energy-based performance measure leads to closed-form expressions for optimal open-loop parameters in terms of feedback gains. The control problem is then completed by optimizing these feedback parameters numerically. Methods to determine optimal terminal time of control are also presented. Furthermore, techniques are established to optimize the response and performance of a system experiencing a known feedback time delay. Different types of optimization are proposed and, for each one, the trade-off between performance and computational efficiency is fully discussed.

Equivalent stiffness and damping of sandwich piezoelectric beams submitted to activecontrol

In this paper, a modal approach is realized to define the damping properties ofpiezoelectric/elastic/piezoelectric beams. This method is based on the classical laminatedbeam theory and some simple assumptions about the electric fields. This leads to anelectromechanical beam constitutive law. The piezoelectric layers play the role of sensorand actuator and two feedback control laws are considered. The equivalent stiffness,eigenfrequencies and loss factors of the whole system beam/control device are obtained.

Thermoelastic Damping in Nanomechanical Resonators with Finite Wave Speeds

ABSTRACT The operation of micro-/nanobeams vibrating at very high frequencies, such as encountered in micro-/nanoelectromechanical systems (MEMS/NEMS), hinges on the minimization of intrinsic material losses. We study the associated thermoelastic damping in such beams from the standpoint of a generalized theory of thermoelasticity with one relaxation time. Some of our results relate to: (i) the cooling (instead of heating) in the compressed surface of the beam; (ii) the existence of not one damping peak appearing in the classical theory, but many peaks, with a decreasing amplitude as the frequency tends to infinity; (iii) the relevance of thermoelasticity with finite wave speeds for frequencies on the order of 1012 Hz.

Vibrations of stretched damped beams under non-ideal boundary conditions

A simply supported damped Euler-Bernoulli beam with immovable end conditions are considered. The concept of non-ideal boundary conditions is applied to the beam problem. In accordance, the boundaries are assumed to allow small deflections and moments. Approximate analytical solution of the problem is found using the method of multiple scales, a perturbation technique.