Damped Beam Research Articles

Study of damping in composite beams

The vibrations produced in the structures or components are the primary source of problems in machine tools, aircraft and automobile structures. Vibrations can be overcome by introducing damping in these structures. A recent method to improve the damping ability of structures is by fabricating these with composites in order to enhance their service life.. In addition, composites are found to have superior strength, stiffness, improved corrosion and fatigue resistance.. The damping of FR composites depends on the structure, diameter and orientation of fiber in matrix. Damping at the fiber-matrix interface dissipates a significant amount of energy. In the current investigation, damping of structural composite beams has been studied to estimate the damping ratios and natural frequencies. The various composite beams used are Glass Fibre Reinforced Epoxy (GFE), Glass Fibre Reinforced Polyester (GFP) and Carbon Black Filled Epoxy (CBFE). Logarithmic decrement method is used to evaluate the damping parameters from the obtained amplitude versus time plots. The damping ratio and natural frequency values are found to be 0.2281 and 1664.94 rad/s respectively for CBFE. CBFE has highest damping property as compared to GFE and GFP.

Proposal of a Viscous Model for Nonviscously Damped Beams Based on Fractional Derivatives

Viscoelastic materials are widely used in structural dynamics for the control of the vibrations and energy dissipation. They are characterized by damping forces that depend on the history of the velocity response via hereditary functions involved in convolution integrals, leading to a frequency‐dependent damping matrix. In this paper, one‐dimensional beam structures with viscoelastic materials based on fractional derivatives are considered. In this work, the construction of a new equivalent viscous system with fictitious parameters but capable of reproducing the response of the viscoelastic original one with acceptable accuracy is proposed. This allows us to take advantage of the well‐known available numerical tools for viscous systems and use them to find response of viscoelastic structures. The process requires the numerical computation of complex frequencies. The new fictitious viscous parameters are found to be matching the information provided by the frequency response functions. New mass, damping, and stiffness matrices are found, which in addition have the property of proportionality, so they become diagonal in the modal space. The theoretical results are contrasted with two different numerical examples.

A generalized model for thermoelastic damping in beams with mid-plane stretching nonlinearity

This work presents a general model to accurately capture thermoelastic damping in beams with mid-plane stretching nonlinearity. The model consists of a nonlinear Timoshenko beam model coupled with a two-dimensional heat conduction model. The heat conduction model accounts for heat transfer in both the transverse and the axial directions of the beam. Numerical solutions are obtained by discretizing the beam using spectral element method (SEM). Using the proposed model, the thermoelastic damping (QTED) quality factor is obtained for beams actuated by electrostatic force using the eigenfrequency analysis and the energy approach. The eigenfrequency analysis conducted on the beams under the electrostatic actuation showed significant difference from the results published in literature. A detailed numerical analysis is performed in order to understand the discrepancy. It is shown that the discrepancy primarily occurs due to the differences in the methods to couple the displacement and thermal fields, with the model presented in the paper being the more general one.

Heating of heavy plasma species by damping electron beam in beam-generated plasma

The heating of heavy species (both ions and neutrals) in the beam-generated plasma by damping electron beam is analyzed using a self-consistent one-dimensional Particle-in-Cell Monte Carlo collisions model. It is observed that the damping of the electron beam leads to the excitation of a wide spectrum of electrostatic waves. These waves lead to the heating not only of the thermal plasma electrons but also of the plasma ions. The first less efficient mechanism of the ion heating is the ponderomotive force due to the generation of non-homogeneous high-frequency electric field which is excited by the damping electron beam. The second more efficient mechanism of the ion heating obtained in the simulations is the acceleration by slow plasma waves. This mechanism is responsible for the acceleration of ions to the velocities few times higher than the ion thermal velocity.

AN EXISTENCE PRINCIPLE OF OPTIMAL ENERGY CONTROL FOR A DAMPED SANDWICH BEAM

A damped sandwich beam system described by partial differentialequations with boundary conditions is investigated in this paper. The systemis transferred to an abstract evolution equation interms of the system operatorsdefined on appropriate Hilbert spaces. A variable structural control problemfor the system is proposed, and an equivalent control theorem for the systemare obtained by means of semigroup of linear operators. Finally, a simulationresult is presented.

Analysis of smart damping of laminated composite beams using mesh free method

This paper is concerned with the development of mesh free model for the performance analysis of active constrained layered damping (ACLD) treatments on smart laminated composite beams. The overall structure is composed of a substrate laminated composite beam integrated with a viscoelastic layer and a piezoelectric layer attached partially or fully at the top surface of the substrate beam. The piezoelectric layer acts as the active constraining layer of the smart beam and the viscoelastic layer acts as the constrained layer. A layer wise displacement theory has been used to derive the models. Both symmetric cross-ply and antisymmetric angle-ply laminated beams are considered for the numerical analysis. It is observed that ACLD treatment significantly improves the active damping properties of the substrate beam. The numerical results also reveal that the triangular ACLD treatment is more effective than the rectangular ACLD treatment of same thickness and volume for active damping of smart composite beams.

Application of genetic algorithm-support vector regression model to predict damping of cantilever beam with particle damper

The performance of particle damper is strongly nonlinear, and the energy dissipation is derived from a combination of mechanisms including plastic collisions and friction between the particles and the walls and between the particles themselves. An optimized support vector regression model is built to predict the damping ratio of cantilever beam with particle damper. Then, the optimal parameters are adopted to construct the support vector regression models. In addition, genetic algorithm is used to select the optimal variables so as to improve the predictive ability of the models. Cross validation combined with support vector regression is used in this research and is compared with the genetic algorithm-support vector regression method. Genetic algorithm-support vector regression as research object to compare with the combination of cross validation and support vector regression. The experimental results demonstrate that the proposed genetic algorithm-support vector regression model provides better prediction capability. Therefore, the genetic algorithm-support vector regression model is proven to be an effective approach to predict the damping ratio of cantilever beam with particle damper.

Fabrication and vibration characterization of electrically triggered shape memory polymer beams

Shape memory polymers (SMP) exhibit temperature, frequency and strain rate dependent properties which may be manipulated by various types of external stimuli to achieve desirable response characteristics. In recent years, the emphasis has been on designing SMPs which do not require external stimuli (such as a heat source) and have a rapid response time with large homogenous and reversible deformation characteristics. In this research, the fabrication process and dynamic vibration testing of an electrically activated SMP are presented. It is shown that conductive SMP beams can be fabricated to achieve tunable stiffness and damping with a reasonable thermal gradient generated by electrical triggering. This can allow the tuning of a range of frequency bandwidth and damping properties of SMPs for vibration control applications. The experimentation yielded modal properties (natural frequencies and damping) of the SMP beams. These parameters were validated against values obtained from the estimated performance of these beams based on the complex modulus parameters obtained using dynamic mechanical analysis (DMA). For a modest 20 °C temperature range in an epoxy based SMP, a resulting shift of approximately 7% in the natural frequency and 100% change in the damping ratio of a rectangular beam was successfully attained. These results recommend SMPs as being tunable materials that can enhance vibrational performance and expand the operational envelope of structures.

A thermography-based method for fatigue behavior evaluation of coupling beam damper

Under cyclic load, local fatigue damage will occur in the metal damper widely used in the shear wall. This will deteriorate the stiffness of damper and weaken the hysteresis behaviour. The present paper proposed a new and easy method to manufacture kinds of coupling beam dampers. A thermography-based experiment was used to study the energy dissipation and damage accumulation during fatigue process of the metal damper. Based on the temperature variation related to fatigue damage process, the relationship between the plastic deformation and thermal energy dissipation was quantitatively established. Besides, the relationships between the temperature increase to damage accumulation and mechanical load were analyzed systematically.

Damage identification of beam structures using free response shapes obtained by use of a continuously scanning laser Doppler vibrometer system

Spatially dense operating deflection shapes and mode shapes can be rapidly obtained by use of a continuously scanning laser Doppler vibrometer (CSLDV) system, which sweeps its laser spot over a vibrating structure surface. This paper introduces a new type of vibration shapes called a free response shape (FRS) that can be obtained by use of a CSLDV system, and a new damage identification methodology using FRSs is developed for beam structures. An analytical expression of FRSs of a damped beam structure is derived, and FRSs from the analytical expression compare well with those from a finite element model. In the damage identification methodology, a free-response damage index (FRDI) is proposed, and damage regions can be identified near neighborhoods with consistently high values of FRDIs associated with different modes; an auxiliary FRDI is defined to assist identification of the neighborhoods. A FRDI associated with a mode consists of differences between curvatures of FRSs associated with the mode in a number of half-scan periods of a CSLDV system and those from polynomials that fit the FRSs with properly determined orders. A convergence index is proposed to determine the proper order of a polynomial fit. One advantage of the methodology is that the FRDI does not require any baseline information of an undamaged beam structure, if it is geometrically smooth and made of materials that have no stiffness and mass discontinuities. Another advantage is that FRDIs associated with multiple modes can be obtained using free response of a beam structure measured by a CSLDV system in one scan. The number of half-scan periods for calculation of the FRDI associated with a mode can be determined by use of the short-time Fourier transform. The proposed methodology was numerically and experimentally applied to identify damage in beam structures; effects of the scan frequency of a CSLDV system on qualities of obtained FRSs were experimentally investigated.

Coupling dynamic behaviors of spatial flexible beam with weak damping

SummaryAs a typical high‐dimensional nonlinear dynamic problem, the difficulties of the dynamic analysis on the spatial flexible damping beam mostly result from the coupling between the spatial motion and the transverse vibration. Considering the coupling effect and the weak structure damping, the dynamic behaviors of the spatial flexible beam are investigated by a complex structure‐preserving method in this paper. Based on the variational principle, the dynamic model of the spatial flexible damping beam is established, which can be decoupled into two parts approximately, one controls the spatial motion and another mainly controls the transverse vibration. For the first part, the classic fourth‐order Runge–Kutta method can be used to discrete it expediently. For the latter part, based on the generalized multi‐symplectic idea, the approximate symmetric form as well as the generalized multi‐symplectic conservation law is formulated, and a 15‐point scheme equivalent to the Preissmann scheme is constructed. Numerical iterations are performed for six typical initial cases between the two parts to study the dynamic behaviors of the spatial flexible beam. In the numerical experiments, the effects of the damping factor and the initial conditions (including the initial radial velocity and the initial attitude angle) on the dynamic behaviors of the spatial flexible beam are investigated in detail. From the numerical results, it can be concluded that the damping effect on the long‐time dynamic behaviors of the spatial flexible beam could not be neglected even if it is weak; the numerical method proposed in this paper owns the tiny numerical dissipation as well as the excellent long‐time numerical stability. Copyright © 2016 John Wiley & Sons, Ltd.

Optimization Design of Coupling Beam Metal Damper in Shear Wall Structures

The coupling beam damper is a fundamental energy dissipation component in coupling shear wall structures that directly influences the performance of the shear wall. Here, we proposed a two-fold design method that can give better energy dissipation performance and hysteretic behavior to coupling beam dampers. First, we devised four in-plane yielding coupling beam dampers that have different opening types but the same amount of total materials. Then the geometry parameters of each opening type were optimized to yield the maximum hysteretic energy. The search for the optimal parameter set was realized by implementing the Kriging surrogate model which iterates randomly selected input shape parameters and the corresponding hysteretic energy calculated by the infinite element method. By comparing the maximum hysteretic energy in all four opening types, one type that had the highest hysteresis energy was selected as the optimized design. This optimized damper has the advantages of having a simple geometry and a high dissipation energy performance. The proposed method also provided a new framework for the design of in-plane coupling beam dampers.

Model order reduction for dynamic simulation of slender beams undergoing large rotations

In this paper a model order reduction technique for the dynamic simulation of beams undergoing large rotations is presented. The finite element model for the motion of such beams is based on the corotational formulation. The trajectory piecewise linear model order reduction (TPWLMOR) method with second order Krylov subspace is used to obtain the reduced order model from the finite element model. Improvements are suggested to improve the accuracy and the computational efficiency of the TPWLMOR model. Several numerical examples which include forced undamped and damped beams are presented to validate the proposed method.

Experimental evaluation of the effects of reverse cyclic loading rate on the mechanical behavior of reinforced SCC beams

In this paper, the effects of cyclic loading rate on the mechanical behavior of reinforced SCC beams are investigated experimentally. Based on the results of cyclic loading tests of nine beam specimens, hysteresis loops, stiffness and strength degradation, energy dissipation capacity and equivalent viscous damping of SCC beams were analyzed with various rates before and after the yielding limit. Cyclic loading was displacement-controlled and a uniform loading procedure was selected for this test. Test results indicate that SCC beams demonstrate different behavior before and after the yielding zone among the increase of the loading rate.

Propagation of a beam of hot electrons through solar wind plasma with generalized ( r , q ) $(r,q)$ distribution function

The background plasma is assumed to have generalized $(r, q)$ distribution for the electrons in the solar wind. The propagation of a beam of hot electrons through solar wind plasma with generalized $(r,q)$ distribution and the generation of Langmuir waves are simulated using quasilinear equations. It is shown that spectral indices $r$ and $q$ affect the quasilinear dynamics of the beam and Langmuir waves. The damping of beam generated waves increases in $(r,q)$ distributed plasma. As indices $r$ and $q$ increase the system shows quasilinear behavior which is more similar to the Maxwellian distribution function. The value of average velocity of the beam increases in a plasma with $(r, q)$ distribution function and as the values of $r$ and $q$ increase, the average velocity of the beam decreases. It is also shown that the gas-dynamical parameters of the beam are functions of spectral indices $r$ and $q$ . The upper boundary of the plateau, and local velocity spread are increasing functions while the lower boundary and height of plateau are decreasing functions of $r$ and $q$ . The local velocity shows smooth behavior with respect to spectral indices $r$ and $q$ , and for all indices at given time and position has approximately same values.

Air damping models for micro- and nano-mechanical beam resonators in molecular-flow regime

Researches on energy dissipation of micro- and nano-mechanical beam resonators into the surrounding air are classified into two models, air drag force damping of the resonant beam vibrating in free space and squeeze film air damping of the resonant beam vibrating nearby walls. Appropriate application of these two models are crucial to the design of high-Q micro- and nano-mechanical beam resonators. In this paper, we will present a detail analysis on appropriate application range of these two models for evaluating air damping in micro- and nano-mechanical beam resonators. The two models are applied for investigating damping in the cantilevered and bridged beam resonators. The obtained analysis results show the significance difference between the two models for estimating the quality factor of the micro- and nano-mechanical beam resonators. The criterion for the distinction between the two models is derived by using the ratio factor of damping coefficients of micro- and nano-mechanical beam resonators.

Hybrid material and foundation damping of Timoshenko beams

In this paper, vibration suppression of shear deformable beams with hybrid material-foundation viscoelastic damping is studied. The precise behavior of viscoelastic polymeric structures is taken into consideration by Boltzmann superposition integral and utilizing dynamic mechanical analysis (DMA) results while foundation viscoelasticity is adapted by Kelvin-Voigt model. The integro-partial differential equations of motion are derived based on the Timoshenko theory via Hamilton principle. Assessment of the solution procedure is carried out by comparison with elastic Timoshenko and viscoelastic Euler-Bernoulli cases. Transient response, natural frequency and modal loss factor are attained by Fourier transform, weighted residual method and numerical iterative algorithm. Influences of hybrid viscoelastic damping, foundation properties, geometrical parameters and boundary conditions on vibration characteristics and dynamic response are investigated via comprehensive parametric study. Due to the absence of similar results in the literature, this paper is likely to fill a gap in the state of the art of this problem.

Active Vibration Control of Smart Functionally Graded Beams

This work is devoted to examine the performance of the constraining layer of the active constrained layer damping (ACLD) treatment made of the active fiber composites (AFC) materials for vibration control of functionally graded (FG) beams. The task of investigating the performance of the active constrained layer damping (ACLD) treatment has been accomplished to demonstrate the use of AFCs as the materials for distributed actuators. Finite Element (FE) model is developed to describe the open loop and closed loop dynamics of the FG beams integrated with the patches of the ACLD treatment. The closed loop frequency response functions computed by the FE models reveal that the ACLD treatment with its constraining layer composed of AFC material significantly enhances the damping characteristics of the FG beams.The performance of the ACLD treatment when the orientation angle of the piezoelectric fibers of the AFC constraining layer is varied has also been investigated. Such variation of piezoelectric fiber orientation angle significantly affects the controllability of the ACLD treatment for active damping of the FG beams. The investigations carried out here may be useful for further experimental verifications and suggest the potential use of AFC materials for developing new distributed actuators of light-weight smart structures.

Variational principles for self-adjoint operator functions arising from second-order systems

Variational principles are proved for self-adjoint operator functions arising from variational evolution equations of the form \[ \langle\ddot{z}(t),y \rangle + \mathfrak{d}[\dot{z} (t), y] + \mathfrak{a}_0 [z(t),y] = 0. \] Here $\mathfrak{a}_0$ and $\mathfrak{d}$ are densely defined, symmetric and positive sesquilinear forms on a Hilbert space $H$. We associate with the variational evolution equation an equivalent Cauchy problem corresponding to a block operator matrix $\mathcal{A}$, the forms \[ \mathfrak{t}(\lambda)[x,y] := \lambda^2\langle x,y\rangle + \lambda\mathfrak{d}[x,y] + \mathfrak{a}_0[x,y], \] where $\lambda\in \mathbb C$ and $x,y$ are in the domain of the form $\mathfrak{a}_0$, and a corresponding operator family $T(\lambda)$. Using form methods we define a generalized Rayleigh functional and characterize the eigenvalues above the essential spectrum of $\mathcal{A}$ by a min-max and a max-min variational principle. The obtained results are illustrated with a damped beam equation.

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