Undamped Beam Research Articles

Analysis of modified structures by Rayleigh quotient

In structural vibrations, the Rayleigh quotient may be used to calculate the natural frequency (or eigenvalue) of a structure given the corresponding mode shape (or eigenvector). Previous works have shown that the eigenvalue may be calculated relatively accurately using the Rayleigh quotient as long as an appropriate guess is made for the eigenvector. Typically, the Rayleigh quotient is used to predict the eigenvalues when the structure does not change. However, in this work the structure will be modified and the change in the eigenvalue will be predicted using the Rayleigh quotient. In order to use the Rayleigh quotient a guess for the eigenvector must be made. Here, the displacement vector of the nominal structure forced near a resonance will be used as the guess. Analysis and computations will be done for an undamped beam that is harmonically forced. To modify the structure, the stiffness elements will be scaled. Using this approach, it will be demonstrated that changes in the natural frequencies can be predicted relatively quickly for a modified structure. In addition, this method provides insight into which elements should be scaled to get the greatest frequency change. [Work supported by ONR under Grant N00014-19-1-2100.]

On unique determination of an unknown spatial load in damped Euler–Bernoulli beam equation from final time output

Abstract In this paper, we discuss the role of the damping term μ ⁢ u t {\mu u_{t}} in unique determination of unknown spatial load F ⁢ ( x ) {F(x)} in a damped Euler–Bernoulli beam equation u t ⁢ t + μ ⁢ u t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x = F ⁢ ( x ) ⁢ G ⁢ ( t ) {u_{tt}+\mu u_{t}+(r(x)u_{xx})_{xx}=F(x)G(t)} from the measured final time displacement u T ⁢ ( x ) := u ⁢ ( x , T ) {u_{T}(x):=u(x,T)} or velocity ν T ⁢ ( x ) := u t ⁢ ( x , T ) {\nu_{T}(x):=u_{t}(x,T)} . In [A. Hasanov Hasanoğlu and V. G. Romanov, Introduction to Inverse Problems for Differential Equations, Springer, Cham, 2017] it was shown that the unique determination of the spatial load F ⁢ ( x ) {F(x)} in the undamped wave equation u t ⁢ t - ( k ⁢ ( x ) ⁢ u x ) x = F ⁢ ( x ) ⁢ G ⁢ ( t ) {u_{tt}-(k(x)u_{x})_{x}=F(x)G(t)} from final data (displacement or velocity) is not possible. This result is also valid for the undamped beam equation u t ⁢ t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x = F ⁢ ( x ) ⁢ G ⁢ ( t ) {u_{tt}+(r(x)u_{xx})_{xx}=F(x)G(t)} . We show that in the presence of the damping term, the spatial load can be uniquely determined from the final time output under some acceptable conditions with respect to the final time T > 0 {T>0} and the damping coefficient μ > 0 {\mu>0} . The cases of pure spatial load G ⁢ ( t ) ≡ 1 {G(t)\equiv 1} , and the exponentially decaying load G ⁢ ( t ) = e - η ⁢ t {G(t)=e^{-\eta t}} are analyzed separately, leading also to sufficient conditions for the uniqueness of the inverse problem solution. The results presented in this paper clearly demonstrate the key role of the damping term in the unique determination of the unknown spatial load F ⁢ ( x ) {F(x)} in the damped Euler–Bernoulli beam equation.

Unique recovery of unknown spatial load in damped Euler–Bernoulli beam equation from final time measured output

In this paper we discuss the unique determination of unknown spatial load F(x) in the damped Euler–Bernoulli beam equation from final time measured output (displacement, u T (x) ≔ u(x, T) or velocity, ν t,T (x) ≔ u t (x, T)). It is shown in [Hasanov Hasanoglu and Romanov 2017 Introduction to Inverse Problems for Differential Equations (New York: Springer)] that the unique determination of F(x) in the undamped wave equation from final time output is not possible. This result is also valid for the undamped beam equation . We prove that in the presence of damping term μu t , the spatial load can be uniquely determined by the final time output, in terms of the convergent singular value expansion (SVE), as , under some acceptable conditions with respect to the final time T > 0, the damping coefficient μ > 0 and the temporal load G(t) > 0. As an alternative method we propose the adjoint problem approach (APA) and derive an explicit gradient formula for the Fréchet derivative of the Tikhonov functional . Comparative analysis of numerical algorithms based on SVE and APA methods are provided for the harmonic loading G(t) = cos(ωt), ω > 0, as a most common dynamic loading case. The results presented in this paper not only clearly demonstrate the key role of the damping term μu t in the inverse problems arising in vibration and wave phenomena, but also allows us, firstly, to find admissible values of the final time T > 0, at which a final time measured output can be extracted, and secondly, to reconstruct the unknown spatial load F(x) in the damped Euler–Bernoulli beam equation from this measured output.

Dynamic Response of a Damped Euler–Bernoulli Beam Having Elastically Restrained Boundary Supports

While doing vibration analysis in terms of the dynamic response of a beam, the end conditions, in most of the cases, are assumed to be fixed, simply supported, free or sliding imposing an ideal condition of end support displacement. But, in real engineering structures, the end supports are non-ideal (for example welded, riveted etc.) and allow certain degree of translational or rotational motion of the support ends. To simulate the effect of real engineering boundary states, a combination of linear and torsional springs has been considered at the two boundaries of the beam structure. The theoretical analysis of free undamped beam structure with elastically restrained end conditions has been undertaken using approximation of the modal displacement as a Fourier cosine series to generate natural frequencies and corresponding mode shape equations. The effect of variation in stiffness values of the boundary springs on the vibration characteristics (natural frequency, mode shapes and dynamic response) has been presented for vibration conditions for the free undamped beam structure. Thereafter, forced undamped and damped (taking combined effects of both viscous and structural damping) beam structures have been theoretically analysed to generate response equation. Finally, numerical assessment has been undertaken to present the effect of damping ratio, forcing frequency and force magnitude on the response of a nearly clamped beam and has been compared with the response of an undamped beam under similar forcing and boundary condition. The results indicate considerable effects of real boundary condition, on the vibration characteristics and dynamic response of beam structures thus cannot be ignored in design of real engineering structures. Further, the presence of damping does not change the vibration characteristics beam structures, but is useful in achieving steady-state condition of response for a harmonic force input. The technique presented provides a simplified and convenient tool for obtaining vibration response for real engineering boundary supports. Further, the beam model can be extended for analysis of plate structures considering non-ideal boundary conditions.

Analysis of the critical velocity of a load moving on a beam supported by a finite depth foundation

A detailed analysis of the critical velocity of a uniformly moving load is carried out in this paper. It is assumed that the load is traversing an infinite beam supported by a finite depth foundation under plane strain condition. The problem is solved analytically in the Fourier domain, respecting two formulations of the interface condition. The critical velocity is determined as the velocity under which the undamped beam deflection tends to infinity, i.e. by identifying double poles in the Fourier image of the beam deflection situated on the real Fourier variable axis. Results obtained are compared with the previously published results of this author, where simplifying assumptions were imposed on the shear contribution of the foundation. Based on the results, an addition to the previous formula for critical velocity estimate is proposed. It is confirmed that there are several critical velocities, but the lowest one, which is the dominating one, smoothly changes from the classical value to the lowest wave-velocity of propagation in the foundation according to the mass ratio defined as the square root of the fraction of the foundation mass to the beam mass. Particular situations, where the first double pole cannot be formed in this model, are discussed in detail. In such cases a significant increase of deflection is observed in the vicinity of the expected location, defining thus a pseudo-critical velocity. Attention is paid to the cases where there are two or more very close critical velocities, which causes an increased risk for practical applications. In addition, several options of damping are compared.

Improving the linear stability of the Beck׳s beam by added dashpots

The effects of lumped dashpots on the linear stability of the Beck׳s beam, internally and externally damped, are studied, with the aim to improve the mechanical performances of the system. A perturbation analysis is carried out, in which the sensitivities of the simple eigenvalues of the undamped and sub-critically loaded beam are evaluated. The analysis provides a simple and very efficient tool for damping design. It is proved that there exists a region along the beam axis assuring that the added dashpots, of extensional and flexural type, increase the critical load of the beam under distributed damping, thus mitigating the detrimental effect of the destabilization paradox. It is also shown that, if the flexural dashpot is located at the tip of the beam, it has a beneficial effect at low level of load, but, counter-intuitively, it is detrimental when the load approaches the critical value of the undamped beam, thus extending analogous findings of the literature concerning the extensional dashpot.

A finite difference scheme for solving the undamped Timoshenko beam equations with both ends free

In this paper, a boundary feedback system of a class of non-uniform undamped Timoshenko beam with both ends free is considered. A linearized three-level difference scheme for the Timoshenko beam equations is derived by the method of reduction of order on uniform meshes. The unique solvability, unconditional stability and convergence of the difference scheme are proved by the discrete energy method. The convergence order in maximum norm is of order two in both space and time. The validity of this theoretical analysis is verified experimentally.

Influence of the nonlinear aerodynamic force components on a vibrating beam, simply supported or clamped at both ends, exposed to a high supersonic airflow along its axial direction

In this work we analyze the case of a vibrating beam, simply supported or clamped at both ends, underthe effect of a high supersonic airflow along its axial direction. A complete aerodynamic model ofthe piston theory, which also takes into account the nonlinear components of the distributed aerody-namic transversal force, is used. The postcritical flutter behavior and its influence on the vibration statesolutions of a fluttering beam without aerodynamic damping have been studied. This paper focusesparticularly on the effects of these nonlinear aerodynamic forces on three frequencies, which are usefulin characterizing the postcritical flutter solution set of the undamped beam in the whole frequency range:the minimum frequency, the frequency where the change of the modal shape with lower amplitude occurs,and the frequency corresponding to the solution with minimum amplitude of the vibration mode. Specialattention has been given to the influence on the solution of the vibrating undamped beam with minimummodal amplitude, whose frequency is the most important among the three mentioned above; in fact, inthe neighborhood of this particular solution, there exists the flutter state of the vibrating damped beamin limit cycle conditions.Three different schemes, two of them semianalytical (based on the classical and well known Rayleigh–Ritz and Galerkin methods) and one of them numerical (based on the finite element method), have beenherein exploited, as in the author’s previous papers, where beam flutter models with linear aerodynamicanalysis were used. The good agreement between the results obtained by the three methods corroboratestheir effectiveness.More sophisticated models have been herein set up, considering that a more accurate analysis isnecessary than in previous cases, where the aerodynamic numerical model was limited to within theframework of the quasisteady linearized piston theory, both for the coupling component between oddand even order vibrating modes, and for the aerodynamic damping component.The results obtained enable us to assess quantitatively the influence of these nonlinear aerodynamicforces on the postcritical beam flutter behavior, and particularly on the undamped beam solution withminimum amplitude of the vibration mode.

Modal power flow with application to damage detection

Features of power flow of an undamped beam at resonance are studied in the present paper. It is found that when an undamped beam undergoes free vibration at one of its natural frequencies, the active component of the power flow becomes zero while the reactive component is of modal pattern, whose characteristic frequency is twice of the natural frequency. The power flow in this case can thus be termed as modal power flow. The instantaneous energy density associated with the vibration mode consists of a static component and a dynamic component, related to the mean total and Lagrangian energy densities, respectively. The modal power flow is relevant to the latter but independent of the former. Potential application of modal power flow to structural damage detection is investigated. Two typical damages, transverse cracks and delaminations, are considered. A damage index based on the modal power flow is proposed, and compared with the damage indices based on the slope, the bending strain, and the strain energy through numerical examples. The imperfection of boundary conditions is also considered. It is shown that the proposed damage index is sensitive to both types of damage, thus can be used as an universal damage indicator.

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.

Periodic solutions of a periodically forced and undamped beam resting on weakly elastic bearings

We study the problem of existence of periodic solutions to a partial differential equation modelling the behavior of an undamped beam subject to an external periodic force. We assume that the ordinary differential equation associated to the first two modes of vibration of the beam has a symmetric homoclinic solution. By using methods borrowed by dynamical systems theory we prove that, if the period is non resonant with the (infinitely many) internal periods of the PDE, the equation has a weak periodic solution of the same period as the external force. In particular we obtain continua of periodic solutions for the undamped beam in absence of external forces. This result may be considered as an infinite dimensional analogue of a result obtained in [16] concerning accumulation of periodic solutions to homoclinic orbits in finite dimensional reversible systems.

Vibration absorption by an undamped beam element

Through two complementary approaches, using modal response and wave propagation, this presentation will show the conditions under which a decaying impulse response, or a nearly irreversible energy trapping, takes place in a linear conservative continuous system. The results show that the basic foundation of near-irreversibility or pseudo-damping rests upon the presence of singularity points in the modal density of the systems. To illustrate the concept of pseudo-damping in detail, a simple undamped beam is modified to introduce a singularity point in its modal density distribution. Simulations show that by a suitable application of a compressive axial force to an undamped beam on an elastic foundation, the impulse response of a beam shows the characteristics of a nearly irreversible energy trap and a decaying impulse response. Attaching such a modified beam to a single degree of freedom oscillator shows energy trapping by the beam producing pseudo damping. [Research carried out while AA served at NSF.]

Effect of Boundary Conditions on Impact Stresses of Beams

A theoretical analysis based on the numerical solution of the beam impact integral equation is carried out to determine the impact force and deflection time histories, the strain energy absorbed by the beams and the maximum bending moment. Effect of beam boundary conditions on impact response of beam is also discussed. The theoretical results obtained in the present analysis are compared with experimental and theoretical works previously done. A good agreement is found between theoretical and experimental results. This indicates that the impact resistance of relatively large beams may be predicted by using the theoretical approach based on equation of undamped beam vibration. All the derivations required to predict the effect of boundary conditions are performed for both forced and free vibrations. For the same falling mass and the same applied kinetic energy (height of drop) for all cases, the maximum central deflection and the maximum impact force are affected by the boundary conditions of the beams

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.

Boundary feedback stabilization of the undamped Timoshenko beam with both ends free

In this paper, we study a boundary feedback system of a class of nonuniform undamped Timoshenko beam with both ends free. We give some sufficient conditions and some necessary conditions for the system to have exponential stability. Our method is based on the operator semigroup technique, the multiplier technique, and the contradiction argument of the frequency domain method.

Near-irreversibility in a conservative linear structure with singularity points in its modal density

Through two complementary approaches, using modal response and wave propagation, the analyses presented here show the conditions under which a decaying impulse response, or a nearly irreversible energy trapping, takes place in a linear conservative continuous system. The results show that the basic foundation of near-irreversibility or apparent damping rests upon the presence of singularity points in the modal density of dynamic systems or, analogously, in the wave-stopping properties associated with these singularities. To illustrate the concept of apparent damping in detail, a simple undamped beam is modified to introduce a singularity point in its modal density distribution. Simulations show that a suitable application of a compressive axial force to an undamped beam placed on an elastic foundation attenuates its impulse response with time and develops the characteristics of a nearly irreversible energy trap.

Discussion on “A Robust Approximation Scheme for the LQG Control of an Undamped Flexible Beam with a Tip Mass”

Discussion on “A Robust Approximation Scheme for the LQG Control of an Undamped Flexible Beam with a Tip Mass”

Wave Characteristics and Vibration Control of Translating Beams by Optimal Boundary Damping

The transverse motion of a translating tensioned Euler-Bernoulli beam is controlled by passive or active damping applied at a boundary. Even for an undamped beam with a symmetric boundary configuration, the interaction between the translating continuum and the stationary or moving boundary leads to energy variation in free motion. With the time-varying energy chosen as a Lyapunov functional, boundary control laws are designed based on Lyapunov’s second method. For various types of translating beams, energy dissipation by boundary damping is quantified using the method of traveling waves. The optimal value of damping, maximizing the energy dissipation, is also explicitly represented by system parameters. The analytical results are compared with numerical simulations using the finite difference scheme.

LOCALIZATION OF VIBRATION IN DISORDERED MULTI-SPAN BEAMS WITH DAMPING

The combined effects of disorder and structural damping on the dynamics of a multi-span beam with slight randomness in the spacing between supports are investigated. A wave transfer matrix approach is chosen to calculate the free and forced harmonic responses of this nearly periodic structure. It is shown that both harmonic waves and normal modes of vibration that extend throughout the ordered, undamped beam become spatially attenuated if either small damping or small disorder is present in the system. The physical mechanism which causes the attenuation, however, is one of energy dissipation in the case of damping but one of energy confinement in the case of disorder. The corresponding rates of spatial exponential decay are approximated by applying statistical perturbation methods. It is found that the effects of damping and disorder simply superpose for a multi-span beam with strong inter-span coupling, but interact less trivially in the weak coupling case. Furthermore, the effect of disorder is found to be small relative to that of damping in the case of strong inter-span coupling, but of comparable magnitude for weak coupling between spans. The adequacy of the statistical analysis to predict accurately localization in finite disordered beams with boundary conditions is also examined.

Analytical Formulation Of Dynamic Stiffness

The dynamic stiffness matrix method enables one to model an infinite number of natural modes by means of a small number of unknowns. The method has been extended to skeletal structures with uniform or non-uniform, straight or curved, damped or undamped beam members. For two-dimensional structures, if one of the dimensions can be eliminated by means of the Kantorovich method, the method still applies. However, for more complicated systems, analytical formulation of the dynamic stiffness is tedious. A computer assisted analytical method is introduced here for any structural members the differential governing equations of which are expressible in matrix polynomial form. Complex arithmetics are used to cater for all possible classification of the characteristic roots. Numerical examples are given and are compared with existing results.