Undamped Case Research Articles

Vibrations and energy distribution in inhomogeneous rods with elastic and viscous boundary conditions

Functionally graded materials have broad engineering applications including mechanical engineering, electronics, chemistry, and biomedical engineering. One notable advantage of such materials is that their stiffness distribution can be optimized to avoid stress concentration. A novel approach for solving the equations describing the longitudinal vibration of functionally graded rods with viscous and elastic boundary conditions is proposed. The characteristic equation of the system is derived for the solution of the undamped case for the constant stiffness rod. Then, a homotopy method is applied to compute the eigenvalues and mode shapes of graded rods for viscoelastic boundary conditions. The changes of the eigenvalues and mode shapes as function of the damping parameters are investigated. The optimal damping of the system is computed. It is shown that the qualitative behavior depends on the relation between the actual damping and the optimal damping of the system. The energy density distribution of graded rods is also discussed. An energy measure, the mean scaled energy density distribution is introduced to characterize the energy distribution along the rod in the asymptotic time limit. The significance of such a measure is that it reveals how the energy tends to distribute along the rod. It is shown that the energy distribution can be manipulated by changing the damping parameters. Qualitative changes depending on the relation between the actual damping and the optimal damping are highlighted.

APPLICATION OF HE’S FREQUENCY FORMULA TO NONLINEAR OSCILLATORS WITH GENERALIZED INITIAL CONDITIONS

This paper focuses on the vibration periodic property of Duffing oscillator with generalized initial conditions. Firstly, the undamped case is solved by Ji-Huan He’s frequency formulation; Secondly, the formulation is extended to the damped case. Numerical verification shows that the frequency formulation is mathematically simple and physically insightful and practically applicable. This paper paves a novel way for engineers to use the formulation to study nonlinear vibration system with ease and reliability.

Research on Vibration Damping Analysis of UHV Suspended Converter Valve Model

The UHV (ultra high voltage) suspended converter valve is hinged to the top of the valve hall, so the equipment has a long structural period, the suspended converter valve is displaced under the action of earthquakes, and the valve tower is susceptible to damage due to interaction with the important equipment connected to it. The high cost, large size, and weight of the real type of UHV suspended converter valve make it difficult to carry out shaking table test studies of the real type of equipment. In this paper, a numerical model of a UHV-suspended converter valve scaling device and its damping structure model are established, and a damping analysis study with damping at the top of the converter valve model is carried out. The analysis showed that the maximum displacement of the converter tower was 136.3 mm under the action of a 0.4 g El-Centro earthquake. With horizontal damping at the top, the maximum displacement of the converter tower is 62.9 mm, a reduction of 53.9% compared to the undamped case. This study proposes a vibration damping design for UHV suspended converter valve, laying the foundation for subsequent shaker test studies of the scaled-down model vibration damping structure of the UHV suspended converter valve and the productization of vibration dampers for the equipment.

Analysis of the response of damped and parametrically driven, strongly anharmonic Klein-Gordon chain - Part 2: Beat-waves

This paper continues a series of publications devoted to the analysis of strongly nonlinear wave phenomena, emerging in genuinely anharmonic, Klein-Gordon cyclic chains, subjected to a traveling-wave parametric excitation. In this study, we analyze a special type of nonlinear waves, which will be further referred to as beat-waves. These are spatially extended, and strongly modulated waves manifested by the extreme amplitude modulation, which can be of stationary or nonstationary type. Stationary beat-waves of Klein-Gordon chains propagate with constant group velocity, that depends on the beat-wave amplitude. In the vicinity of 2:1 parametric resonance, we derive the parametrically driven, discrete p-Schrödinger (PD-DpS) model, which depicts the slow modulation of the response of a parametric oscillator chain. Special coordinates' transformation applied on the PD-DpS model allows the exact reduction of extremely modulated beat-waves to an invariant manifold of this slow-flow system. The system dynamics reduced to the beat-wave invariant manifold is depicted by three collective coordinates. In the undamped case, the analysis of the slow-flow system dynamics on the invariant manifold reveals a special family of nonlinear beat-waves characterized by an extreme amplitude modulation. Among these beat-wave solutions lying on the manifold, our analysis identifies some special solutions manifested by the solitary-pulse like, temporal evolution of the beat-wave amplitude which can either decay to zero or to a constant nonzero value. We derive the exact analytical solutions for these special types of waves as well as obtain an explicit analytical approximation for the beat-waves of other types. In the damped case, analysis of the slow-flow system dynamics on the invariant manifold fully recovers the mechanism of system synchronization on the special, stationary beat-wave solutions characterized by the harmonically modulated envelope. Importantly, the derived reduced order model accurately captures the transient evolution stage as well as the steady-state stage of the collective beat-wave response of PD-DpS model, initialized on the beat-wave invariant manifold. A detailed bifurcation analysis of the stationary undamped as well as the stationary damped beat-waves recovers their entire bifurcation structure and enables to fully characterize the zones of system parameters guaranteeing the existence of this special type of non-linear wave solutions. The results of analytical study of various types of beat wave solutions are in excellent agreement with the results of numerical simulation of both PD-DpS as well as the original, parametrically driven, genuinely an-harmonic, discrete Klein-Gordon model.

Symmetry analysis and hidden variational structure of Westervelt’s equation in nonlinear acoustics

Westervelt’s equation is a nonlinear wave equation that is widely used to model the propagation of sound waves in a compressible medium, with one important application being ultra-sound in human tissue. Two fundamental aspects of the general dissipative version of Westervelt’s equation – symmetries and conservation laws – are studied in the present work by modern methods. Numerous results are obtained: new conserved integrals; potential systems yielding hidden symmetries and nonlocal conservation laws; mapping of Westervelt’s equation in the undamped case into a linear wave equation; hidden variational structures, including a Lagrangian and a Hamiltonian; a recursion operator and a Noether operator; contact symmetries; higher-order symmetries and higher-order conservation laws.

Exponential stability estimates for an axially travelling string damped at one end

We study the small vibrations of an axially travelling string with a dashpot damping at one end. The string is modelled by a wave equation in a time-dependent interval with two endpoints moving at a constant speed . For the undamped case, we obtain a conserved functional equivalent to the energy of the solution. We derive precise upper and lower exponentially decaying estimates for the energy with explicit constants. These estimates do not seem to be reported in the literature even for the non-travelling case .

On second-order [formula omitted]-sub-step explicit algorithms with controllable dissipation and adjustable bifurcation point for second-order hyperbolic problems

This paper proposes a self-starting, second-order accurate, composite s-sub-step explicit method, within which the first five explicit members are developed, analyzed, and compared. Each member attains maximal stability bound, reaching 2×s, where s denotes the number of sub-steps. Identical diagonal elements in the amplification matrix are required in the undamped case, and algorithmic dissipation is controlled and measured at the bifurcation point. Further, if the bifurcation point is regarded as a free parameter during the dissipation analysis, the methods can achieve simultaneously controllable algorithmic dissipation and adjustable bifurcation point. Apart from these, the optimization of low-frequency dissipation is performed in the four- and five-sub-step members. As the number of sub-steps increases, the proposed explicit members can embed better numerical characteristics. This paper provides two approaches to constructing the updating velocity schemes in all sub-steps. The first approach enlarges the stability in the damped case, while the second one enhances the solution accuracy for damping problems. Numerical examples are solved to validate the proposed explicit members’ numerical performance. The novel algorithms are often more accurate and efficient than the existing methods.

Fast structure-preserving difference algorithm for 2D nonlinear space-fractional wave models

In this article, we present an efficient structure-preserving difference scheme for two-dimensional nonlinear space fractional wave models. We use the Crank-Nicolson scheme to discretize in the time direction, and adopt a shifted convolution quadrature (SCQ) formula to approximate the spatial direction. We show and prove an important lemma, and then based on this lemma, we rigorously derive the energy dissipation for the damped case and the approximate energy conservation in undamped case for the fully discrete scheme. In addition, we do the detailed analysis for error estimates with the optimal convergence result O ( h x 2 + h y 2 + τ 2 ) , where h x , h y and τ denote the step length in space-time directions. Furthermore, we provide an efficient fast algorithm called the preconditioned fast BiCG (PF-BiCG) to reduce the cost of time and save memory. Finally, we carry out numerical tests by taking three numerical examples to validate the theories, to observe the performance behaviors of wave solutions, and to reveal the relationship between models by changing fractional parameters.

On a Qualitative and Lie Symmetry Analysis for a Pendulum with Two Reaction Wheels

Summary In this article, it is studied the mechanical system formed by a pendulum with two reaction wheels in which the friction torque is assumed to follow a Coulomb law. A qualitative analysis of the system is performed for the damped case. Specifically, the equilibrium points for the unforced pendulum are analyzed. Also, in the forced case, the conditions for which there exist asymptotically stable solutions are determined. In order to study the exact analytical solution of the unforced pendulum, we also perform a Lie symmetry analysis. In this regard, it is shown that the exact general solution of the system for null motor torques can be expressed in terms of the general solution to an Abel equation. In the unforced and undamped case, the exact general solution is obtained in explicit form and expressed in terms of the Jacobi elliptic function by using the Lie symmetry approach.

Sound and soliton wave propagation in homogeneous and heterogeneous mediums with the new two-derivative implicit–explicit Runge–Kutta–Nyström method

This paper derives a new family of implicit–explicit time-marching methods for PDEs with the second-order derivative in time. The present implicit method is based on the two-derivative Runge–Kutta–Nyström methods, which use a third-order time derivative of the solution. Although the current approach is implicit, it does not need to invert the coefficient matrix of the discretized system of equations. The stability properties are assessed using Fourier analysis for the model test problems by considering space–time discretizations together. The present methods are validated by comparing to some of the most widely used time-marching methods available in the literature. In addition, to assess the robustness and efficiency of the present methods, we have also performed numerical simulations of acoustic wave propagation in two- and three-layered heterogeneous media and sine-Gordon solitons for damped and undamped cases. Computed results match very well with the exact and numerical solutions noted in the literature.

An Elementary Solution to a Duffing Equation.

In this work, we study the Duffing equation. Analytical solution for undamped and unforced case is provided for any given arbitrary initial conditions. An approximate analytical solution is given for the damped or trigonometrically forced Duffing equation for arbitrary initial conditions. The analytical solutions are expressed in terms of elementary trigonometric functions as well as in terms of the Jacobian elliptic functions. Examples are added to illustrate the obtained results. We also introduce new functions for approximating the Jacobian and Weierstrass elliptic functions in terms of the trigonometric functions sine and cosine. Results are high accurate.

Effect of dampers on the seismic performance of masonry walls assessed through fragility and demand hazard curves

In a damage avoidance design perspective, seismic rocking of rigid-like blocks, such as masonry walls, precast r.c./timber panels, art objects, etc., can be controlled through viscous elastic dampers. This paper discusses how to perform a seismic risk assessment of masonry walls undamped and damped by viscous-elastic dampers installed at the top of them. Firstly, a preliminary fragility analysis allows understanding the role of the damper, whose coupling with a recentering component attenuates the motion, increases the rotation frequency and, in some cases, impedes overturning as compared to to the undamped case. Secondly, the risk assessment is made by developing a multiple stripe analysis to compute univariate fragility curves combining uplift, overturning probability and probability of exceedance. Thirdly, seismic demand hazard curves are obtained for two walls representative of buildings and monumental structures in an Italian site of high seismicity. The PSHA results show the remarkable risk reduction of the ultimate limit states, for which the annual exceedance rate is respectively one and two orders of magnitude lower than the undamped case.

Two-layer model of the railway track: Analysis of the critical velocity and instability of two moving proximate masses

In this paper, the widely used two-layer model of the railway track is analysed with focus on the critical velocity and instability of moving masses. The model is assumed to be infinite with no changes in properties in the longitudinal direction. As far as the new demonstrations and conclusions: it is shown that in the undamped case, the onset of instability matches the critical velocity, and this is valid for one or more moving masses. However, in a damped case, the situation is more complicated. Instability of one moving mass occurs always in supercritical velocity range, but when two proximate masses are moving, then under certain conditions, the dynamic interaction between them together with the damping can shift the onset of instability into the subcritical velocity range. This newly derived negative effect has already been identified for Pasternak viscoelastic foundation in previous author's work. All results in this paper are presented in closed-form semi-analytical way for dimensionless parameters.The results presented are validated by eigenvalue expansion method on long finite model. For this purpose, vibration modes and orthogonality condition are derived. Modal equations are coupled and therefore rearrangement of the terms involved is introduced to save computational time. An excellent agreement between the results obtained on the infinite and finite models is achieved, which validates the newly derived formulas and the conclusions drawn from them.In order to simplify the expressions derived, continuous supports are implemented, because differences in behaviour would only be noticeable under high frequencies. This was checked by finite element analysis in commercial software. Unlike the more common usage of the two-layer model, shearing springs are added providing bases for further generalisations.

Regularization of central forces with damping in two and three dimensions

Regularization of damped motion under central forces in two and three-dimensions are investigated and equivalent, undamped systems are obtained. The dynamics of a particle moving in $\frac{1}{r}$ potential and subjected to a damping force is shown to be regularized a la Levi-Civita. We then generalize this regularization mapping to the case of damped motion in the potential $r^{-\frac{2N}{N+1}}$. Further equation of motion of a damped Kepler motion in 3-dimensions is mapped to an oscillator with inverted sextic potential and couplings, in 4-dimensions using Kustaanheimo-Stiefel regularization method. It is shown that the strength of the sextic potential is given by the damping co-efficient of the Kepler motion. Using homogeneous Hamiltonian formalism, we establish the mapping between the Hamiltonian of these two models. Both in 2 and 3-dimensions, we show that the regularized equation is non-linear, in contrast to undamped cases. Mapping of a particle moving in a harmonic potential subjected to damping to an undamped system with shifted frequency is then derived using Bohlin-Sudman transformation.

Dynamic interaction and instability of two moving proximate masses on a beam on a Pasternak viscoelastic foundation

The study analyzes the dynamic interaction of proximate masses using a new form of semi-analytical results for the moving mass problem developed by the author. This paper presents the results for a particular case of two moving masses of equal value acted upon by constant forces of equal values. The interaction level was demonstrated to be substantially high, making it impossible to superimpose the results obtained for one mass unless the masses were located at a significantly large distance apart. In the undamped case, the onset of instability occurred at the critical velocity, as with one moving mass; however, the onset of instability could be shifted significantly into the subcritical velocity range in the damped case. The critical distance between the moving masses was determined as the value at which the exponential increase in the amplitudes was severe. The implementation of dimensionless parameters revealed that this distance was slightly dependent on the damping. Moreover, the limiting moving mass ratio for such unexpected instability was derived.The results were validated by eigenmode expansion analysis on long finite beams, for which computational time savings were proposed by the rearrangement of the terms involved. Excellent agreement between the results was obtained, validating the new formulae. Furthermore, extension to more general cases and additional moving masses can easily be achieved.

Novel solutions to the (un)damped Helmholtz-Duffing oscillator and its application to plasma physics: Moving boundary method

Novel analytical and numerical solutions to the (un)damped Helmholtz-Duffing (H-D) equation for arbitrary initial conditions are derived. Both the analytical (for undamped case) and approximate analytical (for damped case) solutions, are obtained in the form of Weierstrass elliptic function. Also, the soliton solution to the undamped H-D equation is obtained in detail. The semi-analytical solution to the damped H-D equation is compared to the fourth-order Runge–Kutta (RK4) numerical solution. The obtained solution shows an excellent agreement with the numerical simulations but sometimes (according to the values of initial conditions) not on all time interval. Thus, the moving boundary method is utilized to improve the semi-analytical solution. It is found that the improved solution gives good results with high accuracy in the whole time domain. As a realistic application, the obtained solutions are applied to the study of nonlinear oscillations in an electronegative non-Maxwellian dusty plasma. Finally, we conclude that our novel solutions could help us to understand the dynamics of various nonlinear oscillations in engineering and in different branches of sciences such as oscillations in different plasma models.

Innovative negative-stiffness inerter-based mechanical networks

This paper presents the optimal design for novel negative-stiffness non-traditional inerter-based dynamic vibration absorbers called NS-NIDVAs. The underlying idea of negative-stiffness inerter-based mechanical networks (NS-IMNs) was inspired by the negative-stiffness technology concept. Firstly, the non-dimensional mathematical models for NS-NIDVAs are established considering the harmonic force excitation case, and then the dimensionless frequency response functions of the primary structure are computed for optimization purposes. Therefore, a dynamic stability analysis was conducted in order to determine the allowable boundary on negative stiffness variable. Moreover, the Extended Fixed Points Technique (EFPT) has been proposed as an optimal calibration procedure taking in account the undamped system case. In order to ensure a trade-off among the invariant points of the primary structure's FRFs, a constraint equation was added to the EFPT's optimality approach. It was proved that the allowable negative stiffness solution computed by the constraint equation is in accordance with that provided by the Routh- Hurwitz stability criterion. The short optimal explicit functions revealed that the NS-NIDVA-C4 has a similar optimal tuning magnitude as the NS-NIDVA-C6 and C3, with the devices providing the same dynamic performance. In view of this dynamic characteristic, the H∞ performance measure is computed for both NS-NIDVAs devices considering a practical application mass ratio range. When the NS-NIDVA-C4’s performance is compared to the other passive high-performance devices, an improvement of more than 50% in DMF reduction is obtained. Additionally, the NS-NIDVA-C4 can broaden the effective operating bandwidth from 46% to 82% when compared to the classic DVA. Therefore, it has been proved that NS-NIDVA-C4 has the versatility of yielding high-performance with a small physical mass of the DVA, which is the significant finding of this work.

Energetic boundary element method for accurate solution of damped waves hard scattering problems

The paper deals with the numerical solution of 2D wave propagation exterior problems including viscous and material damping coefficients and equipped by Neumann boundary condition, hence modeling the hard scattering of damped waves. The differential problem, which includes, besides diffusion, advection and reaction terms, is written as a space–time boundary integral equation (BIE) whose kernel is given by the hypersingular fundamental solution of the 2D damped waves operator. The resulting BIE is solved by a modified Energetic Boundary Element Method, where a suitable kernel treatment is introduced for the evaluation of the discretization linear system matrix entries represented by space–time quadruple integrals with hypersingular kernel in space variables. A wide variety of numerical results, obtained varying both damping coefficients and discretization parameters, is presented and shows accuracy and stability of the proposed technique, confirming what was theoretically proved for the simpler undamped case. Post-processing phase is also taken into account, giving the approximate solution of the exterior differential problem involving damped waves propagation around disconnected obstacles and bounded domains.

A priori bounds of the solution of a one point IBVP for a singular fractional evolution equation

A priori bounds constitute a crucial and powerful tool in the investigation of initial boundary value problems for linear and nonlinear fractional and integer order differential equations in bounded domains. We present herein a collection of a priori estimates of the solution for an initial boundary value problem for a singular fractional evolution equation (generalized time-fractional wave equation) with mass absorption. The Riemann–Liouville derivative is employed. Results of uniqueness and dependence of the solution upon the data were obtained in two cases, the damped and the undamped case. The uniqueness and continuous dependence (stability of solution) of the solution follows from the obtained a priori estimates in fractional Sobolev spaces. These spaces give what are called weak solutions to our partial differential equations (they are based on the notion of the weak derivatives). The method of energy inequalities is used to obtain different a priori estimates.

Dissipation-preserving Galerkin–Legendre spectral methods for two-dimensional fractional nonlinear wave equations

In this paper, we consider implicit and linearized dissipation-preserving Galerkin–Legendre spectral methods to solve space fractional nonlinear damped wave equation in two dimensions. The full discrete schemes preserve dissipation of energy in damped case and conservation of energy in undamped case. Moreover, the stability and convergence analysis of the full discrete schemes are rigorously given. Furthermore, we get that two methods are convergent with second-order accuracy in time and optimal error estimates in space. In order to reduce the computational cost, we adopt matrix diagonalization approach to solve the resulting algebraic systems in numerical implementation. Numerical results are presented to validate the theoretical analysis.