A successive integration technique for the solution of the Duffing oscillator equation with damping and excitation

In this paper, a successive integration technique is suggested for solving the Duffing oscillator equation with damping and excitation. In the technique, one performs integrations from some initial values over a wider range. Some of them may even be divergent; however, some of them will give a convergent result such that the periodic condition of motion is satisfied. In fact, the convergent result represents a stable periodic solution for the motion. If a convergent result or a stable periodic solution is obtained, the stability test is passed. A harmonic balance method in conjunction with the successive integration technique (abbreviated as HBMSIT) is also suggested. In the method, the initial values are obtained from the harmonic balance method. Therefore, the HBMSIT belongs to the successive integration techniques. Many examples with calculated results are presented.     

Improved stability techniques for the solution of Navier-Stokes equations

When solving the Navier-Stokes equations for transient incompressible viscous flow problems, one normally encounters a decrease in numerical stability of the time integration algorithm with an increase in Reynolds number. This instability cannot be easily overcome due to the non-linearities present. The present paper, using the finite element method to integrate the equations in the spacial dimension, incorporates a time-staggered semi-implicit fractional step technique to improve stability at the higher Reynolds numbers. Unlike the upwind or directional differencing schemes normally employed to increase numerical stability, the present scheme does not introduce numerical damping or artificial viscocity, and becomes more stable as the Reynolds number increases. Results for this scheme are compared with various explicit integration schemes for the case of flow around a circular cylinder at Reynolds numbers of 100 to 400. For comparable accuracy the time-staggered semi-implicit fractional step technique was found to be up to 25 times more efficient than the other explicit integration schemes.     

Influence of the hydraulic fluid pipe on the dynamic stability of a shift gearbox

Shift gearboxes are used in vehicle drive trains in order to transmit the driving torque of the motor and to shift between different transmission ratios. Because of the occurrence of sliding friction forces and -torques during the shifting period, unwanted vibrations can arise. Several mechanisms which lead to a destabilization of the stationary state in shift gearboxes are conceivable: variable friction coefficient [1], wobbling clutch disc [2], or vibrations because of coupled DoFs due to the helical gearing [3]. The latter effect results in translational oscillations in axial and radial direction of the clutch disc. It was observed that damping forces influence the boundary of the region of stability in the parameter space - though it is not entirely clear where such forces originate from. For this reason, in this work the effect of the fluid of the hydraulic actuation of the clutch system is analyzed. The fluid pipe is situated in the center of the gear unit input shaft and forces the fluid to oscillate when the system becomes unstable. In return, the fluid implies shear stress and pressure on the adjacent mechanical parts (shaft, pressure plates). The analysis of the stationary state of the coupled system reveals a clear effect of fluid properties on the stability: both the mass density and the kinematic viscosity are able to change the location of the border between stable and instable regions in the parameter space. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stabilization and destabilization in non-conservative gyroscopic systems

Stability of a linear autonomous non-conservative system in presence of potential, gyroscopic, dissipative, and nonconservative positional forces is studied. The cases when the non-conservative system is close to a gyroscopic system or to a circulatory one, are examined. It is known that the marginal stability of gyroscopic and circulatory systems can be destroyed or improved up to asymptotic stability due to action of small non-conservative positional and velocity-dependent forces. The present contribution shows that in both cases the boundary of the asymptotic stability domain of the perturbed system possesses singularities such as “Dihedral angle” and “Whitney umbrella” that govern stabilization and destabilization. Approximations of the stability boundary near the singularities and estimates of the critical gyroscopic and circulatory parameters are found in an analytic form. In case of two degrees of freedom these estimates are obtained in terms of the invariants of matrices of the system. As an example, the asymptotic stability domain of the modified Maxwell-Bloch equations is investigated with an application to the stability problems of gyroscopic systems with stationary and rotating damping, such as the Crandall gyropendulum, tippe top and Jellet's egg. An instability mechanism in a system with two degrees of freedom, originating after discretization of models of a rotating disc in frictional contact and possessing the spectral mesh in the plane ‘frequency’ versus ‘angular velocity’, is described in detail and its role in the disc brake squeal problem is discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exponential Decay Rates for Structural Acoustic Model with an Overdamping on the Interface and Boundary Layer Dissipation

We study uniform stability properties of a strongly coupled system of Partial Differential Equations of hyperbolic/parabolic type, which arises from the analysis and control of acoustic models with structural damping on an interface. A challenging feature of the present model is the presence of additional strong boundary damping which is responsible for lack of uniform stability of the free system ( overdamping phenomenon). It has been shown recently that by applying full viscous damping in the interior of the domain and suitable static damping on the interface, then the corresponding feedback system is uniformly stable. In this article we prove that uniform decay rates of solutions to the system can be achieved even if viscous damping is active just in an arbitrary thin layer near the interface.     

How far can relaxation functions be increasing in viscoelastic problems?

It is known that when we add a viscoelastic damping to a frictional damping acting in the domain we might lose the property of exponential stability of the system. Moreover, a necessary condition for a system to be sub-exponentially stable is that the kernel itself must be sub-exponentially decaying to zero. Having this in mind, a natural question to be asked is that of when this necessary condition is also sufficient. We prove that this is the case for a fairly large class of kernels.     

Exponential and polynomial decay for a laminated beam with Fourier's law of heat conduction and possible absence of structural damping

We study the well-posedness and decay properties of a onedimensional thermoelastic laminated beam system either with or without structural damping, of which the heat conduction is given by Fourier's law effective in the rotation angle displacements. We show that the system is wellposed by using the Lumer-Philips theorem, and prove that the system is exponentially stable if and only if the wave speeds are equal, by using the perturbed energy method and Gearhart-Herbst-Prüss-Huang theorem. Furthermore, we show that the system with structural damping is polynomially stable provided that the wave speeds are not equal, by using the second-order energy method. When the speeds are not equal, whether the system without structural damping may has polynomial stability is left as an open problem.     

Stability Analysis of A Floating Caliper Disc Brake

In this paper, the onset of the dynamic instability in a .oating caliper disc brake system is studied. The disc is modelled as a .exible rotating plate, the pad is taken as a small mass and a distributed sti.ness, and the motion of the caliper is also taken into consideration. The linearized equations of motion about the equilibrium position are derived assuming a constant braking force. The equations are subsequently discretized with the help of appropriate shape functions. The eigenvalue problem is then solved to detect the onset of instability. The effects of damping on the stability of motion are investigated as well.     

Exponential stability of a kind of wave equation with boundary feedback control

The problem of exponential stability of a kind of wave equation with damping and boundary output feedback control is investigated. The spectral structure of the system operator is analyzed and it is shown that the c0-semigroup generated by the system operator is exponential stable if only the coefficients viscous damping and boundary feedback control are not zeros simultaneously.     

Stabilization of nonlinear systems with a dither control

In this paper a new approach is taken to analyze stabilization of a general nonlinear system with a dither input. Given the original system with a control, an autonomous relaxed system is constructed. It is shown that if the relaxed system is stable, then the original system with dither control would be stable in the finite time. An algorithm is given for constructing the dither control. The technique used here is general and does not have the limitations of the Dual Input Describing Function technique. Furthermore, in many cases it is possible to guarantee global contractive stability as well. Two examples are solved in detail using computer simulations for demonstration of the technique.     

动态问题速度有限元法

分析了N.M.Newmark和E.L.Wilson等按位移作变量逐步积分法的主要特点.提出以速度为变量求解动力学问题的速度元法.针对无阻尼系统,构造了一种简化格式,讨论了稳定性.由于该格式在无阻尼和拟静力阻尼情况下为显式,每个时刻,不求解代数方程组,其计算量与Newmark等方法比较,显著减少.对非线性动态问题,该计算格式可作为取得较好迭代初值的一个办法.文中,就任意阻尼系统,列出了速度元法的推广形式.相应非线性情况,提供了速度增量迭代格式并证明了收敛性.文末,附录了典型问题的数值检验结果.     

Control and identification of chaotic systems by altering their energy

We developed the control technique for (non)linear oscillators when repellors are stabilized by adjusting the system to energy levels corresponding to their stable counterparts. The technique does not require knowledge of the system equations. Two control strategies are possible. Following the first one, we simply test the systems by changing the feedback strength. This strategy does not require any computation of the control signal, and, hence, can be useful for control as well as identification of unknown systems. If the desired target can be identified (say, from the system time series), one can use another strategy based on goal-oriented control of the desired target. We analyze how the perturbation shape can be tuned so as to preserve the system natural response and discuss how to calculate the minimal strength of the perturbation required for stabilization of a priori chosen orbit. Generally, the control represents addition of an extra nonlinear damping to the system. In two limits of the perturbation slope, it manifests itself in (i) changing the oscillator natural damping; (ii) suppressing (enhancing) the external driving force. In the case of large deviations between phases of the chaotic oscillator and the driving force, only first scenario holds. Generalization of the technique to the case of oscillator networks and 3D autonomous dynamical systems is considered.     

Energy decay to Timoshenko system with indefinite damping

We consider the classical Timoshenko system for vibrations of thin rods. The system has an indefinite damping mechanism, ie, it has a damping function a=a(x) possibly changing sign, present only in the equation for the vertical displacement. We shall prove that exponential stability depends on conditions regarding of the indefinite damping function a and a nice relationship between the coefficient of the system. Finally, we give some numerical result to verify our analytical results.     

A second-order accurate three sub-step composite algorithm for structural dynamics

In this paper, a novel three sub-step composite algorithm with desired numerical properties is developed. The proposed method is a self-starting, unconditionally stable and second-order accurate implicit algorithm without overshoot. Particularly, the second-order accuracy in time is achieved in its final form, but it is not required in each sub-step. Its unique algorithmic parameter is analyzed to achieve the unconditional stability and it shares the identical effective stiffness matrix inside three sub-steps to save the computational cost in linear analyses. The same as the Bathe algorithm, the proposed algorithm is always L-stable, meaning that the spurious high-frequency modes can be effectively eliminated. Three numerical examples are simulated to illustrate the superiority of the proposed algorithm over some existing implicit algorithms. The first numerical simulation, solving a linear single-degree-of-freedom system, shows less period elongation errors and the second-order accuracy of the present scheme. The second one, a clamped-free bar excited by the end load, shows the ability of effectively damping out the unexpected high-frequency modes. The last example solves the nonlinear mass-spring system with variable degree-of-freedoms and illustrates that the composite sub-step algorithm can save more computational cost than the traditional implicit algorithm when the integration step size is selected appropriately.     

Mathematical modeling of stability in rough elliptic bore misaligned journal bearing considering thermal and non-Newtonian effects

The paper briefly introduces a finite difference method based mathematical model to predict the stability of a finite journal bearing. The proposed method is used to integrate the geometrical irregularities of bearing such as non-circularity and surface roughness with the operational error like misalignment to represent more accurate film thickness. The bearing bore is assumed elliptic with longitudinal or transverse type wave pattern of roughness. A combined solution of Reynolds equation and Energy equation is made using effective influence Newton–Raphson method of error convergence. The non-Newtonian behavior of lubricant is addressed based on Power Law model. Thermal effect is considered adiabatic. Further to this analysis, the steady state and related perturbed pressures are estimated using linearization of bearing reaction. The dimensionless spring and damping coefficients are evaluated to find the critical mass and whirl ratio. Finally, the effect of misalignment bore ellipticity and roughness pattern on stability of such journal bearing is discussed in detail.     

Optimal damping ratio for linear second-order systems

In this paper, it is shown that the optimal damping ratio for linear second-order systems that results in minimum-time no-overshoot response to step inputs is of bang-bang type. The optimal damping ratio is zero at the outset and is switched to some maximum value at an appropriate instant of time. The switching time is shown to be a function of the maximum damping ratio and the system natural frequency. Furthermore, it is shown that the larger the maximum damping ratio is, the shorter it takes for the system to reach the desired set point. Finally, it is shown that, if the optimal damping ratio is switched as a function of the system state, then the minimum-time no-overshoot criterion is satisfied, irrespective of the magnitude of the uncertainty in the value of the system natural frequency.     

Riesz basis property, exponential stability of variable coefficient Euler-Bernoulli beams with indefinite damping

We study damped Euler–Bernoulli beams that have nonuniformthickness or density. These nonuniformfeatures result in variablecoefficient beam equations. We prove that despite the nonuniformfeatures, the eigenfunctions of the beam form a Riesz basisand asymptotic behaviour of the beam system can be deduced withoutany restrictions on the sign of the damping. We also providean answer to the frequently asked question on damping: ‘howmuch more positive than negative should the damping be withoutdisrupting the exponential stability?’, and result ina criterion condition which ensures that the system is exponentiallystable.     

Non-uniform asymptotic stability for the damped linear oscillator

Sufficient conditions are established for non-uniform asymptotic stability of a linear oscillator with damping term. The obtained results clarify a difference between the uniform asymptotic stability and the asymptotic stability. Some simple examples are included to illustrate the results. Especially, Bessel’s differential equations are taken up and it is proved that the equilibrium is asymptotically stable, but it is not uniformly asymptotically stable.     

Nonlinear dynamics and stability analysis of piezo-visco medium nanoshell resonator with electrostatic and harmonic actuation

In this paper, nonlinear dynamics, vibration and stability analysis of piezo-visco medium nanoshell resonator (PVM-NSR) based on functionally graded (FG) cylindrical nanoshell integrated with two piezoelectric layers subjected to visco-pasternak medium, electrostatic and harmonic excitations is investigated. Nonclassical method of the electro-elastic Gurtin–Murdoch surface/interface theory with von-Karman–Donnell's shell model as well as Hamilton's principle, the assumed mode method combined with Lagrange–Euler's are considered. Complex averaging method combined with arc-length continuation is used to achieve a numerical solution for the steady state vibrations of the system. The stability analysis of the steady state response is performed. The parametric studies such as the effects of different boundary conditions, different geometric ratios, structural parameters, electrostatic and harmonic excitation on the nonlinear frequency response and stability analysis are studied. The results indicate that near the natural frequency of the nanoshell, it will lead to resonance and will have large motion amplitude and near the resonant frequency, the nanoshell shows a softening type of nonlinear behavior, and the nanoshell bandwidth increases due to nonlinear factors. In this range, nanoshell has three different ranges of motion, of which two are stable and the other unstable, and so the jump phenomenon and saddle-node bifurcation are visible in the behavior of the system. Also piezoelectric voltage influences on static deformation and resonant frequency but has no significant effect on nonlinear behavior and bandwidth and also system very sensitive to the damping coefficient and due to decrease of nano shell stiffness, natural frequency decreases. And also, increasing or decreasing of some parameters lead to increasing or decreasing the resonance amplitude, resonant frequency, the system's instability, nonlinear behavior and bandwidth.     

Fractional derivative and time delay damper characteristics in Duffing–van der Pol oscillators

In this paper, we investigate the damping characteristics of two Duffing–van der Pol oscillators having damping terms described by fractional derivative and time delay respectively. The residue harmonic balance method is presented to find periodic solutions. No small parameter is assumed. Highly accurate limited cycle frequency and amplitude are captured. The results agree well with the numerical solutions for a wide range of parameters. Based on the obtained solutions, the damping effects of these two oscillators are investigated. When the system parameters are identical, the steady state responses and their stability are qualitatively different. The initial approximations are obtained by solving a few harmonic balance equations. They are improved iteratively by solving linear equations of increasing dimension. The second-order solutions accurately exhibit the dynamical phenomena when taking the fractional derivative and time delay as bifurcation parameters respectively. When damping is described by time delay, the stable steady state response is more complex because time delay takes past history into account implicitly. Numerical examples taking time delay and fractional derivative are respectively given for feature extraction and convergence study.