Analytical solution of the damped Helmholtz–Duffing equation

In this paper, we derive a class of analytical solution of the damped Helmholtz–Duffing oscillator that is based on a recently developed exact solution for the undamped case. Our solution procedure indicates that this solution holds for specific system parametric choice values.     

Analytic expansion of the lyapunov exponent associated to the SchrÖDinger operator with random potential

The invariant measure and Lyapunov exponent associated to the one–dimensional Schrödinger operator with a random potential (or, in other words, to the damped linear oscillator with random restoring force) are studied for small real noise (diffusions). Analytic expression are given via perturbation expansion. As a by-product, the well-known positivity of the Lyapunov exponent (in the undamped case) is reproved     

Observation and investigation of a non-monotonic period function for an impact oscillator

A mathematical model for a bench-top impact oscillator (a steel pendulum and an aluminum barrier) that incorporates Hertzian contact is analyzed, and predictions derived from our model are compared with experimental data. We report our discovery of a new effect: the existence of a non-monotone period function for the annulus of periodic orbits in the phase plane surrounding the rest point that corresponds to the downward vertical position of the pendulum in the unforced, undamped case. From this non-monotonicity, we predict novel resonance response behavior for the forced, damped oscillator and verify our predictions by experiment.     

Dynamic magnification factors of SDOF oscillators under harmonic loading

The magnification factor for the steady-state response of a SDOF system under harmonic loading is described in many structural dynamics textbooks; the well-known analytical solution is easily obtained from the solution to the damped equation of motion for harmonic loading. The complete and steady-state solutions can differ significantly. An analytical expression for the maximum response to the complete solution (steady state plus transient) remains elusive; however, a simple analytical expression is identified herein for the undamped case. Differences in the magnification factors obtained for the two solutions are discussed.     

Impulsive periodic and quasi-periodic orbits of coupled oscillators with essential stiffness nonlinearity

We study the impulsive responses of a grounded linear oscillator coupled to a light nonlinear attachment through an essentially nonlinear (nonlinearizable) stiffness. We analyze the periodic and quasi-periodic dynamics of the undamped system forced by a single impulse on the linear oscillator and being initially at rest, by considering separately low-, moderate- and high-energy impulsive motions. The motivation for studying the impulsive dynamics of this system centers on passive targeted energy transfer properties of the corresponding weakly damped one, that is, of the possibility of one-way, irreversible transfer of energy from the linear oscillator to the nonlinear attachment. A rather surprising aspect of this work is the complexity of the analysis required to study the impulsive dynamics of this system, due to its high degeneracy, as it undergoes a co-dimension three bifurcation.     

Comparison between the shifted-Laplacian preconditioning and the controllability methods for computational acoustics

Processes that can be modelled with numerical calculations of acoustic pressure fields include medical and industrial ultrasound, echo sounding, and environmental noise. We present two methods for making these calculations based on Helmholtz equation. The first method is based directly on the complex-valued Helmholtz equation and an algebraic multigrid approximation of the discretized shifted-Laplacian operator; i.e. the damped Helmholtz operator as a preconditioner. The second approach returns to a transient wave equation, and finds the time-periodic solution using a controllability technique. We concentrate on acoustic problems, but our methods can be used for other types of Helmholtz problems as well. Numerical experiments show that the control method takes more CPU time, whereas the shifted-Laplacian method has larger memory requirement.     

Stochastic Schrodinger Equation for a Quantum Oscillator with Dissipation

In this paper, we construct an exact solution of the stochastic Schrodinger equation for a quantum oscillator with possible dissipation of energy taken into account. Using the explicit form of the solution, we calculate estimates for the characteristic damping time of free damped oscillations. In the case of forced oscillations, we obtain formulas for the Q-factor of the system and for the variance of the coordinate and momentum of a quantum oscillator with dissipation. We obtain the quantum analog of the classical diffusion equation and explicitly show that the equations of motion for the mean value of the momentum operator following from the solution of the stochastic Schrodinger equation play the role of the quantum Langevin equation describing Brownian motion under the action of a stochastic force.     

On the Green’s function for the Helmholtz operator in an impedance circular cylindrical waveguide

This paper addresses the problem of finding a series representation for the Green’s function of the Helmholtz operator in an infinite circular cylindrical waveguide with impedance boundary condition. Resorting to the Fourier transform, complex analysis techniques and the limiting absorption principle (when the undamped case is analyzed), a detailed deduction of the Green’s function is performed, generalizing the results available in the literature for the case of a complex impedance parameter. Procedures to obtain numerical values of the Green’s function are also developed in this article.     

GENERALIZED RICCATI TRANSFORMATION AND OSCILLATION FOR LINEAR DIFFERENTIAL EQUATIONS WITH DAMPING

Using generalized Riccati transformation, some new oscillation criteria for damped linear differential equations are established. These results improve and generalize some known oscillation criteria due to A.Wintner, I.V.Kamenev for the undamped linear differential equations, and Sobol, J.S.W.Wong for the damped linear differential equations.     

On the relationship between love modes and the set of supercritically reflected waves

Considering the Love problem as an example, we derive relations connecting the following two exact integral representations of its solution: one explicitly involving both damped and undamped modes (residues at the roots of the dispersion equation of the problem) and the other based on expanding the interference field into a series of a geometric progression. In the latter case, to each such summand a generalized ray of a wave of certain multiplicity propagating in the layer can be put into correspondence. By using the methods of contour integrals, a correspondence between the set of multiple waves and interference modes is established. Bibliography: 1 title. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 214, 1994, pp. 200–209. Translated by T. N. Surkova.     

Remarks on stabilization of second-order evolution equations by unbounded dynamic feedbacks

In this paper, we consider second-order evolution equations with unbounded dynamic feedbacks. Under a regularity assumption, we show that observability properties for the undamped problem imply decay estimates for the damped problem. We consider both uniform and non-uniform decay properties.     

偏Riccati型不等式和具阻尼项p-Laplacian型PDE的振动定理

本文建立了具阻尼项的p-Laplacian型PDE的偏Riccati型不等式,并利用该不等式得到了振动定理,推广了Philos线性ODE的结果,并改进了已有的无阻尼p-Laplacian型PDE的振动准则.     

From resolvent estimates to damped waves

In this paper, we show how to obtain decay estimates for the damped wave equation on a compact manifold without geometric control via knowledge of the dynamics near the undamped set. We show that if replacing the damping term with a higher-order complex absorbing potential gives an operator enjoying polynomial resolvent bounds on the real axis, then the “resolvent” associated to our damped problem enjoys bounds of the same order. It is known that the necessary estimates with complex absorbing potential can also be obtained via gluing from estimates for corresponding non-compact models.     

A mapping strategy for the identification of structural systems

In this paper an alternative approach for identification problems is discussed. Unlike existing methods, this new approach combines in a general way finite differences and function approximation and is herein used for the identification of a particular system in structural dynamics, that is the damped Duffing oscillator subject to a swept-sine excitation. The solution obtained by means of the proposed method has been compared with the one obtained by a neural network. The present method gives better results at a low computational cost, with the advantage of solutions in explicit form. Besides, it is possible to prove that the solutions are stable and that from this new approach one can deduce, as a particular case, the approximation previously proposed by other authors.     

Integral representations in quaternionic analysis related to the helmholtz operator

As the Laplacian also the Helmholtz operator can be factorized in Clifford analysis. Using the fundamental solution of the Helmholtz equation, a fundamental solution can be constructed for the factors of the Helmholtz operator. These are used in the case of quaternions to prove Cauchy–Pompeiu type representation formulas in terms of powers of the factors as well as in terms of powers of the Helmholtz operator.     

Integrated mathematical forms for update of physical parameter matrices of damped dynamic system

Mathematical modelling and updating of damped dynamic systems that involve some modelling errors and subsequent analysis based on those errors will lead to inaccuracy in the results. Because measured and analytical data are unlikely to be identical due to measurement noise and model inadequacies, it is necessary to estimate more accurate parameter matrices for design and analysis. By minimizing a cost function expressed as the sum of the norms of the difference between analytical and experimental parameter matrices, this study directly derives the integrated mathematical expressions for updated physical parameter matrices. In the derivation process, the eigenfunction of a damped dynamic system is utilized as a constraint equation for the updating. It is illustrated that the proposed methods take more explicit forms and can be widely utilized in the damped and undamped systems. Based on the comparison with other methods, the validity of the proposed methods is demonstrated in numerical applications.     

分数阶振子方程基于变分迭代的近似解析解序列

在粘弹性介质中的阻尼振动中引入分数阶微分算子,建立分数阶非线性振动方程.使用了分数阶变分迭代法（FVIM）,推导了Lagrange乘子的若干种形式.对线性分数阶阻尼方程,分别对齐次方程和正弦激励力的非齐次方程应用FVIM得到近似解析解序列.以含激励的Bagley-Torvik方程为例,给出不同分数阶次的位移变化曲线.研究了振子运动与方程中分数阶导数阶次的关系,这可由不同分数阶次下记忆性的强弱来解释.计算方法上,与常规的FVIM相比,引入小参数的改进变分迭代法能够大大扩展问题的收敛区段.最后,以一个含分数导数的Van der Pol方程为例说明了FVIM方法解决非线性分数阶微分问题的有效性和便利性.     

Energy decay for the modified Kawahara equation posed in a bounded domain

Studied here is the eventual dissipation of solutions to initial–boundary value problems for the modified Kawahara equation with and without a localized damping term included. It is shown that solutions of undamped problems posed on a bounded interval may not decay if the length of the interval is critical. In contrast, the energy associated to the locally damped problems is shown to be exponentially decreased independently of the interval length.     

Sharp energy estimates for nonlinearly locally damped PDEs via observability for the associated undamped system

We consider the problem of sharp energy decay rates for nonlinearly damped abstract infinite-dimensional systems. Direct methods for nonlinear stabilization generally rely on multiplier techniques, and thus are valid under restrictive geometric conditions compared to the optimal geometric optics condition of Bardos et al. (1992) [10]. We prove sharp, simple and quasi-optimal energy decay rates through an indirect method, namely an observability estimate for the corresponding undamped system. One of the main advantage of these results is that they allow to combine optimal geometric conditions, as for instance that of Bardos et al. (1992) [10] and the optimal-weight convexity method of the first author (Alabau-Boussouira, 2010 [6], Alabau-Boussouira, 2005 [2]) to deduce very simple and quasi-optimal energy decay rates for nonlinearly locally damped systems. We also show that using arguments based on Russell's principle (Russell, 1978 [24]), one can deduce sharp energy decay rates from the exponential stabilization of the linearly damped system. Our results extend to nonlinearly damped systems, those of Haraux (1989) [14] and Ammari and Tucsnak (2001) [9] which concern linearly damped systems.