The Extended Non-Elementary Amplitude Functions as Solutions to the Damped Pendulum Equation, the Van der Pol Equation, the Damped Duffing Equation, the Lienard Equation and the Lorenz Equations

In this paper, we define some non-elementary amplitude functions that are giving solutions to some well-known second-order nonlinear ODEs and the Lorenz equations, but not the chaos case. We are giving the solutions a name, a symbol and putting them into a group of functions and into the context of other functions. These solutions are equal to the amplitude, or upper limit of integration in a non-elementary integral that can be arbitrary. In order to define solutions to some short second-order nonlinear ODEs, we will make an extension to the general amplitude function. The only disadvantage is that the first derivative to these solutions contains an integral that disappear at the second derivation. We will also do a second extension: the two-integral amplitude function. With this extension we have the solution to a system of ODEs having a very strange behavior. Using the extended amplitude functions, we can define solutions to many short second-order nonlinear ODEs.

An internally damped inertial platform for marine gravimetry and a test case in the South China Sea

To dampen periodic off-levelling motions within an inertial platform while undergoing horizontal accelerations of the same period and to achieve a levelling accuracy of a few tens of arcseconds with that system,an internally damped inertial platform for a marine scalar gravity system was the developed.Methods for attenuating horizontal acceleration and reducing off-levelling error by a satisfactory gyro-levelling loop,which are fundamental to the internally damped inertial platform,were designed and implemented.In addition,phase delays are introduced by the levelling loop.The resulting off-levelling gravity errors were analyzed and modeled.A series of tests on a motion simulator were performed in laboratory for a variety of simulated sea conditions.We found that the motion of the platform is a function of the amplitude and period of the simulated ship motions and ranges between 10 and 40 arcseconds.In addition,the phase lag between platform motion and ship motion is not constant but ranges 180°-270°,depending on the period and amplitude of the motion.Then,the platform,on which a gravimeter was mounted,was installed on the R/V Shiyan 2 to conduct a gravity survey in the South China Sea.Despite rough sea conditions,it was shown that in short periods of 2-30 s,the off-levelling angle was less than 30 arcseconds,and the phase lagged the horizontal acceleration by 230°-260°.From a repeated survey line and intersecting survey points,the estimated errors of gravity measurements were between 1.3 and 1.7 mGal.The marine measurements results were compared with those of satellite altimetry data and show a mean value of 0.5 mGal in a standard deviation of 1.5 mGal.

THE ANALYTICAL APPROACH FOR LARGE SCALE NON-CLASSICALLY DAMPED DYNAMIC SYSTEM

This paper gives a dynamic decoupling approach for the analysis of large scale non-classically damped system, in which the complex variable computations were completely avoided not only in solving for the eigenvalue problem but also in the calculation of the dynamic response. The analytical approaches for undamped gyroscopic system, non-classically damped system, including the damped gyroscopic system were unified. Very interesting and useful theoretical results, practical algorithms were obtained which are applicable to both non-defective and defective systems.

Damping characteristics of friction damped braced frame and its effectiveness in the mega-sub controlled structure system

Based on energy dissipation and structural control principle, a new structural configuration, called the megasub controlled structure （MSCS） with friction damped braces （FDBs）, is first presented. Meanwhile, to calculate the damping coefficient in the slipping state a new analytical method is proposed. The damping characteristics of one-storey friction damped braced frame （FDBF） are investigated, and the influence of the structural parameters on the energy dissipation and the practical engineering design are discussed. The nonlinear dynamic equations and the analytical model of the MSCS with FDBs are established. Three building structures with different structural configurations, which were designed with reference to the conventional mega-sub structures such as used in Tokyo City Hall, are comparatively investigated. The results illustrate that the structure presented in the paper has excellent dynamic properties and satisfactory control effectiveness.

Dynamic characteristics of a novel damped outrigger system

This paper presents exact analytical solutions for a novel damped outrigger system, in which viscous dampers are vertically installed between perimeter columns and the core of a high-rise building. An improved analytical model is developed by modeling the effect of the damped outrigger as a general rotational spring acting on a Bernoulli-Euler beam. The equivalent rotational spring stiffness incorporating the combined effects of dampers and axial stiffness of perimeter columns is derived. The dynamic stiffness method(DSM) is applied to formulate the governing equation of the damped outrigger system. The accuracy and effi ciency are verifi ed in comparison with those obtained from compatibility equations and boundary equations. Parametric analysis of three non-dimensional factors is conducted to evaluate the infl uences of various factors, such as the stiffness ratio of the core to the beam, position of the damped outrigger, and the installed damping coeffi cient. Results show that the modal damping ratio is signifi cantly infl uenced by the stiffness ratio of the core to the column, and is more sensitive to damping than the position of the damped outrigger. The proposed analytical model in combination with DSM can be extended to the study of structures with more outriggers.

A hybrid inversion method of damped least squares with simulated annealing used for Rayleigh wave dispersion curve inversion

Surface wave methods are becoming increasingly popular in many geotechnical applications and in earthquake seismology due to their noninvasive characteristics.Inverse surface wave dispersion curves are a crucial step in most surface wave methods.Many inversion methods have been applied to surface wave dispersion curve inversion,including linearized inversion and nonlinearized inversion methods.In this study,a hybrid inversion method of Damped Least Squares(DLS) with Very Fast Simulated Annealing(VFSA) is developed for multi-mode Rayleigh wave dispersion curve inversion.Both synthetic and in situ fi eld data were used to verify the validity of the proposed method.The results show that the proposed method is superior to the conventional VFSA method in aiming at global minimum,especially when parameter searching space is adjacent to real values of the parameters.The advantage of the new method is that it retains both the merits of VFSA for global search and DLS for local search.At high temperatures,the global search dominates the runs,while at a low temperatures,the local search dominates the runs.Thus,at low temperatures,the proposed method can almost directly approach the actual model.

GLOBAL SOLUTIONS AND FINITE TIME BLOW UP FOR DAMPED KLEIN-GORDON EQUATION

We study the Cauchy problem of strongly damped Klein-Gordon equation. Global existence and asymptotic behavior of solutions with initial data in the potential well are derived. Moreover, not only does finite time blow up with initial data in the unstable set is proved, but also blow up results with arbitrary positive initial energy are obtained.

A method for stochastic seismic response analysis of non-classically damped structures

A method to calculate the stationary random response of a non-classically damped structure is proposed that features clearly-defined physical meaning and simple expression. The method is developed in the frequency domain. The expression of the proposed method consists of three terms, i.e., modal velocity response, modal displacement response, and coupled (between modal velocity and modal displacement response). Numerical results from the parametric study and three example structures reveal that the modal velocity response term and the coupled term are important to structural response estimates only for a dynamic system with a tuned mass damper. In typical cases, the modal displacement term can provide response estimates with satisfactory accuracy by itself, so that the modal velocity term and coupled term may be ignored without loss of accuracy. This is used to simplify the response computation of non-classically damped structures. For the white noise excitation, three modal correlation coefficients in closed form are derived. To consider the modal velocity response term and the coupled term, a simplified approximation based on white noise excitation is developed for the case when the modal velocity response is important to the structural responses. Numerical results show that the approximate expression based on white noise excitation can provide structural responses with satisfactory accuracy.

Dynamic analysis of a deployable/retractable damped cantilever beam

Deployable/retractable damped cantilever beams are a class of time-varying parametric structures which have attracted considerable research interest due to their many potential applications in the intelligent robot field and aerospace.In the present work,the dynamic characteristics of a deployable/retractable damped cantilever beam are investigated experimentally and theoretically.The time-varying damping,as a function of the beam length,is obtained by both the enveloped fitting method and the period decrement method.Furthermore,the governing equation of the deployable/retractable damped cantilever beam is derived by introducing the time-varying damping parameter,and the corresponding closed-form solution and vibration principles are investigated based on the averaged method.The theoretical predictions for transient dynamic responses are in good agreement with the experimental results.The dynamic mechanism analysis on time-varying damping offers flexible technology in mechanical and aerospace fields.

Derivatives of repeated eigenvalues and corresponding eigenvectors of damped systems

A procedure is presented for computing the derivatives of repeated eigenvalues and the corresponding eigenvectors of damped systems. The derivatives are calculated in terms of the eigenvalues and eigenvectors of the second-order system, and the use of rather undesirable state space representation is avoided. Hence the cost of computation is greatly reduced. The efficiency of the proposed procedure is illustrated by considering a 5-DOF non-proportionally damped system.

Sliding Mode Control of Vibration in Single-Degree-of-Freedom Exponentially Damped Oscillator

The purpose is to design the control method for a single-degree-of-freedom(SDOF)exponentially damped oscillator.Based on the Lyapunov stability theory,sliding mode control and adaptive sliding mode control have been proposed.Sliding control laws and adaptive sliding laws are designed for exponentially damped oscillator respectively in cases that the bound of the external exciting force is known or unknown.The viability and effectiveness of the above control designs have been validated by numerical simulations.

Asymptotic Behavior of Stochastic Strongly Damped Wave Equation with Multiplicative Noise

In this paper we study the asymptotic dynamics of the stochastic strongly damped wave equation with multiplicative noise under homogeneous Dirichlet boundary condition. We investigate the existence of a compact random attractor for the random dynamical system associated with the equation.

Uniform Exponential Attractors for Non-Autonomous Strongly Damped Wave Equations

In this paper, we study the existence of exponential attractors for strongly damped wave equations with a time-dependent driving force. To this end, the uniform H?lder continuity is established to the variation of the process in the phase apace. In a certain parameter region, the exponential attractor is a uniformly exponentially attracting time-dependent set in the phase apace, and is finite-dimensional no matter how complex the dependence of the external forces on time is. On this basis, we also obtain the existence of the infinite-dimensional uniform exponential attractor for the system.

Damped Harmonic Oscillator with Arduino

The benefits of using experiments in physics classes are widely discussed in the literature, but sometimes experimental setups are not available. In this paper we present different ways of using experiments in physics classes based on the Arduino board, since it involves low cost materials and can be built by the own students in several cases. In this work we addressed the well known damped harmonic oscillator and performed the data acquisition through the Arduino board, a LDR (Light Dependent Resistor), a infrared photodiode sensor and a computer. The setup of the proposed experiment and the technical details related to assembly are discussed in a clear way in order to be reproduced by anyone interested in the subject. We found a significant difference in the results obtained through the LDR and the photodiode. The later has given better results and has reproduced a regular decay in the amplitude of the oscillator even when the experiment was performed in a highly illuminated room. The Arduino board, alongside the referred peripherals, has shown great potential for building low cost experimental setups to be used in physics classes, both for expositive and hands on approaches.

The Global Attractors for the Higher-Order Kirchhoff-Type Equation with Nonlinear Strongly Damped Term

We investigate the global well-posedness and the global attractors of the solutions for the Higher-order Kirchhoff-type wave equation with nonlinear strongly damping: . For strong nonlinear damping σ and ?, we make assumptions (H1) - (H4). Under of the proper assume, the main results are existence and uniqueness of the solution in proved by Galerkin method, and deal with the global attractors.

The Effect of Mass Ratio and Air Damper Characteristics on the Resonant Response of an Air Damped Dynamic Vibration Absorber

In this paper, it is shown that, a road vehicle 2DOF air damped quartercar suspension system can conveniently be transformed into a 2DOF air damped vibrating system representing an air damped dynamic vibration absorber (DVA) with an appropriate change in the ratio μ of the main mass and the absorber mass i.e. when mass ratio μ >> 1. Also the effect of variation of the mass ratio, air damping ratio and air spring rate ratio, on the motion transmissibility at the resonant frequency of the main mass of the DVA has been dis- cussed. It is shown that, as the air damping ratio in the absorber system increases, there is a substantial decrease in the motion transmissibility of the main mass system where the air damper has been modeled as a Maxwell type. Optimal value of the air damping ratio for the minimum motion transmissibility of the main mass of the system has been determined. An experimental setup has been designed and developed with a control system to vary air pressure in the damper in the absorber system. The motion transmissibility characteristics of the main mass system have been obtained, and the optimal value of the air damping ratio has been determined for minimum motion transmissibility of the main mass of the

The Global and Pullback Attractors for a Strongly Damped Wave Equation with Delays

In this paper, we study the global and pullback attractors for a strongly damped wave equation with delays when the force term belongs to different space. The results following from the solution generate a compact set.

Second order conformal multi-symplectic method for the damped Korteweg–de Vries equation

A conformal multi-symplectic method has been proposed for the damped Korteweg–de Vries(DKdV) equation, which is based on the conformal multi-symplectic structure. By using the Strang-splitting method and the Preissmann box scheme,we obtain a conformal multi-symplectic scheme for multi-symplectic partial differential equations(PDEs) with added dissipation. Applying it to the DKdV equation, we construct a conformal multi-symplectic algorithm for it, which is of second order accuracy in time. Numerical experiments demonstrate that the proposed method not only preserves the dissipation rate of mass exactly with periodic boundary conditions, but also has excellent long-time numerical behavior.

Complex complete quadratic combination method for damped system with repeated eigenvalues

A new response-spectrum mode superposition method, entirely in real value form, is developed to analyze the maximum structural response under earthquake ground motion for generally damped linear systems with repeated eigenvalues and defective eigenvectors. This algorithm has clear physical concepts and is similar to the complex complete quadratic combination （CCQC） method previously established. Since it can consider the effect of repeated eigenvalues, it is called the CCQC-R method, in which the correlation coefficients of high-order modal responses are enclosed in addition to the correlation coefficients in the normal CCQC method. As a result, the formulas for calculating the correlation coefficients of high-order modal responses are deduced in this study, including displacement, velocity and velocity-displacement correlation coefficients. Furthermore, the relationship between high-order displacement and velocity covariance is derived to make the CCQC-R algorithm only relevant to the high-order displacement response spectrum. Finally, a practical step-by-step integration procedure for calculating high-order displacement response spectrum is obtained by changing the earthquake ground motion input, which is evaluated by comparing it to the theory solution under the sine-wave input. The method derived here is suitable for generally linear systems with classical or non-classical damping.

Optimal Control Designs for a Class of Nonviscously Damped Systems

A scheme for optimal control problem formulation of a class of nonviscously damped system using pseudo-state-space formulation is presented in this paper.Taking the exponentially damped system of nonviscously damped system as an example,a new state space representation is applied to the n-order exponentially damped system.The necessary conditions for the optimality of the exponentially damped system are obtained.A numerical example of a single degree of freedom exponentially damped system is considered to illustrate the methodology and technique.