TRANSIENT DYNAMIC RESPONSE ANALYSIS OF ORTHOTROPIC CIRCULAR CYLINDRICAL SHELL UNDER EXTERNAL HYDROSTATIC PRESSURE

Following Flügge's exact derivation for the buckling of cylindrical shells, the equations of motion for transient dynamic loading of orthotropic circular cylindrical shells under external hydrostatic pressure have been formulated. The normal mode theory is used to provide transient dynamic response for the equations of motion. The effect of shell's parameters, external hydrostatic pressure and material properties on the shell response has been studied in detail. A part of tables and figures are given in this paper.     

On the occurrence of flutter in the lateral-torsional instabilities of circular arches under follower loads

The lateral-torsional stability of circular arches subjected to radial and follower distributed loading is treated herein. Three loading cases are studied, including the radial load with constant direction, the radial load directed towards the arch centre, and the follower radial load (hydrostatic load), as treated by Nikolai in 1918. For the three cases, the buckling loads are first obtained from a static analysis. As the case of the follower radial load (hydrostatic load) is a non-conservative problem, the dynamic approach is also used to calculate the instability load. The governing equations for out-of-plane vibrations of circular arches under radial loading are then derived, both with and without Wagner's effect. Flutter instabilities may appear for sufficiently large values of opening angle, but flutter cannot occur before divergence for the parameters of interest (civil engineering applications). Therefore, it is concluded that the static approach necessarily leads to the same result as the dynamic approach, even in the non-conservative case.     

一类含时变间隙的强非线性相对转动系统分岔和混沌

建立一类具有时变间隙的两质量相对转动系统的强非线性动力学方程.应用MLP方法求解出变换参数,并运用多尺度法求解该系统发生1/2亚谐共振时的分岔响应方程,采用奇异性理论分析得到系统稳态响应的转迁集,并且得到系统在非自治情形下的分岔特性以及系统的分岔形态.最后通过数值仿真得到系统在间隙和阻尼参数变化下的分岔和混沌行为,发现随着系统参数变化系统将出现周期运动、倍周期运动以及混沌等多种不同的运动形态.     

一类谐波激励相对转动非线性动力系统的稳定性与近似解

建立具有一般非线性弹性力、广义摩阻力和谐波激励的一类相对转动非线性动力系统的动力学方程. 对相对转动非线性自治系统进行定性分析,通过构造Lyapunov函数研究自治系统奇点的稳定性. 运用多尺度法求解谐波激励下非自治系统在几种不同共振响应下的近似解,同时分析了主振系统稳态运动的稳定性. 关键词： 相对转动 非线性动力系统 Lyapunov函数 稳定性     

A vibration isolation system in low frequency excitation region using negative stiffness structure for vehicle seat

This paper designs and fabricates a vibration isolation model for improving vibration isolation effectiveness of the vehicle seat under low excitation frequencies. The feature of the proposed system is to use two symmetric negative stiffness structures (NSS) in parallel to a positive stiffness structure. Here, theoretical analysis of the proposed system is clearly presented. Then, the design procedure is derived so that the resonance peak of frequency-response curve drifts to the left, the load support capacity of the system is maintained, the total size of the system is reduced for easy practical application and especially, the bending of the frequency-response curve is minimized. Next the dynamic equation of the proposed system is set up. Then, the harmonic balance (HB) method is employed to seek the characteristic of the motion transmissibility of the proposed system at the steady state for each of the excitation frequency. From this characteristic, the curves of the motion transmission are predicted according to the various values of the configurative parameters of the system. Then, the time responses to the sinusoidal, multi frequency and random excitations are also investigated by simulation and experiment. In addition, the isolation performance comparison between the system with NSS and system without NSS is realized. The simulation results reveal that the proposed system has larger frequency region of isolation than that of the system without NSS. The experimental results confirm also that with a random excitation mainly spreading from 0.1 to 10 Hz, the isolation performance of the system with NSS is greatly improved, where the RMS values of the mass displacement may be reduced to 67.2%, whereas the isolation performance of the system without NSS is bad. Besides, the stability of the steady-state response is also studied. Finally, some conclusions are given.     

一类非线性相对转动系统的组合谐波分岔行为研究

针对一类具有非线性刚度、非线性阻尼的非线性相对转动系统, 应用耗散系统的拉格朗日原理建立在组合谐波激励作用下非线性相对转动系统的动力学方程. 构造李雅普诺夫函数, 分析相对转动系统的稳定性, 研究自治系统的分岔特性. 应用多尺度法求解相对转动系统的非自治系统在组合激励作用下的分岔响应方程. 最后采用数值仿真方法, 通过分岔图、时域波形、相平面图、Poincaré截面图等研究外扰激励、系统阻尼、 非线性刚度对相对转动系统经历倍周期分岔进入混沌运动的影响. 关键词： 相对转动 组合激励 分岔 混沌     

A comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach

Large-amplitude (geometrically non-linear) vibrations of circular cylindrical shells subjected to radial harmonic excitation in the spectral neighbourhood of the lowest resonances are investigated. The Lagrange equations of motion are obtained by an energy approach, retaining damping through Rayleigh's dissipation function. Four different non-linear thin shell theories, namely Donnell's, Sanders-Koiter, Flügge-Lur’e-Byrne and Novozhilov's theories, which neglect rotary inertia and shear deformation, are used to calculate the elastic strain energy. The formulation is also valid for orthotropic and symmetric cross-ply laminated composite shells. The large-amplitude response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of the lowest natural frequency is computed for all these shell theories. Numerical responses obtained by using these four non-linear shell theories are also compared to results obtained by using the Donnell's non-linear shallow-shell equation of motion. A validation of calculations by comparison with experimental results is also performed. Both empty and fluid-filled shells are investigated by using a potential fluid model. The effects of radial pressure and axial load are also studied. Boundary conditions for simply supported shells are exactly satisfied. Different expansions involving from 14 to 48 generalized co-ordinates, associated with natural modes of simply supported shells, are used. The non-linear equations of motion are studied by using a code based on an arclength continuation method allowing bifurcation analysis.     

On the nonlinear axisymmetric dynamic buckling behavior of clamped functionally graded spherical caps

Here, the dynamic thermal buckling behavior of functionally graded spherical caps is studied considering geometric nonlinearity based on von Karman's assumptions. The formulation is based on first-order shear deformation theory and it includes the in-plane and rotary inertia effects. The material properties are graded in the thickness direction according to the power-law distribution in terms of volume fractions of the material constituents. The effective material properties are evaluated using homogenization method. The governing equations obtained using finite element approach are solved employing the Newmark's integration technique coupled with a modified Newton–Raphson iteration scheme. The pressure load corresponding to a sudden jump in the maximum average displacement in the time history of the shell structure is taken as the dynamic buckling load. The present model is validated against the available isotropic case. A detailed numerical study is carried out to highlight the influences of shell geometries, power law index of functional graded material and boundary conditions on the dynamic buckling load of shallow spherical shells.     

The dynamic analysis of circular plates and shallow spherical shells

In the investigation reported here an attempt has been made to study the influence of Berger's approximation on the non-linear transient response of circular plates and shallow spherical shells. The governing equations of motion obtained from Berger's approximation are solved by using the rapidly converging Chebyshev series spacewise and the Houbolt scheme for integration in the time domain. Results calculated when using Berger's approximation are compared with exact results. It is shown that Berger's method yields very accurate values for plates and shells under transient loading, in the case of immovable edge conditions.     

Non-trivial effect of fast vibration on the dynamics of a class of non-linearly damped mechanical systems

The effect of very high-frequency excitation on the slow dynamics of a class of non-linearly damped mechanical oscillators is considered. Two different models of damping namely, piecewise linear and pth power damping are considered. Fast excitation is modelled as triangular, sinusoidal and random base excitation. The effect of fast excitation is theoretically analyzed using the method of direct partition of motion (MDPM) and direct simulation. The method of numerical averaging is also used, where damping characteristics or excitations are not amenable to analytical techniques. Fast excitation has the non-trivial effect of increasing and decreasing the low-velocity damping of hard and soft dampers, respectively. The effect of fast excitation on the transient and steady state slow dynamics of the system is investigated by direct numerical integration of the equation of motion.     

A method of stability analysis for non-linear vibration of beams

In this paper, a method of stability analysis for the large amplitude, steady state response of a non-linear beam under periodic excitation is presented. The stability problem is investigated by studying the behavior of a small perturbation of the steady state response which results in a coupled Hill-type equation. The problem is transformed by the harmonic balance method into an eigenvalue problem of a non-symmetric matrix. The effectiveness and the accuracy of the proposed method for a Mathieu equation are examined and the application to the stability analysis of the non-linear vibrations of a beam is presented.     

Nonlinear resonant behavior of microbeams over the buckled state

The present study investigates the nonlinear resonant behavior of a microbeam over its buckled (non-trivial) configuration. The system is assumed to be subjected to an axial load along with a distributed transverse harmonic load. The axial load is increased leading the system to lose the stability via a pitchfork bifurcation; the postbuckling configuration is obtained and the nonlinear resonant response of the system over the buckled state is examined. More specifically, the nonlinear equation of motion is obtained employing Hamilton’s principle along with the modified couple stress theory. The continuous system is truncated into a system with finite degrees of freedom; the Galerkin scheme is employed to discretize the nonlinear partial differential equation of motion into a set of ordinary differential equations. This set of equations is solved numerically employing the pseudo-arclength continuation technique; first a nonlinear static analysis is performed upon this set of equations so as to obtain the onset of buckling (supercritical pitchfork bifurcation) and the buckled configuration of the microbeam. The frequency-response and force-response curves of the system are then constructed over the buckled configurations. A comparison is made between the frequency-response curves obtained by means of the modified couple stress and the classical theories. The effect of different system parameters on the frequency-response and force-response curves is also examined.     

耦合相对转动非线性动力系统的稳定性与近似解

研究了一类含三次非线性耦合项的相对转动非线性动力系统的动力学行为. 建立了具有非线性弹性力、广义摩阻力耦合项的系统动力学方程. 运用多尺度法求解谐波激励下耦合非自治系统的近似解,通过讨论系统的主共振和内共振特性,分析了耦合项对系统响应的影响. 应用奇异性理论研究了主振稳态响应分岔方程的稳定性,得到了系统的转迁集和分岔曲线的拓扑结构. 关键词： 相对转动 非线性耦合动力系统 奇异性理论 稳定性     

Vibration with “dynamic friction”

The free and harmonically forced vibration of an idealized straight beam oscillating in its fundamental mode with frictional clamps at each end is modelled by using a singular differential equation. Locking and chatter phenomena are analyzed and slipping motions integrated for the free system and the continuously slipping steady state response to harmonic excitation is qualitatively discussed.     

一类含Mathieu-Duffing振子的相对转动系统的分岔和混沌

建立了一类具有Mathieu-Duffing振子的两质量相对转动系统的非线性动力学方程. 应用多尺度法求解该系统发生主共振-基本参数共振的分岔响应方程,并通过奇异性分析得到系统稳态响应的转迁集. 利用Melnikov方法讨论系统在外激扰动和参激扰动变化下的全局分岔和系统进入混沌状态的可能途径,得到外激和参激幅值变化下系统可能出现多次通向混沌的道路,获得系统发生混沌的必要条件. 最后采用数值方法验证了理论研究的有效性. 关键词： 相对转动 Mathieu-Duffing振子 混沌 Melnikov方法     

The magneto-elastic subharmonic resonance of current-conducting thin plate in magnetic filed

Based on Maxwell equations and corresponding electromagnetic constitutive relations, the electrodynamic equations and electromagnetic force expressions of a current-conducting thin plate in electromagnetic field are deduced. Nonlinear magneto-elastic vibration equations of the thin plate are given. In addition, nonlinear subharmonic resonances of the thin plate with two opposite sides simply supported which is under the mechanic live loads and in constant transverse magnetic field are studied. The corresponding vibration differential equation of Duffing type is deduced by the Galerkin method. The method of multiple scales is used to solve the equation, and the frequency-response equation of the system in steady motion under subharmonic responses is obtained, and the stability of solution is analyzed. According to the Liapunov stability theory, the critical conditions of stability are obtained. By the numerical calculation, the curves of resonance amplitude changing with the detuning parameter, the excitation amplitude and the magnetic intensity and corresponding state planes are obtained. The existing regions of nontrivial solutions and the changing law of stable and unstable solutions are analyzed. The time history response plots, the phase charts and the Poincare mapping charts are plotted. And the effect of the magnetic intensity on the system is discussed, and some complex dynamic performances as period-doubling motion and quasi-period motion are analyzed.     

A simplified method for treating the dynamic response of embedded structures to SH-waves

A simplified method is proposed to treat the dynamic response of rigid embedded structures subjected to plane SH-waves. The soil-structure interface is discretized into a finite number of strips, resulting in an analysis which involves only plane surfaces and plane waves. The interaction force on any strip is determined easily and the contributions from all the strips are superposed to obtain the total force on the structure. The equation of motion then yields the steady state response of the structure. The Green function for the structure is obtained by Fourier synthesis. The transient response of the structure to any other excitation is then obtained by using the Green function and the Duhamel integral. Numerical examples are presented both to validate and illustrate the method.     

谐和与随机噪声联合作用Duffing单边约束系统的响应

研究了Duffing单边约束系统在谐和与随机噪声联合激励下的响应问题. 用谐波平衡法和摄动法分析了系统在确定性谐和激励和随机激励联合作用下的响应,用随机平均法讨论了随机扰动项对系统响应的影响. 在一定条件下,当约束距离较大时对应于不同的初始条件,系统具有两个非碰撞的稳态响应;而当约束距离不大时,对应于不同的初始条件,系统也可以有两个不同的稳态响应,其中一个是发生碰撞的响应,而另外一个则不发生碰撞. 数值模拟表明该方法是有效的. 关键词： Duffing单边约束系统 随机响应 谐波平衡法 摄动法     

一类非线性相对转动系统的混沌运动及多时滞反馈控制

建立一类含非线性粘滑摩擦力的两质量非线性相对转动系统的动力学方程. 研究此非线性相对转动系统在外激励作用下的混沌运动及多时滞反馈控制. 当系统在外激励作用下处于混沌状态时, 考虑引入多时滞反馈对系统的混沌运动进行控制. 应用Melnikov理论给出系统在Smale意义下的混沌临界条件, 研究了多时滞反馈对系统运动及混沌临界值的影响规律. 并结合系统相图、Poincare截面图和功率谱分析多时滞反馈参数对系统混沌运动的控制作用. 关键词： 多时滞 相对转动 控制 数值仿真     

Impacting under harmonic excitation

The steady state response of a one dimensional linear oscillator impacting under harmonic excitation on to rigid obstructions with a coefficient of velocity restitution is investigated. Numerical experiments are discussed and an expression of the steady state velocity for one sided impacting is derived. The variation of impacting velocity with excitation frequency for symmetric two sided impacting steady states is analyzed. It is shown that steady states cannot exist below a minimum frequency whilst for frequencies in the band for which the steady state amplitude exceeds the clearance, a unique steady state can exist. For greater frequencies, two possible steady states exist for a given excitation frequency.