ASYMMETRIC NON-LINEAR FORCED VIBRATIONS OF FREE-EDGE CIRCULAR PLATES. PART 1: THEORY

In this article, a detailed study of the forced asymmetric non-linear vibrations of circular plates with a free edge is presented. The dynamic analogue of the von Kàrmàn equations is used to establish the governing equations. The plate displacement at a given point is expanded on the linear natural modes. The forcing is harmonic, with a frequency close to the natural frequency ω_kn of one asymmetric mode of the plate. Thus, the vibration is governed by the two degenerated modes corresponding to ω_kn, which are one-to-one internally resonant. An approximate analytical solution, using the method of multiple scales, is presented. Attention is focused on the case where one configuration which is not directly excited by the load gets energy through non-linear coupling with the other configuration. Slight imperfections of the plate are taken into account. Experimental validations are presented in the second part of this paper.     

气轨上耦合振子的受迫振动实验

介绍了自行设计的气轨上两耦合振子的受迫振动实验，该实验物理现象非常下观，清晰，便于测量。且实验结果与理论相符，整个装置仅在原有 气垫导轨基础上增加一驱动器即可实现，效果甚佳，易于推广。     

FORCED AND FREE VIBRATIONS OF RECTANGULAR SANDWICH PLATES WITH PARAMETRIC STIFFNESS MODULATION

An active control of the resonant vibrations of a rectangular sandwich plate performed by the parametric stiffness modulation is analyzed. The controlled vibrations are those of the dominantly flexural type excited by the transverse force acting at the first resonant frequency of dominantly flexural vibrations. The stiffness modulation is performed at a comparatively high frequency identified by the resonance of a mode of the dominantly shear type. The method of direct partition of motions is used that predicts an existence of the modal interaction between these two modes of vibrations due to the parametric stiffness modulation. It is shown that such a parametric control can provide a significant shift of the first eigenfrequency of a controlled plate (the one subjected to the stiffness modulation) from its nominal value for an uncontrolled plate. Heavy fluid loading conditions are accounted for as well as the energy dissipation in the material of a plate. It is demonstrated that although heavy fluid loading reduces resonant frequencies of forced vibrations, the suggested mechanism of control remains valid in these cases. Dynamics of an elementary two-degree-of-freedom model mechanical system is considered to illustrate the mechanism of modal interaction, which is involved in the suggested way of an active control of vibrations of sandwich plates.     

NON-LINEAR STEADY STATE VIBRATIONS OF BEAMS EXCITED BY VORTEX SHEDDING

In this paper the non-linear vibrations of beams excited by vortex-shedding are considered. In particular, the steady state responses of beams near the synchronization region are taken into account. The main aerodynamic properties of wind are described by using the semi-empirical model proposed by Hartlen and Currie. The finite element method and the strip method are used to formulate the equation of motion of the system treated. The harmonic balance method is adopted to derive the amplitude equations. These equations are solved with the help of the continuation method which is very convenient to perform the parametric studies of the problem and to determine the response curve in the synchronization region. Moreover, the equations of motion are also integrated using the Newmark method. The results of calculations of several example problems are also shown to confirm the efficiency and accuracy of the presented method. The results obtained by the harmonic balance method and by the Newmark methods are in good agreement with each other.     

ASYMPTOTIC APPROACH FOR NON-LINEAR PERIODICAL VIBRATIONS OF CONTINUOUS STRUCTURES

An asymptotic approach for determining periodic solutions of non-linear vibration problems of continuous structures (such as rods, beams, plates, etc.) is proposed. Starting with the well-known perturbation technique, the independent displacement and frequency is expanded in a power series of a natural small parameter. It leads to infinite systems of interconnected non-linear algebraic equations governing the relationships between modes, amplitudes and frequencies. A non-trivial asymptotic technique, based on the introduction of an artificial small parameter is used to solve the equations. An advantage of the procedure is the possibility to take into account a number of vibration modes. As examples, free longitudinal vibrations of a rod and lateral vibrations of a beam under cubically non-linear restoring force are considered. Resonance interactions between different modes are investigated and asymptotic formulae for corresponding backbone curves are derived.     

PERIODIC SOLUTIONS OF STRONGLY NON-LINEAR OSCILLATORS BY THE MULTIPLE SCALES METHOD

The multiple scales method, developed for the systems with small non-linearities, is extended to the case of strongly non-linear self-excited systems. Two types of non-linearities are considered: quadratic and cubic. The solutions are expressed in terms of Jacobian elliptic functions. Higher order approximations, of solution as well as modulations of amplitude and phase, are derived. Comparisons to numerical simulations are provided and discussed.     

FREE VIBRATIONS OF SIMPLY SUPPORTED AND MULTILAYERED MAGNETO-ELECTRO-ELASTIC PLATES

Analytical solutions are derived for free vibrations of three-dimensional, linear anisotropic, magneto-electro-elastic, and multilayered rectangular plates under simply supported edge conditions. For any homogeneous layer, we construct the general solution in terms of a simple formalism that resembles the Stroh formalism, from which any physical quantities can be solved for given boundary conditions. In particular, the dispersion equation that characterizes the relationship between the natural frequency and wavenumber can be obtained in a simple form. For multilayered plates, we derive the dispersion relation in terms of the propagator matrices. The present solution includes all previous solutions, such as piezoelectric, piezomagnetic, and purely elastic solutions as special cases, and can serve as benchmarks to various thick plate theories and numerical methods used for the modelling of layered composite structures. Typical natural frequencies and mode shapes are presented for sandwich piezoelectric/piezomagnetic plates. It is shown clearly that some of the modes are purely elastic while others are fully coupled with piezoelectric/piezomagnetic quantities, with the latter depending strongly upon the material property and stacking sequence. These frequency and mode shape features could be of particular interest to the analysis and design of various “smart” sensors/actuators constructed from magneto-electro-elastic composite laminates.     

TIME INTEGRATION OF NON-LINEAR DYNAMIC EQUATIONS BY MEANS OF A DIRECT VARIATIONAL METHOD

Non-linear dynamic problems governed by ordinary (ODE) or partial differential equations (PDE) are herein approached by means of an alternative methodology. A generalized solution named WEM by the authors and previously developed for boundary value problems, is applied to linear and non-linear equations. A simple transformation after selecting an arbitrary interval of interest T allows using WEM in initial conditions problems and others with both initial and boundary conditions. When dealing with the time variable, the methodology may be seen as a time integration scheme. The application of WEM leads to arbitrary precision results. It is shown that it lacks neither numerical damping nor chaos which were found to be present with the application of some of the time integration schemes most commonly used in standard finite element codes (e.g., methods of central difference, Newmark, Wilson-θ, and so on). Illustrations include the solution of two non-linear ODEs which govern the dynamics of a single-degree-of-freedom (s.d.o.f.) system of a mass and a spring with two different non-linear laws (cubic and hyperbolic tangent respectively). The third example is the application of WEM to the dynamic problem of a beam with non-linear springs at its ends and subjected to a dynamic load varying both in space and time, even with discontinuities, governed by a PDE. To handle systems of non-linear equations iterative algorithms are employed. The convergence of the iteration is achieved by takingn partitions of T. However, the values of T/n are, in general, several times larger than the usual Δt in other time integration techniques. The maximum error (measured as a percentage of the energy) is calculated for the first example and it is shown that WEM yields an acceptable level of errors even when rather large time steps are used.     

压电陶瓷和金属构成的复合薄圆板的强迫振动——A电压策动

本文报告了电压策动下的复合薄圆板的强迫弯曲振动的严格解和近似解。复合薄圆板是由两片半径为b的压电陶瓷片和半径为a（a≥b）的金属薄圆板构成的。我们把它称作一般复合薄圆板,因为它可简化为各种特殊薄圆板.在压电陶瓷的极性做适当的安排后,交流策动电压将使一般复合薄圆板产生弯曲振动。从哈密顿原理出发推导了该系统的运动微分方程,连接条件和边界条件。并给出了一般解和严格共振频率方程。对于b=a的特殊复合薄圆板作了特殊的讨论。在简支边界情况下,策动电压的作用相当于在边界上加一个策动弯矩。当边界固定时,b=a的复合薄圆板表现为纯弹性,电压策动不能使之发生振动。本文还用里兹方法求出了一般复合圆板在电压策动下的强迫振动的近似解和近似共振频率方程。共振频率的近似值与实验值进行了比较,结果表明近似结果在工程上是可以使用的。     

FORCED AND SELF-EXCITED VIBRATIONS OF PIPES CONTAINING MOBILE BOILING FLUID CLOTS

Numerical modelling of the dynamic behaviour of a pipe containing inner non-homogeneous flows of a boiling fluid has been carried out. Inasmuch as the efforts to solve this problem analytically are confronted by considerable difficulties connected with varying system mass, geometry and discontinuity of equation coefficients, computational techniques for simulating pipe dynamics have been developed based on using of numerical time integration methods and transfer matrix methods together with orthogonalization procedures relating to the space variables. The system vibrations at different values of the parameters of the flow non-homogeneity and its velocity are observed. The possibility of forming stable and unstable flows depending on the character of the non-homogeneity and the velocity of fluid clots has been found.     

CORRECTED SOLVABILITY CONDITIONS FOR NON-LINEAR ASYMMETRIC VIBRATIONS OF A CIRCULAR PLATE

An investigation into non-linear asymmetric vibrations of a clamped circular plate under a harmonic excitation is made. We re-examined a primary resonance studied by Sridhar, Mook and Nayfeh, in which the frequency of excitation is near the natural frequency of an asymmetric mode of the plate. We corrected their solvability conditions and found that in the absence of internal resonance, the steady state response can have not only the form of standing wave but also the form of travelling wave, which is a remarkable contrast to their conclusion.     

NON-LINEAR TRANSVERSE VIBRATIONS OF A SIMPLY SUPPORTED BEAM CARRYING CONCENTRATED MASSES

An Euler-Bernoulli beam carrying concentrated masses is considered to be a beam-mass system. The beam is simply supported at both ends. The non-linear equations of motion are derived including stretching due to immovable end conditions. The stretching introduces cubic non-linearities into the equations. Forcing and damping terms are also included. Exact solutions for the natural frequencies are given for the linear problem. For the non-linear problem, an approximate solution using a perturbation method is searched. Non-linear terms of the perturbation series appear as corrections to the linear problem. Amplitude and phase modulation equations are obtained. Non-linear free and forced vibrations are investigated in detail. The effect of the positions, magnitudes and number of the masses are investigated.     

压电陶瓷和金属构成的复合薄圆板的强迫振动——B:均匀声压策动

本文讨论了一般复合薄圆板在均匀声压策动下的强迫振动。压电陶瓷的电端开路,策动声压是正弦形的。一般复合薄圆板是一个半径为a的金属板,两面对称的粘上半径为b的压电陶瓷片。这种一般复合薄圆板可以简化为各种特殊的复合薄圆板或均匀弹性板。当压电陶瓷的电极极性作适当的安排后,在均匀声压的作用下,一般复合圆板发生弯曲振动,同时在电端产生开路电压。
系统的最大弹性能和最大动能与电压策动的情形大致相同。此时最大弹性能里的电压不再是常量,变分时对它也要实行变分。另外我们把声压对一般复合薄圆板所作的功形式上看作系统势能的一部分。它和最大弹性能相加得到最大势能。利用哈密顿原理得到该系统的运动微分方程、连接条件和边界条件。并给出了简支边界和固定边界的一般解和严格共振频率方程。由这个共振频率方程得到的频率是电阻抗的并联共振频率。
对b=a的三层特殊复合薄圆板作了特殊的讨论。当边界固定时,一般解和共振频率方程在形式上与均匀等厚薄圆板的相同,只是参量有所不同。同时也求出了简支边界的一般解和共振频率方程。
为了工程上的方便还用瑞利一里兹方法给出了均匀声压作用下一般复合薄圆板的强迫振动的近似解和共振频率方程。此外还给出了有效机电耦合系数的公式。     

VIBRATION AND STABILITY OF ANNULAR PLATES IN NON-LINEAR CREEP CONDITIONS

This paper presents a study on the influence of the physical non-linearity of material in creep conditions on vibration and stability of non-uniform annular plates subjected to a follower force radially distributed at the outer edge. The stability analysis requires first the calculation of a membrane stress distribution and then the application of the kinetic stability criterion, making use of small superposed vibrations. The differential equations of the membrane and vibration states have been integrated by means of the transfer matrix method which allowed the determination of the relationships between the real and imaginary parts of the complex frequency and compressive force (characteristic curves). The results have been presented in numerous figures.     

EXPERIMENTAL MEASUREMENTS BY AN OPTICAL METHOD OF RESONANT FREQUENCES AND MODE SHAPES FOR SQUARE PLATES WITH ROUNDED CORNERS AND CHAMFERS

Most of the work done on vibration of plates published in the literature includes analytical and numerical studies with few experimental results available. In this paper, an optical system called the amplitude-fluctuation electronic speckle pattern interferometry for the out-of-plane displacement measurement is employed to investigate the vibration behavior of plates with rounded corners and with chamfers. The boundary conditions are traction free along the circumference of the plate. Based on the fact that clear fringe patterns will appear only at resonant frequencies, both resonant frequencies and corresponding mode shapes can be obtained experimentally using the present method. Numerical calculations by finite element method are also performed and the results are compared with the experimental measurements. Good agreements are obtained for both results. It is interesting to note that the mode number sequences for some resonant modes are changed. The transition of mode shapes from the square plate to the circular plate is also discussed.     

NON-LINEAR VIBRATIONS OF SHELL-TYPE STRUCTURES: A REVIEW WITH BIBLIOGRAPHY

The aim of this paper is to provide a contemporarily relevant survey of studies on non-linear vibrations of shell-type structures. The effects of geometrical non-linearity, and specific difficulties encountered in non-linear dynamic analysis of shell-type structures are presented and discussed. Studies on non-linear vibrations of shells are categorized by different shell configurations (shapes) in a chronological order. Also, the most commonly used methods of modelling and solution are reviewed and commented. Published reviews on non-linear vibrations of shell-type structures including complicating effects of anisotropy, initial stress, added mass, elastic foundation, stiffeners, open geometry (singly and doubly curved), transverse shear deformations, torsion, and interaction with fluid are also surveyed. Comments on the previous non-linear works are presented and some orientations for future research are suggested. Another purpose of this paper is to provide engineers, scientists and researchers with a list of 175 references, which should be very useful for locating relevant existing literature quickly.