On the static and kinetic methods of conservative system stability,and Rayleigh's principle: A reexamination

The relation between the static and kinetic variational methods of the stability of equilibrium analysis of conservative systems and the corresponding static and kinetic Rayleigh's principles is reexamined. Specifically made explicit are the connections between (i) the virtual work principle (for the adjacent equilibrium configuration) and Rayleigh's principle of extremum critical loads and buckled modes, and (ii) Hamilton's principle (for the adjacent non-equilibrium configuration) and Rayleigh's principle of extremum frequencies and mode shapes, through simple familiar examples. These connections are found by considering, in addition to the familiar mode amplitude variations, variations of the load which in turn produce variations in the space (i) and time (ii) domain lengths, respectively; one is thus led to a variable endpoints variational problem (instead of the customary fixed endpoints one) which, by invoking the energetics of these adjacent configurations, is simplified and finally brought to the standard Rayleigh's principle form.     

Coupled out of plane vibrations of spiral beams for micro-scale applications

An analytical method is proposed to calculate the natural frequencies and the corresponding mode shape functions of an Archimedean spiral beam. The deflection of the beam is due to both bending and torsion, which makes the problem coupled in nature. The governing partial differential equations and the boundary conditions are derived using Hamilton’s principle. Two factors make the vibrations of spirals different from oscillations of constant radius arcs. The first is the presence of terms with derivatives of the radius in the governing equations of spirals and the second is the fact that variations of radius of the beam causes the coefficients of the differential equations to be variable. It is demonstrated, using perturbation techniques that the derivative of the radius terms have negligible effect on structure’s dynamics. The spiral is then approximated with many merging constant-radius curved sections joined together to approximate the slow change of radius along the spiral. The equations of motion are formulated in non-dimensional form and the effect of all the key parameters on natural frequencies is presented. Non-dimensional curves are used to summarize the results for clarity. We also solve the governing equations using Rayleigh’s approximate method. The fundamental frequency results of the exact and Rayleigh’s method are in close agreement. This to some extent verifies the exact solutions. The results show that the vibration of spirals is mostly torsional which complicates using the spiral beam as a host for a sensor or energy harvesting device.     

The buckling and frequency of flexural vibration of rectangular isotropic and orthotropic plates using Rayleigh's method

A simple approximate formula for the natural frequencies of flexural vibration of isotropic plates, originally developed by Warburton using characteristic beam functions in Rayleigh's method, is modified to apply to specially orthotropic plates and extended to include the effect of uniform, direct inplane forces. The initial buckling problem is treated simply by equating the frequency expression to zero. The approach permits the ready determination of reasonably accurate natural frequencies and/or buckling loads for a given plate involving any combination of free, simply supported or clamped edges, without requiring the aid of a sophisticated calculating device or a knowledge of plate, vibration or buckling theory. To illustrate the applicability and accuracy of the approach, numerical results for a number of specific plate problems are presented.     

Vibration analysis of circular cylindrical double-shell structures under general coupling and end boundary conditions

In this paper, the free and forced vibration analysis of circular cylindrical double-shell structures under arbitrary boundary conditions is presented. This is achieved by employing the improved Fourier series method based on Hamilton’s principle. In the formulation, each displacement component of the cylindrical shells and annular plates is invariantly expanded as the superposition of a standard Fourier series with several supplementary functions introduced to remove the potential discontinuities of the original displacement and its derives at the boundaries. With the introduction of four sets of boundary springs at the coupling interfaces and end boundaries of the shell–plate combination, both elastic and rigid coupling and end boundary conditions can be easily obtained by assigning the stiffnesses of the artificial springs to certain values. The natural frequencies and mode shapes of the structures as well as frequency responses under forced vibration are obtained with the Rayleigh–Ritz procedure. The convergence of the method is validated by comparing the present results with those obtained by the finite element method. Several numerical results including natural frequencies and mode shapes are presented to demonstrate the excellent accuracy and reliability of the current method. Finally, a number of parameter studies concerning various end and coupling boundary conditions, different dimensions of shells and annular plates are also performed.     

A general theory of harmonic wave propagation in linear periodic systems with multiple coupling

A general theory is presented of harmonic wave propagation in one-dimensional periodic systems with multiple coupling between adjacent periodic elements. The motion of each element is expressed in terms of a finite number of displacement coordinates. The nature and number of different wave propagation constants at any frequency are discussed, and the energy flow associated with waves having real, complex or imaginary propagation constants is investigated. Kinetic and potential energy functions are derived for the propagating waves and a generalized Rayleigh's Quotient and Rayleigh's Principle for the complex wave motion have been found. This is extended to yield a generalized Rayleigh-Ritz method of finding approximate, yet accurate, relationships between the frequencies and propagation constants of the propagating waves. The effect of damping is also considered, and a special class of “damped forced waves” is postulated for hysteretically damped periodic systems. An energy definition for the loss factor of these waves is found. Briefly considered is the two-dimensional multi-coupled periodic system in which a simple wave motion analogous to a plane wave propagates across the whole system.     

Harmonic balance approach to the periodic solutions of the (an)harmonic relativistic oscillator

The first-order harmonic balance method via the first Fourier coefficient is used to construct two approximate frequency-amplitude relations for the relativistic oscillator for which the nonlinearity (anharmonicity) is a relativistic effect due to the time line dilation along the world line. Making a change of variable, a new nonlinear differential equation is obtained and two procedures are used to approximately solve this differential equation. In the first the differential equation is rewritten in a form that does not contain a square-root expression, while in the second the differential equation is solved directly. The approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first due to the fact that, in the second procedure, application of the harmonic balance method produces an infinite set of harmonics, while in the first procedure only two harmonics are produced. Both approximate frequencies are valid for the complete range of oscillation amplitudes, and excellent agreement of the approximate frequencies with the exact one are demonstrated and discussed. The discrepancy between the first-order approximate frequency obtained by means of the second procedure and the exact frequency never exceeds 1.6%. We also obtained the approximate frequency by applying the second-order harmonic balance method and in this case the relative error is as low 0.31% for all the range of values of amplitude of oscillation A.     

Higher accuracy analytical approximations to a nonlinear oscillator with discontinuity by He's homotopy perturbation method

He's homotopy perturbation method is used to calculate higher-order approximate periodic solutions of a nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(x). We find He's homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate period of less than 1.56% for all values of oscillation amplitude, while this relative error is 0.30% for the second iteration and as low as 0.057% when the third-order approximation is considered. Comparison of the result obtained using this method with those obtained by different harmonic balance methods reveals that He's homotopy perturbation method is very effective and convenient.     

A High-Order Projection Method for Tracking Fluid Interfaces in Variable Density Incompressible Flows

We present a numerical method for computing solutions of the incompressible Euler or Navier–Stokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties. The method is based on a second-order projection method for variable density flows using an “approximate projection” formulation. The boundary between the fluids is tracked with a second-order, volume-of-fluid interface tracking algorithm. We present results for viscious Rayleigh–Taylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm. We also present computational results for the Rayleigh–Taylor instability in air-helium and for bubbles and drops in an air–water system without surface tension to demonstrate the behavior of the algorithm on problems with large density and viscosity contrasts.     

基于声弛豫频率的可激发气体压强合成算法

声弛豫频率是声吸收谱峰值点的频率,包含可激发气体成分、环境温度和压强信息.利用声弛豫频率线性正比气体压强的特性,提出一种通过两频点声吸收系数和声速测量值计算声弛豫频率,并通过查表方式合成气体压强的算法.算法的声弛豫频率测量误差具有随声测量值误差线性变换的特性,且当两频点的声吸收测量误差相等时,压强的合成误差为零.对于一定温度下的甲烷及其混合气体,仿真计算证明算法的有效性和声测量误差的稳健性.提供一种简单、稳健性好、可实时连续在线检测可激发气体腔体压强的声学方法.     

The high frequency, transient response of a panel with attached masses

The general, high-frequency response of a panel with attached masses is approximated using a transient form of asymptotic modal analysis (AMA). This method is derived by applying asymptotic simplifications to classical solutions in both the time and frequency domains. These relations are applied to a panel with one or more attached masses that is excited by impulsive loads. Predictions are made of the mean-squared, transverse displacement histories as well as localized responses near the added masses. It is shown that the latter compare well with experimental data when the masses are separated by more than the mean wavelength of the frequency band. The approximate solutions are shown to require relatively little computational time and memory and are applicable to general forms of excitation.     

An efficient general approach to modal analysis of frame resonators with applications to support loss in microelectromechanical systems

The design of high-Q resonators such as Xylophone Bar Resonators (XBRs) capable of being fabricated using Micro-Electro-Mechanical Systems (MEMS) processes is of considerable interest in light of the widespread and rapidly growing use of systems dependent on their availability and performance. This paper is concerned with vibration analysis and Q optimisation of an XBR, with the method extending directly to other planar frames and straightforwardly to more complex structures. The Rayleigh–Ritz method is discussed in some detail, first treating the discrete case, followed by developing and applying a kinematical procedure to an L-frame structure. Attention is given to geometric interpretation of the Rayleigh–Ritz procedure and to developing an intuitive understanding the method before turning to the XBR case. Having developed an approximation for system dynamics, the results are used in conjunction with an analytical model of elastic wave propagation in the substrate to obtain an estimate for the support Q factor. Natural frequencies, mode shapes, and support Q values are presented and compared to Finite Element models of the same problem, with excellent agreement observed at substantially lower computational cost. For the first time in the literature, the geometric impedance tuning principle underlying the XBR design is validated and quantified, including sensitivity to manufacturing error.     

Thermal effect on frequencies of coupled vibrations of a rotating beam of linearly varying semi-circular cross-section

An analysis is presented of the effect of a constant thermal gradient on coupled vibrations of a beam of linearly varying semi-circular cross-section attached to a rotating disc. A method based on Rayleigh's quotient is used to obtain upper bounds of the frequencies corresponding to the first three modes of vibrations. The frequencies for various values of cross-section variation, hub-radius and temperature gradient are obtained.     

NATURAL FREQUENCIES AND MODE SHAPES OF A FREE-FREE BEAM WITH LARGE END MASSES

An analytical solution is presented for the natural frequencies, mode shapes and orthogonality condition, of a free-free beam with large off-set masses connected to the beam by torsion springs. Results are given for a range of masses with various fixed orientations and the validity of the method is confirmed against established results for natural frequencies of beams with five different boundary conditions. The study lays the foundation for investigations into the dynamics and vibration control of multi-link articulated systems such as the Space Shuttle Remote Manipulator.     

A generalized Dunkerley-Graeffe procedure for complex vibrating systems

The combined use of the principle underlying Dunkerley's rule for the approximate determination of the gravest eigenfrequency of a multi-degree-of-freedom elastic system and of the root-squaring process suggested by Graeffe provide a generalized procedure for the approximate solution of complex frequency equations, i.e., for the determination of any number of eigenvalues at the desired accuracy level, provided that the eigenfunctions involved can be expressed in series expansion. By simple algebraic means, the method yields solutions in cases for which sophisticated computing facilities would otherwise be necessary and provides the means for checking complicated computer outputs as well as approximate results for preliminary design purposes.     

A note on the vibrations of restrained beams and rods with point masses

This paper is concerned with the approximate determination of the fundamental frequency and first mode shape of a beam or rod to which springs and point masses are attached. After formulating the general eigenvalue problem which forms the basis of the analysis, the theory is applied to three examples and it is shown that the results are quite satisfactory.     

Coupling effects on a cantilever subjected to a follower force

In this investigation a solution methodology is presented for studying the stability of a uniform cantilever having a translational and rotational spring at its support, carrying two concentrated masses, one at the support and the other at its tip, and subjected to a follower compressive force at its free end. The analysis is based on Timoshenko's beam theory by considering the cantilever as a continuous elastic system. The coupling effects on the flutter load are fully assessed for a variety of parameters such as translational and rotational springs at the support, translational and rotational inertia of the concentrated masses, and cross-sectional shape, as well as transverse shear deformation and rotatory inertia of the mass of the column.     

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

The nonlinear oscillations of a Duffing-harmonic oscillator are investigated by an approximated method based on the ‘cubication’ of the initial nonlinear differential equation. In this cubication method the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude, A. The replacement of the original nonlinear equation by an approximate Duffing equation allows us to obtain explicit approximate formulas for the frequency and the solution as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function, respectively. These explicit formulas are valid for all values of the initial amplitude and we conclude this cubication method works very well for the whole range of initial amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones is demonstrated and discussed and the relative error for the approximate frequency is as low as 0.071%. Unlike other approximate methods applied to this oscillator, which are not capable to reproduce exactly the behaviour of the approximate frequency when A tends to zero, the cubication method used in this Letter predicts exactly the behaviour of the approximate frequency not only when A tends to infinity, but also when A tends to zero. Finally, a closed-form expression for the approximate frequency is obtained in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean as well as Legendre's formula to approximately obtain this mean are used.     

Vibrational energy flow models for the Rayleigh–Love and Rayleigh–Bishop rods

Energy Flow Analysis (EFA) has been developed to predict the vibrational energy density of the system structures in the medium-to-high frequency range. The elementary longitudinal wave theory is often used to describe the longitudinal vibration of a slender rod. However, for relatively large diameter rods or high frequency ranges, the elementary longitudinal wave theory is inaccurate because the lateral motions are not taken into account. In this paper, vibrational energy flow models are developed to analyze the longitudinally vibrating Rayleigh–Love rod considering the effect of lateral inertia, and the Rayleigh–Bishop rod considering the effect not only of the lateral inertia but also of the shear stiffness. The derived energy governing equations are second-order differential equations which predict the time and space averaged energy density and active intensity distributions in a rod. To verify the accuracy of the developed energy flow models, various numerical analyses are performed for a rod and coupled rods. Also, the EFA results for the Rayleigh–Love and Rayleigh–Bishop rods are compared with the analytical solutions for these models, the traditional energy flow solutions, and the analytical solutions for the classical rod.     

Shape Analysis and Solution to a Class of Nonlinear Wave Equation with Cubic Term

In this paper, we analyze the relation between the shape of the bounded traveling wave solutions and dissipation coefficient of nonlinear wave equation with cubic term by the theory and method of planar dynamical systems. Two critical values which can characterize the scale of dissipation effect are obtained. If dissipation effect is not less than a certain critical value, the traveling wave solutions appear as kink profile; while if it is less than this critical value, they appear as damped oscillatory. All expressions of bounded traveling wave solutions are presented, including exact expressions of bell and kink profile solitary wave solutions, as well as approximate expressions of damped oscillatory solutions. For approximate damped oscillatory solution, using homogenization principle, we give its error estimate by establishing the integral equation which reflects the relations between the exact and approximate solutions. It can be seen that the error is an infinitesimal decreasing in the exponential form.     

一种基于warping变换的浅海脉冲声源被动测距方法

针对浅海波导中脉冲声源被动测距问题, 提出了一种利用接收信号的能量密度函数进行warping变换的声源被动测距方法. 对于浅海波导, 接收信号的能量密度函数中不同号简正波相干部分, 经warping变换后输出结果的频谱中包含与声源和接收器位置无关的不变性频率特征. 这些特征频率在数值上等于理想波导中相干的两号简正波的截止频率差, 与海底参数无关, 因此仅需已知海水中的平均声速和海水深度便可计算出特征频率值. 当声源距离未知时, 利用特征频率的提取值与真实特征频率之间的关系可以实现快速测距, 极大地提高了计算速度. 为了验证方法的有效性, 对2011年11月黄海海域水声实验的接收脉冲数据进行了处理, 测距结果与实测距离符合良好, 平均测距误差在8%以内.