Vibration and parametric excitation in asymmetric circular plates under moving loads

The natural frequencies and modes of transverse vibration of circular plates containing small imperfections are determined through a perturbation method. Incision of equally spaced, equal-size radial slots at the rim of the plate creates asymmetry in some, but not all, of the vibration modes, and it causes the repeated natural frequencies of these modes in the symmetric plate to split into two distinct values. These vibration modes are called the split modes, and those associated with the repeated natural frequencies are called the repeated modes. A relationship identifying the split and repeated modes for any configuration of slots is presented. The vibration of a plate containing any number of thin slots cut into it at the rim and with any number of rotating linear springs is analyzed. Parametric instability can be excited in the split modes of the plate by the springs rotating below critical speed, but it cannot be excited in the repeated modes. The response of the plate in forms such as traveling or standing waves at parametric resonance is discussed. The theoretical predictions of split and repeated vibration modes and of the excitation of parametric instability are confirmed by experiments.     

LINEAR VIBRATION CHARACTERISTICS OF CABLE-BUOY SYSTEMS

A theoretical model for the linear vibration of a cable tensioned by a subsurface buoy is developed. The equilibrium of the cable-buoy system subject to drag is evaluated using an approximate closed-form solution whose range of validity is confirmed through comparison with numerical solutions. The three-dimensional equations of cable-buoy motion are linearized about this equilibrium and then used to assess vibration characteristics. The characteristic equations for the natural frequencies of both in-plane and out-of-plane vibration modes are derived. The in-plane natural frequency spectrum exhibits the curve veering phenomena due to asymmetry of the associated mode shapes. Parameter studies reveal the dependencies of the in-plane and out-of-plane vibration modes on the cable tension, the buoy mass, and the current velocity.     

THE EFFECT OF ROTATION UPON THE NATURAL FREQUENCIES OF A MASS-SPRING SYSTEM

When a mass-spring system vibrates it does so with frequencies characteristic of the system. If the system as a whole now undergoes a rotational motion then these characteristic frequencies will change from their non-rotational values. It is the purpose of this paper to show how these changes may be calculated for a specified system and, in particular, to investigate the role in these changes of both the system and the rotational parameters. A system of N masses linked sequentially by springs in tension is allowed to vibrate about an equilibrium configuration both radially and transversely upon a smooth turntable. If the turntable is stationary then the radial and transverse vibrations are independent of each other, provided the amplitudes of vibration are sufficiently small. There are then N natural frequencies of vibration for each mode. However, when the turntable rotates then the Coriolis effects give rise to an interaction between the two modes of vibration, and there are now 2 N natural frequencies for the combined vibrations. If the rate of rotation is “small” then the two modes are almost separated and it is possible to discuss the “essentially radial” or “essentially transverse” mode of vibration each of which has N natural frequencies. It is these natural frequencies which are considered in this work, in particular their dependence upon the rotation rate and upon the tension in the springs (when in the static configuration). In a previous paper, it was shown that if only radial vibrations are allowed (by admitting say a guide rail) then all the natural frequencies decrease, with increasing rotation rate, from their static values. It is shown that the opposite is the case here in that the “essentially radial” natural frequencies increase with increasing rotation rate. This is due to the Coriolis interaction with the transverse vibrations. The “essentially transverse” frequencies are also found and the nature of their dependence discussed. Also included in the analysis is the effect on the frequencies of the (weak) coupling between the motion of the masses and the rotation of the turntable as a consequence of the conservation of angular momentum. In addition to treating N being finite the limiting case of an infinite number of masses is considered to determine the natural frequencies of vibration of a continuous stretched string undergoing rotation.     

Nonlinear free transverse vibration of an axially moving beam: Comparison of two models

Nonlinear free transverse vibration of an axially moving beam is investigated. A partial-differential equation governing the transverse vibration is derived from the Newton's second law. Under the assumption that the tension of beam can be replaced by the averaged tension over the beam, the partial-differential reduces to a widely used integro-partial-differential equation for nonlinear free transverse vibration. The method of multiple scales is applied directly to two equations to evaluate nonlinear natural frequencies. Numerical examples are presented to demonstrate the analytical results and to highlight the difference between two models. Two models yield the essentially same results for the weak nonlinearity, the small axial speed and the low mode, while the difference between two models increases with the nonlinear term, the axial speed, and the order of mode.     

Free vibration of thin walled open section beams with constrained damping treatment

Free vibration characteristics of a thin walled, open cross-section beam, with constrained damping layers at the flanges, are investigated. Both uncoupled transverse vibration and the coupled bending-torsion oscillations, of a beam of a top-hat section, are considered. Numerical results are presented for natural frequencies and modal loss factors in the first two modes of simply supported and clamped-clamped beams. For the uncoupled mode the constrained damping treatment is more effective than an unconstrained one, but for the coupled mode the effect is just the opposite.     

Vibration characteristic analysis of buried pipes using the wave propagation approach

A theoretical analysis for the free vibration of simply supported buried pipes has been investigated using the wave propagation approach. The pipe modeled as a thin cylindrical shell of linear homogeneous isotropic elastic material buried in a linear isotropic homogeneous elastic medium of infinite extent. The vibrations of the pipe are examined by using Flüggle shell equation. The natural frequencies are obtained for the pipes surrounded by vacuo or elastic medium. The results are compared with those available in the literature and agreement is found with them. It is found that the free vibration frequency of the pipe does not appear for some of the axial or circular vibration modes and the real natural frequencies of the pipe are significantly dependent on the rigidity of the surrounding medium.     

Transient vibration analysis of a completely free plate using modes obtained by Gorman's superposition method

This paper shows that the transient response of a plate undergoing flexural vibration can be calculated accurately and efficiently using the natural frequencies and modes obtained from the superposition method. The response of a completely free plate is used to demonstrate this. The case considered is one where all supports of a simply supported thin rectangular plate under self weight are suddenly removed. The resulting motion consists of a combination of the natural modes of a completely free plate. The modal superposition method is used for determining the transient response, and the natural frequencies and mode shapes of the plates used are obtained by Gorman's superposition method. These are compared with corresponding results based on the modes using the Rayleigh-Ritz method using the ordinary and degenerated free-free beam functions. There is an excellent agreement between the results from both approaches but the superposition method has shown faster convergence and the results may serve as benchmarks for the transient response of completely free plates.     

A distributed-mass approach for dynamic analysis of Bernoulli-Euler plane frames

The exact dynamic analysis of plane frames should consider the effect of mass distribution in beam elements, which can be achieved by using the dynamic stiffness method. Solving for the natural frequencies and mode shapes from the dynamic stiffness matrix is a nonlinear eigenproblem. The Wittrick-Williams algorithm is a reliable tool to identify the natural frequencies. A deflated matrix method to determine the mode shapes is presented. The dynamic stiffness matrix may create some null modes in which the joints of beam elements have null deformation. Adding an interior node at the middle of beam elements can eliminate the null modes of flexural vibration, but does not eliminate the null modes of axial vibration. A force equilibrium approach to solve for the null modes of axial vibration is presented. Orthogonal conditions of vibration modes in the Bernoulli-Euler plane frames, which are required in solving the transient response, are theoretically derived. The decoupling process for the vibration modes of the same natural frequency is also presented.     

Free vibration analysis of circular and annular sectorial thin plates using curve strip Fourier p-element

A curve strip Fourier p-element for free vibration analysis of circular and annular sectorial thin plates is presented. The element transverse displacement is described by a fixed number of polynomial shape functions plus a variable number of trigonometric shape functions. The polynomial shape functions are used to describe the element's nodal displacements and the trigonometric shape functions are used to provide additional freedom to the edges and the interior of the element. With the additional Fourier degrees of freedom (dof) and reduce dimensions, the accuracy of the computed natural frequencies is greatly increased. Results are obtained for a number of circular and annular sectorial thin plates and comparisons are made with exact, the curve strip Fourier p-element, the proposed Fourier p-element and the finite strip element. The results clearly show that the curve strip Fourier p-element produces a much higher accuracy than the proposed Fourier p-element, the finite strip element.     

Vibration isolation using extreme geometric nonlinearity

A highly deformed, slender beam (or strip), attached to a vertically oscillating base, is used in a vibration isolation application to reduce the motion of a supported mass. The isolator is a thin strip that is bent so that the two ends are clamped together, forming a loop. The clamped ends are attached to an excitation source and the supported system is attached at the loop midpoint directly above the base. The strip is modeled as an elastica, and the resulting nonlinear boundary value problem is solved numerically using a shooting method. First the equilibrium shapes of the loop with varying static loads and lengths are studied. The analysis reveals a large degree of stiffness tunability; the stiffness is dependent on the geometric configuration, which itself is determined by the supported mass, loop length, and loop self-weight. Free vibration frequencies and mode shapes are also found. Finally, the case of forced vibration is studied, and the displacement transmissibility over a large range of forcing frequencies is determined for varying parameter values. Experiments using polycarbonate strips are conducted to verify equilibrium and dynamic behavior.     

Mode spectrum of noncollinear electroacoustic boundary-guided waves in a ferroelectric with a moving strip domain

The mode spectrum of electroacoustic boundary waves guided by a strip domain uniformly moving in a 4-mm ferroelectric is considered in the quasi-static approximation. The motion of the strip domain is found to cause the wave vector of the electroacoustic wave to be noncollinear with the guiding boundaries. The frequency dependences of the phase velocity are presented for the symmetric and antisymmetric modes of the electroacoustic wave. These dependences are compared in the reference system fixed to the strip domain and in the laboratory reference system. It is shown that, at low and moderate frequencies, the symmetric mode of the electroacoustic wave is more efficiently localized by a moving strip domain than by a single domain wall.     

Study on rectangular ultrasonic radiator of flexural resonant vibration in high modes

Based on the theory of coupling vibration and flexural vibration of thin rod of rectangular cross section, the fiexural vibration of rectangular thin plate was studied. The frequency equation was derived under the condition of freeboundaries. The normal modes and the relation between the normal modes and the resonant frequency were obtained. Experiments showed that the calculated resonant frequencies agree well with the measured results, and the rectangular thin plate in flexural vibration has abundant resonant frequencies. The radiator of flexural vibration used in ultrasonic cleaning and other techniques has the advantages of large acoustic radiating area, uniform acoustic field and easy adjustment of resonant frequencies, proving that it is a promising ultrasonic source.     

Vibration of curved plate assemblies subjected to membrane stresses

In a previous paper [1] the finite strip method was applied to the prediction of the natural frequencies of vibration of longitudinally invariant, rigidly connected assemblies of circularly curved and flat strips having diaphragm end supports. This work is extended here to include the presence of an initial membrane stress field. An individual curved strip may be subjected to a biaxial direct stress field comprising a uniform stress acting in the circumferential direction and a non-uniform stress acting in the longitudinal direction. The presence of the membrane stress field is accommodated in the analysis by the inclusion of an initial stress or geometric stiffness matrix. A further extension included here is a facility to delete in-surface inertia terms. Results are presented for the application of the strip method in predicting the frequencies of vibration of a circular cylinder subjected to a complicated membrane stress system.     

Dynamic ground compliance in multistorey buildings

A study is presented of the changes in the characteristics of the natural modes of vibration for multistorey structures which are founded on flexible foundations. First a standard eigenvalue problem is formulated for the proportionally damped case. Then general relationships of changes in natural frequencies and mode shapes are derived for the linear vibration theory. By means of an example problem it is demonstrated, however, that only the first mode obeys the predicted changes of frequencies and mode shapes over a wide range of foundation stiffness. The higher modes are shown to deviate substantially from the linear behaviour. This deviation is ascribed to geometric changes in mode shapes.     

Free vibration of an elliptic ring membrane clamped along two confocal ellipses

Free vibration of an elliptical ring membrane clamped along two confocal ellipses is studied analytically, and the natural frequencies are tabulated for the first four modes of vibration.     

NON-LINEAR FREE VIBRATION ANALYSIS OF A STRING UNDER BENDING MOMENT EFFECTS USING THE PERTURBATION METHOD

In this paper, the non-linear free vibration of a string with large amplitude is considered. The initial tension, lateral vibration amplitude, cross-section diameter and the modulus of elasticity of the string have main effects on its natural frequencies. Increasing the lateral vibration amplitude makes the assumption of constant initial tension invalid. Therefore, it is impossible to use the classical equation of transverse motion assuming a small amplitude. On the other hand, by increasing the string cross-sectional diameter, the bending moment effect will increase dramatically, and it will act as an impressive restoring moment. Considering the effects of the bending moments, the non-linear equation governing the large amplitude transverse vibration of a string is derived. The time-dependent portion of the governing equation has the form of the Duffing equation. Due to the complexity and non-linearity of the derived equation and the fact that there is no established exact solution method, the equation is solved using the perturbation method. The results of the analysis are shown in appropriate graphs, and the natural frequencies of the string due to the non-linear factors are compared with the natural frequencies of the linear vibration of a string without bending moment effects.     

Free vibration of a point-supported membrane stretched by inextensible strings

An analysis is presented for the free vibration of a point-supported rectangular membrane with uniform tension stretched by inextensible strings along the edges. The membrane is transformed into a square membrane of unit length by a transformation of variables. The transverse deflection of the square membrane is expressed in a series of the products of the deflection functions of strings parallel to the edges, and the frequency equation is derived by the Ritz method. This method is applied to point-supported membranes symmetrical with respect to the center lines, and the natural frequencies and the mode shapes are calculated numerically up to higher modes.     

Purely axial vibration of thin cylindrical shells with shear-diaphragm boundary conditions

This work presents the free vibration characteristics of a thin walled cylindrical shell at the zeroth axial mode number. The cylindrical shell has shear-diaphragm boundary conditions at each end. The thin shell equations by Flügge are used as these equations of motion lead to more accurate results at low frequencies. The zeroth axial mode number is found to occur at the cut-on of the second class of waves. The mode shapes at these natural frequencies result in a purely axial displacement of the middle surface of the shell. High modal density for the first class of waves occurs before the cutting-on of the second class of waves. Beyond this frequency, the dynamic response is dominated by the latter modes.     

Free vibration of thin circular rings on periodic radial supports

Natural frequencies and normal modes are obtained for in-plane, inextensional vibrations of a thin circular ring with equi-spaced, identical radial supports. A wave approach is used. Natural frequencies are determined from the propagation constants of the ring by considering it as an endless periodic structure. Normal modes are obtained by superposition of a pair of opposite-going free wave groups. Numerical results have been presented for both rigid and circumferentially guided supports. It has been shown that at certain frequencies two different natural modes can exist. This has been verified experimentally.     

Galerkin methods for natural frequencies of high-speed axially moving beams

In this paper, natural frequencies of planar vibration of axially moving beams are numerically investigated in the supercritical ranges. In the supercritical transport speed regime, the straight equilibrium configuration becomes unstable and bifurcate in multiple equilibrium positions. The governing equations of coupled planar is reduced to two nonlinear models of transverse vibration. For motion about each bifurcated solution, those nonlinear equations are cast in the standard form of continuous gyroscopic systems by introducing a coordinate transform. The natural frequencies are investigated for the beams via the Galerkin method to truncate the corresponding governing equations without nonlinear parts into an infinite set of ordinary-differential equations under the simple support boundary. Numerical results indicate that the nonlinear coefficient has little effects on the natural frequency, and the three models predict qualitatively the same tendencies of the natural frequencies with the changing parameters and the integro-partial-differential equation yields results quantitatively closer to those of the coupled equations.