Shear and rotatory inertia effects on large amplitude vibration of skew plates

The large amplitude free flexural vibration of elastic, isotropic skew plates is investigated, the effects of transverse shear and rotatory inertia being included. By use of Galerkin's method and the extended Berger approximation, solutions are obtained on the basis of an assumed vibration mode. The non-linear period vs. amplitude behavior is of the hardening type and the non-linear period is found to increase when the effects of transverse shear and rotatory inertia are considered in the analysis. The influence of these effects on aspect ratios and skew angles of thin and moderately thick skew plates is investigated both at small and large amplitudes.     

Free vibration analysis of stiffened plates using finite difference method

Free vibration characteristics of rectangular stiffened plates having a single stiffener have been examined by using the finite difference method. A variational technique has been used to minimize the total energy of the stiffened plate and the derivatives appearing in the energy functional are replaced by finite difference equations. The energy functional is minimized with respect to discretized displacement components and natural frequencies and mode shapes of the stiffened plate have been determined as the solutions of a linear algebraic eigenvalue problem. The analysis takes into consideration inplane deformation of the plate and the stiffener and the effect of inplane inertia on the natural frequencies and mode shapes. The effect of the ratio of stiffener depth to plate thickness on the natural frequencies of the stiffened plate has also been examined.     

Study on the Langevin piezoelectric ceramic ultrasonic transducer of longitudinal-flexural composite vibrational mode

In this paper, the Langevin longitudinal-flexural composite mode piezoelectric ultrasonic transducer is studied. This type of transducers consists of slender metal rods and longitudinally polarized piezoelectric ceramic rings. The resonance frequency equations for the longitudinal and flexural vibrations in the transducer are derived. By correcting the length of the metal slender rods, the simultaneous resonance of the longitudinal and flexural vibrations in the transducer is acquired. The experimental results show that the measured resonance frequencies of the transducers are in good agreement with the computed ones, and the measured resonance frequencies of the longitudinal and the flexural vibrations in the composite transducers are also in good agreement with each other.     

Free vibration analysis of corner-supported rectangular plates with symmetrically distributed edge beams

Analytical type solutions are obtained for the free vibration frequencies and mode shapes of thin corner-supported rectangular plates with symmetrically distributed reinforcing beams, or strips, attached to the plate edges. The method of superposition is employed. Equations governing reactions at plate-beam interfaces are developed in dimensionless form. The approach is comprehensive in that both lateral and rotational stiffness, and inertia, of the beam are incorporated into the analysis. For illustrative purposes computed eigenvalues and mode shapes are presented for two plate-beam systems of realistic geometries. It is shown that the method is easily extended to cover the case where the edge beams do not have a symmetrical distribution. This appears to be the first comprehensive analytical study of this problem of industrial interest.     

Flexural waves on narrow plates

Flexural wave speeds on beams or plates depend upon the bending stiffnesses which differ by the well-known factor (1 - nu2). A quantitative analysis of a plate of finite lateral width displays the plate-to-beam transition, and permits asymptotic analysis that shows the leading order dependence on the width. Orthotropic plates are analyzed using both the Kirchhoff and Kirchhoff-Rayleigh theories, and isotropic plates are considered for Mindlin's theory with and without rotational inertia. A frequency-dependent Young's modulus for beams or strips of finite width is suggested, although the form of the correction to the modulus is not unique and depends on the theory used. The sign of the correction for the Kirchhoff theory is opposite to that for the Mindlin theory. These results indicate that the different plate and beam theories can produce quite distinct behavior. This divergence in predictions is further illustrated by comparison of the speeds for antisymmetric flexural, or torsional, modes on narrow plates. The four classical theories predict limiting wave speeds as the plate width vanishes, but the values are different in each case. The deviations can be understood in terms of torsional waves and how each theory succeeds, or fails, in approximating the effect of torsion. Dispersion equations are also derived, some for the first time, for the flexural edge wave in each of the four "engineering" theories.     

Dynamics and instability of current-carrying microbeams in a longitudinal magnetic field

The dynamics and instability of current-carrying slender microbeams immersed in a longitudinal magnetic field is investigated by considering the material length scale effect of the microbeam. On the basis of modified couple stress theory, a theoretical model considering the effect of Lorentz forces is developed to analyze the free vibration and possible instability of the microbeam. Using the differential quadrature method, the governing equations of motion are solved and the lowest three natural frequencies are determined. The obtained results reveal that the electric current and the longitudinal magnetic field tend to reduce the microbeam's flexural stiffness. It is therefore shown that the lowest natural frequencies would decrease with increasing magnetic field parameter. The mode shapes of the microbeam are found to be generally three-dimensional spatial in the presence of the longitudinal magnetic field. It is interesting that buckling instability would concurrently occur in the first mode or in the higher-order modes when the magnetic field parameter becomes sufficiently large.     

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.     

Equivalent circuits and directivity patterns of air-coupled ultrasonic transducers

Air-coupled transducers for producing ultrasonic radiation in gases are studied. The transducer consists of a circular thin plate in flexural vibration and a sandwich longitudinal electromechanical vibrator that is attached to the center of the plate. The lowest-order axially symmetric flexural vibrational mode of a circular thin plate is analyzed. The equivalent circuits of the circular plate in flexural vibration and the compound transducer are presented and the frequency equation is derived. The radiated ultrasonic field of the circular thin plate in flexural vibration is calculated and the directivity pattern is obtained theoretically. Some transducers of this type are designed according to the frequency equation, and their resonance frequencies are measured. The measured resonance frequencies are in good agreement with the theoretical results, and the calculated radiation ultrasonic field is also in good agreement with the measured results of a previous work.     

Effects of large amplitude and transverse shear on vibrations of triangular plates

In this paper an analytical investigation of large amplitude free flexural vibrations of isotropic and orthotropic moderately thick triangular plates is carried out. The governing equations are expressed in terms of the lateral displacement, w, and the stress function, F, and are based on an improved non-linear vibration theory which accounts for the effects of transverse shear deformation and rotatory inertia. Solutions to the governing equations are obtained by using a single-mode approximation for w, Galerkin's method and a numerical integration procedure. Numerical results are presented in terms of variations of non-linear frequency ratios with amplitudes of vibrations. The effects of transverse shear, rotatory inertia, material properties, aspect ratios, and thickness parameters are studied and compared with available solutions wherever possible. Present results are in close agreement with those reported for thin plates. It is believed that all of the results reported here that are applicable for moderately thick plates are new and therefore, no comparison is possible.     

Pitch change during reverberant decay

The natural frequencies and mode shapes of a number of box beams are calculated by using the finite element displacement method. The structures are considered as assemblages of plates, and in general it is necessary to consider both the in-plane and transverse motion of the plates. A method of representing these two types of motion in the analysis of the vibrations of box beams is presented. A number of box beams of varying sectional parameters are analysed as systems of plates and the results compared with the predictions of Euler and Timoshenko beam theories. The comparisons show that for short beams constructed of thin plates, the new method can successfully represent the localized plate deformations, which cannot be described by beam theory.     

FREE VIBRATIONS OF SELF-STRAINED ASSEMBLIES OF BEAMS

This paper examines the natural frequencies and modes of transverse vibration of two simple redundant systems comprising straight uniform Euler-Bernoulli beams in which there are internal self-balancing axial loads (e.g., loads due to non-uniform thermal strains). The simplest system consists of two parallel beams joined at their ends and the other is a 6-beam rectangular plane frame. Symmetric mode vibration normal to the plane of the frame is studied. Transcendental frequency equations are established for the different systems. Computed frequencies and modes are presented which show the effect of (1) varying the axial loads over a wide range, up to and beyond the values which cause individual members to buckle (2) pinning or fixing the beam joints (3) varying the relative flexural stiffness of the component beams. When the internal axial loads first cause any one of the component beams to buckle, the fundamental frequency of the whole system vanishes. The critical axial loads required for this are determined. A simple criterion has been identified to predict whether a small increase from zero in the axial compressive load in any one member causes the natural frequencies of the whole system to rise or fall. It is shown that this depends on the relative flexural stiffnesses and buckling loads of the different members. Computed modes of vibration show that when the axial modes reach their critical values, the buckled beam(s) distort with large amplitudes while the unbuckled beam(s) move either as rigid bodies or with bending which decays rapidly from the ends to a near-rigid-body movement over the central part of the beam. The modes of the systems with fixed joints change very little (if at all) with changing axial load, except when the load is close to the value which maximizes or minimizes the frequency. In a narrow range around this load the mode changes rapidly. The results provide an explanation for some computed results (as yet unpublished) for the flexural modes and frequencies of flat plates with non-uniform thermal stress distributions.     

Secondary critical speed of flexible rotors with inertia slots

Rotors of two-pole generators have longitudinal slots for the electric windings and thus have dual flexural rigidity. Second order (or twice per revolution) forced vibrations are excited by the weight of the rotor and the problem of secondary critical speed, at half of the normal critical speed, arises. To overcome this difficulty transverse saw cuts or inertia slots are made in the pole faces in order to restore equality of the flexural rigidity of the rotor. In this paper, the critical speeds of rotors with inertia slots are calculated by using the transfer matrix method. The flexural rigidity of the element used in the transfer matrix method is determined by a three-dimensional finite element method. The secondary critical speeds of asymmetric rotors with inertia slots were measured experimentally thus demonstrating the validity of the present analysis.     

Study on the prestressed sandwich piezoelectric ceramic ultrasonic transducer of torsional-flexural composite vibrational mode

Based on the classical torsional and flexural vibrational theory of a slender rod, the prestressed sandwich torsional-flexural composite mode piezoelectric ceramic ultrasonic transducer is studied. This type of transducer consists of the slender metal rods and the longitudinally and tangentially polarized piezoelectric ceramic rings. The resonance frequency equations for the torsional and flexural vibrations in the transducers are derived. The simultaneous resonance of the torsional and flexural vibrations in the transducer is acquired by correcting the length of the metal slender rods resulting from the piezoelectric ceramic elements. The experimental results show that the measured resonance frequencies of the transducers are in good agreement with the computed ones, and the measured resonance frequencies of the torsional and the flexural vibrations in the composite transducers are also in good agreement with each other.     

一种多频局域共振型声子晶体板的低频带隙与减振特性

设计了一种多频局域共振型声子晶体板结构, 该结构由一薄板上附加周期性排列的多个双悬臂梁式子结构而构成. 由于多个双悬臂梁式子结构的低频振动与薄板振动的相互耦合作用, 这种局域共振型板结构可产生多个低频弯曲波带隙(禁带); 带隙频率范围内的板弯曲波会被禁止传播, 利用带隙可以实现对薄板的多个目标频率处低频减振. 本文针对这种局域共振型板结构进行了简化, 并基于平面波展开法建立了其弯曲波带隙计算理论模型; 基于该模型, 结合具体算例进行了带隙特性理论分析. 设计、制备了一种存在两个低频弯曲波带隙的局域共振型板结构样件, 通过激光扫描测振仪测试证实该结构存在两个低频带隙, 在带隙频率范围的板弯曲振动被显著衰减.     

The effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates. Part I: iterative and explicit analytical solution for non-linear transverse vibrations

The effects of large vibration amplitudes on the first two axisymmetric mode shapes of clamped thin isotropic circular plates are examined. The theoretical model based on Hamilton's principle and spectral analysis developed previously by Benamar et al. for clamped-clamped beams and fully clamped rectangular plates is adapted to the case of circular plates using a basis of Bessel's functions. The model effectively reduces the large-amplitude free vibration problem to the solution of a set of non-linear algebraic equations. Numerical results are given for the first and second axisymmetric non-linear mode shapes for a wide range of vibration amplitudes. For each value of the vibration amplitude considered, the corresponding contributions of the basic functions defining the non-linear transverse displacement function and the associated non-linear frequency are given. The non-linear frequencies associated to the fundamental non-linear mode shape predicted by the present model were compared with numerical results from the available published literature and a good agreement was found. The non-linear mode shapes exhibit higher bending stresses near to the clamped edge at large deflections, compared with those predicted by linear theory. In order to obtain explicit analytical solutions for the first two non-linear axisymmetric mode shapes of clamped circular plates, which are expected to be very useful in engineering applications and in further analytical developments, the improved version of the semi-analytical model developed by El Kadiri et al. for beams and rectangular plates, has been adapted to the case of clamped circular plates, leading to explicit expressions for the higher basic function contributions, which are shown to be in a good agreement with the iterative solutions, for maximum non-dimensional vibration amplitude values of 0.5 and 0.44 for the first and second axisymmetric non-linear mode shapes, respectively.     

Large amplitude flexural vibration of stiffened plates

The large amplitude free flexural vibration of thin, elastic orthotropic stiffened plates is studied. The boundary conditions considered are either simply supported on all edges or clamped on all edges and the in-plane edge conditions are either immovable or movable. The governing dynamic equations are derived in terms of non-dimensional parameters describing the stiffening achieved, and the solutions are obtained on the basis of an assumed one-term vibration mode shape for various stiffener combinations. In all cases, the non-linearity is found to be of the hardening type (i.e., the period of non-linear vibration decreases with increasing amplitude). Some interesting conclusions are drawn as to the effect of the stiffening parameters on the non-linear behaviour. A simple method of predicting the postbuckling and static large deflection behaviour from the results obtained in this analysis is included.     

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.     

Bending wave propagation of carbon nanotubes in a bi-parameter elastic matrix

This article studies transverse waves propagating in carbon nanotubes (CNTs) embedded in a surrounding medium. The CNTs are modeled as a nonlocal elastic beam, whereas the surrounding medium is modeled as a bi-parameter elastic medium. When taking into account the effect of rotary inertia of cross-section, a governing equation is acquired. A comparison of wave speeds using the Rayleigh and Euler-Bernoulli theories of beams with the results of molecular dynamics simulation indicates that the nonlocal Rayleigh beam model is more adequate to describe flexural waves in CNTs than the nonlocal Euler-Bernoulli model. The influences of the surrounding medium and rotary inertia on the phase speed for single-walled and double-walled CNTs are analyzed. Obtained results turn out that the surrounding medium plays a dominant role for lower wave numbers, while rotary inertia strongly affects the phase speed for higher wave numbers.     

Flexural vibration band gaps in thin plates with two-dimensional binary locally resonant structures

The complete flexural vibration band gaps are studied in the thin plates with two-dimensional binary locally resonant structures, i.e. the composite plate consisting of soft rubber cylindrical inclusions periodically placed in a host material. Numerical simulations show that the low-frequency gaps of flexural wave exist in the thin plates. The width of the first gap decreases monotonically as the matrix density increases. The frequency response of the finite periodic thin plates is simulated by the finite element method, which provides attenuations of over 20dB in the frequency range of the band gaps. The findings will be significant in the application of phononic crystals.     

Experimental investigation of vibration power flow in thin technical orthotropic plates by the method of vibration intensity

Vibration intensity technique is used to measure vibration power transmission in thin single layer technical orthotropic plates for flexural waves. Measurement of flexural wave power is carried out in far-field conditions. All measurements are undertaken in the frequency domain using the cross-spectra of acceleration signals, facilitating the use of FFT analyzer. The two-transducer technique applicable to these plates is used for these measurements. Technical orthotropic (rectangular corrugation) plates of steel are used for the measurements. One isotropic plate of steel is also considered for comparison. Method of elastic equivalence technique is used. Both input power and vibration power transmission through the plates are estimated. Far-field power is normalized with the input power for flexural wave. Influence of flexural rigidity on vibration energy transfer is also investigated.