Free vibration of a circular plate elastically restrained along some radial segments

An analysis is presented for the free vibration of a circular plate restrained against deflection along radial segments. With the reaction forces acting on the segments regarded as unknown harmonic loads, the stationary response of the plate to these loads is expressed by the use of the Green function. The force distributions along the segments are expanded into Fourier series with unknown coefficients, and the homogeneous equations for the coefficients are derived by restraint conditions on the supports. The natural frequencies and the mode shapes of the plate are determined by calculating the eigenvalues and eigenvectors of the equations. The method is applied to circular plates supported along several radial segments located at equal angular intervals, the natural frequencies and the mode shapes of the plates are calculated numerically and the effect of the supports is discussed.     

The steady state out-of-plane response of a Timoshenko curved beam with internal damping

The steady state out-of-plane response of a Timoshenko curved beam with internal damping to a sinusoidally varying point force or moment is determined by use of the transfer matrix approach. For this purpose, the equations of out-of-plane vibration of a curved beam are written as a coupled set of the first order differential equations by using the transfer matrix of the beam. Once the matrix has been determined by numerical integration of the equations, the steady state response of the beam is obtained. The method is applied to free-clamped non-uniform beams with circular, elliptical, catenary and parabolical neutral axes driven at the free end; the driving point impedance and force or moment transmissibility are calculated numerically and the effects of the slenderness ratio, varying cross-section and the function expressing the neutral axis on them are studied.     

A note on the damping of large amplitude beam vibrations

This paper presents a new series-type method for solving the eigenvalue problems of irregularly shaped plates clamped at all edges. An irregularly shaped plate is formed on a simply supported rectangular plate by rigidly fixing several segments. With the reaction forces and moments acting on all edges of an actual plate of irregular shape regarded as unknown harmonic loads, the stationary response of the plate to these loads is expressed by the use of the Green function. The force and moment distributions along the edges are expanded into Fourier series with unknown coefficients, and the homogeneous equations for the coefficients are derived by restraint conditions on the edges. The natural frequencies and the mode shapes of the actual plate are determined by calculating the eigenvalues and eigenvectors of the equations. The method is applied to a cross-shaped, an I-shaped and an L-shaped plate clamped at all edges, the natural frequencies and the mode shapes of the plates are calculated numerically and the effect of the shape is discussed.     

The steady state out-of-plane response of an internally damped ring supported by springs in some bays

The steady state out-of-plane response of an internally damped ring supported by springs in some bays to a sinusoidally varying point force or moment is determined by use of the transfer matrix technique. For this purpose, the equations of out-of-plane vibration of a uniform circular ring based upon the Timoshenko beam theory are written as a coupled set of first order differential equations by using the transfer matrix of the ring. The matrix is obtained analytically and the steady state response of the ring is determined by the product of the matrices in free bays and those in supported bays. In this case, the elastic moduli of the ring and springs with internal damping are assumed to be complex quantities. The method is applied to rings supported against deflection and torsion in some bays of the same length located at equal angular intervals; the driving point impedance, transfer impedance and the distributions of the deflection, angular rotation, force and moment are calculated numerically, and the effects of the number, the stiffness and the length of supporting springs on them are studied.     

Semi-Analytical Solution for Functionally Graded Solid Circular and Annular Plates Resting on Elastic Foundations Subjected to Axisymmetric Transverse Loading

In this paper, the static analysis of functionally graded (FG) circular plates resting on linear elastic foundation with various edge conditions is carried out by using a semi-analytical approach. The governing differential equations are derived based on the three dimensional theory of elasticity and assuming that the mechanical properties of the material vary exponentially along the thickness direction and Poisson's ratio remains constant. The solution is obtained by employing the state space method (SSM) to express exactly the plate behavior along the graded direction and the one dimensional differential quadrature method (DQM) to approximate the radial variations of the parameters. The effects of different parameters (e.g., material property gradient index, elastic foundation coefficients, the surfaces conditions (hard or soft surface of the plate on foundation), plate geometric parameters and edges condition) on the deformation and stress distributions of the FG circular plates are investigated.     

Determination of the steady state response of viscoelastically point-supported rectangular specially orthotropic plates by an energy-based finite difference method

A method based on a variational procedure in conjunction with a finite difference method is used to examine the free vibration characteristics and steady state response to a sinusoidally varying force applied at the center of a viscoelastically point-supported orthotropic elastic plate of rectangular shape. Using the energy-based finite difference method, the problem is reduced to the solution of a system of algebraic equations. The influence of the mechanical properties, and of the damping of the supports to the mode shapes and to the steady state response of viscoelastically point-supported rectangular plates is investigated numerically for a concentrated load at the center for various values of the mechanical properties characterizing the anisotropy of the plate material and for various damping ratios. The results are given for the frequencies and mode shapes of the first three symmetrical modes. Convergence studies are made. The validity of the present approach is demonstrated by comparing the results with other solutions based on the Kirchhoff-Love plate theory.     

热力联合作用弹性薄圆板的弯曲与屈曲

 导出了热力联合作用下弹性薄圆板的弯曲动力响应控制方程,讨论了其弯曲变形特点及影响失效的因素。分析表明在短时热能沉积作用下,热屈曲是弹性薄板失效的主要方式之一;反鼓包或反冲塞是热屈曲的后继行为;增加外载和热能沉积功率水平都将加速热屈曲的发生;材料的温度相关性与热能沉积的时空分布对薄板的力学行为都有重要影响,同为产生和影响热剪切失效(反冲塞)的重要因素。     

Response of an elastically supported plate strip to a moving load

A theoretical analysis is presented of the steady state response of a plate strip constrained elastically along its edges against rotation and translation under the action of a moving transverse line load. Within the classical plate theory the solutions are obtained by using Fourier and Laplace transformation methods with respect to space variables. Numerical results are given for a plate strip with both edges identically constrained and a normal line load of constant intensity travelling along the plate strip with a constant speed. The first five speeds of the applied load for which a resonance effect occurs in the system are plotted as functions of the edge constraint parameters. The profiles of the displacement and the moment of the plate are also shown graphically for several values of the load speed and the edge constraint parameters.     

The axisymmetrical response of a circular cylindrical double-shell system with internal damping

The axisymmetrical response of a circular cylindrical double-shell system with internal damping to a time-dependent surface load is determined by the matrix analysis method. For this purpose, the equations of vibration of the system based upon the Goldenveizer-Novozhilov theory are written as a coupled set of first order differential equations by the use of the state vector of the system. Once the vector has been determined by quadrature of the equations, the steady state response is calculated numerically together with the natural frequencies in terms of the elements of the transfer matrix of the system under any combination of boundary conditions. By the application of the method, the dynamic response and the resonant frequencies (the natural frequencies) are calculated numerically for a double-shell system simply supported at the edges.     

Dynamic response of a prestressed,orthotropic thick plate strip to a moving line load

This paper is a study of the steady state response of an orthotropic plate strip to a moving line load. The plate is of infinite length and subjected to initial in-plane stresses parallel and perpendicular to the edges. The solution is obtained on the basis of a thick plate theory which takes into account the effects of shear deformation and rotatory inertia. The critical speed of the load which brings about a resonance effect in the system is determined. Further, the bending moment in the plate is calculated for several values of the load speed and the initial stress parameters and shown graphically as a function of the space variable moving with the load.     

Starting to move through a granular medium

We explore the process of initiating motion through a granular medium by measuring the force required to push a flat circular plate upward from underneath the medium. In contrast with previous measurements of the drag and penetration forces, which were conducted during steady state motion, the initiation force has a robust dependence on the diameter of the grains in the medium. We attribute this dependence to the requirement for local dilation of the grains around the circumference of the plate, as evidenced by an observed linear dependence of the initiation force on the plate diameter.     

Response of a mindlin plate with trunnion

The vibratory response of a circular plate with a central trunnion is considered. A harmonic force is allowed to act on the trunnion in a plane parallel to the surface of the plate. The model allows for arbitrary location of the center of mass of the trunnion and the line of action of the exciting force. The plate equations include the effects of transverse shear deformations and rotatory inertia, which makes the analysis useful for either thick or thin plates at acoustic frequencies. Application of the model in the control of noise and vibration of rotating machinery is illustrated.     

Response of a large plate-liquid system to a moving pressure step. Transient and stationary aspects

An unbounded system of a plate in contact with a liquid is presented. The coupling equations are set, accounting for the compressibility of the liquid and the Mindlin-Reissner plate theory. A numerical solution, founded on an explicit scheme, is validated. The resolution is applied to a strip loaded by a unit pressure step spreading uniformly along the strip axis. The simulation for every speed of loading ranging among the characteristic velocities of the coupled system points out how a steady state response can emerge from the transient one. The steady state solution is theoretically established, which agrees well with the numerical prevision. The form of the response is different for every region limited by the characteristic velocities. For a load speed greater than the speed of acoustic waves in the liquid, the pressure propagates along a straight line and the existence of a steady state response is confirmed. On the contrary, for a lower load speed, no pressure front is present and the response always keeps its transient character. The solution shows that displacements obtained in the transient part increase incessantly with time and become quickly larger than in the stationary part. Concerning the stresses, the study reveals that the amplitudes are of the same order in the two parts. The solution can be extended easily to the case of a pressure spreading with a cylindrical symmetry, which can correspond to the real conditions of detonation loading.     

Vibration of L-shaped plates under a deterministic force or moment excitation: a case of statistical energy analysis application

Analytical and closed form solutions are presented in this paper for the vibration response of an L-shaped plate under a point force or a moment excitation. Inter-relationships between wave components of the source and the receiving plates are clearly defined. Explicit expressions are given for the quadratic quantities such as input power, energy flow and kinetic energy distributions of the L-shaped plate. Applications of statistical energy analysis (SEA) formulation in the prediction of the vibration response of finite coupled plate structures under a single deterministic forcing are examined and quantified. It is found that the SEA method can be employed to predict the frequency averaged vibration response and energy flow of coupled plate structures under a deterministic force or moment excitation when the structural system satisfies the following conditions: (1) the coupling loss factors of the coupled subsystems are known; (2) the source location is more than a quarter of the plate bending wavelength away from the source plate edges in the point force excitation case, or is more than a quarter wavelength away from the pair of source plate edges perpendicular to the moment axis in the moment excitation case due to the directional characteristic of moment excitations. SEA overestimates the response of the L-shaped plate when the source location is less than a quarter bending wavelength away from the respective plate edges owing to wave coherence effect at the plate boundary.     

An analytical and experimental comparison of optimal actuator and error sensor location for vibration attenuation

Feedforward active control of the flexural waves in a single and L-shaped plate has been analytically and experimentally investigated. The plates are simply supported along two parallel edges, and free at the other two ends. Point forces were used to generate the primary and secondary plate excitations. The plate flexural displacement is described by a combination of a travelling wave solution and a modal expansion. The flexural wave coefficients were determined using the boundary conditions, continuity equations at the driving force locations, and continuity equations at the corner junction for the L-shaped plate. The control actuator and error sensor are optimally located in order to achieve the best control performance.     

Determination of the steady state response of a Timoshenko beam of varying cross-section by use of the spline interpolation technique

The steady state response of an internally damped Timoshenko beam of varying cross-section to a sinusoidally varying point force is determined by use of the spline interpolation technique. For this purpose, with the beam divided into small elements, the response of each element is expressed by a quintic spline function with unknown coefficients. The response is obtained by determining these coefficients so that the spline function satisfies the equation of motion of the beam at each dividing point and also satisfies the boundary conditions at both ends. In this case, the slope due to pure bending of the beam is conveniently adopted as the function essentially expressing the response, from which the transverse deflection, driving point impedance, transfer impedance and force transmissibility of the beam are derived. The method is applied to cantilever beams with linearly, parabolically and exponentially varying rectangular cross-sections; these responses of the beams are calculated numerically and the effects of the varying cross-section on them are studied.     

The effect of the number of nodal diameters on non-linear interactions in two asymmetric vibration modes of a circular plate

In order to investigate the effect of the number of nodal diameters on non-linear interactions in asymmetric vibrations of a circular plate, a primary resonance of the plate is considered. The plate is assumed to have an internal resonance in which the ratio of the natural frequencies of two asymmetric modes is three to one. The response of the plate is expressed as an expansion in terms of the linear, free oscillation modes, and its amplitude is considered to be small but finite, and the method of multiple scales is used. In view of the corrected solvability conditions for the responses, it has been found that in order for the modes to interact, the ratio of the numbers of nodal diameters of two modes must be either three to one or one to one. In this study the one-to-one case, in which the modes have the same number of nodal diameters, is examined. The non-linear governing equations are reduced to a system of autonomous ordinary differential equations for amplitude and phase variables by means of the corrected solvability conditions. The steady state responses and their stability are determined by using this system. The result shows very complicated interactions between two modes by telling existence of non-vanishing amplitudes of the mode not directly excited.     

Generation of waves in an elastic plate by a torsional moment and a horizontal force

An approximate solution is determined for the motion of an infinite elastic plate, excited by a torsional moment (with the axis of the moment normal to the plate) and by a horizontal force (parallel to the plate). The driving moment and force are sinusoidal in time and applied to a small rigid indenter with a circular base, fixed to the plate. The solution is obtained from a three-dimensional approach but is evaluated only for low frequencies, where the wavelengths of the quasi-longitudinal, tranverse and bending waves are much larger than the thickness of the plate. For the case of excitation with a torsional moment, the solution contains two parts, one describing a travelling transverse wave and the other a local reaction. The local reaction is built up of two infinite sums of Love waves with imaginary wave numbers. The driving-point admittance due to the local reaction is larger than the admittance due to the tranverse wave when the diameter of the indenter is smaller than about twice the thickness of the plate. For the case of excitation with a horizontal force, the solution contains three parts that describe travelling waves (quasi-longitudinal, transverse and bending) and two parts that describe the local reaction (infinite sums of Lamb and Love waves). The admittances due to the three types of travelling waves are all of the same order of magnitude. The admittance due to the local reaction is of importance when the frequency is relatively high and the diameter of the identer much smaller than the thickness of the plate. For both cases of excitation, the admittance due to the local reaction increases with increasing thickness of the plate and tends to the value found for a semi-infinite medium.     

Vibration and stability of a circular plate of unidirectionally varying thickness

An analysis is presented for the vibration and stability of an elastically restrained circular plate of unidirectionally varying thickness subjected to an in-plane force. For this purpose, the transverse deflection of a circular plate of variable thickness is written in a series of the deflection functions of a uniform circular plate without the action of a force. The dynamical energies of the plate are evaluated analytically, and the frequency equation of the plate is derived by the Ritz method. The analysis is applied to circular plates of unidirectionally tapered or stepped thickness; the natural frequencies and the divergence loads are calculated numerically, and the effects of the varying thickness and edge conditions on the vibration and stability are studied.     

Navier-Stokes-like equations applicable to adaptive cruise control traffic flows

Under the scenario in which, within a traffic flow, each vehicle is controlled by adaptive cruise control (ACC), and the macroscopic one-vehicle probability distribution function fits the Paveri-Fontana hypothesis, a set of reduced Paveri-Fontana equations considering the ACC effect is derived. With the set, by maximizing the specially defined informational entropy deviating from a certain reference homogeneous steady state, the Navier-Stokes-like equations considering ACC are introduced. For a homogeneous steady traffic flow in a single circular lane, when the steady velocity or density is perturbed along the lane, numerical simulations indicate that ACC-controlled vehicles require less time for re-equilibration than manually driven vehicles. The re-equilibrated steady densities for ACC and manually driven traffic flows are all close to the original values; the same is true for the re-equilibrated steady velocity for manually driven traffic flows. For ACC traffic flows, the re-equilibrated steady velocity may be higher or lower than the original value, depending upon a parameter ω (introduced to solve the distribution function of the reference steady state), and the headway time (introduced in ACC models). Also, the simulations indicate that only an appropriate parameter set can ensure the performance of ACC; otherwise, ACC may result in low traffic running efficiency, although traffic flow stability becomes better.