This is done numerically via the computer program in appendix a. It can be seen that m12, n12, is sufficient for converged result. The candidate displacement function satisfies the geometric boundary conditions. Boundary conditions for a clamped edge are considered for most of the paper, and the very similar case of a hard simplysupported plate is discussed brie y at the end. If the plate is clamped at an end by being sandwiched between two rigid surfaces, it is customary to model this constraint i. The natural frequency increases with decrease in ba ratio. Rectangular plate, clamped on all edges, triangular loading. Uniformly loaded rectangular thin plates with symmetrical. Following are the various cases of the plate which are considered for present study.
Vibration modes of a single plate with general boundary. Simply supported pzxy simply supported differential equation boundary conditions on x ony 0 w0,y wx, 0 pzx, y ox oy oy on x ony a b wx, b only even derivatives in the boundary value problem. First a flat rectangular plate is considered with arbitrary boundary conditions. Determining the solution of the plates differential equation with partial derivatives that satisfies all the kinematic and static conditions on the considered boundary cannot always be rigurously achieved. For clamped rectangular thin plates no accurate results appear to be available. Simply supported plate an overview sciencedirect topics.
Balch division of mechanics and computation department of mecanical engineering stanford university stretching and bending of plates fundamentals introduction a plate is a structural element which is thin and. Specifications of boundary conditions for plate bending 98 5 here p px, y is the applied pressure load. Let us treat a plate clamped along part of the boundary and simply supported along the remainder, requiring only two different boundary conditions for the numerical procedure fig. In general each edge may be simply supported s, clamped c or free f so there are 21 different, possible combinations of boundary conditions, which are listed in table 1, where the reddy notation of the boundary condition is adoptedin which the consecutive pair of. Nonlinear questions in clamped plate models 3 for the navier boundary value problem 1. The vibration of plates is a special case of the more general problem of mechanical vibrations. Generally, the most severe test is for problems where all boundary parameters are restrained.
Both natural and clamped boundary cubic splines will be constructed and plotted against the given data for comparative purposes. Now equate the total kinetic energy with the total strain energy per rayleighs method, equation 3. Zero stiffness corresponds to a simply supported plate, whereas very high stiffness comes close to the case of a clamped plate. The current work discusses the fundamental natural frequency of an elastic rectangular plate having fixed or clamped corners using the rayleigh method. Relevant formulas valid in this natural system can be found in the appendix. An exact solution for the deflection of a clamped rectangular plate under uniform load. So, there are infinitely many possibilities for boundary conditions in 3d elasticity that correspond to a clamped edge integral average for a plate, including the boundary condition for zero. Analysis bending solutions of clamped rectangular thick plate.
Introduction when developing new nite elements for solution of plate problems based on the reissnermindlin theory it is necessary to check for locking at the thin plate limit to ensure proper behavior. Moody division of design denver, colorado united states department of the interior. Brenner and collaborators thirupathi gudi and shiyuan gu and michael neilan and liyeng sung and kening wang and in. The basic governing equations used for analysis are based on mindlins higherorder shear deformation plate theory. Plate fixed along three edges, moment and reaction. As for short, they are called by cccc rectangular plate, ccsc rectangular plate.
The derivation of plate governing equations of free vibration modes is introduced. Using a new function, the three coupled governing equations have been modified to independent partial differential equations that can be solved separately. It is clamped on all edges and it is subjected to uniform loading on the top and the maximum deformation is determined by considering trial functions and applying the boundary conditions. Furthermore, the slopes at the plate center x y 0 vanishes, as they should due to symmetry. The boundary conditions are combinations of 0and 0 w w n. There are many additional complications in the case of more general boundary conditions, and so the analysis of 1 is not easily extended to the. Plate calculators, rectangular and circular plates with various boundary and loading conditions.
In this paper, we first start by introducing the problem of the plate with clamped corners and resulting boundary conditions. Dynamic response to moving masses of rectangular plates. Circular plate, clamped over half its boundary and loaded by a 100n transverse. Pdf uniformly loaded rectangular thin plates with symmetrical. The geometry has been exerted to the harmonic wave propagation in different modes of vibration. A boundary condition correction for the clamped constraint of elastic. The boundary conditions that are needed to solve the equilibrium equations of plate theory can be obtained from the boundary terms in the principle of virtual work. Exact solutions for freevibration analysis of rectangular. But it does not satisfy the moment, twist, and shear boundary conditions.
For centuries, however, an exact solution for a fully clamped rectangular plate has not yet been obtained, and it is currently considered that an exact solution is not achievable for the rectangular plate problem of this type. The results of examination of the effect of lengthwidth of the plate ratio ab on buckling loads of the plates subjected to external pressure and modelled with simple boundary conditions for supports are presented in table iii and table iv. The natural frequency of a rectangular plate with fixedfreefixedfree boundary conditions by tom irvine email. For the clamped cubic spline, the boundary conditions are s0,t0 ft0. These coefficients are to be determined from the condition that the slope at the boundaries is zero. The maximum amplitude is at center and is equal to ca8 w o. Other boundary conditions will be treated in a forthcoming paper. The determination of natural frequencies of rectangular. Buckling stability assessment of plates with various. Examples demonstrating the importance of this effect are given for a cantilever and a clamped clamped plate beam and also for a clamped circular plate. The proposed nonideal boundary model is applied to the free vibration analyses of eulerbernoulli beam and timoshenko beam. A boundary condition correction for the clamped constraint of. Of all the tested boundary conditions natural frequency is highest at all sides clamped plate cccc. Influence of boundary conditions on sound radiation.
Pdf solutions of dynamic and acoustic responses of a. For small strains and small rotations, the boundary conditions are. The free vibration analysis of the eulerbernoulli beam is carried out. Solution of clamped rectangular plate problems university of. How can i express a clamp boundary condition in terms of. This paper investigates a boundary control aiming at vibration suppression for a kirchhoff plate with clampedclampedclampedfree boundary conditions. The transverse deflection of the plate is w and p is the pressure load. The stress resultants m,, m, and t, in figure 2 are defined with respect to coordinate directions n and s, normal to and along the boundary curve.
The robustness and reliability of the present approach are tested by a number of numerical experiments. The vibration of circular plates was also analyzed with clamped boundary conditions 26. For a rectangular plate clamped at all its edges, the boundary conditions are given by. The bending solutions of rectangular thick plate with all edges clamped and supported were investigated in this study. The resonant frequencies along with the damping frequencies are determined and dispersion curves are drawn. In this research work, the effect of varying thickness of the plate on its deflection and bending stress is studied. For centuries, however, an exact solution for a fully clamped rectangular plate has not yet been obtained, and it is currently considered that an exact solution is not achievable for. A boundary condition correction for the clamped constraint of elastic platebeam theory. Partial differential equation toolbox cannot directly solve this fourthorder plate equation. Bending analysis of simply supported and clamped circular. Both isotropic and orthotropic material systems were considered. Free vibration analysis of beams with nonideal clamped. Boundary control of a thin rectangular plate with clamped.
Such a situtation leads to a need for highly accurate solutions with all boundary points set for clamped conditions. The equations governing the motion of plates are simpler than those for general threedimensional objects because one of the dimensions of a plate is much smaller than the other two. Introduction to the theory of plates stanford university. Vibration analysis of composite plate at different. In general each edge may be simply supported s, clamped c or free f so there are 21 different, possible combinations of boundary conditions, which are listed in table 1, where the. The flow around a solid, however,cannot be treated in such a manner because of viscous friction. Specifications of boundary conditions for reissnermindlin. Boundary layer if the movement of fluid is not affected by its viscosity, it could be treated as the flow of ideal fluid, therefore its analysis would be easier.
An exact solution for the deflection of a clamped rectangular. In realistic domains, open boundary conditions can be extremely difficult to get right. Of all the tested boundary conditions natural frequency is lowest for a cantilever plate cfff. We use the deflection, deflection rate, slope, and slope rate at the free edge as feedback signals and show that the closedloop system is marginally stable by using the lyapunovs direct method. Solutions of dynamic and acoustic responses of a clamped rectangular plate in thermal environments. Taking the origin of coordinates at the center of the plate and x and yaxes parallel, to the side a and b of the plate, and it is supposed throughout the paper that a. Your data need not be evenly spaced, but must be ordered a t0 boundary conditions. An exact solution for the deflection of a clamped rectangular plate. Specifications of boundary conditions for reissnermindlin plate. The left is a clamped, b simply supported, and c free. It will be analyzed by utilizing simple platebeam theory with a modified boundary condition. Reissner, mindlin, plate, boundary layer amsmos subject classi cations.
Clamped, square isotropic plate with uniform pressure load. This rotational stiffness scales with the product of the shear modulus and the square of the plate thickness and further depends upon poissons ratio. Free vibration analysis of thin circular and annular plate. Vibrations analysis of rectangular plates with clamped. Nevertheless, the solution of the general plate problem can be obtained by the bem according to the procedure developed for the biharmonic operator 6,10. There are situations in which incoming flow and outgoing flow happen along the same boundary or even at different depths at the same horizontal location.
Bending analysis of simply supported and clamped circular plate. Vibration analysis of composite plate at different boundary. A displacement function is assumed which satisfies the geometric boundary conditions. In previous work 1, we gave an analysis of the boundary layer for the reissnermindlin model of hard clamped and hard simply supported plates. The sliding boundary conditions will convert the eigenvalue problem into the equilibrium problem and therefore are not considered in the buckling analysis of plates. The influence of boundary conditions as shown in figure 4, five rectangular plates with different boundary conditions are studied in this section. The boundary conditions for the clamped boundaries are w 0 and w. Length of an edge of the rectangular plate parallel to the y or tt axis. Constants in the solution of the differential equation for the levytype rectangular plates and are to be obtained from the boundary conditions along the edges parallel to the x axis. A boundary condition correction for the clamped constraint. The integral of the spline is also computed and printed. A nonideal boundary condition is modeled as a linear combination of the ideal simply supported and the ideal clamped boundary conditions with the weighting factors k and 1k, respectively.
Thus, the kinematic boundary conditions are satis ed identically for any value of the unknown constant c. Examples demonstrating the importance of this effect are given for a cantilever and a clampedclamped platebeam and also for a. Extremely accurate dsc results were obtained for beam bending, vibration and buckling33,34. If the plate is clamped in the range 0 boundary parameters are restrained. The first five natural modes of flexural vibrations for free, clamped and simply plates are shown in fig.