Linear Plate Equation Research Articles

Thin vibrating plates: long time existence and convergence to the von Kármán plate equations

AbstractThe asymptotic behavior of the solutions of three‐dimensional nonlinear elastodynamics in a thin plate is studied, as the thickness h of the plate tends to zero. We discuss the long time existence and convergence to solutions of the time‐dependent von Kármán and linear plate equation under appropriate scalings of the applied force and of the initial values in terms of h (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Lateral Vibrations as Related to Structural Stability

Abstract The apparently different physical problems of lateral vibration and elastic stability are limiting cases of a single phenomenon, the more general expression being the mode of vibration with end thrust. It is shown that the square of the frequency of lateral vibration is approximately linearly related to the end load. The linear relationship is exact if the mode of free vibrations is identical to the buckling mode. In all cases, the load corresponding to zero frequency is the critical buckling load. Experimental tests were conducted on elastically restrained columns in the form of rigid rectangular frames. It was found that the relationship between the square of the frequency and the load is practically linear, and that the extrapolated load corresponding to zero frequency coincides with the buckling load. Rigid-joint trusses also were tested. In some cases the relationship between the square of the frequency and the load deviates considerably from linearity, but the approximate buckling load can be predicted. In the case of thin flat plates, tests showed that the buckling load cannot be predicted from vibration tests in practice. It is indicated that this is probably because the linear plate equations are not valid owing to initial curvatures in the plate.