Analysis of transverse vibrations of centrally clamped spinning membrane

SUMMARY Let us consider a circular membrane extending over a domain D defined by a < r < b. It is uniform, homogeneous, isotropic and has an angular velocity 0 (Fig. 2.3). The membrane on elastic foundation which is homogeneous is taken to be rigidly clamped on a circle of radius a and free at the outer edge, r - b. By taking an infinitesimal element of the membrane, equilibrium equations and in- plane stresses are obtained. The resultant expressions are same as those which are used in the studies of Barasch-Chen and Eversman. Then the equation of motion is derived by taking again an infinitesimal element in rectangular coordinate, and the equation of motion is transfered from (x, y) set to (r, 9) set. So the governing partial differantial equation is reached. In order to solve the governing differantial equation, separation of variable is used. Finally we reach an eigenvalue problem. After a comparison function is selected, stiffness and mass matrices are constructed by means of Rayleigh-Ritz method, and the governing differantial equation is numerically solved. The results are presented by using graphics and shapes. Later the membrane is considered to subject the harmonic excitation. It is analyzed in the condition that its boundary at r = a is subjected to translational harmonic motion in the form of A sin wt. The governing differantial equation is modified and numerically solved by modal analysis. Again the results are presented graphically. Finally a different approach to the problem is made. In order to solve the problem without elastic foundation a finite element software package, called "Applied Structure", is used. The stress and strain distributions, and mode shapes are presented graphically. In this study it is shown that the frequencies of a centrally clamped spinning membrane are affected from the angular velocity and the clamped radius seriously. Mode shapes obtaining in the second chapter are similar to those obtaining by using Applied Structure software. This justifies that the results are accurate enough. Using both FTN77 and AutoLISP programming languages, the vibration phenomena is succesfully simulated in this thesis. VI