Nonlinear transverse vibrations of centrally clamped rotating circular disks
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A study is presented of the instability mechanisms of a damped axisymmetric circular disk of uniform thickness rotating about its axis with constant angular velocity and subjected to various transverse space-fixed loading systems. The natural frequencies of spinning floppy disks are obtained for various nodal diameters and nodal circles with a numerical and an approximate method. Exploiting the fact that in most physical applications the thickness of the disk is small compared with its outer radius, we use their ratio to define a small parameter. Because the nonlinearities appearing in the governing partial-differential equations are cubic, we use the Galerkin procedure to reduce the problem into a finite number of coupled weakly nonlinear second-order equations. The coefficients of the nonlinear terms in the reduced equations are calculated for a wide range of the lowest modes and for different rotational speeds. We have studied the primary resonance of a pair of orthogonal modes under a space-fixed constant loading, the principal parametric resonance of a pair of orthogonal modes when the disk is subject to a massive loading system, and the combination parametric resonance of two pairs of orthogonal modes when the excitation is a linear spring. Considering the case of a spring moving periodically along the radius of the disk, we show how its frequency can be coupled to the rotational speed of the disk and lead to a principal parametric resonance. In each of these cases, we have used the method of multiple scales to determine the equations governing the modulation of the amplitudes and phases of the interacting modes. The equilibrium solutions of the modulation equations are determined and their stability is studied.
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