TOEPLITZ JACOBIAN MATRIX FOR NONLINEAR PERIODIC VIBRATION

The main difference between a linear system and a nonlinear system is in the nonuniqueness of solutions manifested by the singular Jacobian matrix. It is important to be able to express the Jacobian accurately completely, and efficiently in an algorithm to analyze a nonlinear sys tem. For periodic response, the incremental harmonic balance (IHB) met hod is widely used. The existing IHB methods, however, requiring doubl e summations to form the Jacobian matrix, are often extremely time-con suming when higher order harmonic terms are retained to fulfill the co mpleteness requirement. A new algorithm to compute the Jacobian is to be introduced with the application of fast Fourier transforms (FFT) an d Toeplitz formulation. The resulting Jacobian matrix is constructed e xplicitly by three vectors in terms of the current Fourier coefficient s of response, depending respectively on the synchronizing mass, dampi ng, and stiffness functions. The part of the Jacobian matrix depending on the nonlinear stiffness is actually a Toeplitz matrix. A Toeplitz matrix is a matrix whose k, r position depends only on their differenc e k-r. The other parts of the Jacobian matrix depending on the nonline ar mass and clamping are Toeplitz matrices modified by diagonal matric es. If the synchronizing mass is normalized in the beginning, we need only two real vectors to construct the Toeplitz Jacobian matrix (TJM), which can be treated in one complex fast Fourier transforms. The pres ent method of TJM is found to be superior in both computation time and storage than all existing IHB methods due to the simplified explicit analytical form and the use of FFT.