Free Vibration of Two-Dimensional Frameworks

A set of three partial difference equations is derived which describes exactly the in-plane free vibrational motion of the joints of a regular, orthogonal, two-dimensional framework having rigid joint connections. The equations include concentrated masses and inertias at the joints. Two cases are treated in detail; one in which the joints have rotational freedom and one in which rotation and horizontal displacement are permitted. For specific boundary conditions, transcendental equations for the free vibration frequencies and solutions for the modal functions are obtained in the form of discrete trigonometric functions. A method is outlined for finding solutions to the frame having three degrees-of-freedom at all interior joints but with highly specialized conditions along the vertical boundaries. For comparison, the governing difference equations for two lumped parameter models are obtained as special cases of the general equations. Numerical results for specific frames illustrate typical frequency spectra and indicate application of the results to evaluating the accuracy of approximate models.