202 4. Vibration of Continuous Systems TABLE 4.2. Frequency equations and mode shapes of the longitudinal vibrations E cal ,A,-constant c Boundary conditions Frequency equation Mode shapes cos 0 4-A, sinx 02- (2)- 1)ne 21 Fixed-Free Free-Free Fixed-Fixed sin -0 - A, cos x jac 03=> (x) col sin--0 c M y tan y Fixed-Mass M MX) ME Fixed-Spring (x1) where- 0,- A, sinx M y tan y M k -A, sin M Free-Mass (x) m 7 coty=- = A, cos M where= m WE M y coty k b = A, cos x M Free-Spring