22 m k, mg II. As shown in the figure, consider a free vibration of 2DOF linear system consisting of point masses of mas

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22 M K Mg Ii As Shown In The Figure Consider A Free Vibration Of 2dof Linear System Consisting Of Point Masses Of Mas 1 (106.38 KiB) Viewed 26 times

22 m k, mg II. As shown in the figure, consider a free vibration of 2DOF linear system consisting of point masses of mass mi, m2, and two springs of spring constant ki, k2. Left end of the system is fixed. Displacements of two point masses are 11, 12 respectively. Answer the following questions. (25 points) (1) Using = [31 ra] as a displacement vector, derive the equation of motion of free vibration of this system in matrix form Mä+Kr=0. = Here M and K denote the mass and stiffness matrices respectively and []? means transpose. (2) Derive the characteristic equation (frequency equation) of this system and solve this characteristic equation for eigenvalue w2. (3) Find two natural angular frequencies w1,w2 and two natural modes X1, X2 of this system when my = m2 = 1kg, ky = ka = 1N/m. Normalize the natural modes so that the first element of each mode is equal to one. (4) Under the condition of mı = m2 = m, and kn = akı(a is a positive real constant), find the eigenvalue wa using m,kı, a. (5) Under the same condition as in (4), and besides, ki > ka, find the natural angular frequencies wi, wy and discuss the effect of the condition of ki > ky on the motion of the system at wi and w, respectively.