## Mechanical math

Question 1
Consider the mechanical system with three degrees of freedom shown below. The positions of the particles are measured from their equilibrium positions. The system has a normal mode eigenvector (6.57, b, 6.57)T.
If all particles start from their equilibrium positions and the leftmost and rightmost particles are given a velocity of 34.22 ms-1, the velocity of the middle particle is -47 ms-1, the system will oscillate in a normal mode. Determine the value of b, giving your answer to 3 decimal places.

Document Preview:

Question 1
Consider the mechanical system with three degrees of freedom shown below. The positions of the particles are measured from their equilibrium positions. The system has a normal mode eigenvector (6.57, b, 6.57)T.
If all particles start from their equilibrium positions and the leftmost and rightmost particles are given a velocity of 34.22 ms-1, the velocity of the middle particle is -47 ms-1, the system will oscillate in a normal mode. Determine the value of b, giving your answer to 3 decimal places.
Question 2
An external sinusoidal force is applied to an oscillating system which can be modelled by a model spring and a damper. The general solution of the equation of the motion is given by,
x = 3 + 3.183cos(2t – ) + 3.18exp(-7.72t) cos(t + )
where , , and  are some constants . Determine the amplitude of the oscillation when steady-state is reached. Give your answer correct to 3 decimal places.