Mechanical, Damping force

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

Consider two particles of masses m1 and m2 joined to each other and to two fixed walls at both ends by three identical model springs of stiffness k and natural length lo. The matrix equation of motion for the mechanical system is shown below .
For certain choices of m1, m2 and k, a normal mode of the system is given by (x1(t), x2(t))T where a, b, ω and  are some constants. If a = -3.77, b = 15, determine how far (in cm) to the left of its equilibrium position does m1 have be initially displaced if m2  is initially 4.09 cm to the right of its equilibrium position, in order for the system to oscillate as a normal mode. Give your answer correct to 3 decimal places.


Question 3

A mass m of 44.59 kg is travelling towards the left, on a straight frictionless, horizontal track onto buffers at a constant speed of 2.23 ms-1 when at t = 0, it hits the buffer. The buffers are to be modelled by a model spring, with stiffness 135.97 Nm-1, together with a model damper with damping constant 207.73 Nsm-1. The x-axis is chosen directed away from the buffers down the track (in the direction opposite to the incoming mass), with origin at the fixed end of the model spring. The natural length of the model spring is 0.53 m.
The equation of motion is given by, where a, b and c are some constants.

where a, b and c are some constants.
When m = 44.59, determine is the value of c? Give your answer to 3 decimal place.
The system of model damper and model spring is shown below.


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