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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.

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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.
Answer:
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.
Answer:
Question 3
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, ? 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.
Answer:
Question 4
A particle A of mass 2.22 kg collides with another particle B of mass 1.3 kg. Their initial velocities are u1 = 1.55i + 0.88j and u2 = 0.21i + 0.94j just before impact. After collision, they both merged and becomes a composite particle, which travel with a velocity of V = 0.67 i + 0.99j. Calculate the change in kinetic energy. Give your…

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