Damping Forces, Normal Mode (Physics+Maths)

Question is in the attachment.

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 (19.13, b, 19.13)T.

 If all particles start from their equilibrium positions and the leftmost and rightmost particles are given a velocity of 5.77 ms-1, the velocity of the middle particle is -48.16 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 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 =  6 + 7.311cos(2t – ) + 4.37exp(-3.72t) 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.

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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, ω and  are some constants. If a = -3.42, b = 3.93, determine how far (in cm) to the left of its equilibrium position does m1 have be initially displaced if m2  is initially 11.89 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.

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Question 4

A particle A of mass 3.03 kg collides with another particle B of mass 2.1 kg. Their initial velocities are u1 = 2.47i + 0.71j and u2 = 0.16i + 1.19j just before impact. After collision, they both merged and becomes a composite particle, which travel with a velocity of V = 0.86 i + 0.93j.
Calculate the change in kinetic energy. Give your answer to 3 decimal places.
(Note that if there is a loss in kinetic energy, the answer will be negative.)

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Question 5

The equation of motion of a forced vibration problem is given by

Given the values m = 6.24, r = 54.67, k = 396.58, P = 9.28 and ω = 5.11.
Determine the steady-state solution, xp(10.74) of the differential equation, giving your answer correct 3 decimal places.

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Question 6
Two particles A (of mass m) and B (of mass 4m) are connected by a model string which passes over a frictionless pulley as shown. A move downwards while B moves horizontally on a frictionless surface, to the right. Both particles move with a common acceleration.
When m = 7.22, calculate the tension, in N, in the model string, giving your answer to 3 decimal places. Take the magnitude of the acceleration due to gravity to be 9.81 ms-2.

Hint
: Apply Newton’s 2nd Law to each of the two particles A and B.

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Question 7
A particle A, of mass 1 kg, moves along a frictionless horizontal track. The particle is attached to a fixed point O by a model damper, and to another point B by a model spring as shown in the diagram. The two points O and B are a distance 2 metres apart on the track. The damping constant is 18.69 Ns m-1. The spring has stiffness 17.67 Nm-1 and natural length 1.026 metre. The displacement of the particle, x, is measured from O.
Calculate the equilibrium position of the particle, measured from O, giving your answer to 3 decimal places.

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Question 8
A particle A of mass 2.84 kg collides with another particle B of mass 3.99 kg. Their initial velocities are u1 = 1.55i + 1.48j and u2 = 0.87i + 1.19j just before impact. After collision, they both merged and become a composite particle, which travel with a velocity of V.
Calculate the speed |V|, a scalar, in ms-1. Give your answer to 3 decimal places.
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Question 9

A mass m of 39.57 kg is travelling towards the left, on a straight frictionless, horizontal track onto buffers at a constant speed of 0.9 ms-1 when at t = 0, it hits the buffer. The buffers are to be modelled by a model spring, with stiffness 106.5 Nm-1, together with a model damper with damping constant 196.46 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 0.68 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 = 39.57, 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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Question 10
Two particles X and Y, joined by a rigid rod of negligible mass, move along a smooth horizontal plane. Particle X has mass 2.99 kg and is pushed by an external force Fx of magnitude 57 N. Particle Y has mass 7.13 kg and is pulled along the plane by an external force Fy of magnitude 15.29 N. It is found that the two particles accelerate along the plane a fixed distance apart.
Calculate the common acceleration of the particles. Give your answer to 3 decimal places.

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