Download Question 1 Consider the mechanical system with three degrees of

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(2t – ) + 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, 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.
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
answer to 3 decimal places.(Note that if there is a loss in kinetic energy, the answer
will be negative.)
Answer:
Question 5
The equation of motion of a forced vibration problem is given by
𝑚
𝑑2 𝑥
𝑑𝑥
+𝑟
+ 𝑘𝑥 = 𝑃𝑐𝑜𝑠(𝜔𝑡)
2
𝑑𝑡
𝑑𝑡
Given the values m = 7.9, r = 65.49, k = 351.4, P = 8.7 and ω = 5.03.Determine the
steady-state solution, xp(14.99) of the differential equation, giving your answer
correct 3 decimal places.
Answer:
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 = 9.52, 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.
Answer:
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 15.73 Ns m-1. The spring has stiffness
11.17 Nm-1 and natural length 0.684 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.
Answer:
Question 8
A particle A of mass 1.4 kg collides with another particle B of mass 2.18 kg. Their
initial velocities are u1 = 1.65i + 0.71j and u2 = 0.54i + 1.33j 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.
Answer:
Question 9
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.
𝑑2𝑥
𝑑𝑥
𝑚 2 +𝑎
+ 𝑏𝑥 = 𝑐
𝑑𝑡
𝑑𝑡
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.
Answer:
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 6.31 kg and is pushed by an external force Fx
of magnitude 42.73 N. Particle Y has mass 7.35 kg and is pulled along the plane by
an external force Fy of magnitude 43.95 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.
Answer: