Tutorial + Correction Mechanical Vibration Chapter 3
Tutorial + Correction Mechanical Vibration Chapter 3
Tutorial + Correction Mechanical Vibration Chapter 3
Exercise 1
An oscillator has the equation of motion:
Exercise 2
A block of 25 kg weight is mounted on a support rubber (with mass negligible), which is
compressed under the weight of 6.1cm. When the block vibrate freely, the positions of the
mass was recorded after moving 5cm from its equilibrium position (see figure below). Since
the rubber can be represented by spring stiffness damper k associated with a damping
coefficient of viscous friction of α, calculate the coefficients k and α
Exercise 3
A mass of 5 kg is suspended on a spring and set oscillating. It is observed that the amplitude
reduces to 5% of its initial value after 2 oscillations. It takes 0.5 seconds to do them.
Calculate the following.
1-The damping ratio. 2- The natural frequency. 3-The actual frequency. 4- The spring
stiffness. 5- The critical damping coefficient. 6-The actual damping coefficient.
1
2
Exercise 4
A mass of 5 kg is suspended on a spring of stiffness 4000 N/m. The system is fitted with a
damper with a damping ratio of 0.2. The mass is pulled down 50 mm and released. Calculate
the following.
i. The damped frequency.
ii. The displacement, velocity and acceleration after 0.3 seconds.
Exercise 5
An under damped shock absorber is to be design for a motorcycle of mass 200 kg. When the
shock absorber is subjected to an initial vertical velocity due to a road bump, the resulting
displacement-time curve is to be as shown. Find the necessary stiffness and damping
constants of the shock absorber if the damped period is to be 2 s and the amplitude x1 is to
be reduced to one-fourth in one half cycle (i.e., x1.5 = x1/4 ). Also find the minimum initial
velocity that leads to a maximum displacement of 250 mm
Exercise 6
The power winch W was mounted on the truss T as shown in Figure below (a). To analyze
the vibrations of the power winch the installation was modelled by the one degree of
freedom physical model (b). In this figure the equivalent mass, stiffness and damping
coefficient are denoted by m, k and C respectively. Origin of the axis x coincides with the
centre of gravity of the weight m when the system rests in its equilibrium position. To
identify the unknown parameters m, k, and c, the following experiment was carried out. The
winch was loaded with the weight equal to M1 =1000kg. Then the load was released
allowing the installation to perform the vertical oscillations in x direction. Record of those
oscillations is presented in the curve below. Calculate the parameters m, k, and c.
2
3
Exercise 7.
We propose to study the vibration behavior of rubber materials for their use in structural
construction. We assimilate the material elasticity to a spring with a stiffness k and the
energetic friction losses to those occurring in a friction damper with coefficient (see figure
1.a and 1.b).
When placing a mass of 1000 kg on the rubber block, the latter will compressed by a
distance d at the equilibrium. After further compression, the mass is released and the system
starts to oscillate around the equilibrium position. Measuring the time interval between the
1st and the 6th maximum, we found t= 0.2s. The decrease of the corresponding amplitude is
60%.
2.1. Apply the second Newton’s law and establish the equation of motion of the mass m
(differential equation) (3pts)
2.2. Give the general form of the solution y(t) (3pts)
3
4
Exercise 8
The mechanical system shown below is su bjected to the initial excitation: 𝑥(0) =
0.01𝑚; 𝑥̇ (0) = 0, where x is taken relative to the equilibrium position of the 25 kg.
Compute:
a- The undamped natural frequency
b- The dimensionless damping ratio
c- The time required to return 95% of the distance back to the static equilibrium position
d- The value of the damping coefficient C that would make the system critically damped
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Free damped systems: Applications
Problem I
Problem II