Kafr Elsheikh University
Faculty of Engineering
Mech. Eng. Dept.
Working sheet and assignment No. (4)
Two degree of freedom systems
Third Year Mech. Eng.
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Instructor : Dr. Maged Elhefnawey
Dr. Maged Elhefnawey
1- Two masses m1 and m2, each connected by two springs of stiffness k, are connected
by a rigid massless horizontal rod of length l as shown in Figure 1. (a) Derive the
equations of motion of the system in terms of the vertical displacement of the C.G.
of the system, x(t), and the rotation about the C.G. of the system, ϴ(t), (b) Find the
natural frequencies of vibration of the system for m₁ = 50 kg, m₂ = 200 kg, and k =
1000 N/m.
Figure 1
Dr. Maged Elhefnawey
2- A two-mass system consists of a piston of mass m1, connected by two elastic
springs, that moves inside a tube as shown in Figure 2. A pendulum of length l and
end mass m₂ is connected to the piston as shown in Figure2. (a) Derive the equations
of motion of the system in terms of x1(t) and ϴ(t). (b) Derive the equations of motion
of the system in terms of the X1(t) and x2(t). (c) Find the natural frequencies of
vibration of the system.
Figure 2
Dr. Maged Elhefnawey
3- Set up the differential equations of motion for the double pendulum shown in Figure
3, using the coordinates x1and x2 and assuming small amplitudes. Find the natural
frequencies, the ratios of amplitudes, and the locations of nodes for the two modes
of vibration when m1= m2=m and l1 = l2 = l.
Figure 3
Dr. Maged Elhefnawey
4- One of the wheels and leaf springs of an automobile, traveling over a rough road, is
shown in Figure 4. For simplicity, all the wheels can be assumed to be identical, and
the system can be idealized as shown in Figure 4. The automobile has a mass of m₁
= 1000 kg, and the leaf springs have a total stiffness of k₁ = 400 kN/m. The wheels
and axles have a mass of m2 =300 kg, and the tires have a stiffness of k2= 500 kN/m.
If the road surface varies sinusoidally with an amplitude of Y = 0.1 m and a period
of l= 6 m, find the critical velocities of the automobile.
Figure 4
Dr. Maged Elhefnawey
5- For the system shown in Figure 5, m₁ = 1 kg, m₂ = 2 kg, k1= 2000 N/m, k2= 1000
N/m, k3= 3000 N/m, and an initial velocity of 20 m/s is imparted to mass m₁. Find
the resulting motion of the two masses.
Figure 5
Dr. Maged Elhefnawey
6- Figure 6 shows a system of two masses attached to a tightly stretched string, fixed
at both ends. Determine the natural frequencies and mode shapes of the system for
m₁ = m₂ = m and l1=l2=l3 =l
Figure 6
Dr. Maged Elhefnawey
7- Two identical pendulums, each with mass m and length l, are connected by a spring
of stiffness k at a distance d from the fixed end, as shown in Figure 7 .
a. Derive the equations of motion of the two masses.
b. Find the natural frequencies and mode shapes of the system.
c. Find the free-vibration response of the system for the initial conditions ϴ1(0) = a,
ϴ2(0) = 0,
d. Determine the condition(s) under which the system exhibits a beating
phenomenon.
Figure 7
Dr. Maged Elhefnawey
8- The motor-pump system shown in Figure 8(a) is modeled as a rigid bar of mass m
= 50 kg and mass moment of inertia Jo= 100 kg-m². The foundation of the system
can be replaced by two springs of stiffness k₁ = 500 N/m and k₂ = 200 N/m.
Determine the natural frequencies of the system
Figure 8
Dr. Maged Elhefnawey
9- An airplane standing on a runway is shown in Figure 9. The airplane has a mass m
= 20,000 kg and a mass moment of inertia Jo= 50 × 106 kg-m². If the values of
stiffness and damping constant are k₁ = 10 kN/m and c₁ = 2 KN-s/m for the main
landing gear and k₂ = 5 kN/m and c₂ = 5kN-s/m for the nose landing gear, (a) derive
the equations of motion of the airplane, and (b) find the undamped natural
frequencies of the system
Figure 9
Dr. Maged Elhefnawey
10- A simplified ride model of the military vehicle in Figure 10 (a) is shown in Figure
10 (b). This model can be used to obtain information about the bounce and pitch
modes of the vehicle. If the total mass of the vehicle is m and the mass moment of
inertia about its C.G. is Jo, derive the equations of motion of the vehicle using two
different sets of coordinates, as indicated in Section 5.5.
Figure 10
Dr. Maged Elhefnawey
11- An airfoil of mass m is suspended by a linear spring of stiffness k and a torsional
spring of stiffness k, in a wind tunnel, as shown in Figure 11. The C.G. is located at
a distance of e from point O. The mass moment of inertia of the airfoil about an axis
passing through point O is Jo. Find the natural frequencies of the airfoil.
Figure 11
Dr. Maged Elhefnawey
12- The weights of the tup, frame, anvil (along with the workpiece), and foundation
block in a forging hammer (Figure 12) are 5000 lb, 40,000 lb, 60,000 lb, and
140,000 lb, respectively . The stiffnesses of the elastic pad placed between the anvil
and the foundation block and the isolation placed underneath the foundation
(including the elasticity of the soil) are 6 × 106 lb/in. and 3 x 106 lb/in., respectively.
If the velocity of the tup before it strikes the anvil is 15 ft/sec, find (a) the natural
frequencies of the system, and (b) the magnitudes of displacement of the anvil and
the foundation block. Assume the coefficient of restitution as 0.5 and damping to be
negligible in the system.
Figure 12
Dr. Maged Elhefnawey
13- A centrifugal pump, having an unbalance of me, is supported on a rigid foundation
of mass m2 through isolator springs of stiffness k₁, as shown in Figure 13. If the soil
stiffness and damping are k₂ and c₂, find the displacements of the pump and the
foundation for the following data: mg = 0.5 lb, e =6 in, m1g 800 lb, k₁ = 2000 lb/in.,
m2g 2000 lb, k2= 1000 lb/in, c₂ = 200 lb-sec/in., and speed of pump = 1200 rpm.
Figure 13
Dr. Maged Elhefnawey
14- In the system shown in Figure 14, the mass m1 is excited by a harmonic force having
a maximum value of 50 N and a frequency of 2 Hz. Find the forced amplitude of
each mass for m1 =10 kg, m₂ = 5 kg, k₁ = 8000 N/m, and k₂ = 2000 N/m.
Figure 14
Dr. Maged Elhefnawey
15- Determine the equations of motion and the natural frequencies of the system
shown in Figure 15.
Figure 15
Dr. Maged Elhefnawey
16- A turbine is connected to an electric generator through gears, as shown in Figure
16. The mass moments of inertia of the turbine, generator, gear 1, and gear 2 are
given, respectively, by 3000, 2000, 500, and 1000 kg-m². Shafts 1 and 2 are made
of steel and have diameters 30 cm and 10 cm and lengths 2 cm and 1.0 m,
respectively. Find the natural frequencies of the system.
Figure 16
Dr. Maged Elhefnawey