m + I
r2
x + c ˙x + k x = −¨ M r .
b) The equivalent mass, damping, and stiffness are:
meq= m + I
r2, ceq= c, keq= k.
c) When the system is in equilibrium, the displacement of the disk is:
xeq= −M kr.
1.2 Unsolved Problems
Problem 25:
In the figure, the disk has mass m and radius r. The cable wraps around the disk with no slip and is inextensible, attached to a spring of stiffness k and a damper with coefficient c.
a) Find the equations of motion for this system.
b) What are the natural frequency and damping ratio of the system?
c) If the disk is displaced 12cm down and released from rest find the resulting an-gular displacement of the disk with
m = 3kg, r = 9cm, k = 21N/m, c = 63N · s/m,
(m, r) k c
Problem 26:
In the figure, the disk has mass m, radius r, and moment of inertia IG = mr22 about the mass center G and is assumed to roll without slip.
a) Find the equations of motion for this system.
b) What are the natural frequency and damping ratio for this system?
k
k c
(m, r) r/2
G g
Problem 27:
In the spring-mass-damper system shown, the block slides on a frictionless surface, where the coordinate x measures the dis-placement of the system from the unstretched position of the spring. The mass of the block is m = 2kg, while the spring and damp-ing coefficients are k = 288N/m and b = 16(N · s)/m. Finally, the plane is inclined down by an angle φ = 20◦.
a) Determine the static equilibrium dis-placement of the block from the un-stretched position;
b) Find the resulting solution if the sys-tem is released from the unstretched position with initial velocity ˙x(0) =
−3.0m/s;
c) Identify the damping constant b that gives rise to critical damping;
x
k m b
φ
g
Problem 28:
For the system shown to the right
a) find the equations of motion in terms of the rotation of the disk (do not neglect gravity);
b) determine the deflection of the spring from its unstretched length when the system shown is in equilibrium.
r2
r1
m k2
m I
g
Problem 29:
For the mechanical system shown to the right, the uniform rigid bar has mass m and pinned at point O. For this system:
a) find the equations of motion;
b) Identify the damping ratio and natural frequency in terms of the parameters m, c, k, and ℓ.
c) For:
m = 2.50kg, ℓ = 30cm, c = 1.00N/(m/s), k = 200N/m, find the angular displacement of the bar θ(t) for the following initial conditions:
θ(0) = π/6rad, ˙θ(0) = 0rad/s.
Assume that in the horizontal position the system is in static equilibrium and that all angles remain small.
ℓ
2 ℓ
4
m
O
k
2 m
c θ
Problem 30:
In the figure, the disk and the block have mass m and the radius of the disk is r.
a) Find the equations of motion for this system.
b) What are the natural frequency and damping ratio of the system in terms of m, c, and k?
c) If the block is displaced 18cm to the right and released from rest find the re-sulting angular displacement of the disk with
m = 3kg, r = 9cm, k = 21N/m, c = 63N · s/m,
k c
m
4 k k
(m, r)
Problem 31:
For the system shown to the right, the disk of mass m rolls without slip and x measures the displacement of the disk from the unstretched position of the spring.
a) Find the equations of motion.
b) With
c = 16N/(m/s), m = 2kg, r = 0.10m
for what value of the spring stiffness k is the damping ratio of the system one-half of the critically damped value, so that ζ = ζcr/2?
c) With these parameter values, find the displacement of the disk if it is rolled 20cm to the right (from static equilib-rium) and released from rest.
x
c k
(m, r)
Problem 32:
From the figure shown to the right
a) find the equations of motion in terms of the angular rotation of the disk;
b) what are the damping ratio and natural frequency of the system in terms of the parameters m, b, k1, and k2;
c) can you draw an equivalent spring-mass-damper system?
r
r 2
b
k1 k2
m m
Problem 33:
In the figure, the disk has mass m, radius r, and moment of inertia IG = mr22 about the mass center G and is assumed to roll without slip. The identified coordiante θ measures the rotation of the disk with respect to the equi-librium position.
a) Find the equations of motion for this system.
b) If the disk is released from rest with θ(0) = −π2rad, find the resulting angu-lar displacement θ(t) for
m = 3kg, r = 15cm, k = 36N/m, c = 3N · s/m, c) What is the force in the upper spring
during this motion?
k c
k
(m, r) G
g
Problem 34:
For the spring-mass-damper system shown to the right, x is measured from the static equi-librium position. If the surface is assumed to be frictionless:
a) determine the governing equations of motion;
b) what is the frequency of oscillation of the system;
c) what value of the damping coefficient b corresponds to critical damping?
d) if k = 2N/m, b = 4N/(m/s), and m = 3kg, find the displacement of the mass x(t) when the system is started with the initial conditions:
x(0) = 0.10m, ˙x(0) = 0m/s.
x
m
6 k b
Problem 35:
For the system shown to the right, x is mea-sured from the unstretched position of the spring. Each block has mass m and the disk has moment of inertia I and radius r. If the gravitational constant is g:
a) find the equations of motion which de-termine x(t);
b) what is the period of the free oscilla-tions?
x
m
m (I, r) k
Problem 36:
In the figure, the disk has mass m, radius r, and moment of inertia IG = mr22 about the mass center G.
a) Find the equations of motion for this system assuming that the disk rolls without slip.
b) If the disk is released from rest with initial displacement x(0) = x0, find the minimum value of the coefficient of fric-tion for which the disk does not slip.
(m, r) G
x
k c
g
Problem 37:
For the spring-mass-damper system shown to the right, x is measured from the static equi-librium position. If the surface is assumed to be frictionless:
a) determine the governing equations of motion;
b) what is the period of each oscillation;
c) what value of the damping coefficient b corresponds to critical damping?
d) if k = 1N/m and m = 4kg, find the dis-placement of the mass x(t) if the sys-tem is critically damped and started with the initial conditions x(0) = 0,
˙x(0) = ˙x0;
x
m
6 k b
Problem 38:
For the system shown to the right, x is mea-sured from the unstretched position of the spring. Each block has mass m and the disk has moment of inertia I. If the gravitational constant is g:
a) what is the displacement of the spring at static equilibrium;
b) find the kinetic energy of the system in terms of the coordinate x (and/or its velocity).
x
m
m (I, r) k