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Exercises 4.10: Nonhomogeneous Equations - Forced Oscillations and
Resonance
Shlomo Golombeck
Textbook
Differential Equations - Forced Oscillations - Example
Differential equations + resonance
[+]
[+]
Question 1
1/1 pt
2
97
A mass of one kg stretches a spring 49 cm in equilibrium. It is attached to a dashpot that supplies a
damping force of 4 N for each m/sec of speed. Find the steady state component of its displacement if
it's subjected to an external force F (t) = 8 sin(2t) − 6 cos(2t) N.
Note: The steady state component of the displacement is the solution obtained by setting all derivatives
equal to zero. Therefore, in this case it is of the same form as the right-hand side of the ODE, that is,
y(t) = A cos(2t) + B sin(2t). Just find A and B.
y(t) =
−
1
1
cos ( 2t ) + sin ( 2t )
2
4
Question 2
1/1 pt
2
98
A 10 kg weight is attached to a spring with constant k = 160 kg/m and subjected to an external force
F (t) = 140 cos(3t) . The weight begins at rest in its equilibrium position. Find its displacement for t >
0, with y(t) measured positive upwards.
y(t)
= −2 cos ( 4t ) + 2 cos ( 3t )
Question 3
1/1 pt
1
98
A 6 kg weight is attached to a spring with constant k = 54 kg/m and subjected to an external force
F (t) = 90 sin(2t) . The weight begins at rest in its equilibrium position. Find its displacement for t >
0, with y(t) measured positive upwards.
y(t)
= −2 sin ( 3t ) + 3 sin ( 2t )
Question 4
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1/1 pt
2
99
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A 12 kg weight is attached to a spring with constant k = 48 kg/m and subjected to an external force
F (t) = −1008 cos(5t) . The weight is initially displaced 3 meters above equilibrium and given an
downward velocity of 2 m/s. Find its displacement for t > 0, with y(t) measured positive upwards.
y(t)
= −cos ( 2t ) − sin ( 2t ) + 4 cos ( 5t )
Question 5
1/1 pt
2
99
1/1 pt
2
99
Consider the forced damped oscillator equation
′′
y +1y = F sin(ωt)
For this forced oscillator to exhibit resonance,
ω =
1
Question 6
A 9 kg weight is attached to a spring with constant k = 225 kg/m and subjected to an external force
F (t) = 1800 cos(5t) . The weight begins at rest in its equilibrium position. Find its displacement for t
> 0, with y(t) measured positive upwards.
y(t)
= 20t sin ( 5t )
Question 7
1/1 pt
1
99
A 8 kg weight is attached to a spring with constant k = 72 kg/m and subjected to an external force
F (t) = 576 cos(3t) . The weight begins 1 meters below equilibrium and is pushed upwards at a
velocity of 2 m/s. Find its displacement for t > 0, with y(t) measured positive upwards.
y(t)
2
= 12t sin ( 3t ) + 3 sin ( 3t ) − cos ( 3t )
Question 8
1/1 pt
2
96
An object weighing 40 lbs hangs on the end of a spring, stretching the spring 6 inches. The object is in a
medium that exerts a viscous resistance of 60 lbs when the object has a velocity of 4 ft/sec.
Suppose the object is lifted an additional 4 inches and released.
Find an equation for the object's displacement, u(t), in feet after t seconds.
u(t) =
e
(
√
−6t 1
√ 7 t ) + 7 sin ( 2√ 7 t )
(
cos
2
3
7
)
Remember that, in imperial units, acceleration due to gravity is 32 ft/s2 and that lbs are a
measurement of force. Be sure to convert any measurements necessary so that the units match.
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Question 9
0.5/1 pt
1-3
99
A 11 kg object is attached to a spring with spring constant 99 kg/s2. It is also attached to a dashpot
with damping constant c = 110 N-sec/m. The object is pushed upwards from equilibrium with velocity
5 m/s. Find its displacement and time-varying amplitude for t > 0.
y(t) =
(
5 −t
−9t
e −e
8
)
The motion in this example is
overdamped
critically damped
underdamped
Consider the same setup above, but now suppose the object is under the influence of an outside force
given by F (t) = 19 cos(ωt).
What value for
ω
will produce the maximum possible amplitude for the steady state component of the
solution?
What is the maximum possible amplitude?
Question 10
A mass of 0.75
0/1 pt
kg
is attached to the end of a spring whose restoring force is 160
N
m
medium that exerts a viscous resistance of 48
N
when the mass has a velocity of 6
resistance is proportional to the speed of the object.
3
98
. The mass is in a
m
. The viscous
s
Suppose the spring is stretched 0.05 m beyond the its natural position and released. Let positive
displacements indicate a stretched spring, and suppose that external vibrations act on the mass with a
force of 3 sin(4t) N at time t seconds.
Find an function to express the steady-state component of the object's displacement from the spring's
natural position, in m after t seconds. (Note: This spring-mass system is not "hanging", so there is no
gravitational force included in the model.)
u(t) =
Question 11
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2
99
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Solve
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′′
y +9y = cos(3 ⋅ t)
where
y(0) = 5
and
′
y (0) = 4
t
sin ( 3t ) + 4 sin ( 3t ) + 5 cos ( 3t )
6
3
Question 12
0/1 pt
3
99
Find the transient motion and the steady state solution of a damped mass-spring oscillator system with
m = 1, c = 2, and k = 26 under the influence of an external force F (t) = 82 cos(4t) with x(0) = 6
and x′(0) = 0. Write your steady state solution as a single phase-shifted cosine.
The transient solution is
.
The steady state solution is
Question 13
0/1 pt
3
99
Suppose that a 1 kg mass is attached a spring with constant k = 9 and driven by an external force that is
modeled by F (t) = −64 cos(5t). Find the equation of motion of the mass if x(0) = 3 and x′(0) = 0.
Set up and solve a differential equation. Include both the transient solution and the steady state
solution.
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