Question

A spring-mass system has a spring constant of 3 N/m. A mass of 2 kg is attached to the spring, and the motion takes place in a viscous fluid that offers a resistance numerically equal to the magnitude of the instantaneous velocity.

(a) If the system is driven by an external force of (12 cos 3t − 8 sin 3t) N, determine the steady-state response.

(b) Find the gain function if the external force is f(t) = cos(ωt).

(c) Verify that the amplitude predicted by your work in (a) for an ex- ternal force f(t) = cos(3t) is consistent with the amplitude of the response you found in the body of the problem (that they do not match exactly may be attributable to the phase shift in the forcing).

Answer #1

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A spring-mass system has a spring constant of 3 Nm. A mass of 2
kg is attached to the spring, and the motion takes place in a
viscous fluid that offers a resistance numerically equal to the
magnitude of the instantaneous velocity. If the system is driven by
an external force of 15cos(3t)−10sin(3t) N,determine the
steady-state response in the form Rcos(ωt−δ).
R=
ω=
δ=

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position with an initial velocity of 3 cm/s, formulate the initial
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A mass of 4 Kg attached to a spring whose constant is 20 N / m
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A mass weighing 16 pounds stretches a spring
8
3
feet.
The mass is initially released from rest from a point
3 feet
below the equilibrium position, and the subsequent motion takes
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equal to
1
2
the
instantaneous velocity. Find the equation of motion
x(t)
if
the mass is driven by an external force equal to
f(t) = 20 cos(3t).
(Use
g =
32 ft/s2
for
the acceleration...

A mass weighing 16 pounds stretches a spring 8 3 feet. The mass
is initially released from rest from a point 3 feet below the
equilibrium position, and the subsequent motion takes place in a
medium that offers a damping force that is numerically equal to 1 2
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mass is driven by an external force equal to f(t) = 10 cos(3t).
(Use g = 32 ft/s2 for the acceleration...

Solve the following differential equations.
A spring has a constant of 4 N/m. The spring is hooked a mass of
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resistance equivalent to instantaneous speed. If the system is
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Determine:
a. The position function relative to time in the transient state
or homogeneous solution
b. Position function relative to time in steady state or
particular solution
c....

A mass weighing 16 pounds stretches a spring 8/3 feet. The mass
is initially released from rest from a point 3 feet below the
equilibrium position, and the subsequent motion takes place in a
medium that offers a damping force that is numerically equal to 1/2
the instantaneous velocity. Find the equation of motion x(t) if the
mass is driven by an external force equal to f(t) = 10 cos(3t).
(Use g = 32 ft/s^2 for the acceleration due to...

a 3 kg mass is attached to a spring whose constant is 147 N/m,
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= 0, a force equal to f (t) =
12e−5t cos 2t is applied to
the system. In the absence of damping,
(a)
find the position of the mass when t =
π.
(b)
what is the amplitude of vibrations after a very long
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Consider a damped forced mass-spring system with m = 1, γ = 2,
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a) (8 points) Find the position u(t) of the mass at any time t,
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b) (4 points) Find the transient solution uc(t) and the steady
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