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7.3 Applications  (Page 7/13)

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x ( t ) = c 1 x 1 ( t ) + c 2 x 2 ( t ) + x p ( t ) ,

where c 1 x 1 ( t ) + c 2 x 2 ( t ) is the general solution to the complementary equation and x p ( t ) is a particular solution to the nonhomogeneous equation. If the system is damped, lim t → ∞ c 1 x 1 ( t ) + c 2 x 2 ( t ) = 0 . Since these terms do not affect the long-term behavior of the system, we call this part of the solution the transient solution . The long-term behavior of the system is determined by x p ( t ) , so we call this part of the solution the steady-state solution    .

This website shows a simulation of forced vibrations.

Forced vibrations

A mass of 1 slug stretches a spring 2 ft and comes to rest at equilibrium. The system is attached to a dashpot that imparts a damping force equal to eight times the instantaneous velocity of the mass. Find the equation of motion if an external force equal to f ( t ) = 8 sin ( 4 t ) is applied to the system beginning at time t = 0 . What is the transient solution? What is the steady-state solution?

We have m g = 1 ( 32 ) = 2 k , so k = 16 and the differential equation is

x ″ + 8 x ′ + 16 x = 8 sin ( 4 t ) .

The general solution to the complementary equation is

c 1 e −4 t + c 2 t e −4 t .

Assuming a particular solution of the form x p ( t ) = A cos ( 4 t ) + B sin ( 4 t ) and using the method of undetermined coefficients, we find x p ( t ) = − 1 4 cos ( 4 t ) , so

x ( t ) = c 1 e −4 t + c 2 t e −4 t − 1 4 cos ( 4 t ) .

At t = 0 , the mass is at rest in the equilibrium position, so x ( 0 ) = x ′ ( 0 ) = 0 . Applying these initial conditions to solve for c 1 and c 2 , we get

x ( t ) = 1 4 e −4 t + t e −4 t − 1 4 cos ( 4 t ) .

The transient solution is 1 4 e −4 t + t e −4 t . The steady-state solution is − 1 4 cos ( 4 t ) .

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A mass of 2 kg is attached to a spring with constant 32 N/m and comes to rest in the equilibrium position. Beginning at time t = 0 , an external force equal to f ( t ) = 68 e −2 t cos ( 4 t ) is applied to the system. Find the equation of motion if there is no damping. What is the transient solution? What is the steady-state solution?

x ( t ) = − 1 2 cos ( 4 t ) + 9 4 sin ( 4 t ) + 1 2 e −2 t cos ( 4 t ) − 2 e −2 t sin ( 4 t )
Transient solution: 1 2 e −2 t cos ( 4 t ) − 2 e −2 t sin ( 4 t )
Steady-state solution: − 1 2 cos ( 4 t ) + 9 4 sin ( 4 t )

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Resonance

Consider an undamped system exhibiting simple harmonic motion. In the real world, we never truly have an undamped system; –some damping always occurs. For theoretical purposes, however, we could imagine a spring-mass system contained in a vacuum chamber. With no air resistance, the mass would continue to move up and down indefinitely.

The frequency of the resulting motion, given by f = 1 T = ω 2 π , is called the natural frequency of the system . If an external force acting on the system has a frequency close to the natural frequency of the system, a phenomenon called resonance results. The external force reinforces and amplifies the natural motion of the system.

  1. Consider the differential equation x ″ + x = 0 . Find the general solution. What is the natural frequency of the system?
  2. Now suppose this system is subjected to an external force given by f ( t ) = 5 cos t . Solve the initial-value problem x ″ + x = 5 cos t , x ( 0 ) = 0 , x ′ ( 0 ) = 1 .
  3. Graph the solution. What happens to the behavior of the system over time?
  4. In the real world, there is always some damping. However, if the damping force is weak, and the external force is strong enough, real-world systems can still exhibit resonance. One of the most famous examples of resonance is the collapse of the Tacoma Narrows Bridge on November 7, 1940. The bridge had exhibited strange behavior ever since it was built. The roadway had a strange “bounce” to it. On the day it collapsed, a strong windstorm caused the roadway to twist and ripple violently. The bridge was unable to withstand these forces and it ultimately collapsed. Experts believe the windstorm exerted forces on the bridge that were very close to its natural frequency, and the resulting resonance ultimately shook the bridge apart.

    This website contains more information about the collapse of the Tacoma Narrows Bridge.

    During the short time the Tacoma Narrows Bridge stood, it became quite a tourist attraction. Several people were on site the day the bridge collapsed, and one of them caught the collapse on film. Watch the video to see the collapse.

  5. Another real-world example of resonance is a singer shattering a crystal wineglass when she sings just the right note. When someone taps a crystal wineglass or wets a finger and runs it around the rim, a tone can be heard. That note is created by the wineglass vibrating at its natural frequency. If a singer then sings that same note at a high enough volume, the glass shatters as a result of resonance.

    The TV show Mythbusters aired an episode on this phenomenon. Visit this website to learn more about it. Adam Savage also described the experience. Watch this video for his account.
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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