Damped Harmonic Oscillator & Mechanical Energy

In summary, the conversation discusses a damped harmonic oscillator and its loss of energy per cycle. It presents a question about the difference in frequency from the natural frequency and the number of periods it takes for the amplitude to decrease to 1/e of its original value. The conversation then goes on to discuss equations and methods for finding the solution to these questions, including using the amplitude and energy equations.
  • #1
e(ho0n3
1,357
0
Question: A damped harmonic oscillator loses 5.0 percent of its mechanical energy per cycle. (a) By what percentage does its frequency differ from the natural frequency [itex]\omega_0 = \sqrt{k/m}[/itex]? (b) After how may periods will the amplitude have decreased to 1/e of its original value?

So, for (a), the answer is [itex]\omega ' / \omega_0[/itex] where

[tex]\omega ' = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}[/tex]

But that leaves me with 3 unknowns, k, m, and b requiring three equations to solve. The only equations I can think of is E = K + U (mechanical energy) and E = 0.95TE0 where T is the number of cycles and E0 is the initial mechanical energy.

What other equation can I use? Or is there a simpler method of finding the solution?
 
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  • #2
e(ho0n3 said:
Question: A damped harmonic oscillator loses 5.0 percent of its mechanical energy per cycle. (a) By what percentage does its frequency differ from the natural frequency [itex]\omega_0 = \sqrt{k/m}[/itex]? (b) After how may periods will the amplitude have decreased to 1/e of its original value?

So, for (a), the answer is [itex]\omega ' / \omega_0[/itex] where

[tex]\omega ' = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}[/tex]

But that leaves me with 3 unknowns, k, m, and b requiring three equations to solve. The only equations I can think of is E = K + U (mechanical energy) and E = 0.95TE0 where T is the number of cycles and E0 is the initial mechanical energy.

What other equation can I use? Or is there a simpler method of finding the solution?
The amplitude is given by:

[tex]A = A_0e^{-\gamma t}[/tex] where [itex]\gamma = b/2m[/itex] 1

Work out the value for [itex]\gamma[/itex] given that A^2 decreases to .95A_0^2 in the first [itex]T = 2\pi /\omega '[/itex] seconds.

Then find [itex]\omega '[/itex] in terms of [itex]\omega_0[/itex] using:
[itex]\omega '^2 = \omega_0^2 - \gamma^2[/itex]

AM

[Note: 1. The solution to the damped harmonic oscillator is:

[tex]x = A_0e^{-\gamma t}sin(\omega 't + \phi)[/tex]

where [itex]\omega ' = \sqrt{\omega^2 - \gamma^2}[/itex] ]
 
  • #3
Andrew Mason said:
Work out the value for [itex]\gamma[/itex] given that A^2 decreases to .95A_0^2 in the first [itex]T = 2\pi /\omega '[/itex] seconds.
How do you know the square of the amplitude decreases to .95A_0^2 in the first T seconds?
 
  • #4
e(ho0n3 said:
How do you know the square of the amplitude decreases to .95A_0^2 in the first T seconds?
Energy is proportional to the square of the amplitude (all energy is potential energy at maximum amplitude: [itex]E = \frac{1}{2}kx^2[/itex]). If the system loses 5% of its energy in one cycle, the square of the amplitude will decrease to 95% of the square of the original amplitude.

AM
 

1. What is a damped harmonic oscillator?

A damped harmonic oscillator is a type of mechanical system that undergoes repetitive oscillations or vibrations, where the amplitude of the oscillations gradually decreases over time due to the presence of an external damping force. It can be described using the equation d^2x/dt^2 + 2λdx/dt + ω0^2x = 0, where λ represents the damping coefficient and ω0 is the natural frequency of the system.

2. How does the damping coefficient affect the behavior of a damped harmonic oscillator?

The damping coefficient, represented by λ, determines the rate at which the amplitude of the oscillations decreases. A higher damping coefficient results in a faster decrease in amplitude and a shorter period of oscillations. On the other hand, a lower damping coefficient leads to a slower decrease in amplitude and a longer period of oscillations.

3. What is mechanical energy in the context of a damped harmonic oscillator?

Mechanical energy in a damped harmonic oscillator refers to the sum of the kinetic and potential energies of the system. As the oscillator undergoes repetitive oscillations, the mechanical energy is constantly being converted between these two forms. In a damped oscillator, the mechanical energy gradually decreases due to the presence of external forces such as friction or air resistance.

4. How does the mechanical energy change over time in a damped harmonic oscillator?

In a damped harmonic oscillator, the mechanical energy decreases over time due to the presence of external damping forces. As the amplitude of the oscillations decreases, the kinetic energy also decreases. This results in a decrease in the total mechanical energy of the system.

5. What are some real-life examples of damped harmonic oscillators?

Some common examples of damped harmonic oscillators in real life include a swinging pendulum, a car suspension system, and a tuning fork. In these systems, the oscillations gradually decrease due to the presence of external damping forces such as air resistance or friction.

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