Weak Damping - how to relate the amplitude and phase difference

In summary: Thanks for the correction!In summary, the relationship between ##x_{max}##, ##A_{+}##, ##A_{-}##, and ##\phi## is represented by the equation ##x_{max}(cos(\omega_d t) cos(\phi) - sin(\omega_d t)sin(\phi)) = 2A_{+} cos(\omega_d)##. This can be derived by setting the imaginary parts of the two given equations equal to each other and using the fact that they hold for all t.
  • #1
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Homework Statement


Derive the relationship bewteen x_{max}, A_{+}, A_{-} and \phi

Homework Equations


x(t) = e^{\gamma t}(A_{+}e^{i \omega_d t} + A_{-}e^{-i \omega_d t})
x(t) = x_{max} e^{\gamma t} cos(\omega_d t + \phi)

The Attempt at a Solution


I know the e^{\gamma t} cancels and for the imaginary parts to cancel, A_{+} = A_{-} but then i get x_{max}(cos(\omega_d t) cos(\phi) - sin(\omega_d t)sin(\phi)) = 2A_{+} cos(\omega_d) and i can't work out how to simplify it further
 
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  • #2
GayYoda said:

Homework Statement


Derive the relationship bewteen ##x_{max}##, ##A_{+}##, ##A_{-}## and ##\phi##

Homework Equations


##x(t) = e^{\gamma t}(A_{+}e^{i \omega_d t} + A_{-}e^{-i \omega_d t})##
##x(t) = x_{max} e^{\gamma t} cos(\omega_d t + \phi)##

The Attempt at a Solution


I know the ##e^{\gamma t}## cancels and for the imaginary parts to cancel, ##A_{+} = A_{-}## but then i get ##x_{max}(cos(\omega_d t) cos(\phi) - sin(\omega_d t)sin(\phi)) = 2A_{+} cos(\omega_d)## and i can't work out how to simplify it further
First, fixing up the LaTeX by inserting double hash as necessary.

You seem to have dropped a factor t.
After reinstating that, consider that this is supposed to be an identity valid for all t.
How can you use that?
 
Last edited:
  • #3
You dropped ##\phi## as well.
 
  • #4
vela said:
You dropped ##\phi## as well.
That's not how I read it. One side has Φ, the other does not, and the value is to be deduced.
 
  • #5
Ah, makes sense. I misunderstood what the OP was doing.
 

1. What is weak damping?

Weak damping refers to a physical system where the damping force is relatively small compared to the restoring force. This results in the system exhibiting oscillatory behavior with decreasing amplitude over time.

2. How is weak damping different from strong damping?

Strong damping occurs when the damping force is significantly larger than the restoring force, causing the system to quickly come to rest without oscillating. Weak damping, on the other hand, allows for oscillations to occur before the system eventually comes to rest.

3. How can the amplitude of a weakly damped system be related to the phase difference?

The amplitude of a weakly damped system is related to the phase difference through the equation A = A0 * e-δt * cos(ωt + Φ), where A is the amplitude at time t, A0 is the initial amplitude, δ is the damping constant, ω is the angular frequency, and Φ is the phase difference.

4. Why is it important to understand the relationship between amplitude and phase difference in weak damping?

Understanding the relationship between amplitude and phase difference in weak damping is important for predicting the behavior of physical systems, such as oscillating springs or pendulums. It also allows for the analysis of energy dissipation in the system and can help in designing systems with desired damping characteristics.

5. How can weak damping be beneficial in certain applications?

In some applications, weak damping can be desirable because it allows for sustained oscillations. This can be seen in musical instruments, where the strings or air columns are weakly damped, allowing for prolonged vibrations and producing sound. Weak damping can also be used in shock absorbers to provide a smoother ride by reducing abrupt movements.

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