Damped Simple Harmonic Motion

In summary, the conversation discusses a problem with finding the solution for damped simple harmonic motion and determining the values of C_1 and C_2 in order to show that x' is approximately equal to -w'Asinphi at t = 0. The conversation also mentions using the exponential function to rearrange the coefficients for cos x and i sinx.
  • #1
runevxii
7
1
I'm having trouble with this problem. I want to get it into a form with cos but I'm stumped.
The solution for damped simple harmonic motion is given by
x = (e^(-rt/2m))(C_1*e^(iw't)+C_2*e^(-iwt))
If x = Acos phi at t = 0, find the values of C_1 and C_2 to show that x'=(approx) -w'Asinphi at t = 0 only if r/m is very small or phi =(approx) pi/2.

Where w = omega and phi = angle phi and i=complex variable x' = 1st derivative

Any ideas or help would be really appreciated.
 
Last edited:
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  • #2
exp (ix) = cos x + i sin x, and exp (-ix) = cos x - i sinx

Expand the exponentials and then rearrange the coefficents for cos x and i sinx.
 

1. What is damped simple harmonic motion?

Damped simple harmonic motion is a type of oscillatory motion where the amplitude of the motion decreases over time due to the presence of an external force, such as friction or air resistance. It is characterized by a decreasing amplitude and a period that remains constant.

2. What causes damped simple harmonic motion?

Damped simple harmonic motion is caused by the presence of an external force that opposes the motion, such as friction or air resistance. These forces act to dissipate the energy of the system, resulting in a decrease in amplitude over time.

3. How is damped simple harmonic motion different from simple harmonic motion?

The main difference between damped simple harmonic motion and simple harmonic motion is the presence of an external force. Simple harmonic motion occurs in the absence of any external forces, resulting in a constant amplitude and period. Damped simple harmonic motion, on the other hand, is influenced by an external force, causing the amplitude to decrease over time.

4. What are the applications of damped simple harmonic motion?

Damped simple harmonic motion has many applications in the real world, such as in the suspension systems of cars or in the oscillation of springs in watches. It is also used in musical instruments, where the damping effect can be controlled to produce specific sounds.

5. How is damped simple harmonic motion mathematically represented?

In mathematics, damped simple harmonic motion can be represented by the equation x = Ae^(-kt)cos(ωt + ϕ), where x is the displacement, A is the initial amplitude, k is the damping constant, ω is the angular frequency, and ϕ is the phase angle. This equation takes into account the effects of both the external force and the natural frequency of the system.

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