How do I find the frequency of oscillation for a damped harmonic oscillator?

In summary, the conversation discussed finding the formula for the frequency of oscillation in a system with a freely falling object suspended by a spring. The conversation mentioned using a second order, homogeneous, linear, differential equation with constant coefficients and finding the roots of its characteristic equation to solve the problem. The conclusion was reached that the system was under-damped and the frequency could be expressed as -g/2v_t +/- sqrt(g^2/(4v^2)+ g/a).
  • #1
Slightly
29
0

Homework Statement


The terminal speed of a freely falling object is v_t (assume a linear form of air resistance). When the object is suspended by a spring, the spring stretches by an amount a. Find the formula of the frequency of oscillation in terms of g, v_t, and a.

Homework Equations



the linearly damped force = -cx'

The Attempt at a Solution



I started by writing the equation,

x'' + cx'/m + kx/m - mg = 0

Is this the correct way to start the problem and where would I go to next? I know that undamped frequency is equal to sqrt(k/m) but I'm not sure what to do with the equation. I'm stuck!
 
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  • #2
Slightly said:

Homework Statement


The terminal speed of a freely falling object is v_t (assume a linear form of air resistance). When the object is suspended by a spring, the spring stretches by an amount a. Find the formula of the frequency of oscillation in terms of g, v_t, and a.

Homework Equations



the linearly damped force = -cx'

The Attempt at a Solution



I started by writing the equation,

x'' + cx'/m + kx/m - mg = 0

Is this the correct way to start the problem
Almost. In your "mg" term, you forgot to divide by m like you did in your other terms.

Perhaps more importantly though, this mg is a uniform force across all x and it does not depend on x or any of its derivatives.

I think it might be fine to simply remove the "mg" term altogether. It doesn't affect the frequency of oscillation, in this problem. It will affect the center of oscillation, but you're not asked to find that. So if I were you I'd just get rid of it.

Can you express k in terms of a, m and g?

and where would I go to next? I know that undamped frequency is equal to sqrt(k/m) but I'm not sure what to do with the equation. I'm stuck!

So at the moment, you have a second order, homogeneous, linear, differential equation with constant coefficients. Single solutions of this type of equation come in the form [itex] x(t) = Ae^{\lambda t} [/itex] The general solution may involve the linear sum of two, single solutions. (Here, [itex] \lambda [/itex] can be, and generally is, a complex number.)

The first step in solving that is to write down its characteristic equation. Then find the roots of the characteristic equation.

You might wish to consult your textbook/course material on how to solve such differential equations. If not, even a quick Google search on "second order, homogeneous, linear, differential equation with constant coefficients" should produce all you need.
 
  • #3
collinsmark said:
. (Here, [itex] \lambda [/itex] can be, and generally is, a complex number.)

Can you explain to me why you came to this conclusion? How is this system under-damped?
 
  • #4
Slightly said:
Can you explain to me why you came to this conclusion? How is this system under-damped?
I never said it was necessarily an imaginary number. Just a complex number (all real numbers are also complex).
 
  • #5
That's funny.

So when I get the solution to the DE, where will the frequency come from?
 
  • #6
Slightly said:
That's funny.

So when I get the solution to the DE, where will the frequency come from?
The fact that "frequency of oscillation" was even mentioned, I take that as implying the system is under-damped. I mean, if the system were critically damped or over-damped, there wouldn't be any oscillations at all.

An under-damped system has a sinusoidal function associated with it (like sine or cos). And in the case of a damped harmonic oscillator, it involves the sinusoidal function multiplied by an exponentiation function. It might be helpful to recall [itex] \cos{ \left( \omega_1 t \right)} = \frac{1}{2} \left( e^{i \omega_1 t} + e^{-i \omega_1 t} \right) [/itex]. Here, [itex] \omega_1 [/itex] is an angular frequency (having units such as RAD/s).

So you might want to put the general solution in a form like [itex] x(t) = A e^{- \alpha t} \left( e^{ i \omega_1 t} + e^{- i \omega_1 t} \right) [/itex]
 
  • #7
I went through the DE and was able to obtain the characteristic equation to be

r^2+cr/m + k/m = 0

I said k = - mg/a

and c = mg/v_t

so... r^2+gr/v_t - g/a = o

In this case, can I assume r is the frequency?

so, using the quadratic formula,

r = -g/2v_t +/- sqrt(g^2/(4v^2)+ g/a) is this right?
 
  • #8
I was able to figure it out! Thanks.
 
  • #9
Slightly said:
I was able to figure it out! Thanks.
Great! :smile:
 

What is a damped harmonic oscillator?

A damped harmonic oscillator is a system in which a mass attached to a spring oscillates back and forth due to the restoring force of the spring, but is also subject to a damping force that decreases the amplitude of the oscillations over time.

What causes damping in a harmonic oscillator?

Damping in a harmonic oscillator can be caused by external factors such as air resistance or friction, or it can be inherent in the system itself, such as internal friction within the spring.

How is the motion of a damped harmonic oscillator described?

The motion of a damped harmonic oscillator is described by a sinusoidal function with a decreasing amplitude and a phase shift due to the damping force.

What is the difference between underdamped, critically damped, and overdamped oscillations?

Underdamped oscillations occur when the damping force is less than the critical damping value, resulting in oscillations with a decreasing amplitude. Critically damped oscillations occur when the damping force is equal to the critical damping value, resulting in the fastest decay of the oscillations without any overshooting. Overdamped oscillations occur when the damping force is greater than the critical damping value, resulting in a slow decay and no oscillations.

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

Some real-life examples of damped harmonic oscillators include a swinging pendulum with air resistance, a car suspension system, and a guitar string that gradually loses its vibration.

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