How to find formula for resonant frequency of a forced oscillator.

[alimon.cioro]

In a damped forced harmonic oscillator the amplitude is determined by a series of paramenters according to :

A = (Fo/m)/ (sqrt( (wo^2-w^2)^2+(wy)^2) ).

where

Fo= driving force,
m=mass of spring
wo=natural frequency of system.
w=driving frequency
y=damping constant.

Now my question is how do you find the driving angular frequency w at which A is maximum, which should be the resonante frequency (that is not exactly Wo).

The resonant frequency formula is :

Wres = sqrt(Wo^2-(y^2)/2)) .

I though that differentiating the formula for A in terms of dA/dw and equating it to zero should give me an answer but the maths look to convoluted for such a simple answer. Any ideas of how to get the resonante frequency?

 

[nrqed]

In a damped forced harmonic oscillator the amplitude is determined by a series of paramenters according to :

A = (Fo/m)/ (sqrt( (wo^2-w^2)^2+(wy)^2) ).

where

Fo= driving force,
m=mass of spring
wo=natural frequency of system.
w=driving frequency
y=damping constant.

Now my question is how do you find the driving angular frequency w at which A is maximum, which should be the resonante frequency (that is not exactly Wo).

The resonant frequency formula is :

Wres = sqrt(Wo^2-(y^2)/2)) .

I though that differentiating the formula for A in terms of dA/dw and equating it to zero should give me an answer but the maths look to convoluted for such a simple answer. Any ideas of how to get the resonante frequency?

That's precisely what you must do. The equation you get is actually easy to solve.
Note that you get something over ( (wo^2-w^2)^2+(wy)^2)^(3/2). You may multiply both sides of the equation by ( (wo^2-w^2)^2+(wy)^2)^(3/2) and you are left with a simple expression equal to zero, which is then easy to solve.

 

[AlephZero]

The easy way is to see that if A is a maximum, 1/A is a mimimum, and 1/A² is also a minimum.

Differentiating d (1/A²) / dw is easy.

BTW the formula you gave in the OP for the resonant frequency is wrong. If can't possibly be right to subtract a frequency squared and a damping corefficient squared, they don't have the same units!

 

[bgc]

The damping coefficient is s^-1 Rather convoluted to show. OTOH, note the equation for the position. X = A*exp(-dc*t)*cos(etc.) for the simple harmonic damped oscillator.

bc

 

[sardar]

I can provide some suggestions on how to find the formula for resonant frequency of a forced oscillator. First, it is important to understand the concept of resonance in a forced oscillator. Resonance occurs when the driving frequency is equal to the natural frequency of the system, resulting in a maximum amplitude response.

To find the resonant frequency, we can start by looking at the formula for amplitude (A) in terms of the driving frequency (w). This formula includes the natural frequency (wo) and the damping constant (y), which are known parameters in the system.

Next, we can differentiate the amplitude formula with respect to the driving frequency (w) and set it equal to zero. This will give us the value of w at which the amplitude is maximum, which is the resonant frequency. However, as you mentioned, the resulting equation may be complex and difficult to solve.

Another approach is to plot the amplitude (A) as a function of the driving frequency (w) and look for the peak value, which corresponds to the resonant frequency. This can be done using a graphing tool or by hand if the equation is simple enough.

Additionally, we can use the formula provided in the question, Wres = sqrt(Wo^2-(y^2)/2)), which gives the resonant frequency directly in terms of the natural frequency and damping constant. This formula can also be derived using the concept of resonance and the condition for maximum amplitude.

In conclusion, there are different ways to find the formula for resonant frequency of a forced oscillator. It is important to have a good understanding of the concept of resonance and the parameters involved in the system. By using different approaches, we can arrive at the same formula for resonant frequency and confirm its validity.