# Finding the undamped natural frequency of 2nd order system

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1. Mar 24, 2015

### engnrshyckh

the following 2nd order differential equation is given:
2y'' + 4y' +8y=8x.....................................(1)
i want to find damping ratio, undamped natural frequency, damping ratio coefficient and time constant for the above system.
solution:
comparimg (1) with general system equaion

(veriable can be exchanged)
{where: x(t) = Response of the System,
u(t) = Input to the System,
z = Damping Ratio,
wn=Undamped Natural Frequency,
Gdc= The DC Gain of the System.}
damping ratio z or zeta:

2zw=2
w=2 so z=2/4=0.5

undamped natural frequency w or omega:
w=2 but correct ans is 1. any help?

2. Mar 25, 2015

### Simon Bridge

Using: http://en.wikipedia.org/wiki/Harmonic_oscillator

Putting the DE familiar form: $\ddot x + 2\dot x + 4x = 4t$ would be the equivalent right?
Compare with $\ddot x + 2\zeta \omega_0 \dot x + \omega_0^2 x = f(t)$ I get $2\zeta\omega =2$ like you did, and $\omega_0^2=4\implies \omega_0=2 \implies \zeta = 1/2$ ...

Are you sure the answer you quote as "w" is the undamped frequency?

3. Mar 25, 2015

### engnrshyckh

http://www.facstaff.bucknell.edu/mastascu/eControlHTML/SysDyn/SysDyn2.html yes it is undamped natural frequency

4. Mar 25, 2015

### engnrshyckh

5. Mar 25, 2015

### engnrshyckh

another way is to use laplace transformation as:

• Then, Laplace transforming both sides and solving for the transfer function - the ratio of the transform of the output to the transform of the input, we find the transfer function to be.

but you still get wn=2

6. Mar 25, 2015

### Tom_K

I agree the undamped w = 2
Why do you think the correct answer is .1?
Taking damping into consideration w = 1.73

7. Mar 25, 2015

### engnrshyckh

please tell me how you find w=1.73....
w=0.1 ans is given in book Electronics and communication engg (OT) by Handa

8. Mar 25, 2015

### Tom_K

The auxiliary equation is: 2m^2 + 4m + 8 = 0
Use the quadratic formula to solve for the roots = -1 +/- i 1.73
That leads to the general solution form of e^-t*(A Cos 1.73t + B Sin 1.73t)
A damped oscillation where w = 1.73
To solve for the undamped case just disregard the coefficient of the m term which represents the damping resistance. The roots then are +/- i 2 purely imaginary
An undamped oscillation where w = 2.0

I don't know where that 0.1 could have come from, a typo maybe?

9. Mar 25, 2015

### engnrshyckh

ty for the help. can you please tell me about damping co-efficient and time for this particular question?

10. Mar 25, 2015