# Contradiction of bandwidth and damping

I am studying control theory and mechanical vibrations.

From control theory aspect, I know that bandwidth and response time is inversely proportional, which means bandwidth and damping as well.

But in my mechanical vibrations textbook the definition of quality factor states that damping and bandwidth (specifically half maximum power bandwidth , $${w_1 - w_2}$$ ) are not inversely but directly proportional.

So which one is true? Is bandwidth directly or inversely proportional to damping ?

What am I missing? Can someone please explain?

$$Q \simeq \frac{1}{2\xi} \simeq \frac{w_n}{w_1 - w_2}$$

• "...which means bandwidth and damping as well". Is there a reference for this claim ? This is probably the wrong assumption in your question. Moreover, the the response mentioned in the vibrations textbook probably applies to band-pass type responses while the one mentioned in control theory usually applied to low-pass type responses. Please add more details to the question.
– AJN
Apr 24 at 9:14
• I did that assumption based on assumption that response time and damping directly proportional,which leads to ''means bandwidth and damping as well". Yes it can be seen as bandpass filter(frequency response of 1DOF mass-damp-spring system).But does it matters ? Does it mean that lowpass and bandpass filters have totally opposite behaviour when bandwith is altered ?
– Tym
Apr 24 at 9:27
• I am not sure; I think it is the assumption that response time and damping are directly proportional. I would think that response time has a smaller dependence on damping when compared to bandwidth.
– AJN
Apr 24 at 12:46

Assuming bandwidth is the frequency where the magnitude response is 0.7079 times the DC response, I don't think damping and bandwidth are neither proportional or inversely proportional. Strictly, if they are proportional, then there exists a linear relation between then. Using the general 2nd order system $$G(s)=\frac{K}{s^2+2\zeta\omega_ns+\omega_n^2}$$ the bandwidth is $$\omega_{bw}=\omega_n\sqrt{(1-2\zeta^2)+\sqrt{4\zeta^4-4\zeta^2+2}}, \hspace{1.5cm} (**)$$ Clearly, the relation between them is nonlinear. The equation for $$\omega_{bw}$$ is taken from Nise's book "Control Systems Engineering". The figure shows the relation between damping and MATLAB's bandwidth function using the 2nd order transfer function with $$\omega_n=10$$. You get the exact same plot when using the equation $$(**)$$.