Bode plot implies that for a linear system, if the input is a sinusoidal signal, output of the system will also be a sinusoidal signal with same frequency but different amplitude and phase. Knowing input, we can estimated steady state output based on bode plot. Note that it states input and output have same frequency. But I found an example which seems doesn't match this rule.
The system is a simple mass-spring system,
For simplicity, set m=1kg, k=1N/m, then equation of motion is
If system input is F=sin(2t),
the output is x=(2/3)*sin(t)-(1/3)*sin(2t) by solving differential equation.
As can be seen, input has only one frequency component, but output has two frequency components, one is the natural frequency of the system, the other is the same as input frequency, and the two components are steady state rather transient. Obviously, in this case, the output can't be estimated using bode plot of the system.
I am not sure if people have realized this phenomenon, I was always sure about that input and steady-state output should have same frequency before I found this. Can this example prove that bode plot is not valid in some situations?
Best Answer
Great question. I think a subtle mistake has led you to this paradoxical conclusion.
When you solved the second order differential equation, you would have arrived at a general solution of:
$$ x(t) = c_1sin(t) + c_2cos(t) - \frac{1}{3}sin(2t) $$
where \$c_1\$ and \$c_2\$ are constants determined by initial conditions. It looks like you've assumed initial conditions of:
$$ x(0) = 0,\, \dot{x}(0) = 0 $$
to arrive at your \$\frac{2}{3}sin(t)\$ term. In fact, if that was the state of the system at initial time, then there would be a slight impulse and hence, a tiny broadband frequency component in the input to excite the resonance.
Instead, if you assume that the external \$F\$ is the only input to the system, then the initial conditions would be:
$$ x(0) = 0,\, \dot{x}(0) = -\frac{2}{3} $$
Plugging those initial conditions in give you constants of \$c_1=0\$ and \$c_2=0\$ and thus a system response of just:
$$ x(t) = - \frac{1}{3}sin(2x) $$
which only has frequency components that exist in the input.
We tend to gloss over the derivation and importance of the initial conditions when studying this material, which is why I think this question has caused some confusion. So thank you for bringing it up!