With damping included it does not really have a natural frequency!
First ignore the damping term. Then there is natural frequency, which depends on the oscillation amplitude. There are interesting ways to compute this natural frequency approximately, but you may also recognize the solution to be an elliptic function, and determine the frequency from this solution.
However, with damping included the amplitude decreases with time, and hence the system can be said to acquire a time dependent local natural frequency (a meaningful concept at least for weak damping). For long times the oscillation amplitude becomes small, and you may linearize the motion around an appropriate point of minimum potential.
Thank you so much for your response. First let's consider that the damping term is eliminated, then i have the nonlinear term in term of T^3, so how can i find the natural frequency with this nonlinear term? If i linearize the equation, then will eliminate the nonlinear term also. So finally i will get equation like T(double dot)-T(pi^2*)(c^2-1)=0, or T(double dot)+T(pi^2)(1-c^2)=0, so linear natural frequency will be Square root((pi^2)*(1-c^2)). Am i right, i need your valuable comment on this, and if i do not linearize the equation. then how i can find the natural frequency with this nonlinearity??
You need to be more careful when linearising the equation. You don't just ignore the T^3 term - you need to use the Taylor series expansion, and consider just the linear term there. This includes the derivative, which will not be zero. If you have
y=ax^3
Then the linearised form of this at x=x0 is
y=-2a(x0)3+3a(x0)2x
The natural frequency will depend on the dampening term, so you need to include this in the equation. The requirement is that the system be underdamped in order to have oscillations - the weaker the dampening the better.
Sorry for the delay--I haven't been back to RG since I last posted.
In the region where 4>c2>1, you want to linearize the equation about the fixed points I mentioned in the first post. You'll wind up with a stable linearized equation there (as long as \mu>0), and can get a frequency for that oscillation.
The next question you want to look at is what region(s) of the (T, dT/dt) phase space converge to one of the fixed points, and which one each area converges to.
Out of curiosity, are you a professor or a student at King Fahd University?
Cool. You're studying some interesting stuff. I haven't done much in nonlinear dynamics in a while, and it was fun to go through some the techniques I remember using for these problems.
Were these questions from classwork, or part of a research project?
you can look into kreyszig phase plane and stability chapter to see how to represent higher order differential equations into first order differential equations. Then find the equilibrium points in state space by setting the right hand side of the first order representation of the system to zero. And then linearising the rhs at these equilibrium points. resulting In xdot=Ax+Bu type of equation and then finding the eigenvalues of the state Matrix A. if These eigenvalues are complex then the imaginary part of these eigenvalues are your natural frequencies. The real part should be negative for stable equilibrium points and positive for unstable equilibrium points. If the eigenvalues are real only then the natural frequency of the system near the equilibrium point is zero.
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