Your differential equation seems to be non-autonomous, since the oscillator is driven and there is an explicit dependance by the time, at the cosinus phase term "wt". One approach is to treat the phase as an additional variable φ, which has a first derivative of w:
x' = y
x"= y' = f cos (φ) - kx - cy - alpha x^3
φ’=w
Now the equations are transformed in the autonomous form, however this method needs some care: the phase variable will continuously increase and therefore achieving periodic motion with this convention will be problematic, so consult the literature before (books of Nayfeh, Hagedorn, Guckenheimer & Holmes, Khali and many others). There are other methods as well, and also analytical approximations if am not wrong.
You can anyway start with the free system (without external forces) and gain a first understading of the system features. Good luck
Last edit: lanast 2020-08-12
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From: Sibi vivek sibi@users.sourceforge.net
Sent: Wednesday, August 12, 2020 1:11 PM
To: [matcont:discussion]
Subject: [matcont:discussion] How do we get Frequency Response Function (FRF) of duffing and van der pol oscillators in Matcont ?
Thank you sir for your explanation and time
could you please explain the further steps to solve and acheive freque
I have not worked with these kind of equations, therefore I suggest you refer to the most recent MatCont manual of the developers, it is very comprehensive. Best wishes
If you would like to refer to this comment somewhere else in this project, copy and paste the following link:
I have to plot the frequency vs amplitude curve for duffing and van der pol oscillators. how should formulate the standard equations of motions?
Duffing equation
mx"+cx'+kx+alpha x^3 = f cos (wt)
i have converted this equation to two first order DE's and solved for transient response.
x' = y
x"= y' = f cos (wt) - kx - cy - alpha x^3
How could i get the steady state response for this equation using MatCont
can anyone give step by step procedure?
thanks in advance
Last edit: Sibi vivek 2020-08-08
Your differential equation seems to be non-autonomous, since the oscillator is driven and there is an explicit dependance by the time, at the cosinus phase term "wt". One approach is to treat the phase as an additional variable φ, which has a first derivative of w:
Now the equations are transformed in the autonomous form, however this method needs some care: the phase variable will continuously increase and therefore achieving periodic motion with this convention will be problematic, so consult the literature before (books of Nayfeh, Hagedorn, Guckenheimer & Holmes, Khali and many others). There are other methods as well, and also analytical approximations if am not wrong.
You can anyway start with the free system (without external forces) and gain a first understading of the system features. Good luck
Last edit: lanast 2020-08-12
Thank you sir for your explanation and time
could you please explain the further steps to solve and acheive frequency vs amplitude curve after transforming to autonomous form.
thanks in advance
Last edit: Sibi vivek 2020-08-12
I have placed a full tutorial on this here: https://wwwhome.ewi.utwente.nl/~meijerhge/FrequencyResponse.pdf
From: Sibi vivek sibi@users.sourceforge.net
Sent: Wednesday, August 12, 2020 1:11 PM
To: [matcont:discussion]
Subject: [matcont:discussion] How do we get Frequency Response Function (FRF) of duffing and van der pol oscillators in Matcont ?
Thank you sir for your explanation and time
could you please explain the further steps to solve and acheive freque
thanks in advance
How do we get Frequency Response Function (FRF) of duffing and van der pol oscillators in Matcont ?
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I have not worked with these kind of equations, therefore I suggest you refer to the most recent MatCont manual of the developers, it is very comprehensive. Best wishes