My Math Forum (http://mymathforum.com/math-forums.php)
-   Differential Equations (http://mymathforum.com/differential-equations/)
-   -   Differential equations (http://mymathforum.com/differential-equations/35653-differential-equations.html)

 Felpudio April 30th, 2013 04:53 AM

Differential equations

1 Attachment(s)
I need some help to solve this problem.

 TeM May 7th, 2013 06:20 AM

Re: Differential equations

This is a simple harmonic motion with sinusoidal forcing. From the second law of motion, we have $m\,\ddot{s}(t)= - k\,s(t) + F(t)$ where placing $\omega^2\, := \,\frac{k}{m}$ is obtained $\ddot{s}(t) \,+\, \omega^2\,s(t) \,= \,\frac{A}{m}\,\cos^3\left(\Omega\,t\right)$, a differential equation of second order with constant coefficients, linear, non-homogeneous. Remembering, for example, the cosine's triple angle formula $\left[\cos(3\,\alpha) \,= \,4\,\cos^3\alpha \,- \,3\,\cos\alpha \right]$ it has $\ddot{s}(t) \,+\, \omega^2\,s(t)\,=\, \frac{A}{4\,m}\,\left( \cos\left(3\,\Omega\,t\right) \,+ \, 3\,\cos\left(\Omega\,t\right) \, \right)$ that thanks to the linearity of the equation of motion proffers $s(t)=c_1\cos\left(\omega\,t\right)\,+\,c_2\sin\lef t(\omega\,t\right)\,+\,\frac{A}{4\,m}\,\left(\frac {1}{\omega^2-\left(3 \, \Omega\right)^2}\,\cos\left(3\,\Omega\,t\right) \, + \, \frac{3}{\omega^2-\Omega^2}\,\cos\left(\Omega\,t\right) \, \right)$. Therefore, the first stationary solution is resonant for $\omega= 3\,\Omega$ whilst the second is for $\omega= \Omega$ (values ??for which cancel out the two denominators). So, to find the pulse of resonance of the system, in effects, you do not need to solve the differential equation, but only to decompose the forcing $\cos^3\left( \Omega\, t \right)$ into a sum of sinusoidal functions harmonics $\cos\left( 3 \, \Omega \, t \right)$ e $\cos\left(\Omega\,t\right)$. In numbers, it's concluded that $\Omega= 1.\bar{3} \, Hz \; or \; \Omega = 4.0 \; Hz$. I hope I was clear :wink:

 All times are GMT -8. The time now is 10:22 AM.