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April 30th, 2013, 04:53 AM   #1
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Differential equations

I need some help to solve this problem.
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 May 7th, 2013, 06:20 AM #2 Newbie   Joined: May 2013 Posts: 5 Thanks: 0 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

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