The vibration of strings
Many papers have previously used the duffing equation, which is obtained from study on the motion of pendulum at large angles, to model the vibration of strings. The duffing force 𝑤 0 2 𝑥(1+𝜀𝑥 2 ) is being introduced. It suggests that the restoring force deviates linearly even without damping. The addition of this nonlinear term provides an intermediate state to chaotic motion, where 𝜀 controls the strength of the nonlinearity. An approximated analytical solution for damped duffing equation can be achieved using several methods, including Poincare- Lindsted method, Taylor’s method and Fourier expansion. For small nonlinearity 𝜀 , we only study the most fundamental leading order solution, since higher order terms, also known as the overtone terms, will not periodically affect the oscillation. Therefore, these three methods provide the same solutions to the equation. Here, we do a Fourier representation to obtain the leading order solution 𝐴 𝑠𝑖𝑛(𝑤𝑡 + 𝜑) which must have the same frequency as the driving frequency but with a phase shift 𝜑 . Considering small nonlinearities, the cubed term in the restoring force can then be approximated by trig identities,
𝑥 3 = 𝐴 3 𝑠𝑖𝑛 3 (𝑤𝑡 +𝜑)
3 4 𝑠𝑖𝑛(𝑤𝑡 + 𝜑) −
1 4 𝑠𝑖𝑛(3𝑤𝑡 + 3𝜑)]
= 𝐴 3 [
= 𝐴 3 3 4
𝑠𝑖𝑛(𝑤𝑡 + 𝜑)
Here, again we drop the 𝑠𝑖𝑛(3𝑤𝑡 + 3𝜑) term since it does not have a trajectory with a period of 𝑤𝑡 .
Frequency Response
If we plug the solution into the duffing equation, we can obtain the complex amplitude equation.
3 4 𝐺 2 )−𝑤 2 +𝑗𝑤𝑤
2 (1+𝜀
ⅈ(𝑤𝑡+𝜑) = 𝐹ⅇ ⅈ𝑤𝑡
[𝑤 0
0 𝛽] 𝐺ⅇ
Extracting the modulus on both sides, the real equation is
9 16
3 2
4 𝜀 2 𝐺 6 −
2 𝜀(𝑤 2 −𝑤 0
2 )𝐺 4 +[(𝑤 2 −𝑤 0
2 ) 2 +𝑤 2 𝑤 0
2 𝛽 2 ]𝐺 2 = 𝐹 2
𝑤 0
𝑤 0
9 16
The 𝑤 0 4 𝜀 2 𝐺 6 term suggests the possibility of having three amplitudes at a given magnitude of the driving force, with two stable sates and one unstable state. We will later analyse the stability or instability of these three amplitudes.
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