The vibration of strings
The tilted frequency response curve indicates that a string vibrating in a plane should exhibit hysteresis due to the coexistence of two stable states, and that a discontinuous transition between two states is possible when the parameter is slowly varied. In other words, the phenomenon of frequency jump can be observed when slowly increasing or decreasing the driving frequency. When increasing the frequency, the stable solution jumps from point D vertically downwards to the new stable solution. On the other hand, when the driving frequency is slowly decreased, the amplitude switch up from point U. In the region between U and D, stable orbits and unstable orbit all coexist. The periodic solution in this region is extremely sensitive to the initial condition which can be driven by the vector field.
Stability and instability
Periodic solution can be described by an equation in the form
𝑥(𝑡)= 𝑥 𝑜 (𝑡) + 𝑞(𝑡)
where 𝑥 𝑜 is the periodic solution and 𝑞 represents a small displacement between 𝑥 and 𝑥 𝑜 . By setting 𝑤 0 𝛽 and 𝑤 0 to be one and plugging this equation to the original ODE, we have
3 + 𝑞̈ + 𝑞̇ + 3𝜀𝑥 2 𝑞 = 𝑓
𝑥̈ 0 +𝑥̇+𝜀𝑥
0 sin(𝑤𝑡)
𝐴 2 2
2 =
(1 + 𝑐𝑜𝑠(2𝑤𝑡))
Where 𝑥 0
For small displacement q, by setting the periodic solution equal to the driving term and plugging 𝑥 0 2 into the equation, we have
3𝜀𝐴 2 2
3𝜀𝐴 2 2
2 𝑞 =1+
1+3𝜀𝑥 0
+
cos(2𝑤𝑡)
Therefore, the ODE for displacement q is just the Mathieu equation. Stability and instability regions can thus be analysed using the solution to the Mathieu equation.
With the stability diagram, solutions can be classified using the region on the diagram. By plugging in boundary values for stable solutions, we can obtain the frequency response curve that is been shown before. Where the stable and unstable point meet is the saddle-node bifurcation point.
Experiments
Here we used the same method as Marcello Carla once used when illustrating modelling nonlinear oscillation of a steel guitar string. A schematic of the experimental setup is shown below. Current I, flowing in the string, interacts with the field B produced by a magnetic bar placed alongside the string of length L. Current is controlled by the pc using 2 DAC and an amplifier (A1). A neodymium magnet
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