The nonlinear behaviour of a periodically forced guitar string
Shiwei Zhang
Introduction
The simple pendulum and the mass-spring system are the first examples of a harmonic oscillator. Closed analytical solutions can be admitted by a second order differential equation as long as the system is linear, ignoring viscosity damping, forced vibration and nonlinearities. However, in the real world, nonlinear dynamics are typically involved, which eventually leads to the chaotic nature of mechanical systems. The purpose of this paper is to introduce complicated factors to a simple physics phenomenon, damped periodically forced oscillation of a steel guitar string, to gain insight into one dimensional nonlinear system. Normally, the guitar string is put into vibration on its natural frequency by plucking it, producing an exponentially damped oscillation, in which the harmonic content depends on the starting position of the finger. When periodic driving force is applied, the phase resonance frequency of the vibration is dependent on the driving frequency and can be observed by sweeping the driving frequency. The stability or instability of equilibrium amplitude is being discussed in this paper. Moreover, with a relatively large driving force, the nonlinear behaviour of the guitar string becomes more obvious as the trajectory is gradually dominated by other small trajectories instead of the original simple harmonic trajectory. Studies have carried out many different approaches to measure the motion of the string. A very convenient experimental method is to use a metallic string where an alternating current flows. In the presence of a magnetic field, as generated by a magnetic bar properly positioned on the side of the string, the interaction of the current with the magnetic field excites the string oscillation via the Lorentz force. This experimental approach was used to excite and study periodic and chaotic motion, while the string movement was measured using optical means.
Approximation to the complete solution
The linear damped harmonic oscillator is described by a differential equation that can be written in the form
2 𝑥 = 𝐹 sin(𝑤𝑡)
𝑥̈ +𝑤 0 𝛽𝑥̇+𝑤 0
where 𝑤 0 is the natural angular frequency of the guitar string, 𝑤 is the driving frequency and 𝛽 is a half of the damping ratio. Assuming the driving force is sinusoidal, general solution and particular solution can be easily achieved to obtain the obvious relationship between amplitude and the ratio of driving frequency and natural frequency. The restoring force in this case deviates linearly due to the damping term, thus an exponential decay in the amplitude can be seen through the transient state of the general solution. To be more realistic, the well-known duffing equation is used to add nonlinearities to the oscillator.
2 𝑥(1+𝜀𝑥 2 ) = 𝐹 sin(𝑤𝑡)
𝑥̈+𝑤 0 𝛽𝑥̇+𝑤 0
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