Oscillateurs harmoniques: Fréquence, Période, Amplitude et Amortissement

Study Notes

Harmonic Oscillators: Frequency, Period, Amplitude, and Damping

Introduction

A harmonic oscillator is a simple system that oscillates with an angular frequency [latex] \omega [/latex]. In physics, many physical systems can be described as harmonic oscillators. However, in the real world, oscillations seldom follow true simple harmonic motion (SHM). Friction of some sort usually acts to dampen the motion so it dies away, or needs more force to continue. Damped harmonic oscillators have non-conservative forces that dissipate their energy. In this article, we will focus on the subtopics of frequency, period, amplitude, and damping in the context of damped harmonic oscillators.

Frequency and Period

The frequency of a harmonic oscillator is the rate at which it oscillates. It is measured in Hertz (Hz), which represents the number of oscillations per second. The period of a harmonic oscillator is the time it takes to complete one oscillation. It is the reciprocal of the frequency, calculated as [latex] T=1/\omega [/latex]. For a damped harmonic oscillator, the angular frequency for damped harmonic motion becomes [latex] \omega = \sqrt{\omega_{0}^{2} - \left(\frac{b}{2m}\right)^{2}} [/latex], where [latex] \omega [/latex] is the angular frequency for damped harmonic motion, [latex] \omega_{0} [/latex] is the undamped angular frequency, [latex] b [/latex] is the damping coefficient, and [latex] m [/latex] is the mass of the oscillator.

Amplitude

Amplitude refers to the maximum displacement of a harmonic oscillator from its equilibrium position. In a damped harmonic oscillator, the amplitude of vibration decreases over time due to damping forces. When the damping constant is small, the system oscillates while the amplitude of the motion decays exponentially. This system is said to be underdamped, as in curve (a). Many systems are underdamped, and oscillate while the amplitude decreases exponentially, such as the mass oscillating on a spring. The damping may be quite small, but eventually the mass comes to rest.

Damping

A damped harmonic oscillator is affected by damping forces, such as frictional forces that slow down the motion of an object. Critical damping returns the system to equilibrium as fast as possible without oscillating. Driven harmonic oscillators are further affected by an externally applied force. The resonance effect occurs only in the underdamped systems. For strongly underdamped systems, the value of the amplitude can become quite large near the resonance frequency.

Conclusion

In summary, harmonic oscillators are described by their frequency, period, amplitude, and damping. Although many physical systems can be modeled as harmonic oscillators, the presence of damping forces often affects their behavior. Damped harmonic oscillators exhibit a decaying amplitude, and the damping constant plays a crucial role in determining the system's oscillation behavior. Understanding these properties is essential for modeling and analyzing various physical phenomena in the context of harmonic oscillators.

Description

Découvrez les subtilités des oscillateurs harmoniques, y compris la fréquence, la période, l'amplitude et l'amortissement. Comprenez comment les forces d'amortissement affectent le comportement des oscillateurs et l'évolution de l'amplitude. Explorez les concepts clés essentiels pour modéliser et analyser divers phénomènes physiques liés aux oscillateurs harmoniques.