Lesson 7.2: Damping, Forced Oscillations and Resonance
Introduction
Welcome to Lesson 7.2! In this lesson, we will explore the fascinating concepts of damping, forced oscillations, and resonance. By the end of this lesson, you, students, will be able to:
- Distinguish between free and forced oscillations and identify natural frequency.
- Understand light, heavy, and critical damping, along with their displacement-time graphs.
- Explain resonance and the resonance curve, and assess the effect of damping on the peak.
- Apply these concepts to real-world examples such as Barton’s pendulums, bridges, and musical tuning.
So, let’s get started!
Free versus Forced Oscillations
Free Oscillations
A free oscillation occurs when a system oscillates naturally after being displaced from its equilibrium position. The system has a specific frequency known as its natural frequency, denoted by $ f_n $. The formula for natural frequency of a simple pendulum is:
$$
$ f_n = \frac{1}{2\pi} \sqrt{\frac{g}{L}} $
$$
where $ g $ is the acceleration due to gravity and $ L $ is the length of the pendulum.
Example: Pendulum Movement
Imagine a swing that you push gently. Once you let go, it swings back and forth at its natural frequency until friction stops it. There are no external forces acting on it, hence it’s a free oscillation!
Forced Oscillations
In contrast, forced oscillations occur when a periodic force is applied to the system. Unlike free oscillations, forced oscillations do not rely on the system’s natural frequency alone but rather the driving frequency from an external force.
Example: Child on a Swing
If you were to regularly push a child on a swing at intervals, you are adding energy to the system at a specific frequency. If you match the swing’s natural frequency while pushing, you create a more extensive oscillation.
Damping: Light, Heavy, and Critical
Damping is a process that reduces the amplitude of oscillations in a system over time. There are three primary types of damping:
1. Light Damping
In light damping, the oscillations gradually decrease in amplitude, allowing many complete cycles. The damping ratio is less than 1.
Displacement-Time Graph
This graph shows that the oscillations continue for many cycles, gradually reducing in amplitude.
2. Heavy Damping
When heavy damping occurs, the system returns to equilibrium quickly without completing a full oscillation cycle. The damping ratio is greater than 1.
Displacement-Time Graph
You can see that instead of oscillating, the amplitude decays quickly without oscillating.
3. Critical Damping
Critical damping is the threshold at which the system returns to equilibrium in the shortest time without oscillating, with a damping ratio exactly equal to 1.
Displacement-Time Graph
In this graph, the system returns to equilibrium as fast as possible, but without overshooting.
Resonance and the Resonance Curve
What is Resonance?
Resonance occurs when an external force is applied at the system's natural frequency. The mechanical advantage is that the amplitude of oscillation reaches significantly higher levels, leading to extraordinary effects.
The Resonance Curve
The resonance curve is a graph that shows how the amplitude of oscillation varies with the frequency of the applied force.
Characteristics of the Resonance Curve:
- At the natural frequency, the amplitude peaks dramatically.
- Damping affects the height of the peak: higher damping reduces the peak amplitude.
Example: Tuning a Guitar
When tuning a guitar, the strings resonate at specific frequencies. When you strum a string at its natural frequency, the amplitude increases, creating a beautiful sound characterized by resonance.
Conclusion
In conclusion, understanding the differences between free and forced oscillations, types of damping, and the concept of resonance is crucial in physics. These concepts not only help us describe mechanical systems but also explain many phenomena in everyday life.
Study Notes
- Free oscillations occur naturally; forced oscillations require an external force.
- Natural frequency is the frequency at which systems oscillate freely.
- Light damping allows oscillations to persist longer, while heavy damping suppresses them quickly.
- Critical damping returns a system to equilibrium quickly without oscillation.
- Resonance amplifies oscillations at natural frequency but can be affected significantly by damping.
- Real-world examples include pendulums, bridges, and musical instruments.