168. Damped Harmonic MotionLearning Objectives
A guitar string stops oscillating a few seconds after being plucked. To keep a child happy on a swing, you must keep pushing. Although we can often make friction and other nonconservative forces negligibly small, completely undamped motion is rare. In fact, we may even want to damp oscillations, such as with car shock absorbers. For a system that has a small amount of damping, the period and frequency are nearly the same as for simple harmonic motion, but the amplitude gradually decreases as shown in Figure 2. This occurs because the nonconservative damping force removes energy from the system, usually in the form of thermal energy. In general, energy removal by nonconservative forces is described as $${W}_{\text{nc}}=\mathrm{\Delta}(\text{KE}+\text{PE})\text{,}$$where ${W}_{\text{nc}}$ is work done by a nonconservative force (here the damping force). For a damped harmonic oscillator, ${W}_{\text{nc}}$ is negative because it removes mechanical energy (KE + PE) from the system. If you gradually increase the amount of damping in a system, the period and frequency begin to be affected, because damping opposes and hence slows the back and forth motion. (The net force is smaller in both directions.) If there is very large damping, the system does not even oscillate—it slowly moves toward equilibrium. Figure 3 shows the displacement of a harmonic oscillator for different amounts of damping. When we want to damp out oscillations, such as in the suspension of a car, we may want the system to return to equilibrium as quickly as possible Critical damping is often desired, because such a system returns to equilibrium rapidly and remains at equilibrium as well. In addition, a constant force applied to a critically damped system moves the system to a new equilibrium position in the shortest time possible without overshooting or oscillating about the new position. For example, when you stand on bathroom scales that have a needle gauge, the needle moves to its equilibrium position without oscillating. It would be quite inconvenient if the needle oscillated about the new equilibrium position for a long time before settling. Damping forces can vary greatly in character. Friction, for example, is sometimes independent of velocity (as assumed in most places in this text). But many damping forces depend on velocity—sometimes in complex ways, sometimes simply being proportional to velocity. Example 1: Damping an Oscillatory Motion: Friction on an Object Connected to a SpringDamping oscillatory motion is important in many systems, and the ability to control the damping is even more so. This is generally attained using nonconservative forces such as the friction between surfaces, and viscosity for objects moving through fluids. The following example considers friction. Suppose a 0.200kg object is connected to a spring as shown in Figure 4, but there is simple friction between the object and the surface, and the coefficient of friction ${\mu}_{k}$ is equal to 0.0800. (a) What is the frictional force between the surfaces? (b) What total distance does the object travel if it is released 0.100 m from equilibrium, starting at $v=0$? The force constant of the spring is $k=\text{50}\text{.}\mathrm{0\; N/m}\text{}$. Strategy This problem requires you to integrate your knowledge of various concepts regarding waves, oscillations, and damping. To solve an integrated concept problem, you must first identify the physical principles involved. Part (a) is about the frictional force. This is a topic involving the application of Newton’s Laws. Part (b) requires an understanding of work and conservation of energy, as well as some understanding of horizontal oscillatory systems. Now that we have identified the principles we must apply in order to solve the problems, we need to identify the knowns and unknowns for each part of the question, as well as the quantity that is constant in Part (a) and Part (b) of the question. Solution a
Discussion a The force here is small because the system and the coefficients are small. Solution b Identify the known:
Discussion b This is the total distance traveled back and forth across $x=0$, which is the undamped equilibrium position. The number of oscillations about the equilibrium position will be more than $d/X=(1\text{.}\text{59}\phantom{\rule{0.25em}{0ex}}\text{m})/(0\text{.}\text{100}\phantom{\rule{0.25em}{0ex}}\text{m})=\text{15}\text{.}9$ because the amplitude of the oscillations is decreasing with time. At the end of the motion, this system will not return to $x=0$ for this type of damping force, because static friction will exceed the restoring force. This system is underdamped. In contrast, an overdamped system with a simple constant damping force would not cross the equilibrium position $x=0$ a single time. For example, if this system had a damping force 20 times greater, it would only move 0.0484 m toward the equilibrium position from its original 0.100m position. This worked example illustrates how to apply problemsolving strategies to situations that integrate the different concepts you have learned. The first step is to identify the physical principles involved in the problem. The second step is to solve for the unknowns using familiar problemsolving strategies. These are found throughout the text, and many worked examples show how to use them for single topics. In this integrated concepts example, you can see how to apply them across several topics. You will find these techniques useful in applications of physics outside a physics course, such as in your profession, in other science disciplines, and in everyday life. Check Your UnderstandingWhy are completely undamped harmonic oscillators so rare? Show/Hide Solution SolutionFriction often comes into play whenever an object is moving. Friction causes damping in a harmonic oscillator. Check Your UnderstandingDescribe the difference between overdamping, underdamping, and critical damping. Show/Hide Solution SolutionAn overdamped system moves slowly toward equilibrium. An underdamped system moves quickly to equilibrium, but will oscillate about the equilibrium point as it does so. A critically damped system moves as quickly as possible toward equilibrium without oscillating about the equilibrium. Section Summary
Conceptual QuestionsExercise 1Give an example of a damped harmonic oscillator. (They are more common than undamped or simple harmonic oscillators.) Exercise 2How would a car bounce after a bump under each of these conditions?
Exercise 3Most harmonic oscillators are damped and, if undriven, eventually come to a stop. How is this observation related to the second law of thermodynamics? Problems & ExercisesExercise 1The amplitude of a lightly damped oscillator decreases by $3\text{.}\mathrm{0\%}\text{}$ during each cycle. What percentage of the mechanical energy of the oscillator is lost in each cycle?
