Damped Harmonic Motion*
In simple harmonic motion, an object oscillates with a constant amplitude, because there is no mechanism for dissipating energy. In reality, however, friction or some other energy-dissipating mechanism is always present. In the presence of energy dissipation, the oscillation amplitude decreases as time passes, and the motion is no longer simple harmonic motion. Instead, it is referred to as damped harmonic motion, the decrease in amplitude being called “damping.”
The physics of a shock absorber.
One widely used application of a damped harmonic motion is in the suspension system of an automobile. Figure 10.22a shows a shock absorber attached to a main suspension spring of a car. A shock absorber is designed to introduce damping forces, which reduce the vibrations associated with a bumpy ride. As part b of the drawing shows, a shock absorber consists of a piston in a reservoir of oil. When the piston moved in response to a bump in the road, holes in the piston head permit the piston to pass through the oil. Viscous forces that arise during this movement cause the damping.
Figure 10.23 illustrates the different degrees of damping that can exist. As applied to the example of a car’s suspension system these graphs show the vertical position of the chassis after it has been pulled upward by an amount A0 at time t0 = 0 s and then released. Part a of the figure compares undamped or simple harmonic motion in curve 1 (red) to slightly damped motion in curve 2 (green). In damped harmonic motion, the chassis oscillates with decreasing amplitude and eventually comes to rest. As the degree of damping is increased from curve 2 to curve 3 (gold), the car makes fewer oscillations before coming to a halt.
One widely used application of a damped harmonic motion is in the suspension system of an automobile. Figure 10.22a shows a shock absorber attached to a main suspension spring of a car. A shock absorber is designed to introduce damping forces, which reduce the vibrations associated with a bumpy ride. As part b of the drawing shows, a shock absorber consists of a piston in a reservoir of oil. When the piston moved in response to a bump in the road, holes in the piston head permit the piston to pass through the oil. Viscous forces that arise during this movement cause the damping.
Figure 10.23 illustrates the different degrees of damping that can exist. As applied to the example of a car’s suspension system these graphs show the vertical position of the chassis after it has been pulled upward by an amount A0 at time t0 = 0 s and then released. Part a of the figure compares undamped or simple harmonic motion in curve 1 (red) to slightly damped motion in curve 2 (green). In damped harmonic motion, the chassis oscillates with decreasing amplitude and eventually comes to rest. As the degree of damping is increased from curve 2 to curve 3 (gold), the car makes fewer oscillations before coming to a halt.
Figure 10.23b shows that as the degree of damping it increased further, there comes a point when the car does not oscillate at all after it is released but, rather, settles directly back to its equilibrium position, as in curve 4 (blue). The smallest degree of damping that completely eliminates the oscillations is termed “critical damping,” and the motion is said to be critically damped. Figure 10.23b also shows that the car takes the longest time to return to its equilibrium position in curve 5 (purple), where the degree of damping is above the value for critical damping. When the damping exceeds the critical value, the motion is said to be overdamped. In contrast, when the damping is less than the critical level, the motion is said to be underdamped (curves 2 and 3). Typical automobile shock absorbers are designed to produce underdamped motion somewhat like that in curve 3.
Check Your Understanding
The shock absorbers on a car are badly in need of replacement and introduce very little damping. Does the number of occupants in the car affect the vibration frequency of the car’s suspension system?
Answer: Yes, because the vibration frequency depends on the mass of the car and its occupants.
The shock absorbers on a car are badly in need of replacement and introduce very little damping. Does the number of occupants in the car affect the vibration frequency of the car’s suspension system?
Answer: Yes, because the vibration frequency depends on the mass of the car and its occupants.
* Cutnell, John D., et al. Physics. Wiley, 2015
This text has 30 sentences, with 500 words (17.24 per sentence) with 1.58 syllables per word.
Flesch-Kincaid Grade Level 9.8
This text has 30 sentences, with 500 words (17.24 per sentence) with 1.58 syllables per word.
Flesch-Kincaid Grade Level 9.8
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