15.15
Concept of Resonance and its Characteristics
If a driven oscillator needs to resonate at a specific frequency, then very light damping is required. An example of light damping includes playing pi…
Consider four pendulums tied to a string and suspended from a rigid support. Let the lengths of pendulums A and C be identical, while pendulums B and D have different lengths.
When pendulum A is displaced, it oscillates with a natural frequency, and the energy from pendulum A is transferred to the other pendulums, forcing them to oscillate with a driving frequency.
The given expression relates the amplitude with natural and driving frequency.
On close observation, it is found that pendulums B and D oscillate with driving frequencies quite different from their natural frequencies, resulting in smaller amplitudes.
In contrast, the driving frequency of the pendulums with identical lengths is equal to their natural frequency, causing the amplitude to increase gradually with each oscillation. This condition—when the driving frequency equals the natural frequency—is called resonance.
In a damped harmonic oscillator, under the resonance condition, the amplitude can be altered depending on the damping force. The amplitude is large and narrow for small damping, while it is smaller for heavy damping.
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Q1: What happens when a driving frequency equals an oscillator's natural frequency?
When driving frequency matches natural frequency, the system enters resonance. At this condition, amplitude increases gradually with each oscillation. This occurs because energy transfers efficiently from the driving force into the oscillator, causing constructive interference between successive oscillations.
Q2: How does damping affect resonance amplitude?
Damping significantly alters resonance amplitude. Light damping produces large, narrow amplitude peaks at resonance, ideal for musical instruments like piano strings. Heavy damping reduces amplitude but broadens the frequency response, making the system respond across more frequencies, as seen in car suspension systems.
Q3: Why do identical pendulums oscillate with larger amplitudes than different-length pendulums?
Identical pendulums resonate because their natural frequencies match the driving frequency from the source pendulum. Different-length pendulums have natural frequencies mismatched to the driving frequency, resulting in smaller amplitudes and less efficient energy transfer from the driving force.
Q4: What is the relationship between damping and frequency selectivity in radio tuning?
Radio selectivity depends inversely on damping. Lower damping creates sharper frequency discrimination, allowing the radio to isolate a specific station's broadcast frequency while rejecting others. Higher damping reduces selectivity but broadens the frequency range the radio can receive.
Q5: Why do car suspension systems use heavy damping despite resonance concerns?
Heavy damping reduces amplitude oscillations, providing passenger comfort on normal roads. However, the tradeoff is that shock absorbers respond across multiple frequencies. Speed bumps and corrugated gravel roads still trigger resonance because their frequencies occasionally match the system's response, causing noticeable oscillations.
Q6: How does the amplitude expression relate driving and natural frequencies?
The amplitude expression shows that amplitude depends on both driving and natural frequencies. When these frequencies differ significantly, amplitude remains small. As driving frequency approaches natural frequency, amplitude increases. At exact resonance, the relationship predicts maximum amplitude, modified by damping forces present in real systems.
Q7: What determines whether light or heavy damping is appropriate for an oscillating system?
System purpose determines damping choice. Light damping is required for systems needing strong resonance response at specific frequencies, like musical instruments. Heavy damping suits applications requiring small-amplitude oscillations regardless of frequency, such as vehicle suspensions, where energy conversion to heat prevents excessive motion.