Damped harmonic motion
Damped Harmonic Motion
Damped harmonic motion describes the behavior of a system in which an object is subject to a restoring force that is proportional to its displacement from equilibrium, as well as a damping force that opposes the motion. This damping force typically depends on the velocity of the object and acts to reduce the amplitude of oscillation over time. Damped harmonic motion is a common phenomenon in many physical systems, including mechanical oscillators and electrical circuits.
Understanding Damped Harmonic Motion
The Damping Force
The damping force is usually proportional to the velocity of the object but can also depend on higher powers of velocity or other factors. The simplest form of damping, linear damping, is directly proportional to the velocity:
$$ F_{\text{damping}} = -b \cdot v $$
where:
- $ F_{\text{damping}} $ is the damping force,
- $ b $ is the damping coefficient,
- $ v $ is the velocity of the object.
The Differential Equation
The motion of a damped harmonic oscillator can be described by the following second-order linear differential equation:
$$ m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0 $$
where:
- $ m $ is the mass of the object,
- $ \frac{d^2x}{dt^2} $ is the acceleration,
- $ b $ is the damping coefficient,
- $ \frac{dx}{dt} $ is the velocity,
- $ k $ is the spring constant,
- $ x $ is the displacement from equilibrium.
Solution to the Differential Equation
The solution to this differential equation depends on the value of the damping coefficient $ b $. There are three cases to consider:
- Underdamped ($ b^2 < 4mk $): The system oscillates with a gradually decreasing amplitude.
- Critically damped ($ b^2 = 4mk $): The system returns to equilibrium as quickly as possible without oscillating.
- Overdamped ($ b^2 > 4mk $): The system returns to equilibrium without oscillating, but more slowly than in the critically damped case.
The General Solution
For the underdamped case, the general solution to the differential equation is:
$$ x(t) = e^{-\gamma t} (A \cos(\omega' t) + B \sin(\omega' t)) $$
where:
- $ \gamma = \frac{b}{2m} $ is the damping constant,
- $ \omega' = \sqrt{\frac{k}{m} - \gamma^2} $ is the damped angular frequency,
- $ A $ and $ B $ are constants determined by the initial conditions.
Differences and Important Points
| Aspect | Undamped Motion | Damped Motion |
|---|---|---|
| Restoring Force | $ F = -kx $ | $ F = -kx $ |
| Damping Force | None | $ F_{\text{damping}} = -bv $ |
| Equation of Motion | $ m \frac{d^2x}{dt^2} + kx = 0 $ | $ m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0 $ |
| Frequency | $ \omega = \sqrt{\frac{k}{m}} $ | $ \omega' = \sqrt{\frac{k}{m} - \gamma^2} $ |
| Amplitude | Constant | Decreases over time |
| Energy | Conserved | Dissipated due to damping |
| Types | Not applicable | Underdamped, Critically damped, Overdamped |
Examples
Example 1: Underdamped Oscillator
Consider a mass-spring system with $ m = 1 \text{ kg} $, $ k = 100 \text{ N/m} $, and $ b = 1 \text{ Ns/m} $. The system is underdamped because $ b^2 < 4mk $. The damped angular frequency is:
$$ \omega' = \sqrt{\frac{k}{m} - \left(\frac{b}{2m}\right)^2} = \sqrt{100 - \left(\frac{1}{2}\right)^2} \approx 9.99 \text{ rad/s} $$
If the mass is displaced from equilibrium and released, it will oscillate with a decreasing amplitude due to the damping effect.
Example 2: Critically Damped Oscillator
For a critically damped system, the damping coefficient is set so that $ b^2 = 4mk $. If $ m = 1 \text{ kg} $ and $ k = 100 \text{ N/m} $, then $ b = 20 \text{ Ns/m} $. The system does not oscillate but instead returns to equilibrium as quickly as possible.
Example 3: Overdamped Oscillator
If the damping coefficient is increased further, say $ b = 30 \text{ Ns/m} $, the system becomes overdamped. It will return to equilibrium without oscillating, and the return will be slower than in the critically damped case.
In summary, damped harmonic motion is characterized by the presence of a damping force that causes the amplitude of oscillation to decrease over time. The behavior of the system is determined by the relationship between the damping coefficient, the mass, and the spring constant. Understanding these principles is crucial for analyzing real-world systems where damping plays a significant role, such as in mechanical engineering, automotive suspension systems, and seismology.