15-5 Damped Simple Harmonic Motion#

Prompts

Why does a real oscillator (e.g., pendulum in air) eventually stop? Where does the energy go?
What is the damping force \(F_d = -bv\)? Why does it oppose the velocity? What are typical units of \(b\)?
For damped SHM, how does the amplitude change with time? Write \(x(t)\) and identify the decaying envelope.
Why does damping change the angular frequency? What is the physical intuition? How does the frequency evolve from undamped → underdamped → critical damping → overdamped?
Why is energy damping twice as fast as amplitude damping? (Equivalently: why is the energy relaxation time half of the amplitude relaxation time?)

Lecture Notes#

Overview#

Real oscillators lose energy to friction, air resistance, or other dissipative forces. The motion is damped—amplitude and energy decrease over time.
Damping force \(F_d = -bv\): proportional to velocity, opposes motion. Energy is transferred to thermal energy.
Relaxation time \(\tau = m/b\): time scale for energy decay; \(1/\tau\) sets the decay rate.
Displacement: \(x(t) = x_m e^{-t/(2\tau)}\cos(\omega' t + \phi)\)—a cosine with an exponentially decaying amplitude.

Undamped	Damped
Amplitude constant	Amplitude \(\propto e^{-t/(2\tau)}\)
\(E\) constant	\(E \propto e^{-t/\tau}\)
\(\omega = \sqrt{k/m}\)	\(\omega' = \sqrt{\omega^2 - 1/(4\tau^2)}\)

The damping force#

A common model: the damping force is proportional to velocity and opposes it:

(75)#\[ F_d = -bv \]

\(b\) = damping constant (SI: kg/s). Depends on the vane, fluid, or friction mechanism.
- Dimensional analysis: \([b/m] = \text{(kg/s)}/\text{kg} = 1/\text{s}\)—a time scale.
- Define the relaxation time
  
  (76)#\[ \tau = \frac{m}{b} \]
The minus sign ensures \(F_d\) opposes the direction of motion.

Newton’s second law: \(-bv - kx = ma\), giving

(77)#\[ m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0 \]

Displacement and damped frequency#

The solution (for \(b < 2\sqrt{km}\), underdamped case) is

(78)#\[ x(t) = x_m e^{-t/(2\tau)} \cos(\omega' t + \phi) \]

Amplitude envelope: \(x_m e^{-t/(2\tau)}\)—decays exponentially; \(t/(2\tau)\) is dimensionless.
Damped angular frequency:

(79)#\[ \omega' = \sqrt{\omega^2 - \frac{1}{4\tau^2}} \]
- Undamped: If \(b = 0\): \(\tau \to \infty\), \(\omega' = \omega = \sqrt{k/m}\).
- Lightly damped: \(\tau\) large \(\Rightarrow\) \(1/\tau \ll \omega\) \(\Rightarrow\) \(\omega' \approx \omega\)—period nearly unchanged; only amplitude decays.

Overdamping and critical damping

If \(b \geq 2\sqrt{km}\) (equivalently \(1/\tau \geq 2\omega\)), the motion is overdamped (no oscillation) or critically damped (fastest return to equilibrium without overshoot). The cosine form Eq. (78) applies only when \(b < 2\sqrt{km}\).

Energy decay#

Mechanical energy \(E = K + U\) is not conserved—it decreases as work is done against the damping force.

For small damping, the amplitude is \(x_m e^{-t/(2\tau)}\), so (replacing the constant amplitude in \(E = \frac{1}{2}kx_m^2\)):

(80)#\[ E(t) \approx \frac{1}{2}kx_m^2\, e^{-t/\tau} \]

Time constant for energy decay: \(\tau\) (Eq. (76)). After time \(\tau\), \(E\) drops by factor \(e^{-1} \approx 0.37\).
Time for \(E\) to drop to fraction \(f\) of initial: \(e^{-t/\tau} = f\) \(\Rightarrow\) \(t = \tau\ln(1/f)\).

Poll: Amplitude vs energy decay

A lightly damped oscillator’s amplitude decreases by 2% during each cycle. Approximately what fraction of its mechanical energy is lost per cycle?

(A) 2%
(B) 4%
(C) 1%
(D) Undetermined

Summary#

Damping: \(F_d = -bv\); energy lost to thermal.
\(\tau = m/b\) (relaxation time); \(1/\tau\) is the energy-decay rate.
\(x(t) = x_m e^{-t/(2\tau)}\cos(\omega' t + \phi)\); amplitude decays exponentially.
\(\omega' = \sqrt{\omega^2 - 1/(4\tau^2)}\); for light damping (\(1/\tau \ll \omega\)), \(\omega' \approx \omega\).
\(E(t) \approx \frac{1}{2}kx_m^2\, e^{-t/\tau}\); after time \(\tau\), \(E\) drops by factor \(e^{-1}\).