Damped Harmonic Motion Presentation

Damped Harmonic Motion

Introduction

• Damped harmonic motion occurs when a system experiences a damping force that opposes its motion.
• Key equation for Simple Harmonic Motion (SHM):

\(m \frac{d^2x}{dt^2} = -kx\)

• \(m\): Mass of the object
• \(k\): Spring constant
• \(x\): Displacement
• \(t\): Time

Damping Force

• In damped harmonic motion, a damping force is introduced.
• Damping force (\(F_d = -bv\)), where \(b\) is the damping coefficient.

Equation of Motion

\(m \frac{d^2x}{dt^2} = -kx - bv\)

Combines restoring force (\(-kx\)) and damping force (\(-bv\)).

Derivation of the Differential Equation

1. Start with Newton's second law: \(F = ma\).
2. Sum of forces is mass times acceleration.
3. Include damping force in the equation.

\(m \frac{d^2x}{dt^2} = -kx - bv\)

Solution of the Differential Equation

• Solution to the differential equation:

\(x(t) = A e^{-\frac{b}{2m}t} \cos(\omega_d t + \phi)\)

• \(A\): Amplitude
• \(\omega_d\): Damped angular frequency
• \(\phi\): Phase constant

Damped angular frequency: \(\omega_d = \sqrt{\frac{k}{m} - \left(\frac{b}{2m}\right)^2}\)

Interpretation

• \(e^{-\frac{b}{2m}t}\) represents damping effect, causing amplitude to decrease.
• \(\cos(\omega_d t + \phi)\) represents oscillatory behavior, akin to undamped case.

Conclusion

• Describes how the system oscillates and decays over time in the presence of damping.
• Rate of decay determined by damping coefficient \(b\).
• Oscillatory behavior influenced by mass \(m\) and spring constant \(k\).

Questions?

Feel free to ask if you have any questions or need further clarification!