Final answer:
The damped harmonic oscillator equation describes energy loss in an oscillating system and its solution, assuming underdamping, is given by Euler's formula, resulting in a displacement x(t) = A e^(-α t) cos(θ + ω t), where A is the amplitude and α, θ, and ω are constants related to damping, phase, and frequency.
Step-by-step explanation:
Derivation of the Damped Harmonic Oscillator Solution
The equation of motion for a damped harmonic oscillator is given by:
dx^2/dt^2 + (b/m) dx/dt + (k/m)x = 0
This differential equation describes how an oscillating system loses energy over time due to a damping force. The solution to this equation, assuming underdamping (√k/m > b/2m), can be derived using Euler's formula, eid = cos(d) + i sin(d), where e is the base of the natural logarithms, i is the imaginary unit, and d is the argument of cosine and sine functions. The solution that describes the displacement x(t) in terms of time t for a damped harmonic oscillator is:
x(t) = A e^(-α t) cos(θ + ω t)
Where A represents the amplitude, α is the damping factor (b/2m), θ is the phase angle, and ω is the undamped angular frequency (√k/m). The displacement x(t) could also be derived by applying the initial conditions related to the energy of the system and by using the relationship between energy, mass, and spring constant.
The equation for simple harmonic motion (SHM) can be translated into circular motion, i.e., the characteristics of SHM can be derived from the projection of uniform circular motion.