Notes › MATH 4890: Fourier Analysis Concept
SHM derivation
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SHM Derivation
The restoring force of an ideal spring is $F_s = -ky(t)$. By Newton’s 2nd Law:
$$-ky(t) = my''(t)$$Since $c = \sqrt{\frac{k}{m}}$, the second order ODE becomes:
$$y''(t)+c^2y(t)=0$$with a general solution of:
$$y(t)=a\cos{ct} + b\sin{ct}$$If we are given initial position and velocity ($y(0)$ and $y'(0)$), the unique solution is given by:
$$y(t) = y(0)\cos{ct} + \frac{y'(0)}{c}\sin{ct}$$And there are constants $A > 0$ and $\gamma \in \mathbb{R}$ such that:
$$y(t) = a\cos{ct}+b\sin{ct} = A\cos({ct-\phi})$$where:
- $A = \sqrt{a^2 + b^2}$ and is the amplitude
- $c$ is the frequency
- $\phi$ is the phase
- $\frac{2\pi}{c}$ is the period