Definition: Simple Harmonic Motion
If for some coodinate $q$ the equation of motion follows the form $$\ddot q=-\omega^2q$$ then we say the motion in $q$ is simple harmonic with angular frequency $\omega$. The general form of the soution to $q$ is $$q=A\cos(\omega t+\varphi)$$ where $A$ is the amplitude, $\omega$ is the frequency, and $\varphi$ is the phase shift. The maximum speed of the oscillator can be found as $$v_\text{max}=A\omega$$
Some common frequencies of oscillation include a block-spring system ($\omega=\sqrt{k/m}$), a simple pendulum ($\omega=\sqrt{g/l}$), and a rigid pendulum ($\omega=\sqrt{mgd/I}$). It's good practice to have these three memorized.
Idea: Spring addition
In a lot of simple harmonic motion problems, we will be dealing with systems of springs. For springs $k_1,\cdots,k_n$ connected in parallel, then the effective spring constant $k_{p}$ is found by adding the constants together: $$k_p=\sum_{i=1}^n k_i$$ If the springs are connected in series, then we add their reciprocals: $$\frac 1{k_s}=\sum_{i=1}^n \frac 1{k_i}$$An immediate result from the latter fact is that a spring $k$ cut into a length a fraction $f$ of its original length will have a new spring constant $k/f$.
Idea: Energy method
When solving for the frequency of oscillation of a system, the total energy can be used instead of $F=ma$. If the total energy, in terms of the coordinate $q$, follows the form $$E=\frac12\kappa q^2+\frac12 \mu \dot q^2$$then the oscillation frequency is $$\omega=\sqrt{\frac\kappa\mu}$$
Definition: Normal modes
The normal modes of a system are the patterns of oscillation such that the system moves at a fixed, simple harmonic frequency.
Idea: Coupled oscillators
When finding the equations of motion of a system of coupled oscillators, first find the normal mode frequencies. Then, use a linear combination of the normal mode solutions to find the solution in a given state. See the last example from lecture for a more concrete walk through.