Foundations — Damped oscillations — underdamped, critically damped, overdamped

Before you can read a single line of the damped-oscillation page, you need a small toolbox of ideas. Below I build every symbol and picture from scratch, in an order where each one leans only on the ones before it. If the parent note wrote a symbol, it is defined here first.

1. Time $t$ — the variable everything runs on

Picture the horizontal axis of every graph to come: it is $t$, running left (early) to right (late). Whenever you see $e^{\lambda t}$ or $\cos ({\omega }_{d}t)$ later, the $t$ inside is just this clock, in seconds. Nothing on this page changes with anything but $t$.

2. Position, and its two rates of change

Picture a bead on a horizontal line. The centre mark is $x=0$. To the right is positive, to the left is negative. That single number tells you the whole state of where the mass is at that instant.

But "where" is not enough — we also care about how fast and how the speed itself is changing. Those are the derivatives of $x$ with respect to time $t$.

WHY do we need both dots? A spring cares about where the mass is ($x$, it pulls it back). Damping cares about how fast it moves ($\dot{x}$, it resists speed). And Newton's law is written in terms of acceleration ($\ddot{x}$). So the whole topic is a sentence that mixes all three. You cannot skip any of them.

3. The two forces that act

WHY these two and no others? The spring is what makes it want to oscillate. The damping is what steals the energy. Take away the spring and there's nothing to bounce back; take away the damping and it swings forever. The whole three-regime story is the tug-of-war between exactly these two arrows.

4. Newton's second law — the sentence that ties it together

WHY divide by $m$? Dividing gives $\ddot{x}+\frac{b}{m}\dot{x}+\frac{k}{m}x=0$. This isolates two combinations, $\frac{k}{m}$ and $\frac{b}{m}$, that turn out to be the physically meaningful "frequencies" of the system. This is why the parent renames them into cleaner symbols next.

5. Why an angular frequency needs "radians" — the circle picture

Before we name any frequency, we need to know what "frequency" is measured in. That is what this section builds — so the units in the next section aren't a surprise.

WHY radians and not degrees? Because calculus of sines and cosines is only clean in radians (the slope of $\sin$ is exactly $\cos$ only when the angle is in radians). Since the whole topic runs on derivatives of oscillations, radians are forced on us. You met this idea in Simple Harmonic Motion.

6. The renamed constants: ${\omega }_0$ and $\gamma$

Now that "radians per second" has a meaning, we can name the two key parameters.

WHY the factor 2 in $2\gamma$? Pure bookkeeping. It makes a square root later come out as $\sqrt{{\gamma }^2-{\omega }_0^2}$ with no stray factors. Nothing physical hides in the 2.

7. The exponential $e^{\lambda t}$ — the shape of decay

WHY guess $x=e^{\lambda t}$ to solve the ODE? Because differentiating it just multiplies by $\lambda$. That turns the calculus equation into an ordinary algebra equation for $\lambda$ (the "characteristic equation"). The whole trick of Second-order linear ODEs is that exponentials convert derivatives into multiplication.

This is the hinge of the whole topic: the real part of $\lambda$ becomes the decay $e^{-\gamma t}$, and the imaginary part becomes the wobble $\cos ({\omega }_{d}t)$.

8. The discriminant ${\gamma }^2-{\omega }_0^2$ — the referee

Solving ${\lambda }^2+2\gamma \lambda +{\omega }_0^2=0$ by the quadratic formula gives $\lambda =-\gamma \pm \sqrt{{\gamma }^2-{\omega }_0^2}.$

WHY does the sign matter so much? Because a real square root gives decaying exponentials (no wobble), while an imaginary one gives sines and cosines (wobble). The sign is literally the switch between "swings" and "crawls." This same discriminant idea powers Quality factor & bandwidth and appears in RLC circuits where resistance plays the role of damping.

Prerequisite map

Equipment checklist

Test yourself — reveal only after answering.

Related: Damped Oscillations — Underdamped, Critically Damped, Overdamped · Simple Harmonic Motion · Forced Oscillations & Resonance · Second-order linear ODEs · Quality factor & bandwidth · RLC circuits