Algos, Animation - Natural Motion, Inertial Bounce

Abstract

In physics, a damped harmonic oscillator is a system that exhibits oscillatory motion (like a spring-mass system) but gradually loses its energy over time due to damping. This is similar to, for example, a swinging pendulum or other natural systems gradually coming to rest due to air resistance or a spring-mass system oscillating but eventually stopping due to friction.

Formula

Inertial Bounce

$value+v\cdot 0.1\cdot \sin (4\pi t)\cdot e^{-2t}$

Where:

• A = amplitude (0.1)
• v = initial velocity
• f = frequency (2.0)
• d = decay rate (2.0)
• t = time since keyframe

Explained

amp = .1;
freq = 2.0;
decay = 2.0;
n = 0;
time_max = 4;
if (numKeys > 0){
	n = nearestKey(time).index;
	if (key(n).time > time) {
		n--;
	}
}
if (n == 0) {
	t = 0;
} else {
	t = time - key(n).time;
}
if (n > 0 && t < time_max) {
	v = velocityAtTime(key(n).time - thisComp.frameDuration/10);
	value + v * amp * Math.sin(freq * t * 2 * Math.PI) / Math.exp(decay * t);
} else {
	value
}

where:

1. Variable Initialization:

• amp = .1: Sets the amplitude of a sine wave to 0.1.
• freq = 2.0: Sets the frequency of the sine wave to 2.0.
• decay = 2.0: Sets the decay rate to 2.0.
• n = 0: Initializes the variable n to 0.
• time_max = 4: Sets the maximum time to 4.

2. Keyframe Handling:

• Checks if there are any keyframes (numKeys > 0), if yes, it proceeds.
• Finds the nearest keyframe index (nearestKey(time).index) relative to the current time.
• Adjusts the keyframe index if the keyframe time is greater than the current time.
• Sets t to the difference between the current time and the time of the nearest keyframe.

3. Value Calculation:

• If there are keyframes and t is within a certain range (t < time_max), it proceeds.
• Calculates the velocity at the time of the nearest keyframe (velocityAtTime).
• Calculates the new value using a sinusoidal function with amplitude amp, frequency freq, and a decay factor decay.
• The new value is calculated using the current value plus the velocity adjusted by the sine function and decay factor.
• If there are no keyframes or if t exceeds time_max, it returns the current value.

References

• Gray Machine: https://www.graymachine.com/top-5-effects-expressions