{"category": "C#/Runtime/Utility/Math", "keywords": "Math, Physics, Simple Harmonic Motion, SHM, Second Order System"} An approximate simulator for Simple Harmonic Motion. Note: This is not a precise simulator.

SimpleHarmonicMotion.cs

// Visualizer(In Unity Engine): https://gist.github.com/SolarianZ/a53363f0994aa7858d9e0a08ecf3f014

using System;
using System.Numerics;

// Note: This is not a precise simulator.
public class SimpleHarmonicMotion
{
    /// <summary>
    /// The natural frequency of the system measured in Hz. 
    /// It describes the speed at which the system will respond to changes in the input, 
    /// as well as the frequency the system will tend to vibrate at, 
    /// but does not affect the shape of the resulting motion.
    /// </summary>
    public float Frequency { get; set; }
    /// <summary>
    /// The damping cofficient. It describes how the system comes to settle at the desitination.
    /// When the damping is 0, the system will vibrate indefinitely.
    /// When the damping is between 0 and 1, the system is underdamped, and will vibrate before settling at the desitination.
    /// When the damping greater than 1, the system is overdamped, and will settle at the desitination without vibrating.
    /// </summary>
    public float Damping { get; set; }
    public Vector3 Position { get; set; }
    public Vector3 Velocity { get; set; }


    /// <summary>
    /// Creates a new simple harmonic motion simulator.
    /// </summary>
    /// <param name="frequency">
    /// The natural frequency of the system measured in Hz. 
    /// It describes the speed at which the system will respond to changes in the input, 
    /// as well as the frequency the system will tend to vibrate at, 
    /// but does not affect the shape of the resulting motion.
    /// </param>
    /// <param name="damping">
    /// The damping cofficient. It describes how the system comes to settle at the desitination.
    /// When the damping is 0, the system will vibrate indefinitely.
    /// When the damping is between 0 and 1, the system is underdamped, and will vibrate before settling at the desitination.
    /// When the damping greater than 1, the system is overdamped, and will settle at the desitination without vibrating.
    /// </param>
    /// <param name="initialPosition"></param>
    public SimpleHarmonicMotion(float frequency, float damping, Vector3 initialPosition)
    {
        Frequency = frequency;
        Damping = damping;
        Position = initialPosition;
        Velocity = Vector3.Zero;
    }

    public void ResetPosition(Vector3 initialPosition)
    {
        Position = initialPosition;
        Velocity = Vector3.Zero;
    }

    /// <summary>
    /// Updates the system and returns the new position.
    /// </summary>
    /// <param name="deltaTime">The delta time since the last update.</param>
    /// <param name="destination">The destination of the system.</param>
    /// <returns></returns>
    public Vector3 Tick(float deltaTime, Vector3 destination)
    {
        // Calculate the angular frequency
        float omega = 2 * MathF.PI * Frequency;

        // Calculate the damping ratio
        float zeta = Damping;

        // Calculate the natural response coefficient
        float k1 = 2 * zeta * omega;
        float k2 = omega * omega;

        // Calculate the acceleration based on the current position, velocity, and destination
        Vector3 acceleration = (destination - Position) * k2 - (k1 * Velocity);

        // Update the velocity and position based on the deltaTime
        Velocity += acceleration * deltaTime;
        Position += Velocity * deltaTime;

        return Position;
    }
}