Springs

Spring Forces

To start out, we'll script a spring. Specifically, we'll apply a spring force to every object towards the center of the screen using this function:

const SPRING_CONSTANT = 0.001;
const DAMPING_FACTOR = 0.999;

fn apply_spring_forces(ids) {
    for body in ids {
        draw_spring(
            vec(0.0, body.get_pos().y),
            body.get_pos(),
            body.get_radius() * 2.0,
            body.get_radius() / 2.0,
            10
        );
        let spring_length = (body.get_pos().x + body.get_vel().x);
        body.add_force(vec(SPRING_CONSTANT * -spring_length, 0.0));
        body.set_vel(body.get_vel() * DAMPING_FACTOR);
    }
}

Try to understand this function. The important line is this:

body.add_force(vec(SPRING_CONSTANT * -spring_length, 0.0));

The damping_factor simulates energy lost by the spring. It is common in computational integration to prevent integration error for growing; it is better for the total energy of the system to increase than decrease. In real life, the total energy of a system will generally fall due to air resistance, friction, or other small forces, so damping is a good approximation.

We can use these functions in the update function below. Also, make sure to include the draw_spring function:

set_g(0.0);

let update = |ids, bodies| {
    if (!is_paused()) {
        apply_spring_forces(ids);
    }
};

fn apply_spring_forces(ids) {
    ...
}

fn draw_spring(start, end, width, thickness, coils) {
    let r = end - start;
    let incr = r / coils;
    let offs = incr.rotate(3.1415 / 2.0).normalized * width;

    let pos = start;
    for i in 0..coils {
        let p1 = pos + (incr / 3.0) + (offs / 2.0);
        let p2 = pos + (incr * 2.0/3.0) - (offs / 2.0);
        let p3 = pos + incr;
        draw_line(pos, p1, thickness);
        draw_line(p1, p2, thickness);
        draw_line(p2, p3, thickness);
        pos = p3;
    }
}

Run this script and try instantiating some objects. What happens? Try giving them a bit of vertical velocity.

Total Energy

Next, we'll examine the total energy of the system by calculating it at each step of the simulation. Let's add some code to print the kinetic, potential, and total energy of the simulation. Fill in the question marks below:

set_g(0.0);

let update = |ids, bodies| {
    if (!is_paused()) {
        apply_spring_forces(ids);
    }

    let total_kinetic_energy = 0.0;
    let total_potential_energy = 0.0;
    for body in ids {
        let m = body.get_mass();
        let x = body.get_pos().x;
        let v = body.get_vel();
        let v_mag = v.length;

        // use the SPRING_CONSTANT variable for 
        // potential energy
        let body_kinetic_energy = ???;
        let body_potential_energy = ???;

        // add the body's energy to the total
        total_kinetic_energy += body_kinetic_energy;
        total_potential_energy += body_potential_energy;
    }

    let total_energy = 
        total_kinetic_energy + total_potential_energy;

    print(
        [total_kinetic_energy, total_potential_energy, total_energy]
    );
};

fn apply_spring_forces(ids) {
    ...
}

fn draw_spring(start, end, width, thickness, coils) {
    ...
}

Clear the scene and run the script, and then use the "Create" button in the top left to add a body a few inches left of the center of the screen. Then, try pausing the simulation by pressing space when the body is roughly at the center of the screen. Next, do the same when the body is at the right or left of the screen.

1. When is the potential energy maximized, and when is the kinetic energy maximized?

2. You might notice that the total energy of the system isn't completely constant. Given your knowledge about computational integration, why do you think this is? Try adjusting the damping_factor and delta time, or adding more bodies.

3. Super Bonus: Try playing with https://cloth.mikail-khan.com. This cloth simulator uses spring constraints between every node. However, it doesn't use anything like the apply_spring_forces function above. Instead, it uses a spring constraint solving approach which updates the positions directly. Try implementing this approach in your own script, and see how the energy is preserved. You can find the source code for the cloth simulator here: https://github.com/mkhan45/clothsim