I made a Matlab physics animation

Here's a cool pulley simulation I made visualised on Matlab:

I'm learning to animate simulations on Matlab. So, I decided to animate one of my coolest problems, Simulating Dynamics-2

The actual simulation was done with the help of Karan Chatrath's code solution to my problem. I modified the code to make it more efficient and added some extra features, and more importantly writing the second part of the code, which was animating the data.

The setup above is basically two masses connected by a pulley, with the second mass dragging the first mass with the help of gravity. There is friction between the first mass and the floor.

The equations of motion can be derived using Newton's Laws, here's the system in matrix form:

[m10μcos⁡(θ)0m201001sin⁡(θ)2x28−2y00]\begin{bmatrix} m_1 & 0 & \mu & \cos(\theta) \\ 0 & m_2 & 0 & 1 \\ 0 & 0 & 1 & \sin(\theta)\\ 2x & 28-2y & 0 & 0 \end{bmatrix} [x¨y¨NT]=[0m2gm1g2x˙2+2y˙2]\begin{bmatrix} \ddot{x} \\ \ddot{y} \\ N \\ T \end{bmatrix} = \begin{bmatrix} 0 \\ m_2 g \\ m_1g \\ 2\dot{x}^2 + 2\dot{y}^2 \end{bmatrix}

We can solve for these values simply by multiplying the inverse system matrix by the vector on the right hand side. Once we do this, we have to numerically integrate x¨,y¨\ddot{x},\ddot{y}, the first and second indices of the solution to A−1bA^{-1}b.

For the animation, just draw two squares with the tops at the position of the point masses at x coordinate for the first mass xx and y coordinate for the second mass yy. Then draw lines from the hinge to each mass.

To maintain accuracy, I added a counter to add the values of time, xx, and (y) to arrays. The animation works by while looping through these arrays, drawing the data, deleting the data, and redrawing through the loops.

The values I used for the simulation are m1=3,m2=2,g=10,μ=0.8,l=14mm_1 = 3, m_2 = 2, g = 10, \mu = 0.8, l = 14m

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clear all;
close all;
clc;

%% Initialisation and constants
deltaT= 10^-5;
time = 0;

m1 = 3;
m2 = 2;

g  = 10;
mu = 0.8;

x = -14*cos(asin(8/14));
y = 0;

xDot = 0;
yDot = 0;

massPosition1 = [];
massPosition2 = [];
times = [];

count = 0;

%% Simulation
while xDot >= 0

    if mod(count, 1000) == 0
        times = [times; time];
        massPosition1 = [massPosition1; x];
        massPosition2 = [massPosition2; y];
    end

    %System matrix
    A = [m1 0 mu -x/(y-14);0 m2 0 1;0 0 1 (1 - x^2/(y - 14)^2)^(1/2);2*x 28-2*y 0 0];

    %Vector
    b = [0;m2*g;m1*g;2*yDot^2-2*xDot^2];

    %Solution vector (accelerations, tension, and normal force)
    S = inv(A)*b;



    xDotDot = S(1);
    yDotDot = S(2);

    %Numerical Integration
    xDot = xDot + xDotDot*deltaT;
    yDot = yDot + yDotDot*deltaT;

    x  = x + xDot*deltaT;
    y  = y + yDot*deltaT;

    %Updating time and counter
    time = time + deltaT;
    count = count + 1;
end





%% Plot
axis(gca,"equal");
O = [0 0];
Hinge = [0.2 8];
axis([-12 1.5 -1 9]);
grid on; 


%% Animation
i = 1;


while i <= length(times)
    %animation speed control conditional statement
    if mod(i,5) == 0
        %Hinge circle
        hingeCircle = viscircles(Hinge, 0.2);

        %time
        title("time: " + num2str(ceil(times(i))) + " s");

        %Positions and string line plotting
        Mass_1 = rectangle('Position',[(massPosition1(i)-0.5) -1 1 1]);
        stringToHinge1 = line([massPosition1(i) Hinge(1)-0.15],[0 Hinge(2)+0.2]);

        Mass_2 = rectangle('Position', [0 (7-massPosition2(i)) 1 1]);
        stringToHinge2 = line([0.4 0.4], [8 (8-massPosition2(i))]);

        %pausing for animation
        pause(10^-6);

        %Deleting previous elements to draw new ones in next iteration
        if i < length(times)-5
            delete(Mass_1);
            delete(stringToHinge1);
            delete(Mass_2);
            delete(stringToHinge2);
        end

    end

    %looping through arrays
    i = i+1;
end

%% End

Have a try for yourself, or, if you don't have Matlab, take a look at the GIF above.

#Mechanics

Note by Krishna Karthik
5 months, 2 weeks ago

No vote yet
1 vote

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Comments

@Percy Jackson

@NSCS 747

In case you're interested. I couldn't get Karan Chatrath or Steven Chase's tag though lol.

@Talulah Riley If you're active anymore.

Krishna Karthik - 5 months, 2 weeks ago

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Nice bro! Your mentions are all wrong. I didn't get the notif as they're not blue. I can mention them for you though - @NSCS 747, @Talulah Riley, @Karan Chatrath, @Steven Chase

A Former Brilliant Member - 5 months, 1 week ago

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Yeah; I may have edited the comment. When you edit comments, all your mentions go away. Btw thanks a lot for adding them back :)

Krishna Karthik - 5 months, 1 week ago

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@Krishna Karthik – no prob bro :)

A Former Brilliant Member - 5 months, 1 week ago

Impressive work

Karan Chatrath - 5 months, 1 week ago

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Thanks. Seeing it visualised adds a level of cool to simulations.

Krishna Karthik - 5 months, 1 week ago
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