Plotting Damped Nonlinear Oscillator

Question

Athanasios Paraskevopoulos 2024-5-7

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https://ww2.mathworks.cn/matlabcentral/answers/2116317-plotting-damped-nonlinear-oscillator

回答： Sam Chak 2024-5-8

采纳的回答： Sam Chak

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The given system is a damped nonlinear oscillator with a nonlinear resistance, described by the differential equation:

where

.

I tried to simulate the above problem for different values of

.

As you can see below I have results for

, but the code it gives me nothing for different values of

. What should I change?

% Define common parameters

v = 1; % Damping coefficient

a = 1; % Linear stiffness coefficient

T = 100; % Total simulation time

dt = 0.01; % Time step

t = 0:dt:T; % Time vector

b = (1/20) * a; % Nonlinear stiffness coefficient

x0 = 0.1; % Initial displacement

y0 = 0; % Initial velocity

m_values = [0.1, 1, 10]; % Different mass values

% Initialize a figure

figure(1);

hold on

% Loop through each mass value

for m = m_values

N = length(t); % Number of time steps

% Initialize arrays for displacement and velocity

x = zeros(1, N);

y = zeros(1, N);

% Set initial conditions

x(1) = x0;

y(1) = y0;

% Simulate the oscillator's response

for i = 2:N

% Calculate the acceleration and velocity changes using equations of motion

f = -(v/m) * y(i-1) - (a/m) * x(i-1) + (b/m) * x(i-1)^3;

g = y(i-1);

% Update velocity and displacement using Euler's method

y(i) = y(i-1) + dt * f;

x(i) = x(i-1) + dt * g;

end

% Plot the displacement and velocity for the current mass value

plot(t, x, 'DisplayName', ['x, m = ', num2str(m)]);

plot(t, y, '--', 'DisplayName', ['y, m = ', num2str(m)]);

end

% Add labels, legend, and grid to the plot

xlabel('Time (s)');

ylabel('Amplitude');

title('Damped Nonlinear Oscillator Response for Different Mass Values');

legend('show');

grid on;

hold off;

% Define common parameters

v = 1; % Damping coefficient

m = 1; % Mass

T = 50; % Total simulation time

dt = 0.01; % Time step

t = 0:dt:T; % Time vector

x0 = 0.1; % Initial displacement

y0 = 0; % Initial velocity

a_values = [0.1, 1, 10]; % Different stiffness values

% Initialize a figure

figure(2);

hold on

% Loop through each stiffness value

for a = a_values

b = (1/20) * a; % Nonlinear stiffness coefficient dependent on a

N = length(t); % Number of time steps

% Initialize arrays for displacement and velocity

x = zeros(1, N);

y = zeros(1, N);

% Set initial conditions

x(1) = x0;

y(1) = y0;

% Simulate the oscillator's response

for i = 2:N

% Calculate the acceleration and velocity changes using equations of motion

f = -(v/m) * y(i-1) - (a/m) * x(i-1) + (b/m) * x(i-1)^3;

g = y(i-1);

% Update velocity and displacement using Euler's method

y(i) = y(i-1) + dt * f;

x(i) = x(i-1) + dt * g;

end

% Plot the displacement and velocity for the current stiffness value

plot(t, x, 'DisplayName', ['x, a = ', num2str(a)]);

plot(t, y, '--', 'DisplayName', ['y, a = ', num2str(a)]);

end

% Add labels, legend, and grid to the plot

xlabel('Time (s)');

ylabel('Amplitude');

title('Damped Nonlinear Oscillator Response for Different Stiffness Values');

legend('show');

grid on;

hold off;

% Define common parameters

m = 1; % Mass, kept constant

a = 1; % Linear stiffness coefficient

T = 100; % Total simulation time

dt = 0.01; % Time step

t = 0:dt:T; % Time vector

b = (1/20) * a; % Nonlinear stiffness coefficient

x0 = 0.1; % Initial displacement

y0 = 0; % Initial velocity

v_values = [0.1, 1, 10]; % Different damping coefficients

% Initialize a figure

figure(3);

hold on

% Loop through each damping coefficient value

for v = v_values

N = length(t); % Number of time steps

% Initialize arrays for displacement and velocity

x = zeros(1, N);

y = zeros(1, N);

% Set initial conditions

x(1) = x0;

y(1) = y0;

% Simulate the oscillator's response

for i = 2:N

% Calculate the acceleration and velocity changes using equations of motion

f = -(v/m) * y(i-1) - (a/m) * x(i-1) + (b/m) * x(i-1)^3;

g = y(i-1);

% Update velocity and displacement using Euler's method

y(i) = y(i-1) + dt * f;

x(i) = x(i-1) + dt * g;

end

% Plot the displacement and velocity for the current damping value

plot(t, x, 'DisplayName', ['Displacement, v = ', num2str(v)]);

plot(t, y, '--', 'DisplayName', ['Velocity, v = ', num2str(v)]);

end

% Add labels, legend, and grid to the plot

xlabel('Time (s)');

ylabel('Amplitude');

title('Damped Nonlinear Oscillator Response for Different Damping Values');

legend('show');

grid on;

hold off;

% Define common parameters

v = 1; % Damping coefficient

a = 1; % Linear stiffness coefficient

T = 10; % Total simulation time

dt = 0.01; % Time step

t = 0:dt:T; % Time vector

b = (1/20) * a; % Nonlinear stiffness coefficient

m = 1; % Mass (constant in this case)

x0_values = [0.1, 1, 10]; % Different initial displacements

y0 = 0; % Initial velocity

% Initialize a figure

figure(4);

hold on

% Loop through each initial displacement value

for x0 = x0_values

N = length(t); % Number of time steps

% Initialize arrays for displacement and velocity

x = zeros(1, N);

y = zeros(1, N);

% Set initial conditions

x(1) = x0;

y(1) = y0;

% Simulate the oscillator's response

for i = 2:N

% Calculate the acceleration and velocity changes using equations of motion

f = -(v/m) * y(i-1) - (a/m) * x(i-1) + (b/m) * x(i-1)^3;

g = y(i-1);

% Update velocity and displacement using Euler's method

y(i) = y(i-1) + dt * f;

x(i) = x(i-1) + dt * g;

end

% Plot the displacement and velocity for the current initial displacement value

plot(t, x, 'DisplayName', ['Displacement, x_0 = ', num2str(x0)]);

plot(t, y, '--', 'DisplayName', ['Velocity, x_0 = ', num2str(x0)]);

end

% Add labels, legend, and grid to the plot

xlabel('Time (s)');

ylabel('Amplitude');

%title('Damped Nonlinear Oscillator Response for Different Initial Displacements');

legend('show');

grid on;

hold off;

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Torsten 2024-5-7

在 MATLAB Online 中打开

Seems x0 = 10 is too large.

% Define common parameters

v = 1; % Damping coefficient

a = 1; % Linear stiffness coefficient

T = 10; % Total simulation time

dt = 0.1; % Time step

t = 0:dt:T; % Time vector

b = (1/20) * a; % Nonlinear stiffness coefficient

m = 1; % Mass (constant in this case)

x0_values = [0.1,1,10]; % Different initial displacements

y0 = 0; % Initial velocity

% Initialize a figure

figure(4);

hold on

% Loop through each initial displacement value

for x0 = x0_values

[t,z] = ode45(@(t,z)[z(2);-(v/m) * z(2) - (a/m) * z(1) + (b/m) * z(1)^3],t,[x0;y0]);

x = z(:,1);

y = z(:,2);

% Plot the displacement and velocity for the current initial displacement value

plot(t, x, 'DisplayName', ['Displacement, x_0 = ', num2str(x0)]);

plot(t, y, '--', 'DisplayName', ['Velocity, x_0 = ', num2str(x0)]);

end

Warning: Failure at t=9.360545e-01. Unable to meet integration tolerances without reducing the step size below the smallest value allowed (1.776357e-15) at time t.

% Add labels, legend, and grid to the plot

xlabel('Time (s)');

ylabel('Amplitude');

title('Damped Nonlinear Oscillator Response for Different Initial Displacements');

legend('show');

grid on;

hold off;

Athanasios Paraskevopoulos 2024-5-8

在 MATLAB Online 中打开

You are right for

the code works. Thank you!

% Define common parameters

v = 1; % Damping coefficient

a = 1; % Linear stiffness coefficient

T = 10; % Total simulation time

dt = 0.1; % Time step

t = 0:dt:T; % Time vector

b = (1/20) * a; % Nonlinear stiffness coefficient

m = 1; % Mass (constant in this case)

x0_values = [0.1,0.5,1]; % Different initial displacements

y0 = 0; % Initial velocity

% Initialize a figure

figure(4);

hold on

% Loop through each initial displacement value

for x0 = x0_values

[t,z] = ode45(@(t,z)[z(2);-(v/m) * z(2) - (a/m) * z(1) + (b/m) * z(1)^3],t,[x0;y0]);

x = z(:,1);

y = z(:,2);

% Plot the displacement and velocity for the current initial displacement value

plot(t, x, 'DisplayName', ['Displacement, x_0 = ', num2str(x0)]);

plot(t, y, '--', 'DisplayName', ['Velocity, x_0 = ', num2str(x0)]);

end

% Add labels, legend, and grid to the plot

xlabel('Time (s)');

ylabel('Amplitude');

title('Damped Nonlinear Oscillator Response for Different Initial Displacements');

legend('show');

grid on;

hold off;

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Answer 1

Sam Chak 2024-5-8

1
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2116317-plotting-damped-nonlinear-oscillator#answer_1454112

在 MATLAB Online 中打开

Hi @Athanasios Paraskevopoulos

You could have solved the system's equations to determine the critical points

of the Nonlinear Oscillator and then selected appropriate initial values. Previously, the selected initial displacement

was outside the stability region, which caused the trajectory to diverge. Here is a demo for reference. I have also corrected the information displayed in the legends.

%% Determine the critical points

syms x y

m = 1; % mass

v = 1; % damping coefficient

a = 1; % linear stiffness

b = a/20; % nonlinear stiffness

eq1 = y == 0;

eq2 = (- v*y - a*x + b*x^3)/m == 0;

eqn = [eq1, eq2];

sol = solve(eqn);

xcrit = double(sol.x)

xcrit = 3x1

0 -4.4721 4.4721

%% Damped Nonlinear Oscillator

% Define common parameters

T = 10; % Total simulation time

dt = 0.1; % Time step

t = 0:dt:T; % Time vector

x0 = [1, xcrit(3), 4.4723]; % Different initial displacements

y0 = 0; % Initial velocity

% Initialize a figure

figure(4);

hold on

% Loop through each initial displacement value

for j = 1:length(x0)

[t, z] = ode45(@(t, z) [z(2); ...

-(v/m)*z(2) - (a/m)*z(1) + (b/m)*z(1)^3], t, [x0(j); y0]);

x = z(:,1);

y = z(:,2);

% Plot the displacement and velocity for the current initial displacement value

plot(t, x, 'DisplayName', ['init disp, x_0 = ', num2str(x0(j))]);

plot(t, y, '--', 'DisplayName', ['init vel, y_0 = ', num2str(y0)]);

end

% Add labels, legend, and grid to the plot

xlabel('Time (s)');

ylabel('Amplitude');

title('Responses of Nonlinear Oscillator for different initial values');

legend('show', 'location', 'best');

grid on;

hold off;

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Plotting Damped Nonlinear Oscillator

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Plotting Damped Nonlinear Oscillator

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