Plotting Damped Nonlinear Oscillator
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;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; 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; plot, plotting, mathematics, differential equations MATLAB Answers — New Questions
