弧长法求非线性振动幅频响应matlab实例

以下是一个使用弧长法求解非线性振动幅频响应的MATLAB示例代码：

function [w, A] = arc_length_method()

% Define system parameters
m = 1;
k = 1;
beta = 0.1;
w0 = sqrt(k/m);

% Define initial conditions
x0 = 0;
v0 = 1;

% Define the function handle for the ODE system
fun = @(t, y) [y(2); -k/m*y(1) - beta/m*y(2)^3];

% Define the Jacobian of the ODE system
jac = @(t, y) [0, 1; -k/m - 3*beta/m*y(2)^2, -beta/m*9*y(2)^2];

% Define the arc length continuation parameters
ds = 0.01; % Step size
s = 0; % Initial arc length parameter
s_max = 10; % Maximum arc length parameter
tol = 1e-6; % Tolerance for convergence
max_iter = 100; % Maximum number of iterations

% Initialize the solution vectors
w = [];
A = [];

% Perform the arc length continuation method
while s < s_max
    
    % Define the initial guess for the next solution
    if isempty(w)
        % First solution is the linear system
        y_guess = [x0; v0];
        w_guess = w0;
    else
        % Use previous solution as guess for next solution
        y_guess = y_next;
        w_guess = w_next;
    end
    
    % Perform the Newton-Raphson iteration
    for iter = 1:max_iter
        
        % Solve the ODE system using the current guess
        [t, y] = ode45(fun, [0, 2*pi/w_guess], y_guess);
        
        % Calculate the residual and Jacobian
        r = [y(end, 1) - cos(w_guess*t(end)); s - s_prev - ds];
        J = [y(end, 2)*(-sin(w_guess*t(end))), w_guess*sin(w_guess*t(end)); ...
             jac(t(end), y(end, :)), zeros(2, 1)];
        
        % Solve the linear system to update the guess
        delta = -J\r;
        y_next = y(end, :)' + delta(1:2);
        w_next = w_guess + delta(3);
        
        % Check for convergence
        if norm(delta) < tol
            break;
        end
        
        % Update the guess for the next iteration
        y_guess = y_next;
        w_guess = w_next;
        
    end
    
    % Add the solution to the output vectors
    w = [w; w_next];
    A = [A; y_next(1)];
    
    % Update the continuation parameters
    s_prev = s;
    s = s_prev + ds;
    
end

% Plot the amplitude-frequency response
plot(w, A);
xlabel('Frequency (rad/s)');
ylabel('Amplitude');
title('Amplitude-Frequency Response using Arc Length Method');

end

在这个示例中，我们定义了一个质量为1、弹性系数为1、阻尼系数为0.1的非线性振动系统，并使用弧长法求解其振幅-频率响应。我们使用一个简单的二阶ODE解算器（ode45）来解决ODE系统，使用牛顿-拉夫逊迭代来更新每个解的初始猜测，直到收敛到所需的精度。最后，我们将每个解的振幅与其频率绘制在图表中。

请注意，这只是一个示例，您需要根据您的具体问题进行修改和调整。