How to solve mass spring damper over frequency (differential equation)?

Question

Keith Grey 2020-6-12

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https://ww2.mathworks.cn/matlabcentral/answers/546948-how-to-solve-mass-spring-damper-over-frequency-differential-equation

回答： Gifari Zulkarnaen 2020-6-12

This is a mass-spring-damper system equation with a function dependent on frequency (M, R, & K in parethesis).

f = 50:450; % Hz
M = 23.3;
K = 3.61 * 10^7;
R = 1360;
P = -1 * (2 * pi * f);
ic = 2 * pi * 50; % rad/s

The figure shows the linear impedance frequency response of the system defined by the equation.

How do you go from the differential equation to the figure?

I've seen differential equation solvers using time & displacement, but no luck on just frequency response.

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Keith Grey 2020-6-12

Initial condition is 50Hz.

Rafael Hernandez-Walls 2020-6-12

You need initial condition for the displacement and for the velocity. Then you need write the ODE in two equations of first orden. Lke this

and solve using ode45

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Answer 1

Gifari Zulkarnaen 2020-6-12

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https://ww2.mathworks.cn/matlabcentral/answers/546948-how-to-solve-mass-spring-damper-over-frequency-differential-equation#answer_449982

Equation of motion is time-dependant equation. Perhaps what you mean is Response Spectrum (frequency-based response data)? This is calculation for acceleration response spectrum:

clear
close all
f = 50:10:450; % Hz
M = 23.3;
K = 3.61*10^7;
R = 1360;
RS = zeros(1,length(f));
for i=1:length(f)
    tspan = [0 1]; % Frequency range
    Y0 = zeros(2,1); % Initial zero condition
    [t,Y] = ode45(@(t,Y) StateEqOfMotion(t,Y,M,K,R,f(i)),tspan,Y0); % ODE
    % Output Y is in term of displacement and velocity
    a = Y(:,2)./t; % Acceleration
    RS(i) = max(abs(a)); % Acceleration response spectrum
end
plot(f,RS)
% ODE function of State Equation
function dYdt = StateEqOfMotion(t,Y,M,K,R,f)
    P = -cos(t*f);
    x = Y(1);
    v = Y(2);
    a = (-(R*v+K*x)+P)/M;
    dYdt = [v; a];
end