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

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Keith Grey
Keith Grey 2020-6-12
回答： Swami 2024-9-17
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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Rafael Hernandez-Walls
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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回答（2 个）

Gifari Zulkarnaen
Gifari Zulkarnaen 2020-6-12
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
The result is still different with what your graph, but maybe this can give insight
Swami
Swami 2024-9-17
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

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