How to convert SDOF system to MDOF system of Nonlinear Dynamic analysis of a building

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Chaudhary P Patel 2022-4-19

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评论： Chaudhary P Patel 2022-4-21

Please, anyone help me.

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Alan Stevens 2022-4-19

You need to provide more detail! Upload your code for the SDOF building. Specify exactly how you want to extend it - for example: to lateral as well as vertical movement? - for extra floors? - etc.

Chaudhary P Patel 2022-4-21

@Alan Stevens sir, please reply me.

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Sam Chak 2022-4-21

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@Chaudhary P Patel

I'm not a certified Building Dynamicist. But if the equations of motion of a 4-storey building can described like this:

, with

, such that

, where

function dxdt = f(t, x)

dxdt = zeros(8,1);

m1 = 4000; % kg

m2 = m1;

m3 = m2;

m4 = m3;

k1 = 5000; % N/m

k2 = k1;

k3 = k2;

k4 = k3;

% Mass matrix

M = diag([m1, m2, m3, m4]);

% Stiffness matrix

K = [k1+k2 -k2 0 0; -k2 k2+k3 -k3 0; 0 -k3 k3+k4 -k4; 0 0 -k4 k4];

% Viscous damping matrix

C = K/1e1;

% State Matrix

A = [zeros(length(M)) eye(length(M)); -M\K -M\C];

dxdt = A*x;

end

then you can solve the system using ode45 solver

tspan = [0 30];                     % time span of simulation
x0 = [0.01 0.1 0.2 0.25 0 0 0 0];   % initial displacement with zero velocity
[t, x] = ode45(@(t,x) f(t, x), tspan, x0);
plot(t, x(:, 1:4), 'linewidth', 1.5)
grid on
xlabel('Time, t [sec]')
ylabel({'$x_{1},\; x_{2},\; x_{3},\; x_{4}$'}, 'Interpreter', 'latex')
title('Displacements of the building storeys')
legend({'$x_{1}$', '$x_{2}$', '$x_{3}$', '$x_{4}$'}, 'Interpreter', 'latex', 'location', 'best')