Vibration of Euler-Bernoulli Beam with Moving Load

Question

Chris Thomas 2019-5-3

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https://ww2.mathworks.cn/matlabcentral/answers/460045-vibration-of-euler-bernoulli-beam-with-moving-load

评论： Youssef Elganouny 2021-4-22

Dear fellow MATLABers, I am wanting to produce a graph of u(x,t) [Displacement, y] against the Length [Length of beam, x] for each value of t = 0:100. From the graph I am plotting I would expect to see the max displacement of the beam to shift with the movement of the load going across the beam, instead the mid displacement just grows with each loop.

The equations I am using come from the book 'Vehicle-Bridge Interaction Dynamics : With Applications to High-Speed Railways' in Chapter 2. For convenience I will include the equations uderneath:

(Summing for n=1 to n=5)

(Mode shape for simply supported beam)

(Time response for moving loads)

If anyone wants to know more about the terms I am using feel free to ask and I can show them. The code I am using is shown bellow, any help is much appreciated!

clc
clear
% Vehicle properties
P = 1000; % Gravitational force from moving load [N]
v = 1.5; % Speed [m/s] or 100 [km/hour]
% Bridge properties
L = 150; % Length of the beam [m]
m = 3000; % Mass per unit length [kg/m]
EI = 10^6; % Flexural rigidity [N-m^2]
% Arrays for storage
Xl_n_vec = zeros(1,151);
W_x_t_mat = zeros(5,151);
z = 0:150;
for t = (0:100)/50
    
   for n = 1:5
    
       W_n = (((n^2)*pi^2)/L)*(EI/m)^0.5; % Natural frquecies for n modeshape [rad/sec]
   
       Omega_n = n+pi*v/L; % Exciting frequency from moving load [1/sec]
   
       S_n = Omega_n/W_n; % Speed parameter [NA]
   
        for x = 0:L
       
            Xl_n = 1/n^4*sin(n*pi*x/L); % Mode shapes for n modes
       
            Xl_n_vec(x+1) = Xl_n; % Stores values as vectors
       
        end
         
       T_n = (sin(Omega_n*t)-S_n*sin(W_n*t))/(1-S_n^2); % Time response of beam
              
       W_x_t = T_n*Xl_n_vec; % Total deformation of beam
       
       W_x_t_mat(n,:) = W_x_t; % Stores mode shape vectors in matrix
       
   end
       Sum_W_x_t = sum(W_x_t_mat);
       
       W_X_T = ((2*P*L^3)/(EI*pi^4))*Sum_W_x_t;
         
       figure(1)
       plot(z,W_X_T) % Displacemetn against length of beam
       title('Time')
       xlabel('Position [m]')
       ylabel('Displacement [m]')
       pause(0.05)
    
end

2 个评论
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YIYA 2019-9-27

Hello!Very lucky to read your share!I am currently studying the damage identification of cracked beams under the action of moving loads. I would like to ask you how to add structural damage information in your code, such as element stiffness reduction!Thank you very much

J. Ismail 2020-12-16