Vibration of Euler-Bernoulli Beam with Moving Load
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)
endDear 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 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 engineering, vibration, dynamics, bridge, moving loads, pde MATLAB Answers — New Questions
