Dynamics of Machines - Part II - IFS.pdf
Dynamics of Machines - Part II - IFS.pdf
Dynamics of Machines - Part II - IFS.pdf
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% (3.CASE) Changing the Position <strong>of</strong> the Concentrated Masses<br />
% Data for the mass-spring system<br />
%__________________________________________________<br />
M1=m1+m2; %concentrated mass [Kg] |<br />
M2=m3+m4; %concentrated mass [Kg] |<br />
M3=m5+m6; %concentrated mass [Kg] |<br />
L1= 0.150; %length for positioning M1 [m] |<br />
L2= 0.300; %length for positioning M2 [m] |<br />
L3= 0.450; %length for positioning M3 [m] |<br />
%__________________________________________________|<br />
% Coefficients <strong>of</strong> the Stiffness Matrix [N/m]<br />
K11= (3*E*Iz*L2^3*(L2 - 4*L3))/(L1^3*(L1 - L2)^2*( ...<br />
2*L1*L2 + L2^2 + L1*L3 - 4*L2*L3));<br />
K12= (-3*E*Iz*(-3*L2*(L2 - 2*L3)*L3 + L1*(L2^2 - ...<br />
2*L2*L3 - 2*L3^2)))/(L1*(L1 - L2)^2*(L2 - ...<br />
L3)*(2*L1*L2 + L2^2 + L1*L3 - 4*L2*L3));<br />
K13= (-9*E*Iz*L2^2)/(L1*(L1 - L2)*(L2 - L3)*(...<br />
2*L1*L2 + L2^2 + L1*L3 - 4*L2*L3));<br />
K21= (-3*E*Iz*(-3*L2*(L2 - 2*L3)*L3 + L1*(L2^2 - ...<br />
2*L2*L3 - 2*L3^2)))/(L1*(L1 - L2)^2*(L2 - ...<br />
L3)*(2*L1*L2 + L2^2 + L1*L3 - 4*L2*L3));<br />
K22= (3*E*Iz*(L1 - 4*L3)*(L1 - L3)^2)/((L1 - ...<br />
L2)^2*(L2 - L3)^2*(2*L1*L2 + L2^2 + ...<br />
L1*L3 - 4*L2*L3));<br />
K23= (-3*E*Iz*(L1^2 - 2*L1*L2 - 2*L2^2 - 3*L1*L3 + ...<br />
6*L2*L3))/((L1 - L2)*(L2 - L3)^2*(2*L1*L2 + ...<br />
L2^2 + L1*L3 - 4*L2*L3));<br />
K31= (-9*E*Iz*L2^2)/(L1*(L1 - L2)*(L2 - L3)*(2*L1*L2 ...<br />
+ L2^2 + L1*L3 - 4*L2*L3));<br />
K32= (-3*E*Iz*(L1^2 - 2*L1*L2 - 2*L2^2 - 3*L1*L3 + ...<br />
6*L2*L3))/((L1 - L2)*(L2 - L3)^2*(2*L1*L2 + ...<br />
L2^2 + L1*L3 - 4*L2*L3));<br />
K33= (3*E*Iz*(L1 - 4*L2))/((L2 - L3)^2*(2*L1*L2 + ...<br />
L2^2 + L1*L3 - 4*L2*L3));<br />
%Mass Matrix<br />
M= [M1 0 0; 0 M2 0; 0 0 M3];<br />
%Stiffness Matrix<br />
K= [K11 K12 K13; K21 K22 K23; K31 K32 K33];<br />
%Damping Matrix<br />
D= [0 0 0; 0 0 0; 0 0 0];<br />
%State Matrices<br />
A= [ M D ;<br />
zeros(size(M)) M ] ;<br />
B= [ zeros(size(M)) K ;<br />
-M zeros(size(M))];<br />
%Dynamical Properties <strong>of</strong> the Mass-Spring System<br />
[u,w]=eig(-B,A); %natural frequency [rad/s]<br />
%Dynamical Properties <strong>of</strong> the Mass-Spring System<br />
w=sort(diag(abs(w)))/2/pi %natural frequency [rad/s]<br />
w1=w(1); %first natural frequency [Hz]<br />
w2=w(3); %second natural frequency [Hz]<br />
w3=w(5); %third natural frequency [Hz]<br />
wexp1=1.094 %measured natural frequency [Hz]<br />
%IMPORTANT: Freq resolution 400 lines<br />
wexp2=7.188 %measured natural frequency [Hz]<br />
%IMPORTANT: Freq resolution 400 lines<br />
wexp3=20.25 %measured natural frequency [Hz]<br />
%IMPORTANT: Freq resolution 400 lines<br />
dif1=(w1-wexp1)/wexp1 %error between calculated and measured freq.<br />
dif2=(w2-wexp2)/wexp2 %error between calculated and measured freq.<br />
dif3=(w3-wexp3)/wexp3 %error between calculated and measured freq.<br />
pause;<br />
54<br />
% (4.CASE) Changing the Position and the Values <strong>of</strong> the Concentrated Masses<br />
% Data for the mass-spring system<br />
%__________________________________________________<br />
M1=m1+m4+m5; %concentrated mass [Kg] |<br />
M2=m2+m6; %concentrated mass [Kg] |<br />
M3=m3; %concentrated mass [Kg] |<br />
L1= 0.150; %length for positioning M1 [m] |<br />
L2= 0.300; %length for positioning M2 [m] |<br />
L3= 0.450; %length for positioning M3 [m] |<br />
%__________________________________________________|<br />
% Coefficients <strong>of</strong> the Stiffness Matrix [N/m]<br />
K11= (3*E*Iz*L2^3*(L2 - 4*L3))/(L1^3*(L1 - L2)^2*( ...<br />
2*L1*L2 + L2^2 + L1*L3 - 4*L2*L3));<br />
K12= (-3*E*Iz*(-3*L2*(L2 - 2*L3)*L3 + L1*(L2^2 - ...<br />
2*L2*L3 - 2*L3^2)))/(L1*(L1 - L2)^2*(L2 - ...<br />
L3)*(2*L1*L2 + L2^2 + L1*L3 - 4*L2*L3));<br />
K13= (-9*E*Iz*L2^2)/(L1*(L1 - L2)*(L2 - L3)*(...<br />
2*L1*L2 + L2^2 + L1*L3 - 4*L2*L3));<br />
K21= (-3*E*Iz*(-3*L2*(L2 - 2*L3)*L3 + L1*(L2^2 - ...<br />
2*L2*L3 - 2*L3^2)))/(L1*(L1 - L2)^2*(L2 - ...<br />
L3)*(2*L1*L2 + L2^2 + L1*L3 - 4*L2*L3));<br />
K22= (3*E*Iz*(L1 - 4*L3)*(L1 - L3)^2)/((L1 - ...<br />
L2)^2*(L2 - L3)^2*(2*L1*L2 + L2^2 + ...<br />
L1*L3 - 4*L2*L3));<br />
K23= (-3*E*Iz*(L1^2 - 2*L1*L2 - 2*L2^2 - 3*L1*L3 + ...<br />
6*L2*L3))/((L1 - L2)*(L2 - L3)^2*(2*L1*L2 + ...<br />
L2^2 + L1*L3 - 4*L2*L3));<br />
K31= (-9*E*Iz*L2^2)/(L1*(L1 - L2)*(L2 - L3)*(2*L1*L2 ...<br />
+ L2^2 + L1*L3 - 4*L2*L3));<br />
K32= (-3*E*Iz*(L1^2 - 2*L1*L2 - 2*L2^2 - 3*L1*L3 + ...<br />
6*L2*L3))/((L1 - L2)*(L2 - L3)^2*(2*L1*L2 + ...<br />
L2^2 + L1*L3 - 4*L2*L3));<br />
K33= (3*E*Iz*(L1 - 4*L2))/((L2 - L3)^2*(2*L1*L2 + ...<br />
L2^2 + L1*L3 - 4*L2*L3));<br />
%Mass Matrix<br />
M= [M1 0 0; 0 M2 0; 0 0 M3];<br />
%Stiffness Matrix<br />
K= [K11 K12 K13; K21 K22 K23; K31 K32 K33];<br />
%Damping Matrix<br />
D= [0 0 0; 0 0 0; 0 0 0];<br />
%State Matrices<br />
A= [ M D ;<br />
zeros(size(M)) M ] ;<br />
B= [ zeros(size(M)) K ;<br />
-M zeros(size(M))];<br />
%Dynamical Properties <strong>of</strong> the Mass-Spring System<br />
[u,w]=eig(-B,A); %natural frequency [rad/s]<br />
%Dynamical Properties <strong>of</strong> the Mass-Spring System<br />
w=sort(diag(abs(w)))/2/pi %natural frequency [rad/s]<br />
w1=w(1); %first natural frequency [Hz]<br />
w2=w(3); %second natural frequency [Hz]<br />
w3=w(5); %third natural frequency [Hz]<br />
exp1=1.312 %measured natural frequency [Hz]<br />
%IMPORTANT: Freq resolution 400 lines<br />
wexp2=7.219 %measured natural frequency [Hz]<br />
%IMPORTANT: Freq resolution 400 lines<br />
wexp3=18.000 %measured natural frequency [Hz]<br />
%IMPORTANT: Freq resolution 400 lines<br />
dif1=(w1-wexp1)/wexp1 %error between calculated and measured freq.<br />
dif2=(w2-wexp2)/wexp2 %error between calculated and measured freq.<br />
dif3=(w3-wexp3)/wexp3 %error between calculated and measured freq.<br />
pause;
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