Dynamics of Machines - Part II - IFS.pdf
Dynamics of Machines - Part II - IFS.pdf Dynamics of Machines - Part II - IFS.pdf
The analytical solution can be divided into three steps: (I) homogeneous solution (transient analysis); (II) permanent solution (steady-state analysis) and (III) general solution (homogeneous + permanent), as mentioned in section 1.7.3. Introducing the initial conditions of displacement and velocity zini = { v1ini one gets v2ini v3ini y1ini y2ini y3ini }T z(t) = C1u1e λ1t + C2u2e λ2t + C3u3e λ3t + C4u4e λ4t + C5u5e λ5t + C6u6e λ6t + Ae iωt ⎧ ⎪⎨ ⎪⎩ C1 C2 C3 C4 C5 C6 where λ1 = −ξ1ωn1 − ωn1 1 − ξ2 1 · i and u1 λ2 = −ξ1ωn1 + ωn1 1 − ξ2 1 · i and u2 λ3 = −ξ2ωn2 − ωn2 1 − ξ2 2 · i and u3 λ4 = −ξ2ωn2 + ωn2 1 − ξ2 2 · i and u4 λ5 = −ξ3ωn3 − ωn3 1 − ξ2 3 · i and u5 λ6 = −ξ3ωn3 + ωn3 1 − ξ2 3 · i and u6 ⎫ A = [jωA + B] −1 f ⎪⎬ = [ u1 u2 u3 u4 u5 u6 ] −1 { zini − A} ⎪⎭ 1.8.3 Programming in Matlab – Theoretical Parameter Studies and Experimental Validation 52 (83) (82)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % MACHINERY DYNAMICS LECTURES (72213) % % IKS - DEPARTMENT OF CONTROL ENGINEERING DESIGN % % DTU - TECHNICAL UNIVERSITY OF DENMARK % % % % Copenhagen, February 11th, 2000 % % IFS % % % % 3 D.O.F. SYSTEMS - 4 DIFFERENT EXPERIMENTAL CASES % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Concentred Masses Values m1= 0.191; %[Kg] m2= 0.191; %[Kg] m3= 0.191; %[Kg] m4= 0.191; %[Kg] m5= 0.191; %[Kg] m6= 0.191; %[Kg] %Elastic Properties of the Beam of 600 [mm] E= 2.07e11; %elasticity modulus [N/m^2] b= 0.030 ; %width [m] h= 0.0012 ; %thickness [m] Iz= (b*h^3)/12; %area moment of inertia [m^4] % (1.CASE) Data for the mass-spring system %__________________________________________________ M1=m1; %concentrated mass [Kg] | M2=m2; %concentrated mass [Kg] | M3=m3; %concentrated mass [Kg] | L1= 0.203; %length for positioning M1 [m] | L2= 0.406; %length for positioning M2 [m] | L3= 0.610; %length for positioning M3 [m] | %__________________________________________________| % Coefficients of the Stiffness Matrix [N/m] K11= (3*E*Iz*L2^3*(L2 - 4*L3))/(L1^3*(L1 - L2)^2*( ... 2*L1*L2 + L2^2 + L1*L3 - 4*L2*L3)); K12= (-3*E*Iz*(-3*L2*(L2 - 2*L3)*L3 + L1*(L2^2 - ... 2*L2*L3 - 2*L3^2)))/(L1*(L1 - L2)^2*(L2 - ... L3)*(2*L1*L2 + L2^2 + L1*L3 - 4*L2*L3)); K13= (-9*E*Iz*L2^2)/(L1*(L1 - L2)*(L2 - L3)*(... 2*L1*L2 + L2^2 + L1*L3 - 4*L2*L3)); K21= (-3*E*Iz*(-3*L2*(L2 - 2*L3)*L3 + L1*(L2^2 - ... 2*L2*L3 - 2*L3^2)))/(L1*(L1 - L2)^2*(L2 - ... L3)*(2*L1*L2 + L2^2 + L1*L3 - 4*L2*L3)); K22= (3*E*Iz*(L1 - 4*L3)*(L1 - L3)^2)/((L1 - ... L2)^2*(L2 - L3)^2*(2*L1*L2 + L2^2 + ... L1*L3 - 4*L2*L3)); K23= (-3*E*Iz*(L1^2 - 2*L1*L2 - 2*L2^2 - 3*L1*L3 + ... 6*L2*L3))/((L1 - L2)*(L2 - L3)^2*(2*L1*L2 + ... L2^2 + L1*L3 - 4*L2*L3)); K31= (-9*E*Iz*L2^2)/(L1*(L1 - L2)*(L2 - L3)*(2*L1*L2 ... + L2^2 + L1*L3 - 4*L2*L3)); K32= (-3*E*Iz*(L1^2 - 2*L1*L2 - 2*L2^2 - 3*L1*L3 + ... 6*L2*L3))/((L1 - L2)*(L2 - L3)^2*(2*L1*L2 + ... L2^2 + L1*L3 - 4*L2*L3)); K33= (3*E*Iz*(L1 - 4*L2))/((L2 - L3)^2*(2*L1*L2 + ... L2^2 + L1*L3 - 4*L2*L3)); %Mass Matrix M= [M1 0 0; 0 M2 0; 0 0 M3]; %Stiffness Matrix K= [K11 K12 K13; K21 K22 K23; K31 K32 K33]; %Damping Matrix D= [0 0 0; 0 0 0; 0 0 0]; %State Matrices A= [ M D ; zeros(size(M)) M ] ; B= [ zeros(size(M)) K ; -M zeros(size(M))]; %Dynamical Properties of the Mass-Spring System [u,w]=eig(-B,A); %natural frequency [rad/s] %Dynamical Properties of the Mass-Spring System w=sort(diag(abs(w)))/2/pi %natural frequency [rad/s] w1=w(1); %first natural frequency [Hz] w2=w(3); %second natural frequency [Hz] w3=w(5); %third natural frequency [Hz] 53 wexp1=1.031 %measured natural frequency [Hz] %IMPORTANT: Freq resolution 400 lines wexp2=7.000 %measured natural frequency [Hz] %IMPORTANT: Freq resolution 400 lines wexp3=19.312 %measured natural frequency [Hz] %IMPORTANT: Freq resolution 400 lines dif1=(w1-wexp1)/wexp1 %error between calculated and measured freq. dif2=(w2-wexp2)/wexp2 %error between calculated and measured freq. dif3=(w3-wexp3)/wexp3 %error between calculated and measured freq. pause; % (2.CASE) Increasing the Mass Values % Data for the mass-spring system %__________________________________________________ M1=m1+m4; %concentrated mass [Kg] | M2=m2+m5; %concentrated mass [Kg] | M3=m3+m6; %concentrated mass [Kg] | L1= 0.203; %length for positioning M1 [m] | L2= 0.406; %length for positioning M2 [m] | L3= 0.610; %length for positioning M3 [m] | %__________________________________________________| % Coefficients of the Stiffness Matrix [N/m] K11= (3*E*Iz*L2^3*(L2 - 4*L3))/(L1^3*(L1 - L2)^2*( ... 2*L1*L2 + L2^2 + L1*L3 - 4*L2*L3)); K12= (-3*E*Iz*(-3*L2*(L2 - 2*L3)*L3 + L1*(L2^2 - ... 2*L2*L3 - 2*L3^2)))/(L1*(L1 - L2)^2*(L2 - ... L3)*(2*L1*L2 + L2^2 + L1*L3 - 4*L2*L3)); K13= (-9*E*Iz*L2^2)/(L1*(L1 - L2)*(L2 - L3)*(... 2*L1*L2 + L2^2 + L1*L3 - 4*L2*L3)); K21= (-3*E*Iz*(-3*L2*(L2 - 2*L3)*L3 + L1*(L2^2 - ... 2*L2*L3 - 2*L3^2)))/(L1*(L1 - L2)^2*(L2 - ... L3)*(2*L1*L2 + L2^2 + L1*L3 - 4*L2*L3)); K22= (3*E*Iz*(L1 - 4*L3)*(L1 - L3)^2)/((L1 - ... L2)^2*(L2 - L3)^2*(2*L1*L2 + L2^2 + ... L1*L3 - 4*L2*L3)); K23= (-3*E*Iz*(L1^2 - 2*L1*L2 - 2*L2^2 - 3*L1*L3 + ... 6*L2*L3))/((L1 - L2)*(L2 - L3)^2*(2*L1*L2 + ... L2^2 + L1*L3 - 4*L2*L3)); K31= (-9*E*Iz*L2^2)/(L1*(L1 - L2)*(L2 - L3)*(2*L1*L2 ... + L2^2 + L1*L3 - 4*L2*L3)); K32= (-3*E*Iz*(L1^2 - 2*L1*L2 - 2*L2^2 - 3*L1*L3 + ... 6*L2*L3))/((L1 - L2)*(L2 - L3)^2*(2*L1*L2 + ... L2^2 + L1*L3 - 4*L2*L3)); K33= (3*E*Iz*(L1 - 4*L2))/((L2 - L3)^2*(2*L1*L2 + ... L2^2 + L1*L3 - 4*L2*L3)); %Mass Matrix M= [M1 0 0; 0 M2 0; 0 0 M3]; %Stiffness Matrix K= [K11 K12 K13; K21 K22 K23; K31 K32 K33]; %Damping Matrix D= [0 0 0; 0 0 0; 0 0 0]; %State Matrices A= [ M D ; zeros(size(M)) M ] ; B= [ zeros(size(M)) K ; -M zeros(size(M))]; %Dynamical Properties of the Mass-Spring System [u,w]=eig(-B,A); %natural frequency [rad/s] %Dynamical Properties of the Mass-Spring System w=sort(diag(abs(w)))/2/pi %natural frequency [rad/s] w1=w(1); %first natural frequency [Hz] w2=w(3); %second natural frequency [Hz] w3=w(5); %third natural frequency [Hz] wexp1=0.71875 %measured natural frequency [Hz] %IMPORTANT: Freq resolution 400 lines wexp2=5.125 %measured natural frequency [Hz] %IMPORTANT: Freq resolution 400 lines wexp3=14.312 %measured natural frequency [Hz] %IMPORTANT: Freq resolution 400 lines dif1=(w1-wexp1)/wexp1 %error between calculated and measured freq. dif2=(w2-wexp2)/wexp2 %error between calculated and measured freq. dif3=(w3-wexp3)/wexp3 %error between calculated and measured freq. pause;
- Page 1 and 2: DYNAMICS OF MACHINES 41614 PART I -
- Page 3 and 4: 1 Introduction to Dynamical Modelli
- Page 5 and 6: 1.3 Data of the Mechanical System
- Page 7 and 8: 1.5 Calculating Stiffness Matrices
- Page 9 and 10: 1.6 Mechanical Systems with 1 D.O.F
- Page 11 and 12: Demanding (λ 2 + 2ξωnλ + ω 2 n
- Page 13 and 14: 1 yini − A det λ1 vini − A C2
- Page 15 and 16: 1.6.4 Analytical and Numerical Solu
- Page 17 and 18: (a) y(t) [m] (b) y(t) [m] (c) y(t)
- Page 19 and 20: 1.6.6 Homogeneous Solution or Free-
- Page 21 and 22: (a) Amplitude [m/s 2 ] x Signal 10
- Page 23 and 24: (a) Amplitude [m/s 2 ] (b) Amplitud
- Page 25 and 26: Imag(A(ω)) [m/N] 0 −1 −2 −3
- Page 27 and 28: 1.6.11 Superposition of Transient a
- Page 29 and 30: (a) Amplitude [m/s 2 ] (b) Amplitud
- Page 31 and 32: 1.7 Mechanical Systems with 2 D.O.F
- Page 33 and 34: ⎧ ⎫ ⎪⎨ ˙y1(t) ⎪⎬ ˙y
- Page 35 and 36: zini = U c + A ⇒ c = U −1 {(zin
- Page 37 and 38: 1.7.4 Modal Analysis using Matlab e
- Page 39 and 40: 0.7 0.6 0.5 0.4 0.3 0.2 0.1 First M
- Page 41 and 42: %__________________________________
- Page 43 and 44: ||y 1 (ω)|| [m/N] Phase [ o] Excit
- Page 45 and 46: ||y i (ω)|| [m/N] Phase [ o] 0.8 0
- Page 47 and 48: Imag(y i (ω)/f 1 (ω)) (i=1,2) [m/
- Page 49 and 50: 1.8 Mechanical Systems with 3 D.O.F
- Page 51: which could be verified using Modal
- Page 55 and 56: 1.8.4 Theoretical Frequency Respons
- Page 57 and 58: (a) Amplitude [m/s 2 ] (b) Amplitud
- Page 59 and 60: 4. Vary the number of masses attach
- Page 61 and 62: 1.10 Project 0 - Identification of
- Page 63 and 64: (a) (b) REAL(Acc/force) [(m/s 2 )/N
- Page 65 and 66: 6. Model application - As explained
- Page 67 and 68: changeable unbalanced mass for simu
- Page 69 and 70: %Modal Matrix u with mode shapes %M
- Page 71 and 72: acc [m/s 2 ] acc [m/s 2 ] 0.8 0.6 0
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%<br />
% MACHINERY DYNAMICS LECTURES (72213) %<br />
% IKS - DEPARTMENT OF CONTROL ENGINEERING DESIGN %<br />
% DTU - TECHNICAL UNIVERSITY OF DENMARK %<br />
% %<br />
% Copenhagen, February 11th, 2000 %<br />
% <strong>IFS</strong> %<br />
% %<br />
% 3 D.O.F. SYSTEMS - 4 DIFFERENT EXPERIMENTAL CASES %<br />
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%<br />
%Concentred Masses Values<br />
m1= 0.191; %[Kg]<br />
m2= 0.191; %[Kg]<br />
m3= 0.191; %[Kg]<br />
m4= 0.191; %[Kg]<br />
m5= 0.191; %[Kg]<br />
m6= 0.191; %[Kg]<br />
%Elastic Properties <strong>of</strong> the Beam <strong>of</strong> 600 [mm]<br />
E= 2.07e11; %elasticity modulus [N/m^2]<br />
b= 0.030 ; %width [m]<br />
h= 0.0012 ; %thickness [m]<br />
Iz= (b*h^3)/12; %area moment <strong>of</strong> inertia [m^4]<br />
% (1.CASE) Data for the mass-spring system<br />
%__________________________________________________<br />
M1=m1; %concentrated mass [Kg] |<br />
M2=m2; %concentrated mass [Kg] |<br />
M3=m3; %concentrated mass [Kg] |<br />
L1= 0.203; %length for positioning M1 [m] |<br />
L2= 0.406; %length for positioning M2 [m] |<br />
L3= 0.610; %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 />
53<br />
wexp1=1.031 %measured natural frequency [Hz]<br />
%IMPORTANT: Freq resolution 400 lines<br />
wexp2=7.000 %measured natural frequency [Hz]<br />
%IMPORTANT: Freq resolution 400 lines<br />
wexp3=19.312 %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 />
% (2.CASE) Increasing the Mass Values<br />
% Data for the mass-spring system<br />
%__________________________________________________<br />
M1=m1+m4; %concentrated mass [Kg] |<br />
M2=m2+m5; %concentrated mass [Kg] |<br />
M3=m3+m6; %concentrated mass [Kg] |<br />
L1= 0.203; %length for positioning M1 [m] |<br />
L2= 0.406; %length for positioning M2 [m] |<br />
L3= 0.610; %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=0.71875 %measured natural frequency [Hz]<br />
%IMPORTANT: Freq resolution 400 lines<br />
wexp2=5.125 %measured natural frequency [Hz]<br />
%IMPORTANT: Freq resolution 400 lines<br />
wexp3=14.312 %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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