Dynamics of Machines - Part II - IFS.pdf
Dynamics of Machines - Part II - IFS.pdf Dynamics of Machines - Part II - IFS.pdf
% (3.CASE) Changing the Position of the Concentrated Masses % Data for the mass-spring system %__________________________________________________ M1=m1+m2; %concentrated mass [Kg] | M2=m3+m4; %concentrated mass [Kg] | M3=m5+m6; %concentrated mass [Kg] | L1= 0.150; %length for positioning M1 [m] | L2= 0.300; %length for positioning M2 [m] | L3= 0.450; %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=1.094 %measured natural frequency [Hz] %IMPORTANT: Freq resolution 400 lines wexp2=7.188 %measured natural frequency [Hz] %IMPORTANT: Freq resolution 400 lines wexp3=20.25 %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; 54 % (4.CASE) Changing the Position and the Values of the Concentrated Masses % Data for the mass-spring system %__________________________________________________ M1=m1+m4+m5; %concentrated mass [Kg] | M2=m2+m6; %concentrated mass [Kg] | M3=m3; %concentrated mass [Kg] | L1= 0.150; %length for positioning M1 [m] | L2= 0.300; %length for positioning M2 [m] | L3= 0.450; %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] exp1=1.312 %measured natural frequency [Hz] %IMPORTANT: Freq resolution 400 lines wexp2=7.219 %measured natural frequency [Hz] %IMPORTANT: Freq resolution 400 lines wexp3=18.000 %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;
1.8.4 Theoretical Frequency Response Function (a) y11 [m/N] (d) y21 [m/N] (c) y31 [m/N] 0.1 0.05 Excitation on Point 1 0 0 10 20 30 Frequency [Hz] 0.4 0.3 0.2 0.1 0 0 10 20 30 Frequency [Hz] 0.8 0.6 0.4 0.2 0 0 10 20 30 Frequency [Hz] (b) y12 [m/N] (a) y22 [m/N] (d) y32 [m/N] 0.4 0.3 0.2 0.1 Excitation on Point 2 0 0 10 20 30 Frequency [Hz] 0.8 0.6 0.4 0.2 0 0 10 20 30 Frequency [Hz] 2 1.5 1 0.5 0 0 10 20 30 Frequency [Hz] (c) y13 [m/N] (b) y23 [m/N] (d) y33 [m/N] 0.8 0.6 0.4 0.2 Excitation on Point 3 0 0 10 20 30 Frequency [Hz] 2 1.5 1 0.5 0 0 10 20 30 Frequency [Hz] 3 2 1 0 0 10 20 30 Frequency [Hz] Figure 33: Forced Vibration – Theoretical Frequency Response Function (FRF) of the clampedfree flexible beam when two concentrated masses m33 = m1 + m2 = 0.382 Kg are attached at its free end (L = 0.610 m), two additional masses m22 = m3 + m4 = 0.382 Kg are attached at L = 0.410 m and two additional masses m11 = m5 + m6 = 0.382 Kg are attached at L = 0.210 m – – Natural frequencies of the 3 D.O.F. mass-spring system ”A”: 1.03 Hz, 7.00 Hz and 19.31 Hz. 55
- 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 and 52: which could be verified using Modal
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- Page 57 and 58: (a) Amplitude [m/s 2 ] (b) Amplitud
- Page 59 and 60: 4. Vary the number of masses attach
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- 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
1.8.4 Theoretical Frequency Response Function<br />
(a) y11 [m/N]<br />
(d) y21 [m/N]<br />
(c) y31 [m/N]<br />
0.1<br />
0.05<br />
Excitation on Point 1<br />
0<br />
0 10 20 30<br />
Frequency [Hz]<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
0 10 20 30<br />
Frequency [Hz]<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
0<br />
0 10 20 30<br />
Frequency [Hz]<br />
(b) y12 [m/N]<br />
(a) y22 [m/N]<br />
(d) y32 [m/N]<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
Excitation on Point 2<br />
0<br />
0 10 20 30<br />
Frequency [Hz]<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
0<br />
0 10 20 30<br />
Frequency [Hz]<br />
2<br />
1.5<br />
1<br />
0.5<br />
0<br />
0 10 20 30<br />
Frequency [Hz]<br />
(c) y13 [m/N]<br />
(b) y23 [m/N]<br />
(d) y33 [m/N]<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
Excitation on Point 3<br />
0<br />
0 10 20 30<br />
Frequency [Hz]<br />
2<br />
1.5<br />
1<br />
0.5<br />
0<br />
0 10 20 30<br />
Frequency [Hz]<br />
3<br />
2<br />
1<br />
0<br />
0 10 20 30<br />
Frequency [Hz]<br />
Figure 33: Forced Vibration – Theoretical Frequency Response Function (FRF) <strong>of</strong> the clampedfree<br />
flexible beam when two concentrated masses m33 = m1 + m2 = 0.382 Kg are attached at<br />
its free end (L = 0.610 m), two additional masses m22 = m3 + m4 = 0.382 Kg are attached<br />
at L = 0.410 m and two additional masses m11 = m5 + m6 = 0.382 Kg are attached at<br />
L = 0.210 m – – Natural frequencies <strong>of</strong> the 3 D.O.F. mass-spring system ”A”: 1.03 Hz,<br />
7.00 Hz and 19.31 Hz.<br />
55
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