Dynamics of Machines - Part II - IFS.pdf
Dynamics of Machines - Part II - IFS.pdf Dynamics of Machines - Part II - IFS.pdf
% Lumped Masses m1 = 1.95 % [kg] lowest mass m2 = 1.72 % [kg] above lowest mass m3 = 1.95 % [kg] below highest mass m4 = 3.04 % [kg] highest mass with motor % Beam Properties E=2.0e11 % [N/m^2] elasticity modulus b=0.0291 % [m] width h=0.0011 % [m] thickness I=(b*h^3)/12 % [m^4] area moment of inertia % Position of the Platforms L1=0.122 % [m] Length to Platform 1 L2=0.123 % [m] Length from Platform 1 to Platform 2 L3=0.149 % [m] Length from Platform 2 to Platform 3 L4=0.127 % [m] Length from Platform 3 to Platform 4 • Write the mathematical model (equations of motion) based on Newton’s laws, achieving the mass M, stiffness K and damping D matrices. • How is the structure of the mass matrix M? How to properly calculate the mass coefficients? • How is the structure of the stiffness matrix K? Find two different ways of obtaining the stiffness coefficients. • How is the structure of the damping matrix D? Find three different ways of obtaining the damping coefficients, using proportional damping, D1 = α · M. D2 = β · K. D3 = α · M + β · K. Remember that when you do not know how to exactly model and achieve the damping coefficients of the matrix D, assumptions have to be made. A realistic approximation of the structural damping factor ξ is 0.005, according to the experimental results obtained in equation (46). Try to adjust the proportionality factors α and β so that the damping factor ξ1 related to the first mode shape of the structure is 0.005. Feel free to do your own assumptions regarding damping, if you want! 2. NUMERICAL IMPLEMENTATION – Write a program in Matlab dof4-integration.m, or use the program of your preference. Use as reference the dof2-integration.m, if you want. Calculate the natural frequencies ωi (i = 1, ...,4) of the mechanical model and the damping ratios ξi (i = 1, ...,4) related to the 4 modes shapes, considering the three different damping matrices D1, D2 and D3. %State Matrices A and B A= [ M D ; zeros(size(M)) M ] ; B= [ zeros(size(M)) K ; -M zeros(size(M))]; 68
%Modal Matrix u with mode shapes %Matrix w with damping factors and damped natural frequencies [u,w]=eig(-B,A); 3. ANALYSIS – Neglecting the damping, write the modal matrix with help of your program dof4-integration.m. Based on the information contained in such a matrix describe the mode shapes of the structure with some drawings; 4. ANALYSIS – Without neglecting the damping, write the modal matrix with help of your program dof4-integration.m. Based on the information contained in such a matrix describe the mode shapes of the structure and try to explain the physical meaning of the complex numbers in the modal matrix. 5. EXPERIMENTAL – Try to predominately excite the first mode shape of the building using an appropriate initial condition of displacement. Capture the acceleration signal in time domain and plot it. Based on the logarithmic decrement try to establish the damping factor ξ1 associated to the first mode shape of the building. Please, download the file yyy4−trans.txt from campus net in order to rebuild figure 40. After obtaining the damping ratio ξ1, compare with your assumption of 0.005. acc [m/s 2 ] 3 2 1 0 −1 −2 −3 Acceleration of Mass 4 −4 0 1 2 3 4 time [s] 5 6 7 8 Figure 40: Experimental transient vibration response (acceleration) of mass 4 after initial condition of displacement, which excites only the first mode shape of the structure. 6. EXPERIMENTAL – Obtain 4 frequency response functions in the range of 0 − 40 Hz, when the building is excited on the first mass by magnetic forces. Please, download the files: frf−general.m, xxx1.txt, xxx2.txt, xxx3.txt, xxx4.txt, yyy1.txt, yyy2.txt, yyy3.txt, yyy4.txt (campus net) 7. EXPERIMENTAL – Experimental Modal Analysis (EMA) deals with the determination of natural frequencies, modes shapes, and damping ratios from experimental measurements. The fundamental idea behind modal testing is the resonance. If a structure is excited at resonance, its response exhibits two distinct phenomena: (a) as the excitation frequency 69
- Page 17 and 18: (a) y(t) [m] (b) y(t) [m] (c) y(t)
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- Page 25 and 26: Imag(A(ω)) [m/N] 0 −1 −2 −3
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- Page 33 and 34: ⎧ ⎫ ⎪⎨ ˙y1(t) ⎪⎬ ˙y
- Page 35 and 36: zini = U c + A ⇒ c = U −1 {(zin
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- Page 47 and 48: Imag(y i (ω)/f 1 (ω)) (i=1,2) [m/
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- Page 51 and 52: which could be verified using Modal
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%Modal Matrix u with mode shapes<br />
%Matrix w with damping factors and damped natural frequencies<br />
[u,w]=eig(-B,A);<br />
3. ANALYSIS – Neglecting the damping, write the modal matrix with help <strong>of</strong> your program<br />
d<strong>of</strong>4-integration.m. Based on the information contained in such a matrix describe the mode<br />
shapes <strong>of</strong> the structure with some drawings;<br />
4. ANALYSIS – Without neglecting the damping, write the modal matrix with help <strong>of</strong> your<br />
program d<strong>of</strong>4-integration.m. Based on the information contained in such a matrix describe<br />
the mode shapes <strong>of</strong> the structure and try to explain the physical meaning <strong>of</strong> the complex<br />
numbers in the modal matrix.<br />
5. EXPERIMENTAL – Try to predominately excite the first mode shape <strong>of</strong> the building using<br />
an appropriate initial condition <strong>of</strong> displacement. Capture the acceleration signal in time<br />
domain and plot it. Based on the logarithmic decrement try to establish the damping<br />
factor ξ1 associated to the first mode shape <strong>of</strong> the building. Please, download the file<br />
yyy4−trans.txt from campus net in order to rebuild figure 40. After obtaining the damping<br />
ratio ξ1, compare with your assumption <strong>of</strong> 0.005.<br />
acc [m/s 2 ]<br />
3<br />
2<br />
1<br />
0<br />
−1<br />
−2<br />
−3<br />
Acceleration <strong>of</strong> Mass 4<br />
−4<br />
0 1 2 3 4<br />
time [s]<br />
5 6 7 8<br />
Figure 40: Experimental transient vibration response (acceleration) <strong>of</strong> mass 4 after initial condition<br />
<strong>of</strong> displacement, which excites only the first mode shape <strong>of</strong> the structure.<br />
6. EXPERIMENTAL – Obtain 4 frequency response functions in the range <strong>of</strong> 0 − 40 Hz,<br />
when the building is excited on the first mass by magnetic forces. Please, download the<br />
files: frf−general.m, xxx1.txt, xxx2.txt, xxx3.txt, xxx4.txt, yyy1.txt, yyy2.txt, yyy3.txt,<br />
yyy4.txt (campus net)<br />
7. EXPERIMENTAL – Experimental Modal Analysis (EMA) deals with the determination <strong>of</strong><br />
natural frequencies, modes shapes, and damping ratios from experimental measurements.<br />
The fundamental idea behind modal testing is the resonance. If a structure is excited at<br />
resonance, its response exhibits two distinct phenomena: (a) as the excitation frequency<br />
69
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