Dynamics of Machines - Part II - IFS.pdf
Dynamics of Machines - Part II - IFS.pdf Dynamics of Machines - Part II - IFS.pdf
4. Excite the mass-spring system with just an initial velocity at the third coordinate ( ˙y3ini ) (initial displacements and excitation forces are set zero). Describe the vibration behavior of the points y1, y2 and y3. 5. Compare the three last simulations. Why is the transient behavior so different when the system is perturbed with initial velocity at point y1, y2 or y3. 6. Vary the number of masses attached to the first coordinate of the beam, y1, while exciting the mass-spring system with just an initial velocity at the first coordinate ( ˙y1ini ) (initial displacements and excitation forces are set zero). (a) Explain what happens with the natural frequencies of the system; (b) How many natural frequencies change when you change the mass in just one point of the structure? Explain. 7. Vary the number of masses attached to the second coordinate of the beam, y2, while exciting the mass-spring system with just an initial velocity at the first coordinate ( ˙y1ini ) (initial displacements and excitation forces are set zero). (a) Explain what happens with the natural frequencies of the system; (b) How many natural frequencies change when you change the mass in just one point of the structure? Explain. 8. Set the damping factor ξ = 0.005, while exciting the mass-spring-damping system with just an excitation force of f1 = 0.1 · e j·w·t [N] (initial velocities and initial displacements are set zero). Explain the vibration behavior of the system in terms of amplitudes and frequencies, when: (a) w = 10%wn1; (b) w = 50%wn1; (c) w = 90%wn1; (d) w = wn1; (e) w = 110%wn1; (f) w = 90%wn2; (g) w = wn2; (h) w = 110%wn2; (i) w = 200%wn2; (j) w = 90%wn3; (k) w = wn3; (l) w = 110%wn3; (m) w = 200%wn3. 9. Set the damping factor ξ = 0.05, while exciting the mass-spring-damping system with just an excitation force of f1 = 0.1 · e j·w·t [N] (initial velocities and initial displacements are set zero). Explain the vibration behavior of the system in terms of amplitudes and frequencies, when: (a) w = 10%wn1; (b) w = 50%wn1; (c) w = 90%wn1; (d) w = wn1; (e) w = 110%wn1; (f) w = 90%wn2; (g) w = wn2; (h) w = 110%wn2; (i) w = 200%wn2; (j) w = 90%wn3; (k) w = wn3; (l) w = 110%wn3; (m) w = 200%wn3. 10. Set the damping factor ξ = 0.005, while exciting the mass-spring-damping system with just an excitation force of f3 = 0.1 · e j·w·t [N] (initial velocities and initial displacements are set zero). Explain the vibration behavior of the system in terms of amplitudes and frequencies, when: (a) w = 10%wn1; (b) w = 50%wn1; (c) w = 90%wn1; (d) w = wn1; (e) w = 110%wn1; (f) w = 90%wn2; (g) w = wn2; (h) w = 110%wn2; (i) w = 200%wn2; (j) w = 90%wn3; (k) w = wn3; (l) w = 110%wn3; (m) w = 200%wn3. 11. Set the damping factor ξ = 0.05, while exciting the mass-spring-damping system with just an excitation force of f3 = 0.1 · e j·w·t [N] (initial velocities and initial displacements are set zero). Explain the vibration behavior of the system in terms of amplitudes and frequencies, when: (a) w = 10%wn1; (b) w = 50%wn1; (c) w = 90%wn1; (d) w = wn1; (e) w = 110%wn1; (f) w = 90%wn2; (g) w = wn2; (h) w = 110%wn2; (i) w = 200%wn2; (j) w = 90%wn3; (k) w = wn3; (l) w = 110%wn3; (m) w = 200%wn3; 12. Explain how such a variation of parameters could be useful in a case with a real machine? 60
1.10 Project 0 – Identification of Model Parameters (An Example) GOAL – With the first project the student will face a practical problem of the real life: how to properly choose the coefficients of linear differential equations of second order, aiming at achieving a reliable mathematical model, which can predict the machine dynamics? (a) (b) Figure 36: (a) Offshore platform http : //www.civl.port.ac.uk/comp−prog/offshore−platforms; (b) Laboratory prototype composed of one concentrated mass (foundation and rotor) attached to four flexible beams – An equivalent model of 1 D.O.F. system for analyzing the platform’s linear vibration in the horizontal direction. To represent the 2D-movements of the offshore platform shown in figure 36(a) a laboratory prototype was built, as it can be seen in figure 36(b). This simplified test rig is composed of one concentrated mass (foundation and rotor) attached to four flexible beams. An equivalent model of 1 D.O.F. system can be created with the purpose of analyzing the platform’s linear vibration in the horizontal direction. m0 2.108 [kg] platform mass L0 0.205 [m] beam length b0 0.025 [m] beam width h0 0.001 [m] beam thickness E 1.9 · 10 11 [N/m 2 ] steel elastic modulus Table 5: Main parameters of the test rig (platform) 1. Create a mechanical model of one-degree-of-freedom for describing the horizontal vibration of the test rig. Use Newton’s law and equivalent coefficients of mass m [Kg], viscous damping d [N/(m/s)] and linear stiffness k [N/m]. 2. There are two different ways of experimentally obtaining the forced vibration response of the platform in the frequency domain, i.e. its frequency response functions FRF(ω), namely by means of H1(ω) and H2(ω) functions. Detail about how to experimentally obtain H1(ω) and H2(ω) will be given in the second part of manuscript. Anyway, for now, it is important to relate such experimental functions to the frequency response functions 61
- 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 55 and 56: 1.8.4 Theoretical Frequency Respons
- Page 57 and 58: (a) Amplitude [m/s 2 ] (b) Amplitud
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- Page 65 and 66: 6. Model application - As explained
- Page 67 and 68: changeable unbalanced mass for simu
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- Page 71 and 72: acc [m/s 2 ] acc [m/s 2 ] 0.8 0.6 0
4. Excite the mass-spring system with just an initial velocity at the third coordinate ( ˙y3ini )<br />
(initial displacements and excitation forces are set zero). Describe the vibration behavior<br />
<strong>of</strong> the points y1, y2 and y3.<br />
5. Compare the three last simulations. Why is the transient behavior so different when the<br />
system is perturbed with initial velocity at point y1, y2 or y3.<br />
6. Vary the number <strong>of</strong> masses attached to the first coordinate <strong>of</strong> the beam, y1, while exciting<br />
the mass-spring system with just an initial velocity at the first coordinate ( ˙y1ini ) (initial<br />
displacements and excitation forces are set zero). (a) Explain what happens with the natural<br />
frequencies <strong>of</strong> the system; (b) How many natural frequencies change when you change the<br />
mass in just one point <strong>of</strong> the structure? Explain.<br />
7. Vary the number <strong>of</strong> masses attached to the second coordinate <strong>of</strong> the beam, y2, while exciting<br />
the mass-spring system with just an initial velocity at the first coordinate ( ˙y1ini ) (initial<br />
displacements and excitation forces are set zero). (a) Explain what happens with the natural<br />
frequencies <strong>of</strong> the system; (b) How many natural frequencies change when you change the<br />
mass in just one point <strong>of</strong> the structure? Explain.<br />
8. Set the damping factor ξ = 0.005, while exciting the mass-spring-damping system with just<br />
an excitation force <strong>of</strong> f1 = 0.1 · e j·w·t [N] (initial velocities and initial displacements are set<br />
zero). Explain the vibration behavior <strong>of</strong> the system in terms <strong>of</strong> amplitudes and frequencies,<br />
when: (a) w = 10%wn1; (b) w = 50%wn1; (c) w = 90%wn1; (d) w = wn1; (e) w = 110%wn1;<br />
(f) w = 90%wn2; (g) w = wn2; (h) w = 110%wn2; (i) w = 200%wn2; (j) w = 90%wn3; (k)<br />
w = wn3; (l) w = 110%wn3; (m) w = 200%wn3.<br />
9. Set the damping factor ξ = 0.05, while exciting the mass-spring-damping system with just<br />
an excitation force <strong>of</strong> f1 = 0.1 · e j·w·t [N] (initial velocities and initial displacements are set<br />
zero). Explain the vibration behavior <strong>of</strong> the system in terms <strong>of</strong> amplitudes and frequencies,<br />
when: (a) w = 10%wn1; (b) w = 50%wn1; (c) w = 90%wn1; (d) w = wn1; (e) w = 110%wn1;<br />
(f) w = 90%wn2; (g) w = wn2; (h) w = 110%wn2; (i) w = 200%wn2; (j) w = 90%wn3; (k)<br />
w = wn3; (l) w = 110%wn3; (m) w = 200%wn3.<br />
10. Set the damping factor ξ = 0.005, while exciting the mass-spring-damping system with just<br />
an excitation force <strong>of</strong> f3 = 0.1 · e j·w·t [N] (initial velocities and initial displacements are set<br />
zero). Explain the vibration behavior <strong>of</strong> the system in terms <strong>of</strong> amplitudes and frequencies,<br />
when: (a) w = 10%wn1; (b) w = 50%wn1; (c) w = 90%wn1; (d) w = wn1; (e) w = 110%wn1;<br />
(f) w = 90%wn2; (g) w = wn2; (h) w = 110%wn2; (i) w = 200%wn2; (j) w = 90%wn3; (k)<br />
w = wn3; (l) w = 110%wn3; (m) w = 200%wn3.<br />
11. Set the damping factor ξ = 0.05, while exciting the mass-spring-damping system with just<br />
an excitation force <strong>of</strong> f3 = 0.1 · e j·w·t [N] (initial velocities and initial displacements are set<br />
zero). Explain the vibration behavior <strong>of</strong> the system in terms <strong>of</strong> amplitudes and frequencies,<br />
when: (a) w = 10%wn1; (b) w = 50%wn1; (c) w = 90%wn1; (d) w = wn1; (e) w = 110%wn1;<br />
(f) w = 90%wn2; (g) w = wn2; (h) w = 110%wn2; (i) w = 200%wn2; (j) w = 90%wn3; (k)<br />
w = wn3; (l) w = 110%wn3; (m) w = 200%wn3;<br />
12. Explain how such a variation <strong>of</strong> parameters could be useful in a case with a real machine?<br />
60
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