Dynamics of Machines - Part II - IFS.pdf
Dynamics of Machines - Part II - IFS.pdf Dynamics of Machines - Part II - IFS.pdf
m [kg], damping d [N/(m/s)] and stiffness k [N/m] of the platform and then can not be calculated. However, you can measure the forces applied to the platform and its acceleration response (steady-state response) and are able to experimentally obtain the frequency response function presented in table 1 6 and plotted in figure 37. Remember that the frequency response function is acceleration/force in this case, or FRF(ω) = −ω 2 /m (−ω 2 + ω 2 n + 2 · ξ · ωn · ω · i) = −ω 2 (−m · ω 2 + d · ω · i + k) Elaborate a simple parameter identification procedures based on the Least-Square Method, assuming that the frequency response functions FRF(ω) are known, and identify simultaneously the three parameter, i.e. mass, stiffness and damping: ⎡ ⎢ ⎣ −ω 2 1 1 −ω 2 2 1 −ω 2 3 1 ... ... −ω 2 N 1 ⎤ ⎥ ⎥ m ⎥ ⎦ k ⎧ ⎪⎩ REAL REAL ⎪⎨ = REAL ... REAL ω2 1 FRF(ω1) ω2 2 FRF(ω2) ω2 3 FRF(ω3) ω 2 N FRF(ωN) ⎫ ⎪⎬ ⎪⎭ (84) =⇒ A · x = b (85) x = A T · A −1 · A T · b (86) ⎡ ⎢ ⎣ ω1 ω2 ω3 ... ωN ⎤ ⎧ IMAG IMAG ⎥ ⎪⎨ ⎥ ⎥ d = IMAG ⎦ ... ⎪⎩ IMAG ω2 1 FRF(ω1) ω2 2 FRF(ω2) ω2 3 FRF(ω3) ω 2 N FRF(ωN) ⎫ ⎪⎬ ⎪⎭ =⇒ Ā · ¯x = ¯ b (87) ¯x = Ā T · Ā −1 · Ā T · ¯ b (88) Implement the identification procedure using MAPLE, or MATHEMATICA or MATLAB or MATCAD or another software. Use H1(ω) as well as H2(ω) for identifying the coefficients of mass m [kg], stiffness k [N/m] and damping d [N/(m/s)]. 4. Find theoretical and experimental ways of checking the identified values of mass, stiffness and damping, in order to assure that such values are really the correct mass, stiffness and damping coefficients of the real system. The more ”checking procedures” you can find, the better! Explain them all in details. 5. Model application – On the platform top a rotating machine is mounted, as it can be seen in figure.36(b). Its characteristics are delivered by the manufacturer. The maximum angular velocity is 1, 200 [rpm] (20 [Hz]). It is also known that the machine unbalance (m unb · ε) is 0.00012 [Kg · m]. By using your mathematical model, plot the vibration amplitude of the platform as a function of the machine angular velocity. Determine the maximum vibration amplitude of the platform. 1 It is important to notice that the values of H1(ω) and H2(ω) presented in table 6 are −H1(ω) and −H2(ω). When using the values of such functions to identify the model parameters, they have to be multiplied by -1. 64
6. Model application – As explained in the last item, the motor characteristics are delivered by the manufacturer. The maximum angular velocity is 1, 200 [rpm] (20 [Hz]). It is also known that the machine unbalance (m unb · ε) is 0.00012 [Kg · m]. Based on the dynamic equilibrium between motor torque, and torques associated to inertia, losses and external loading, the motor start-up curve shows a linear variation of angular velocity from 0 to 1, 200 [rpm] in a period of 40 [s]. Based on your mathematical model, please simulate computationally the vibration behavior of the platform during the period of 40 [s], knowing that, when the motor starts, the platform is already deformed due to a constant lateral wind. The platform static deformation is 0.001 [m]. Plot the platform displacement in the time domain, considering two cases: (a) considering the static wind force acting on the platform the whole time; (b) considering the lateral wind force suddenly disappears 30 [s] after the motor start-up. Analyze and discuss the behavior of the plots. What is the maximum vibration amplitude of the platform in both cases? 7. Question – Explain why the test rig can be modelled as an one-degree-of-freedom system in the range of 0 to 10 Hz, if one knows that a rigid body in the space (platform mass of the test rig) should be represented by a mechanical model of six-degrees-of-freedom, i.e. three linear and three angular displacements. 8. Write your final conclusions. (No Technical report!) 65
- 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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6. Model application – As explained in the last item, the motor characteristics are delivered<br />
by the manufacturer. The maximum angular velocity is 1, 200 [rpm] (20 [Hz]). It is also<br />
known that the machine unbalance (m unb · ε) is 0.00012 [Kg · m]. Based on the dynamic<br />
equilibrium between motor torque, and torques associated to inertia, losses and external<br />
loading, the motor start-up curve shows a linear variation <strong>of</strong> angular velocity from 0 to<br />
1, 200 [rpm] in a period <strong>of</strong> 40 [s]. Based on your mathematical model, please simulate<br />
computationally the vibration behavior <strong>of</strong> the platform during the period <strong>of</strong> 40 [s], knowing<br />
that, when the motor starts, the platform is already deformed due to a constant lateral<br />
wind. The platform static deformation is 0.001 [m]. Plot the platform displacement in<br />
the time domain, considering two cases: (a) considering the static wind force acting on<br />
the platform the whole time; (b) considering the lateral wind force suddenly disappears 30<br />
[s] after the motor start-up. Analyze and discuss the behavior <strong>of</strong> the plots. What is the<br />
maximum vibration amplitude <strong>of</strong> the platform in both cases?<br />
7. Question – Explain why the test rig can be modelled as an one-degree-<strong>of</strong>-freedom system in<br />
the range <strong>of</strong> 0 to 10 Hz, if one knows that a rigid body in the space (platform mass <strong>of</strong> the<br />
test rig) should be represented by a mechanical model <strong>of</strong> six-degrees-<strong>of</strong>-freedom, i.e. three<br />
linear and three angular displacements.<br />
8. Write your final conclusions.<br />
(No Technical report!)<br />
65
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