Dynamics of Machines - Part II - IFS.pdf

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Contents 1 Introduction to Dynamical Modelling of Machines and Structures and Experimental Analysis of Mechanical Vibrations based on the Human Senses 3 1.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Description of the Test Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Data of the Mechanical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Calculating Equivalent Stiffness Coefficients – Beam Theory . . . . . . . . . . . 5 1.5 Calculating Stiffness Matrices – Beam Theory . . . . . . . . . . . . . . . . . . . 7 1.6 Mechanical Systems with 1 D.O.F. . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6.1 Physical System and Mechanical Model . . . . . . . . . . . . . . . . . . . 9 1.6.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6.3 Analytical and Numerical Solution of the Equation of Motion . . . . . . . 10 1.6.4 Analytical and Numerical Solution of Equation of Motion using Matlab . 15 1.6.5 Comparison between the Analytical and Numerical Solutions of Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.6.6 Homogeneous Solution or Free-Vibrations or Transient Response - Experimental Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.6.7 Natural Frequency – ωn [rad/s] or fn [Hz] . . . . . . . . . . . . . . . . . . 19 1.6.8 Damping Factor ξ or Logarithmic Decrement β . . . . . . . . . . . . . . . 19 1.6.9 Forced Vibrations or Steady-State Response . . . . . . . . . . . . . . . . 24 1.6.10 Resonance and Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.6.11 Superposition of Transient and Forced Vibrations in Time Domain (Simulation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.6.12 Resonance – Experimental Analysis in Time Domain . . . . . . . . . . . . 28 1.7 Mechanical Systems with 2 D.O.F. . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.7.1 Physical System and Mechanical Model . . . . . . . . . . . . . . . . . . . 31 1.7.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.7.3 Analytical and Numerical Solution of System of Differential Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.7.4 Modal Analysis using Matlab eig-function [u, w] = eig(−B, A) . . . . . . . 37 1.7.5 Analytical and Numerical Solutions of Equation of Motion using Matlab . 40 1.7.6 Analytical and Numerical Results of the System of Equations of Motion . 41 1.7.7 Programming in Matlab – Frequency Response Analysis . . . . . . . . . . 42 1.7.8 Understanding Resonances and Mode Shapes using your Eyes and Fingers 43 1.7.9 Resonance – Experimental Analysis in Time Domain . . . . . . . . . . . . 48 1.8 Mechanical Systems with 3 D.O.F. . . . . . . . . . . . . . . . . . . . . . . . . . . 49 1.8.1 Physical System and Mechanical Model . . . . . . . . . . . . . . . . . . . 49 1.8.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 1.8.3 Programming in Matlab – Theoretical Parameter Studies and Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.8.4 Theoretical Frequency Response Function . . . . . . . . . . . . . . . . . . 55 1.8.5 Experimental – Natural Frequencies . . . . . . . . . . . . . . . . . . . . . 56 1.8.6 Experimental – Resonances and Mode Shapes . . . . . . . . . . . . . . . . 56 1.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 1.10 Project 0 – Identification of Model Parameters (An Example) . . . . . . . . . . . 61 1.11 Project 1 – Modal Analysis & Validation of Models . . . . . . . . . . . . . . . . . 66 2
1 Introduction to Dynamical Modelling of Machines and Structures and Experimental Analysis of Mechanical Vibrations based on the Human Senses 1.1 Summary The aim of this study is to show some theoretical and experimental examples to facilitate the understanding of the physical meaning of the main topics and definitions used in relation to vibrations in machines. Theoretical and experimental studies are led side-by-side, clarifying the definitions of stiffness, flexibility, natural frequency, damping factor, logarithmic decrement, resonance, phase, beating, unbalance, natural mode shape, modal node and so on. The experimental investigations are always carried out in low frequency ranges, aiming at making easy the visualization of the movements by the human eyes and the understanding of the mechanical vibrations without necessarily having to use sensors and electronic devices. 1.2 Description of the Test Facilities Figures 1 and 2 show the simple elements used during the experimental investigations: a flexible beam (ruler), concentrated masses or magnets, a support, an accelerometer, a signal amplifier, a shaker and a signal analyzer. The beam or ruler already has a scale, enabling it to easily achieve the information about the position (length) where the magnets will be mounted. At each position more than one magnet can be mounted, allowing changes in the values of the concentrated masses. The masses or magnets can be easily moved and attached to different positions along the ruler, aiming at investigating changes in the natural frequencies of the magnet-ruler system (mass-spring system). The ruler is very flexible in one plane only due to the characteristic of its cross-section (moment of inertia of area). The beam can easily be mounted with different boundary conditions, as free-free, clamped-free, simply supported in both ends, etc. allowing an investigation of the influence of the boundary conditions on its flexibility, and consequently on the natural frequencies of the system. Regarding rotating machines some analogies can be made between the mass-spring system presented here and a centrifugal compressor, while comparing the ruler (or flexible beam) to the flexible shaft, and the magnets (or concentrated masses) to the impellers or rigid discs. Changes in the montage of shaft into the bearings (boundary conditions) or changes in the positioning of the impellers along the shaft will lead to different critical speeds and mode shapes. As mentioned above, the mass-spring system was designed to have a very flexible behavior with very low natural frequencies. This allows the detection of natural frequencies, modes shapes and resonance using the human eyes as sensors (or sighting senses). Moreover, it is also made possible to excite the structure with human fingers, aiming at understanding the 90 degrees phase between displacement and excitation (while operating around resonance conditions) by means of tactile senses. Accelerometer, amplifier, shaker and signal analyzer are used aiming at confirming what the human senses detect. 3
Page 1: DYNAMICS OF MACHINES 41614 PART I -
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 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 51 and 52: which could be verified using Modal
Page 53 and 54:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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
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