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SIMULATION AND EXPERIMENTAL STUDY FOR VIBRATION ANALYSIS ON ROTATING
MACHINERY
Msc in Mechanical Engineering with Emphasis on Structural Mechanic
MOHD SHAFIQ SHARHAN BIN ZAINAL
Supervisor: Dr Martin Magnevall PhD
Faculty of Engineering, Blekinge Institute of Technology, 371 79
Karlskrona, Sweden
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This thesis is submitted to the faculty of Mechanical Engineering at Blekinge Institute of Technology in partial fulfilment of the requirements for the degree of Master of Science in Mechanical Engineering. The thesis is equivalent to 20 weeks of full-time studies. The authors declare that they are the sole authors of this thesis and that they have not used any sources other than those listed in the bibliography and identified as references. They further declare that they have not submitted this thesis at any other institution to obtain a degree. Contact Information: Author: Mohd Shafiq Sharhan bin Zainal E-mail: [email protected] University advisor: Dr Martin Magnevall, Ph.D Department of Mechanical Engineering Faculty of Engineering Internet: www.bth.se Blekinge Institute of Technology Phone : +46 455 38 50 00 SE-371 79 Karlskrona, Sweden Fax : +46 455 38 50 57
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Abstract This thesis aims to analyze the unbalance on rotating machinery by simulation and experimental. The machinery flywheel rotation is modelled as a Single Degree of Freedom (SDOF) and Multi Degree of Freedom (MDOF) system. The model rotation unbalance is simulated by MATLAB. Then the vibration measurement is taken by experimental. In addition, the tachometer is used to determine the flywheel speed calibration. Finally, the rotating unbalance reduction simulation is performed with different parameter value to determine an optimum level of machinery rotation vibration. Unbalance on rotating machinery causes a harmful influence on the environment and machinery. The root cause of rotating unbalance is determined by the simulation and experimental analysis. The analysis result is used as an indicator for predicting machinery breakdown and estimating the correct predictive maintenance action for the machinery. In this project, the simulation and experimental analysis were carried out on a rotating component of the KICKR Snap Bike Trainer. The simulation and numerical analysis are performed by MATLAB programme. On the experimental part, the vibration measurement method and results were discussed. The suggestion of unbalance reduction were recommended base on measurement and vibration analysis results. Keywords: Multi Degree of Freedom (MDOF), Rotating machinery, Single Degree of Freedom (SDOF), Unbalance, Vibration analysis
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Acknowledgement This thesis is my last step towards a master’s degree in Mechanical Engineering with Emphasis on Structural Mechanics, conducted at Blekinge Institute of Technology (BTH) in Karlskrona, Sweden. I would like to express my sincere gratitude and appreciation towards Dr. Martin Magnevall Ph.D who has been my supervisors during this project. Your impeccable knowledge, support and valuable insights have helped and guided me throughout this thesis, thank you. I would also like to extend my gratitude towards the Swedish Institute Study Scholarship (SISS) for provide me the scholarship to study here. This publication has been produced during my scholarship period at BTH funded by Swedish Institute. Special thanks to my working organization, Royal Malaysian Navy (RMN) for continuous support in letting me pursue my master’s degree here. I will utilize the knowledge on this project for betterment of RMN in future. To Dr. Ansel Berghuvud Ph.D my examiner. I appreciate all of your comments and remarks during this semester. Last but not least, I would like to thank my lovely parent (Zainal and Zawiah) and wonderful wife, Nurul Aini, for always believing in me and supporting me in everything I do. Karlskrona, June 2020 Mohd Shafiq Sharhan bin Zainal
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Contents Abstract ........................................................................................................ ii Acknowledgement ...................................................................................... iii List of Figures ............................................................................................. vi List of Tables ............................................................................................ viii Notation ....................................................................................................... ix
1 Introduction .......................................................................................... 1
1.1 Background .................................................................................... 1 1.2 Problem identification .................................................................... 1
1.3 Aim and scope ................................................................................ 1 1.4 Research Questions ........................................................................ 2 1.5 Methods .......................................................................................... 2
1.6 Main Contribution .......................................................................... 2 1.7 Hypothesis ...................................................................................... 2
1.8 Related work .................................................................................. 3 1.9 Limitation ....................................................................................... 3
2 Theory ................................................................................................... 4
2.1 General ........................................................................................... 4 2.2 Unbalance definition ...................................................................... 4
2.3 Type of unbalance .......................................................................... 5 2.4 Force and motion ............................................................................ 6
2.5 Vibration measurement procedure ................................................. 6
2.6 Vibration transducer - accelerometer ............................................. 7
2.7 Units in vibration............................................................................ 8 2.8 Vibration amplitude measurement ................................................. 8 2.9 Frequency Domain ......................................................................... 9 2.10 Fast Fourier Transform ................................................................ 10
3 Simulation Approach ......................................................................... 11
3.1 Machinery information ................................................................. 11 3.2 Single Degree of Freedom (SDOF) system ................................. 12
3.3 Multi Degree of Freedom (MDOF) system ................................. 15
4 Numerical Approach .......................................................................... 16
4.1 ODE45 method............................................................................. 16
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4.2 Validation SDOF/MDOF with ODE 45 method .......................... 18
5 Experimental Approach .................................................................... 20
5.1 Experimental setup ....................................................................... 20 5.2 Experiment procedure .................................................................. 22
5.2.1 Tachometer - rotation speed measurement ........................... 22
5.2.2 Accelerometer - vibration measurement ............................... 23 5.3 KICKR Snap 2017 programme .................................................... 25 5.4 Kinomap programme ................................................................... 25
6 Result and Analysis ............................................................................ 26
6.1 Simulation result .......................................................................... 26 6.1.1 Simulation 1 – Comparison SDOF/MDOF system .............. 28
6.1.2 Simulation 2 - Simulate with flywheel parameter ................ 32 6.1.3 Simulation 3 - Altering unbalance mass/eccentricity ........... 39
6.2 Experimental result ...................................................................... 45 6.2.1 Type of experiment ............................................................... 45 6.2.2 Calibration test result ............................................................ 46
6.2.3 Unbalance test result ............................................................. 50 6.2.4 Vibration test result ............................................................... 54
7 Conclusion ........................................................................................... 61
7.1 Simulation work conclusion ......................................................... 61
7.2 Experimental work conclusion ..................................................... 61
8 Future Work ....................................................................................... 63
9 Reference ............................................................................................. 64
10 Appendix ......................................................................................... 66
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List of Figures
Figure 2.1: Type of unbalance [5] ................................................................ 5 Figure 2.2: Vibration measurement procedure ............................................. 7
Figure 2.3: Accelerometer diagram [6] ......................................................... 7 Figure 2.4: Vibration measurement unit [7] ................................................. 8 Figure 2.5: Sine wave [9] .............................................................................. 9
Figure 2.6: Vibration signal conversion time domain to frequency domain 9 Figure 3.1: KICKR Snap Bike Trainer [11] ............................................... 11 Figure 3.2: SDOF Model [12]..................................................................... 12
Figure 3.3: Free body diagram for unbalance mass .................................... 12 Figure 3.4: Free body diagram for whole mass system .............................. 13
Figure 3.5: MATLAB script for SDOF model ........................................... 14 Figure 3.6: MDOF model with rider[10] .................................................... 15 Figure 3.7: MATLAB script for SDOF ODE45 method ............................ 17
Figure 4.1: Comparison simulation and numerical SDOF ......................... 18
Figure 4.2: Comparison simulation and numerical MDOF ........................ 19 Figure 5.1: Experimental setup (side view) ................................................ 20 Figure 5.2: Accelerometer position and data acquisition setup .................. 20
Figure 5.3: Accelerometer specification ..................................................... 21 Figure 5.4: Tachometer position and the ratio ............................................ 22
Figure 5.5: Accelerometer with different position ...................................... 23 Figure 5.6: Experimental approach workflow ............................................ 24 Figure 5.7: KICKR Snap 2017 display ....................................................... 25
Figure 5.8: Kinomap display ...................................................................... 25
Figure 6.1: Type of simulation ................................................................... 26
Figure 6.2: MATLAB script for tachometer speed signal .......................... 27 Figure 6.3: MATLAB script for vibration analysis .................................... 27 Figure 6.4: SDOF simulation result ............................................................ 28
Figure 6.5: SDOF ODE45 result ................................................................. 28 Figure 6.6: SDOF result screenshot from MATLAB ................................. 29
Figure 6.7: MDOF simulation result ........................................................... 30 Figure 6.8: MDOF ODE45 result ............................................................... 30 Figure 6.9: MDOF result screenshot from MATLAB ................................ 31
Figure 6.10: Flywheel model ...................................................................... 32 Figure 6.11: Simulation 2 workflow ........................................................... 33 Figure 6.12: Experiment 1 vibration result and time spectral map ............. 34
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Figure 6.13: Harmonic response from tachometer signal ........................... 35
Figure 6.14: Conversion signal from displacement to acceleration ............ 35 Figure 6.15: Acceleration plot between simulation and experimental ....... 36 Figure 6.16: Flywheel model ...................................................................... 37 Figure 6.17: Acceleration plot comparison (frequency domain) ................ 37
Figure 6.18: Experiment 2 vibration result and time spectral map ............. 39 Figure 6.19: Acceleration plot comparison (time domain) ......................... 40 Figure 6.20: Acceleration plot comparison (frequency domain) ................ 40
Figure 6.21: Acceleration plot comparison (time domain) ......................... 41 Figure 6.22: Acceleration plot comparison (frequency domain) ................ 41
Figure 6.23: Acceleration plot comparison (time domain) ......................... 42
Figure 6.24: Acceleration plot comparison (frequency domain) ................ 42 Figure 6.25: Acceleration plot comparison (time domain) ......................... 43
Figure 6.26: Acceleration plot comparison (frequency domain) ................ 43 Figure 6.27: Type of experiment work ....................................................... 45 Figure 6.28: Vibration result - with rider .................................................... 46
Figure 6.29: Vibration result – non rider .................................................... 46 Figure 6.30: Time spectral map – with rider .............................................. 47 Figure 6.31: Time spectral map – non rider ................................................ 47
Figure 6.32: RMS and linear RMS result – with rider ................................ 48
Figure 6.33: RMS and linear RMS result – non rider ................................. 48 Figure 6.34: Tachometer and vibration data (fast cycle) ............................ 50 Figure 6.35: Tachometer and vibration data (normal cycle)....................... 50
Figure 6.36: Tachometer and vibration data (fast cycle) ............................ 51 Figure 6.37: Tachometer and vibration data (normal cycle)....................... 51
Figure 6.38: RMS and linear RMS result (fast speed >170 RPM) ............. 52 Figure 6.39: RMS and linear RMS result (normal speed) .......................... 52 Figure 6.40: Accelerometer position at different place .............................. 54
Figure 6.41: Time waveform on each accelerometer position .................... 55 Figure 6.42: Experiment 1 vibration result (Position 1 and 2) ................... 56 Figure 6.43: Experiment 2 vibration result (Position 3 and 4) ................... 57
Figure 6.44: Experiment 3 vibration result (Position 5 and 6) ................... 58
Figure 6.45: Vibration result summary ....................................................... 59 Figure 7.1: Ranges of vibration severity for various classes of machinery ISO 12372 [14] ........................................................................................... 62
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List of Tables Table 4.1: Parameter on SDOF simulation using MATLAB ..................... 18 Table 4.2: Analytical and numerical result for SDOF model ..................... 18 Table 4.3: Analytical and numerical result for MDOF model .................... 19 Table 5.1: Experiment equipment detail ..................................................... 21
Table 5.2: Ratio between flywheel and tyre ............................................... 22 Table 6.1: SDOF simulation result with different unbalance mass ............ 29
Table 6.2: MDOF simulation result with different unbalance mass ........... 31
Table 6.3: Flywheel parameter ................................................................... 32 Table 6.4: Simulation from experiment 1 result ......................................... 37 Table 6.5: Simulation from experiment 2 result ......................................... 44
Table 6.6: Calibration test result ................................................................. 49 Table 6.7: Unbalance test result .................................................................. 53
Table 6.8: Vibration result between KICKR structure and floor ................ 59 Table 7.1: Comparison of vibration velocity with ISO limit ...................... 63
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Notation Symbol Meaning Unit mo Unbalance mass kg e Eccentricity from the mass centerline m ωn Angular natural frequency rad/s ζ Damping ratio N/m ωr Angular velocity rad/s Φ Phase angle rad K Stiffness N/m c Damping Ns/m m Flywheel mass kg fs Sampling frequency Hz
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1 Introduction 1.1 Background Rotating machinery is commonly use in the modern engineering machinery to generate energy and production. The continuous operation of rotating machinery produce vibration by its nature. Vibration phenomena affects the machinery performance and lead to the component break down. The increasing demand of productivity require a sustainable machinery operation that produce an optimum result with less maintenance. Excessive vibration on machinery will degrade the machinery performance. Reducing the excessive vibration able to reduce the repair and maintenance cost. Therefore, the analysis on the machinery vibration is needed to determine the source of excessive vibration. 1.2 Problem identification 2017 KICKR Snap Bike Trainer widely used for indoor exercise. Continuous usage of machinery flywheel creates rotating unbalance and affecting the machinery performance and leads to machinery breakdown. Reduction on the unbalance rotation is determine by finding the root cause of the unbalance rotation. A recommendation on unbalance rotation reduction made by simulation and experimental. 1.3 Aim and scope The main purpose of this thesis is to develop a simulation to evaluate the unbalance on the machinery rotating part. The rotation unbalance simulation amplitude is validated with numerical and experimental approach. Finally, a suggestion for reducing rotating unbalance is propose from the result on simulation and experimental. The study result is beneficial because the concept is applicable to use in any machinery rotating part in future.
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1.4 Research Questions
1. How to model the machinery as a Single Degree of Freedom (SDOF) or Multi Degree of Freedom (MDOF) system for simulation in MATLAB?
2. How to perform machinery rotating unbalance measurement by experimental?
3. What is the effect of rotating unbalance on machinery to the structure?
1.5 Methods
1. Modelling and simulation the machinery by MATLAB. 2. Vibration measurement on the rotating machinery using
accelerometer and tachometer. 3. Validation of simulation and experimental results.
1.6 Main Contribution
1. Obtain the right simulation modelling for the system. 2. Suggestion on rotating unbalance reduction by simulation. 3. Determine the source of vibration on machinery.
1.7 Hypothesis
1. Rotating unbalance affect the vibration level of machinery. 2. Rotating unbalance affect the machinery performance. 3. Applying the correction mass reduce the rotation unbalance.
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1.8 Related work There are several previous papers that relate with machinery rotating unbalance study. B.Kiran Kumar et al.[1] has determined the unbalance in rotating machines by using the vibration analysis and identifies the system ills by observing the vibrational frequency spectra of the rotating machinery. Viliam Fedak et al.[2], has analyzes the balancing of unbalanced rotors and shafts using GUI MATLAB. The study shows that the GUI MATLAB enables the researcher to change parameters of the mechanical system to observe behavior of the unbalance rotating body. This method also enables to solve machinery rotating balancing by calculation of balancing masses and finding proper position for the fixing. Sawant [3] has deals with experimental setup for determination of damping coefficient and measurement of damped forced vibrations with rotating unbalance of SDOF system. The study shows the theoretical and experimental frequency response of a particular system at any particular frequency is lower for higher value of damping. There is also a slight difference on the result in the theoretical and practical due to inaccuracies in the measurement system. Finally, Md. Abdul Saleem et al.[4] detects the unbalance in rotating machinery using the shaft deflection measurement during its operation. The result on the Deflection Shape Shaft (DSS) is taken as an early warning indicator of unbalance in the rotating component. 1.9 Limitation
1. There is no simulation model on previous study represent this type of machinery. Most of the previous study are using machinery from industry.
2. Expected time consuming on computational and simulation time because of simulation work using non-optimization method.
3. Covid-19 issue restrict student access to university. Therefore, student has limited access to utilize the lab equipment.
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2 Theory 2.1 General Nowadays there are many types of technique to check and monitor critical machines, equipment, and system in a rotating machinery. The inspection is focus on the operational parameter such as temperature, pressure, speed, moment, power, noise, and vibration. Among of these parameters, vibration is the dominant parameter that used to check and monitor machinery conditional. Vibration is favorable parameter because it is the most sensitive and accurate indicator. Unbalance in rotating machinery exist because the centerline of the mass and the geometric centerline do not coincide. Reducing of unbalance is important because unbalance will damage the machinery structure, amplify resonance and exacerbates looseness. The cause of unbalance is due to the dirt build up, wear, cavitation and improper manufacture on the machinery. 2.2 Unbalance definition Unbalance is a defect on the rotating part that creates an unbalanced centrifugal force that transmits into the stationary structure as an oscillating motion. The measurement on unbalance is taken from the oscillating motion on the stationary structure. Forces on the oscillating motion does damage to mechanical parts. The vibration motion measured at a point on the stationary structure depends not only on the centrifugal force itself, but on the machine size and configuration, the bearings, the stiffness of the support structure, and whether resonance is an active player.
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2.3 Type of unbalance Static unbalance Unbalance phenomena when the central principal axis is displaced parallel to the rotating centerline. Spotted by rotor insertion at its point of rotation on each end. The heavy side of the rotor will swing to the bottom. Static unbalance can be modified by adding or removing weight in only one correction plane. Couple unbalance Unbalance when two unbalances exist 180 degrees apart on different planes. Couple unbalance central principal mass axis crossing the rotating centerline. Spotted during machinery rotation and the unbalance phenomena is recognized by comparing the shaft vibration amplitude and phase readings at each end of the rotor. The data obtain from machinery couple unbalance show identical amplitudes of vibration with phase readings which differ by 180 degrees. Dynamic unbalance Unbalance occur due to non-overlap of the rotating centerline and the shaft central principal axis. Dynamic unbalance is combination of static and couple unbalance phenomena. Dynamic unbalance resulting the central principal axis is both tilted and displaced from the rotating centerline. Correction is made by applying correction weight in a minimum of two planes.
Figure 2.1: Type of unbalance [5]
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2.4 Force and motion A pure unbalance creates a centrifugal force that is radial written in Eq. (2.1): Fc = m.r.ꞷ 2 (2.1)
Where Fc is the peak force, m is the imaginary mass, r is the radius of the imaginary mass and ω is the rotating speed in radians per second. The imaginary mass for unbalance is taken because unknown on the exact cause for each specific rotor. The unbalance caused might be from nonuniform material density or eccentricity from poor fits. Placing the correction mass will produce the mass and radius of the correction weight becomes 180 ° opposite from the imaginary mass. Measuring the motion with an accelerometer is calculate from the force of Newton's Second Law is written in Eq. (2.2): F = m.a (2.2)
Where F is the force, m is the mass and a is the acceleration. The estimation of the mass in motion is difficult for a support structure bolted to a concrete foundation since it also has significant stiffness. Thus, for a spring-isolated machine, the estimated mass in motion is taken from all the mass value above the spring coefficient. The unbalance can be calculated by combining Eq. (2.1) and Eq. (2.2): U = 𝑀𝑎
ω2 (2.3)
Where U is the unbalance, M is the mass in motion, a is the acceleration and ω is the circular speed in radians per second. 2.5 Vibration measurement procedure Vibration measurement procedure is start as follows; the motion of vibrating body is converted into an electrical signal by the vibration transducer. In general, a transducer is a device that transforms changes in mechanical quantities (displacement, velocity, acceleration or force) into changes in electrical quantities (voltage or current).
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Figure 2.2: Vibration measurement procedure
The following considerations often dictate the type of vibration measurement instruments to be used in vibration test, expected ranges of the frequencies and amplitudes, sizes of the machine/structure involved, conditions of operation of the machine/structure and type of data processing used (graphical display or the record in digital form for computer processing). 2.6 Vibration transducer - accelerometer An accelerometer is a sensor that measures the dynamic acceleration of a physical device as a voltage.(see Fig. 2.3). Accelerometers are widely used for vibration measurements. Accelerometers are full-contact transducers typically mounted directly on high-frequency elements, such as rolling-element bearings, gearboxes, or spinning blades.
Figure 2.3: Accelerometer diagram [6]
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2.7 Units in vibration The displacement from vibrating object is to measure the vibration amplitude. The displacement is the distance of vibration from a reference position or equilibrium point. Velocity meaning is the rate of change of displacement and measured in units of meter per seconds. While the acceleration is the rate of change of velocity and measured in units of G, or the average acceleration due to gravity at the earth surface.
Figure 2.4: Vibration measurement unit [7]
As a conversion, displacement is the sum of differentiation to produce velocity. Velocity is differentiation to produce acceleration. In contrast, it is possible to integrate from acceleration to velocity to displacement. The relationship between these three parameters displacement, velocity and acceleration for a simple harmonic motion is showed as below:
• Displacement, x = A sin (ωt) • Velocity, v (dx/dt) = Aω cos (ωt) • Acceleration, a (d2x/dt2) = Aω2 sin (ωt)
Where A represent the maximum amplitude and sin (ωt) is the oscillation term. 2.8 Vibration amplitude measurement Figure 2.5 shows the Peak amplitude as the maximum value of the wave from the zero point. While the Peak-to-Peak amplitude is the distance from a negative peak to a positive peak. For the sine wave vibration, the peak-to-
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peak value is twice the peak value because the waveform is symmetrical. Root Mean Square amplitude (RMS) is the square root of the average of the squared values of the waveform.and the RMS value is 0.707 times the peak value on the sine wave. These parameters are taken in quantifying the strength of vibration profile. [8]
Figure 2.5: Sine wave [9]
2.9 Frequency Domain Vibration occurs in time and called as time domain. The illustration of a vibration motion in the time domain is a called time waveform, and it display on an oscilloscope motion. If the waveform is transformed to a spectrum analysis, it is called a spectrum. The spectrum is exit in the frequency domain. Every detailed analysis of machinery vibration data is performed in frequency domain. The transformation from time domain to frequency domain is obtain by using the Fast Fourier Transform (FFT) method.
Figure 2.6: Vibration signal conversion time domain to frequency domain
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2.10 Fast Fourier Transform Time waveform is the total value of a series of sinusoids from diverse frequencies, amplitudes, and phases. A Fourier series is the sequence of sine waves. Fourier analysis or spectrum analysis is utilized to deconstruct a vibration signal into its individual sine wave components. The outcome is acceleration/vibration amplitude as a function of frequency. Fourier analysis generates by analyze the presence of each frequency component. A discrete Fourier transform (DFT) multiplies the raw waveform by sine waves of discrete frequencies to regulate if they suit among each other and their matching amplitude and phase. While a fast Fourier transform (FFT) is similar as DFT and using a more effective algorithm that takes lead of the symmetry in sine waves. The FFT entails a vibration signal span of some power of two for the transform and separations the process into cascading groups of 2 to obtain these symmetries. This process intensely improves computational and calculation processing speed. The Fourier transform (FT) of the function F(x) is the function of F(ω), where: F(ω) = ∫ 𝑓(𝑥)𝑒−𝑖𝜔𝑥𝑑𝑥
∞
−∞ (2.4)
While the inverse Fourier transform is: F(x) = 1
2𝜋 ∫ 𝑓(𝜔)𝑒𝑖𝜔𝑥𝑑𝜔
∞
−∞ (2.5)
Where i = √−1 and e𝑖𝜃= cos 𝜃 + i sin 𝜃
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3 Simulation Approach 3.1 Machinery information KICKR Snap Bike Trainer is an indoor machinery that support the static cycling exercise. The machinery is operating by the rotation of flywheel that creates the same resistance experience when riding bicycle outdoors [10].
Figure 3.1: KICKR Snap Bike Trainer [11]
The machinery specifications as follows: Dimensions : 74cm x 66cm Machinery weight : 38 lbs / 17.2 kg Rear wheel size : 650c RD / 28 inches Flywheel weight : 10.5 lbs / 4.7 kg Flywheel diameter : 71.12cm Maximum power output : 1500 Watts
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3.2 Single Degree of Freedom (SDOF) system
Figure 3.2: SDOF Model [12]
The machinery flywheel is modelled as a rotating unbalance of the Single Degree of Freedom (SDOF) system as shown in Figure 3.2. The free body diagram (FBD) is dividing by 2 parts which represent the unbalance mass and the whole mass system.
Figure 3.3: Free body diagram for unbalance mass
The equation of motion for rotating unbalance mass as follows: Fv = mẍ - moe ωr
2 sin ωr t (3.1) Where mo is the unbalance mass, e is the eccentricity from the mass centerline and ω is the angular velocity. The equation of motion is taken in the term of x.
Fv
m
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The sine defines the x coordinate because in chosen coordinates, x is vertical. If the mass rotates with a constant angular velocity ωr, then the circle that defines its rotation can be described as x(t) = e sin ωr t and y(t) = e cos ωr t. The free body diagram and equation of motion for the whole mass the whole mass system as follows:
Figure 3.4: Free body diagram for whole mass system
(m - mo) ẍ = - kx - cẋ - Fv (3.2) Substituting the Eq. (3.1) into (3.2), yields: (m - mo) ẍ = - kx - cẋ - (mẍ - moe ωr
2 sin ωr t) (3.3) The acceleration is obtained by the second derivative of the expression above with the respect of time. From the expression for x, the equations of motion can be modelled. The acceleration of the mass without the unbalance is ẍ. Count in the effects of the stiffness and damper and collecting x variable then moving the sine term to the other side of the expression: (m - mo) ẍ + mo (ẍ - e ωr
2 sin ωr t) = - kx - cẋ (3.4) mẍ + cẋ + kx = moe ωr
2 sin ωr t (3.5) Dividing by the system mass gives the final equation of: ẍ + 2ζ ωn ẋ + ωn
2 x = moe ωr2 sin ωr t (3.6)
Fv
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Where ωn is the natural frequency, e is the flywheel radius, ζ is the damping ratio, mo is the flywheel mass and ωr is the angular velocity. The final equation for rotating unbalance without free vibration are: xp(t) = X sin (ωr t - Ꝋ) (3.7) where X = 𝑚𝑜𝑒
𝑚 𝑟2
√(1−𝑟2)2+(2𝜁𝑟)2 is the amplitude of steady state, Ꝋ =
𝑡𝑎𝑛−1 2𝜁𝑟
1−𝑟2 is the phase angle of steady state and r = ωn
ωr is the frequency
ratio. Then the homogeneous solution is: xh(t) = A𝑒−ωntsin (ωdt + ϴ) (3.8) Combining the Eq. (3.7) and (3.8), yields the time response for the system: x(t) = xp(t) + xh(t) (3.9) Figure 3.5 illustrate the overall step on time response system from eq. (3.9) into the MATLAB script.
Figure 3.5: MATLAB script for SDOF model
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3.3 Multi Degree of Freedom (MDOF) system Multi Degree of Freedom (MDOF) model is taken by adding the participating mass on top of the rotating part. The participating mass equal with rider mass. The correlation between the participating mass and rotating unbalance part is:
Figure 3.6: MDOF model with rider[10]
The expression with initial value problem for a 2nd order ODE for MDOF is rewrite as: m2ẍ2 + c2 ẋ2 + k2 x2 = m2*g (3.10)
m1ẍ1 + (c1+ c2) ẋ1 - c2 ẋ2 +(k1+ k2) x1 - k2 x2 = m1r ωr2 sin (ωr t)
The system of Eq. (4.8) is transformed into the vector ODE: [𝑚1 00 𝑚2
] {ẍ1ẍ2
} + [𝑐1 + 𝑐2 −𝑐2−𝑐2 𝑐2
] {ẋ1ẋ2
} + [𝑘1 + 𝑘2 −𝑘2−𝑘2 𝑘2
] {x1x2
} = {F1F2
} (3.11) The 2nd order ODE is converted into a system of two 1st order ODE to obtain the two-vector x component. This vector is applied to MATLAB for the numerical analysis. 𝑑1
𝑑𝑡 (ẋ) = 𝑓1(𝑡)
𝑚1 - (𝑐1 + 𝑐2)
𝑚1x2 + 𝑐2
𝑚1x2 -
(𝑘1 + 𝑘2)
𝑚1x1 + 𝑘2
𝑚1x1 (3.12)
𝑑2
𝑑𝑡 (ẋ) = 𝑓2(𝑡)
𝑚2 - 𝑐 2
𝑚2x2 -
𝑘2
𝑚2 x1
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4 Numerical Approach 4.1 ODE45 method For the SDOF system the solver used is MATLABs function ‘ode45’ which is based on an explicit Runge-Kutta method. It is a one-step solver and is considered to be the best method for a first trial. The original 2nd order ODE is: mẍ + cẋ + kx = f(t) (4.1) m𝑑2𝑥
𝑑𝑡2 + c𝑑𝑥
𝑑𝑡 + kx = f(t) (4.2)
where f(t) =moe ωr
2 sin ωr t. The expression with initial value problem for a 2nd order ODE is rewrite: mẍ + cẋ + kx = moe ωr
2 sin ωr t (4.3) Rearranging the ẍ to the left side and introducing the help function, one obtains ẍ = 𝑓(𝑡)
𝑚 - 𝑐
𝑚 ẋ - 𝑘
𝑚 x, where v = 𝑑𝑥
𝑑𝑡 and a =
𝑑𝑣
𝑑𝑡 (v = ẋ and a = ẍ) (4.4)
The equation is transforms to system of equation: 𝑑𝑣
𝑑𝑡 = 𝑓(𝑡)
𝑚 - 𝑐
𝑚 ẋ - 𝑘
𝑚 x (4.5)
Transforming the system of equation into vector ODE, one obtains:
{
𝑑𝑥
𝑑𝑡𝑑𝑣
𝑑𝑡
} = [0 1
−𝑘
𝑚−
𝑐
𝑚
] {𝑥𝑣
} + {0
𝑓(𝑡)
𝑚
} (4.6)
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The 2nd order ODE is converted into a system of two 1st order ODE to obtain the two-vector x component which is x and ẋ by following variable substitution ẋ(t)=x2, x(t)=x1 and 𝑑
𝑑𝑡 (x) = x2 obtains:
𝑑
𝑑𝑡 (ẋ) = 𝐹(𝑡)
𝑚 - 𝑐
𝑚 x2 - 𝑘
𝑚 x1 (4.7)
After the expression 1st order ODE, Eq. (4.7) is applied in the MATLAB programme as follows: [t,x] = ode45(@fname, tspan, xinit, options). (4.8) Figure 3.6 illustrate the overall step on time response system from eq. (4.8) into the MATLAB script.
Figure 3.7: MATLAB script for SDOF ODE45 method
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4.2 Validation SDOF/MDOF with ODE 45 method The validation of SDOF model between analytical and numerical is perform by MATLAB programme using parameter as follows:
Table 4.1: Parameter on SDOF simulation using MATLAB Data Value Data Value
Flywheel mass 4.7 kg RPM 140 Unbalance mass 0.1 kg Frequency sampling 2000Hz
Eccentricity 0.0825m
Figure 4.1: Comparison simulation and numerical SDOF
Table 4.2: Analytical and numerical result for SDOF model
Amplitude RMS SDOF model 4.449 x 10-5m 2.78 x 10-5m
ODE45 4.446 x 10-5m 2.81 x 10-5m
• Figure 4.1 illustrates the similarity on the plotted graph of harmonic response between SDOF and ODE45 method.
• Table 4.2 shows the small difference of RMS and amplitude on the harmonic response between SDOF and ODE45 method.
• The result validates that both method between SDOF and ODE45 are identical.
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MDOF system is a modify model from the SDOF system by adding the value of participating mass. The parameter to insert on MATLAB is similar with SDOF with an addition of rider mass.
Figure 4.2: Comparison simulation and numerical MDOF
Table 4.3: Analytical and numerical result for MDOF model
Amplitude RMS MDOF model 1.09 x 10-6m 6.86 x 10-7m
ODE45 0.99 x 10-6m 6.68 x 10-7m
• Figure 4.3 illustrates the similarity on the plotted graph of harmonic response between MDOF and ODE45 method.
• Table 4.3 shows the small difference of RMS and amplitude on the harmonic response between MDOF and ODE45 method.
• The result validates that both method between MDOF and ODE45 are identical.
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5 Experimental Approach 5.1 Experimental setup Vibration measurement were performed using accelerometer and tachometer. Figure 5.1 and 5.2 shows the detail on experimental setup.
Figure 5.1: Experimental setup (side view)
Figure 5.2: Accelerometer position and data acquisition setup
Equipment used for collecting data was provided by the supervisor and BTH lab. Table 5.1 shows the equipment details and Figure 5.3 shows the accelerometer sensitivity value.
1 2
4
3
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Table 5.1: Experiment equipment detail No. Equipment Manufacturer Model Sensitivity/
Accuracy 1 KICK Snap
Bike Trainer Wahoo Fitness WFBKTR3 ± 3%
2 Laser tachometer
Compact Instrument Limited
PSU/S1 0.02% ± 1 digit
3 Accelerometer Bruel & Kjaer Type 4507 B 004
95.02 mV/g
4 Data acquisition unit
National Instrument
High Speed NI USB 9162
-
Figure 5.3: Accelerometer specification
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5.2 Experiment procedure 5.2.1 Tachometer - rotation speed measurement Laser tachometer is a device to measure rotational or surface speed in non-contact way. During measurement, the important parameter to consider is the operational tachometer range and accuracy of the measurement. Operational area is speed tachometer that can be measured, and the accuracy is typically expressed in unit ± r/min (r/min). The experimental procedures are:
1. Apply reflective tape to flywheel as a marker for tachometer reading.
2. Point the laser tachometer at the flywheel reflective tape. 3. Set flywheel different speed (slow, medium and fast cycle). 4. Record at least 2 readings from each speed laser tachometer.
Figure 5.4: Tachometer position and the ratio
Table 5.2: Ratio between flywheel and tyre No. Component Equipment Diameter Ratio A Driven Flywheel A 16.5cm A:C = 13:1 C Driver Bicycle tyre 71.12 cm
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5.2.2 Accelerometer - vibration measurement Vibration measurement of the machinery is obtained by the data from accelerometer. Vibration due to flywheel imbalance is seen as a peak in spectrum at the vibration frequency. The vibration level on rotational frequency of the rotor signal could be read directly from the signal display. The vibration measurement procedure as follows:
1. Apply the accelerometer at the desired position. The best position is the strong structure near with rotation flywheel.
2. Accelerometer is position as y-axis (vertical) for vibration signal reference.
3. Acquisition of vibration and flywheel rotational frequency signal with difference speed (slow, medium and fast cycling).
4. Processing of vibration signal using FFT.
Figure 5.5: Accelerometer with different position
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The detail of experimental approach workflow is shown in figure 5.6.
Figure 5.6: Experimental approach workflow
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5.3 KICKR Snap 2017 programme Flywheel rotation result can be access through KICKR Snap 2017 apps in the smartphone. The application develop by Wahoo Fitness is a running, cycling, and fitness app that leverages the smartphone to deliver data-driven power that fuels the training and fitness goals. The result of wheel revolution is display in the apps during machinery operation.
Figure 5.7: KICKR Snap 2017 display
5.4 Kinomap programme Kinomap is a complete end-to-end platform for creating, hosting, sharing and using motion videos for user-generated videos. Kinomap allows uploading motion videos shot with a GPS camera or a common camera coupled with GPS track from another device [12]. This is another modern application to monitor the speed and distance of flywheel. On this experiment, Kinomap is use as an additional software to monitor the speed reading.
Figure 5.8: Kinomap display
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6 Result and Analysis 6.1 Simulation result There are 3 type of simulation performed on this project. The details and objective of the simulation as shown in figure 6.1.
Figure 6.1: Type of simulation
A simulation study is carried out using the SDOF and MDOF model that has been establish in Chapter 3 and 4. For the MDOF simulation, the participating mass is taken as rider weight that has been attach on top of the rotating part. Several value of unbalance masses (UM) is applied in MATLAB script to see the unbalance effect. On this project simulation and experimental study, the value of vibration amplitude and root mean square (RMS) is taken as a result. Amplitude is the magnitude of vibration. A machine with large vibration amplitude is one that experiences large, fast, or forceful vibratory movements. The larger the amplitude, the more movement or stress is experienced by the machine, and the more prone the machine is to damage. Vibration amplitude is thus an indication of the severity of vibration.
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RPM of the flywheel is obtained from tachometer speed signal. Figure 6.2 illustrate the overall step to obtain RPM from tachometer speed signal using MATLAB script.
Figure 6.2: MATLAB script for tachometer speed signal
Figure 6.3 illustrate the MATLAB script for the step on vibration analysis. The vibration signal from time domain is convert to frequency domain using the Fast Fourier Transform function. Then flattop window (for harmonic response) and waterfall diagram is used to see the relation of frequency and RMS as a function of time.
Figure 6.3: MATLAB script for vibration analysis
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6.1.1 Simulation 1 – Comparison SDOF/MDOF system Single Degree of Freedom (SDOF) The simulation is performed using SDOF model establish in Chapter 3 on Eq. (3.9). The estimation of unbalance mass (UM) value is 0.1, 0.2 and 0.3kg. The result of time response is shown in same plot to see the amplitude difference by effect of unbalance mass.
Figure 6.4: SDOF simulation result
Figure 6.5: SDOF ODE45 result
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Figure 6.6: SDOF result screenshot from MATLAB
Table 6.1: SDOF simulation result with different unbalance mass
Eccentricity,e (m)
Unbalance mass, UM
(kg)
e*UM Max amplitude (*10-5m)
Max RMS (*10-5m)
SDOF ODE45 SDOF ODE45
0.0825 0.1 0.00825 4.4496 4.4465 2.7868 2.8129 0.2 0.0165 8.8993 8.8991 5.5737 5.6863 0.3 0.0245 0.13348 0.13349 8.6332 8.6332
Discussions
• Figure 6.4 and 6.5 illustrate the harmonic response for the machinery with different value of estimation unbalance mass. The result is compared between simulation model and ODE45 method. The result display between these 2 methods are identical. The screenshot of result from MATLAB is shown in figure 6.6.
• Table 6.1 shows full result on harmonic response amplitude and RMS value. The table shows only small difference value from each method both plotted graphs look identical. The similarity between each method is closer when e*UM = 0.0245 (as highlighted in bold).
• This simulation verified the influence of unbalance mass in the effect on amplitude harmonic response system.
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Multi Degree of Freedom (MDOF) The simulation is performed using MDOF model establish in Chapter 3 on Eq. (3.10). The participating mass of rider weight 90kg is included on top of rotating part. The estimation of unbalance mass value same as applies in SDOF model. The result of time response is shown in same plot to see the difference and effect of unbalance mass.
Figure 6.7: MDOF simulation result
Figure 6.8: MDOF ODE45 result
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Figure 6.9: MDOF result screenshot from MATLAB
Table 6.2: MDOF simulation result with different unbalance mass
Rider mass (kg)
e (m) UM (kg)
e*UM Max amplitude (*10-5m)
Max RMS (*10-5m)
MDOF ODE45 MDOF ODE45
90 0.0825 0.1 0.00825 0.1383 1.3597 0.2208 2.0333 0.2 0.0165 0.2766 2.7293 0.4416 3.9945 0.3 0.0245 0.4149 4.1340 0.6625 5.9554
Discussions
• Figure 6.7 and 6.8 illustrate the MDOF harmonic response for the machinery with different value of estimation unbalance mass. The result is compared between simulation model and ODE45 method. The result show ODE45 method produce little bit high amplitude than MDOF model. The screenshot of result from MATLAB is shown in figure 6.9.
• Table 6.2 shows full result on harmonic response amplitude and RMS value. The ODE45 method produce little bit high amplitude than MDOF model. However, both plotted graphs look identical as unbalance mass affect the harmonic response amplitude. The small difference toward similarity between each method is closer when e*UM = 0.0245 (as highlighted in bold).
• This MDOF model shows rider mass has influence to reduce the harmonic response amplitude. The heavier of rider mass attach on top of rotating part, the lower of amplitude will obtain. However, this applies when the rider mass transmits directly to rotating part without any support from other structure. This simulation also verified the influence of unbalance mass in the effect on amplitude harmonic response system.
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6.1.2 Simulation 2 - Simulate with flywheel parameter The simulation is performed by using the speed signal from tachometer with the measured flywheel parameter. The result of speed signal from tachometer is simulated to obtain the response of the machine. The unbalance mass, mo is determined by numerous trials to obtain the most similar as acceleration signal from experimental. Method
1. Analyze the speed signal from tachometer to obtain the RPM of flywheel and tyre. Time duration for this test is 60 seconds.
2. Insert the RPM result in the formula to obtain angular velocity and frequency ratio. Then insert these values into simulation model.
3. Simulate the response of the machine with different unbalance mass (UM) value. Compare the simulation result with experiment result.
4. There are 2 experiment conducted. The first experiment duration is 60 seconds with variety speed and second experiment duration is 20 seconds with constant speed.
Figure 6.10: Flywheel model
Table 6.3: Flywheel parameter
Data Value Flywheel mass, m 4.7 kg
Eccentricity, e 0.0825m (flywheel radius) Time duration 60 seconds
m
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Simulation 2 workflow is shows in figure 6.11.
Figure 6.11: Simulation 2 workflow
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Figure 6.12: Experiment 1 vibration result and time spectral map
Figure 6.12 shows the vibration acceleration result from experiment 1. The vibration acceleration shows vibration acceleration is quite constant between 0 to 20 seconds, then it is increasing due to faster flywheel speed in between 20 to 40 seconds. Finally, the acceleration reduces to remain constant after 40 to 60 seconds due to slower cycle speed. This experiment result is compared with simulation result to find the similarity.
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Simulation from experiment 1 result Using constant eccentricity e = 0.0825 and 3 different estimation unbalance mass, the simulation is perform using MATLAB to obtain the time response signal.
Figure 6.13: Harmonic response from tachometer signal
Figure 6.13 show time response of the simulation using different unbalance mass UM, with constant value of e. The analysis is continued by converting the signal from displacement to acceleration as shown in figure 6.14.
Figure 6.14: Conversion signal from displacement to acceleration
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The time response signal is converted into velocity by derivative of simulated displacement. The it continues by derivative of simulated velocity to obtain acceleration. The result for simulated acceleration is compared with experimental acceleration to see the difference.
Figure 6.15: Acceleration plot between simulation and experimental
Figure 6.15 shows the simulated acceleration signal with experimental simulated signal. The plotted signal is zoom in to see the difference more precisely.
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Figure 6.16: Flywheel model
Figure 6.16 shows unbalance mass of 0.128kg (Ymax_128g) producing similar maximum amplitude with acceleration from experimental (Ymax_Experimental) on time duration 0 to 20 seconds. Then the time domain signal is converted to frequency domain using fast fourier transform as shown in figure 6.15.
Figure 6.17: Acceleration plot comparison (frequency domain)
Table 6.4: Simulation from experiment 1 result
e (m)
UM (kg)
e*UM Peak in frequency domain max (*10-
3m/s2) Experiment Simulation
0.0825 0.128 0.0104 0.0009493 0.001102
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Discussion
• Figure 6.17 shows the amplitude on experimental work is 0.5x10-3 at 46.78Hz and amplitude on simulation is 1.10.5x10-3 at 15.77Hz. The simulation at 3 difference unbalance mass producing peak at the same frequency.
• After numerous trials on simulation, the most suitable unbalance mass value found is 0.128kg. The result of maximum acceleration value of 0.0234 m/s2 in time duration 0 to 20 seconds is same with maximum acceleration value from experimental work (as shown in figure 6.15).
• The simulation results show this flywheel is having unbalance mass of approximately 128 grams at the end of eccentricity (flywheel radius). The unbalance reduction can be made by applying the correction mass to counter the unbalance then reduce the harmonic response amplitude.
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6.1.3 Simulation 3 - Altering unbalance mass/eccentricity The simulation is similar method as Simulation 2, but it performed by changing the value of unbalance mass, with the fix eccentricity. Then continue with changing the eccentricity with fix unbalance mass. Both of these simulation conduct to see the best parameter represent the time response system. The experiment was performed with constant speed within time duration 20 seconds and the flywheel parameter same as table 6.3.
Figure 6.18: Experiment 2 vibration result and time spectral map
Figure 6.18 shows experiment 2 result. This result will be comparing with simulation work to find the similarity.
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Test 1 – Altering unbalance mass with fix eccentricity (e = radius)
Figure 6.19: Acceleration plot comparison (time domain)
Figure 6.20: Acceleration plot comparison (frequency domain)
Figure 6.19 and 6.20 shows comparison between simulation and experiment result on both time domain and frequency domain. Time domain shows similarity on harmonic response pattern after time duration goes 4 seconds onward. While on frequency domain, the peak amplitude difference between simulation and experimental is in a very small value.
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Test 2 – Altering unbalance mass with fix eccentricity (e = ½ radius)
Figure 6.21: Acceleration plot comparison (time domain)
Figure 6.22: Acceleration plot comparison (frequency domain)
Figure 6.21 and 6.22 shows comparison between simulation and experiment result on both time domain and frequency domain. Time domain shows higher on harmonic response pattern before 2 second and then goes decaying after that. While on frequency domain, the peak amplitude simulation is smaller than experimental.
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Test 3 – Altering eccentricity with fix unbalance mass (UM = 0.1 kg)
Figure 6.23: Acceleration plot comparison (time domain)
Figure 6.24: Acceleration plot comparison (frequency domain)
Figure 6.23 and 6.24 shows comparison between simulation and experiment result on both time domain and frequency domain. Time domain shows similarity on harmonic response pattern before 2 second and then goes decaying after that. While on frequency domain, the peak amplitude simulation is smaller than experimental.
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Test 4 – Altering eccentricity with fix unbalance mass (UM = 0.3 kg)
Figure 6.25: Acceleration plot comparison (time domain)
Figure 6.26: Acceleration plot comparison (frequency domain)
Figure 6.25 and 6.26 shows comparison between simulation and experiment result on both time domain and frequency domain. Time domain shows similarity on harmonic response pattern after time duration goes 4 seconds onward. While on frequency domain, the peak amplitude difference between simulation and experimental is in a very small value.
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Table 6.5: Simulation from experiment 2 result e
(m) UM (kg)
e*UM Peak in frequency domain max (*10-3m/s2)
Experiment Simulation Test 1
0.0825 0.2 0.0165
0.003711 0.002559
0.3 0.02475 0.003839 0.4 0.033 0.003839
Test 2
0.04125 0.2 0.00825
0.003711 0.00192
0.3 0.012375 0.00192 0.4 0.0165 0.00192
Test 3 0.0825
0.1 0.00825
0.003711 0.00128
0.04125 0.004125 0.0006199 0.02 0.002 0.0006199
Test 4 0.0825
0.3 0.02475
0.003711 0.003839
0.04125 0.012375 0.00192 0.02 0.006 0.00192
Discussion
• Table 6.5 shows the summary of result for all the test conducted. The result from test 1 and 4 show a small difference of peak in frequency domain between experimental and simulation (as highlighted in bold). The small difference on peak frequency domain between experimental and simulation is obtained when e*UM = 0.02475 and e*UM = 0.033. These values are same as result obtain in Table 6.1 and 6.2.
• Overall, the result from all simulation shows e*UM = 0.0104 (result from table 6.4), 0.02475 (result from table 6.1 and 6.3) and 0.033 (result from table 6.5) is the best result to obtain time response similarity between simulation and experimental.
• Therefore, the suitable range for modelling the simulation to obtain similar with this machinery is by fulfill the resultant of e*UM = 0.0104 to 0.033.
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6.2 Experimental result 6.2.1 Type of experiment Several types of experiment were performed to determine vibration behavior of this machinery.
1. Calibration test – An experimental to calibrate the machinery by comparing the vibration result with the effect of participating mass (rider weight). The machinery is running with and without the rider.
2. Unbalance test – An experimental to determine the machinery unbalance when operates with the different speed and time duration.
3. Vibration test – An experimental to determine the vibration level of different position of accelerometer. The accelerometer is placed on structure vertical side, horizontal side and floor.
The summary of each experiment is shown in figure 6.27.
Figure 6.27: Type of experiment work
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6.2.2 Calibration test result This test was performed to see the influence of rider mass on the vibration effect. There were 2 type of experiment was conduct which is rotating the flywheel with rider and then compare it with no-rider. The result is plotted in both time and frequency domain to see the details.
Figure 6.28: Vibration result - with rider
Figure 6.29: Vibration result – non rider
Figure 6.28 and 6.29 shows the vibration acceleration in time domain between operational with rider and non-rider. From the both figure, vibration acceleration with rider shows constantly high amplitude rather than non-rider.
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Figure 6.30: Time spectral map – with rider
Figure 6.31: Time spectral map – non rider
Figure 6.30 and 6.31 shows the time spectral map between operational with rider and non-rider.
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Figure 6.32: RMS and linear RMS result – with rider
Figure 6.33: RMS and linear RMS result – non rider
Figure 6.32 and 6.33 shows the RMS and linear RMS spectra between operational with rider and non-rider. The figure shows RMS value when operates with rider mass is slightly higher than non-rider.
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Table 6.6: Calibration test result
Time (s)
Linear RMS (m/s2) RMS (m/s2) With rider mass 90kg
Free running With rider mass 90kg
Free running
20 8.6 x 10-3 7.5 x 10-3 3.605 3.783 Discussion
• Figure 6.32 until 6.33 illustrate the result on time spectral map between machinery operational with rider and non-rider. While Table 6.6 show the of RMS and linear RMS result between cycling with rider and non-rider.
• The result of linear RMS when operational with rider is higher compare than non-rider. While the result of RMS when operational non rider is higher than with rider. However, the result difference between each method is in a very small value.
• This result shows that the rider mass has only a small influence on
the machinery vibration effect. Rider weight not influence very much on experimental work flywheel harmonic response because of the machinery is supported by machinery structure. Weight load and vibration on rider mass is transferred to the structure.
• This condition is different with MDOF modelling that has been establish in Chapter 3 on Eq. (3.10). On that model, the rider mass is attached directly on top of flywheel. Therefore, it affects and influence flywheel harmonic response.
• Thus, this experiment validate that the participating mass did not affecting very much on the rotating unbalance and vibration level on machinery.
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6.2.3 Unbalance test result This test was performed to see the influence of flywheel speed with the effect of vibration level. The flywheel is operating with faster speed on the first test with time duration 120 seconds. Then the flywheel operates with normal speed with time duration 500 seconds. The result is plotted in both time and frequency domain to see the details.
Figure 6.34: Tachometer and vibration data (fast cycle)
Figure 6.35: Tachometer and vibration data (normal cycle)
Figure 6.34 and 6.35 shows the tachometer and vibration reading obtain from flywheel operates with fast speed and normal speed. Time duration for experiment fast speed is 120 seconds while on normal speed is 500 seconds.
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Figure 6.36: Tachometer and vibration data (fast cycle)
Figure 6.37: Tachometer and vibration data (normal cycle)
Figure 6.36 and 6.37 shows the time spectral map between operation with fast speed and normal speed. The time spectral map on fast speed is more visible on frequency amplitude due to changing speed rapidly.
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Figure 6.38: RMS and linear RMS result (fast speed >170 RPM)
Figure 6.39: RMS and linear RMS result (normal speed)
Figure 6.38 and 6.39 shows the RMS and linear RMS spectra between operational with fast speed and normal speed. The figure shows RMS and linear RMS result when operates with fast speed is higher than normal speed.
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Table 6.7: Unbalance test result Speed Time
(s) Vibration Result
Linear RMS (m/s2) RMS (m/s2) Fast (>170 RPM) 120 3.2 x 10-3 2.173
Normal 500 3.0 x 10-3 2.426 Discussions
• The high frequency amplitude is clearly visible when flywheel speed above 20 Kph or 170 RPM. Table 6.5 show the difference of RMS and linear RMS result between each experiment speed.
• RMS and linear RMS value is higher when the flywheel is running with the faster speed compare than the slower speed. The frequency amplitude and RMS maintain low value even though it is running with longer time duration.
• The result shows the machinery is producing a high vibration level during fast speed despite it operates on short time. While the vibration maintains at moderate level despite it operates on longer time duration. Speed of flywheel affect the machinery vibration level rather than time duration.
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6.2.4 Vibration test result This test was performed to see the effect of vibration at different position between machinery structure and floor base. This experiment is performed by placing an accelerometer at different place to obtain different vibration result. The result is plotted in both time and frequency domain to see the details. The result from this experiment will shows which part are most affects when flywheel rotates. and helps to identify the location to implement vibration reduction on future study.
Figure 6.40: Accelerometer position at different place
1
2
5
6
4 3
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Figure 6.41: Time waveform on each accelerometer position
Figure 6.41 illustrate the time waveform of the signal on each accelerometer position.
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Figure 6.42: Experiment 1 vibration result (Position 1 and 2)
Figure 6.42 illustrate the time spectral map and RMS result on position 1 and 2. The vibration at position 1 (structure) is very high compare than position 2 (floor base). The result shows vibration is largely affects the machinery structure rather than floor base.
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Figure 6.43: Experiment 2 vibration result (Position 3 and 4)
Figure 6.43 illustrate the time spectral map and RMS result on position 3 and 4. The vibration at position 2 (structure y axis) is slightly high compare than position 4 (structure x axis). Vibration at y axis is higher because of structure and gravitation load on vertical side. While the vibration on horizontal side is produce because of movement occur during the cycling.
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Figure 6.44: Experiment 3 vibration result (Position 5 and 6)
Figure 6.44 illustrate the time spectral map and RMS result on position 1 and 2. The result is similar with experiment 1 due to vibration affects at the machinery structure rather than floor base.
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Table 6.8: Vibration result between KICKR structure and floor Exp. no.
Acc. position
Location Time (s)
Linear RMS max (m/s2)
RMS Max (m/s2)
1 1 Structure 300 2.8 x 10-3 3.809 2 Floor 300 0.8 x 10-3 0.76
2 3 Structure 300 2.6 x 10-3 3.59 4 Structure (x axis) 300 1.0 x 10-3 1.758
3 5 Structure 300 2.9 x 10-3 3.82 6 Front floor 300 0.7 x 10-3 0.74
Figure 6.45: Vibration result summary
Discussions
• Figure 6.42 until 6.44 illustrate the time spectral map RMS and linear RMS value for each experiment. The plotted graph shows frequency amplitude is high when the flywheel speed is increase at the machinery structure. While at the floor, the frequency amplitude maintains constant and it did not affect by the speed.
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• Table 6.8 shows the RMS and linear RMS value for each experiment.
The high amplitude obtains at the KICKR structure rather than the floor.
• The result shows the highest RMS occur at KICKR structure on
vertical side. This is occurred because the KICKR structure support the whole vibration load during operation. While the vibration at floor is maintain on very minimum level. Machinery rotating unbalance did not affect the floor vibration very much. Therefore, this experiment validates the rotating unbalance vibration more affecting the machinery structure rather than the floor base.
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7 Conclusion This work has investigated the vibration behavior of KICKR Snap 2017 using simulation and experimental for both time and frequency domain. The rotating unbalance was investigated by a theoretical approach using SDOF and MDOF model. These models have provided a steady harmonic response and give a good prediction of the rotating unbalance response behavior. The unbalance mass has been analytically and numerically evaluated on the rotating part for different harmonic response result. 7.1 Simulation work conclusion On the simulation work, vibration amplitude and RMS value are increasing when unbalance mass is applied. Therefore, it shows unbalance mass influence on machinery unbalance increment. The harmonic response is reaching to zero amplitude when the unbalance mass value equal to zero value. The modelling simulation result obtained is reasonable due to unbalance mass enhance the unbalance rotation. The simulation also shows eccentricity distance affecting the rotating unbalance. The higher amplitude time response obtains by longer eccentricity from centerline. The result of imaginary unbalance mass and eccentricity from Simulation 1 and 2 is obtain by numerous trials in MATLAB. However, these trials and error can be reduced by optimization method. Thus, it will save computational time and obtain the best result that represent the machinery time response system. 7.2 Experimental work conclusion On the experimental work, the result shows unbalance symptom is clearly visible when the machinery is running with faster speed. The faster of flywheel speed, it will produce higher vibration amplitude. The highest vibration amplitude obtained from the structure on vertical axis. Experimental result also shows the machinery vibration didn’t affect the floor vibration. The determine the real status of machinery severity, the machinery vibration result is compared with Vibration Severity Chart ISO 12372. From the ISO
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chart, we consider the machinery is fall under class 1 because power output is 1500 Watts. The ISO chart is shows as below:
Figure 7.1: Ranges of vibration severity for various classes of machinery
ISO 12372 [14]
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The resultant of maximum value of vibration velocity and velocity RMS on each position is shown in table 7.1.
Table 7.1: Comparison of vibration velocity with ISO limit Pos. Maximum velocity
(mm/s) Effective velocity RMS
(mm/s) Experiment
result Good ISO
limit Experiment result Good ISO
limit 1 0.4297
0.4
0.2481
0.28
2 0.1774 0.1016 3 0.4276 0.2471 4 0.1755 0.1016 5 0.4283 0.2475 6 0.1668 0.0963
The result shows the vibration level on this machinery is within good vibration limit. However, with continuous usage, the vibration level will be increasing and affect the machinery in future.
8 Future Work As future work one can do any of the following:
1. Further investigations on machinery structure material and suggest the better structure material to reduce the vibration.
2. Placing a new damping beneath the machinery structure to absorb the
vibration. The new damping position put at the highest structure vibration source.
3. Investigate the possibility to use a magnetic correction mass on the
flywheel and investigate the benefit of this.
4. This project using the numerous trials to obtain the best simulation result. Further analysis can be made by applying the optimization method to obtain more precise simulation model to compare with experimental result.
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10 Appendix 1. MATLAB Script simulation SDOF/MDOF model
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2. MATLAB Script experimental work
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