VETOMAC 2010 Analysis of Torsional Vibration Characteristics of a Jaw Coupling D.Martin Sureshbabu 1 *, Dr.P.V.Mohanram 1 , Mr.M.Jeyaselvan 2 , 1 Dept. of Mechanical Engineering, PSG College of Technology, Coimbatore, India. 2 Manager, Technology Development, ELGI Equipments Ltd, Coimbatore, India. * [email protected]Abstract Comfort and convenience enjoyed in this technology driven world is made possible by the numerous rotating machineries. Without which, everything from transportation to manufacturing is a standstill. A repair or replacement of these are very expensive and also time consuming. It is good to see them run for longer period. To get an enhanced lifespan, it is necessary to prevent or predict imminent failures, and its source(s). One such source was the torsional vibration experienced by the system (rotating machinery). Extracting of torsional vibration from the system and signal is considered as a stringent process. Misalignment in a jaw coupling (rotating element) is the minor cause, when unnoticed leads to major breakdown and loss of the system. Fatigue is built on the shafts, due to misalignment induced torsional vibration. It is must to analyse the torsional
Transcript
VETOMAC 2010
Analysis of Torsional Vibration Characteristics of a Jaw Coupling
1Dept. of Mechanical Engineering, PSG College of Technology, Coimbatore, India.2Manager, Technology Development, ELGI Equipments Ltd, Coimbatore, India.
Comfort and convenience enjoyed in this technology driven world is made possible by the numerous rotating machineries. Without which, everything from transportation to manufacturing is a standstill. A repair or replacement of these are very expensive and also time consuming. It is good to see them run for longer period.To get an enhanced lifespan, it is necessary to prevent or predict imminent failures, and its source(s). One such source was the torsional vibration experienced by the system (rotating machinery). Extracting of torsional vibration from the system and signal is considered as a stringent process. Misalignment in a jaw coupling (rotating element) is the minor cause, when unnoticed leads to major breakdown and loss of the system. Fatigue is built on the shafts, due to misalignment induced torsional vibration. It is must to analyse the torsional vibration characteristics of the system with and without misalignment condition. Accuracy, time and cost combined together to adopt a methodology in detecting torsional vibration. Though, a jaw coupling is capable of handling misalignment to some extent, efficiency in transmitting torque is the desired one. The decrease in efficiency is due to the heat lost in the elastomer (spider) during operation. Calculations show that the losses are negligible, but the analysis is a useful one for larger machines and extensive research. Torsional vibration experienced due to misalignment is identified by the instantaneous angular speed of the rotating shaft. With the advent of Digital Signal Processing (DSP) and applying Phase De-modulation, torsional (modulating) component is extracted accurately. Processing the modulating component, the angular displacement, angular velocity and angular acceleration are known to take necessary counter-action.
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1. Introduction
Due to current trends in the design of rotating machinery towards higher speeds, manufacturers are tending to produce machines which operate closer to lateral critical speeds. The need for a better understanding of torsional vibration is becoming a necessity for practical engineers for the purpose of condition monitoring and diagnosing. Most rotating machinery consists of a driver and driven machine element coupled through a mechanical coupling, mainly to transmit torque/power.
Power transmission couplings are widely used for modification of stiffness and damping in power transmission systems, both in torsion and in other directions (misalignment compensation). To make a good coupling design is a very difficult task, because couplings have severe limitations on size and rotational inertia, etc. Torsionally flexible couplings are used for tuning dynamic characteristics (natural frequencies and/or damping) of the drive/transmission by changing their stiffness and damping.
Torsional vibration of a rotating element is the rapid fluctuations of angular shaft velocity. As a machine changes speed, torque is applied to the shaft in one direction or other. When the rotational speed of the machine fluctuates during one rotation of shaft, it is considered torsional vibration. Basic units are either radians or degrees.
The main sources of torsional vibration in rotating machines are; i) Unbalance of rotors ii) Misalignments of shafts iii) Non-uniform driving torque
The above excitations result in dynamic force, which affects the systems’ structural failure.
2. Literature Review
Though many books are available on the topic of torsional vibration, none is described about the analysis of torsional vibration. Wachel & Fred R.Szenasi [1] in their work clearly stated the analysis procedure, causes, effects and remedy. System modeling, forced response analysis for different components, and improving the system reliability were discussed in better way. According to Mark A.Corbo & Stanley B. Malanoski [2] it is difficult to detect the incipient failures due to torsional vibration. They stress the need of thorough torsional vibration analysis of an essential component of the turbo-machinery design process. They provide a practical torsional vibration procedure.
Various couplings and its roles in machine transmission and comparative assessment are formulated by Rivin E.I [3]. At the end an optimized coupling design index is formed. Boris M Klebanov et al [4] details the design and life of the couplings. As the number of technical papers is less in this area, for selection and
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analysis the procedure is adopted using manufacturers data in-line with standards specified by Shigley [5].
The concept of compensating stiffness is derived by Rivin E.I [3]. The force developed due to the misalignment is calculated using analytical concepts. Different types of compensating members are analyzed and compared. Parallel misalignment produces an extra deformation besides the deformation produced by transferring the torque as analysed by Diangui Huang [6]. It equals one time rotating frequency. The 1x rotating frequency sometime accompanied by smaller torsional vibrations at 2x, 3x, 4x, 5x, 6x and natural torsional frequency. Force due to angular misalignment is calculated analytically and also using a FEM model. Dimensionless stability criteria of the non-linear system of differential equations of two misaligned rigid rotors are derived using Liapunov’s direct method. The system kinetic, potential, and dissipation energies are derived by Al-Hussain [7]. The results show that an increase in angular misalignment leads to an increase in the model stability region.
A S Sekhar et al. [8], studies the effects of improper aligning of shafts, which leads to severe vibration. They modelled the misalignment using FEM analysis. They conclude that, the 2x vibration response is cause of misaligned shafts. The applications of flexible couplings for turbo-machinery were discussed by Mancuso [9]. This reference includes reasons for using flexible couplings, difference between gear and flexible element coupling, and the selection of couplings for new applications. Lorenzen et al. [10] introduced a comparison of critical speeds of a high-speed compressor train alternatively equipped with different types of couplings. The unbalance response using different types of couplings was calculated, which led to the conclusion that solid-type couplings can make the system more stable compared to other types of couplings.
Yuhua Li et al. [11] discusses several methods getting the instantaneous angular speed. Jiri Tuma et al. [12][13], describes the several methods possible to measure the angular displacement and in their work gives the details of analog method of measurement. P S Meirelles at al. [14][15] in their work employs the measurement techniques for practical applications. They highlight the modeling and analysis methods and calculations. R Archambanlt [16] explains the sequence of extraction information. The formation of analytic signal using Hilbert transform, and the method to get the modulating components was written nicely.
3. Methodology
3.1 Torsional vibration analysis methodology
The analysis is divided into three sections:1. Theoretical or analytical torsional vibration analysis, and2. Experimental torsional vibration analysis3. Coupling selection and analysis
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Theoretical analysis uses the fundamental concepts, equations and calculations. This helps to avoid misled interpretation of experiment results. The basic design concept built upon the theoretical analysis. If the theoretical outcome is unsatisfactory, the analysis is carried by changing the system parameters. It saves valuable time, costly fabrication of the system.
3.2 Experimental methodolgy
As seen in the literature studies, there exist many measurement techniques to measure angular displacement. The accuracy, time and cost is to be considered in the decision of selection. The two signal acquisition methods:
1. Analog method, and2. Timer method (digital) are considered for the extraction. These methods have some advantages as
well as disadvantages. Exactness of experiment can be verified, by performing both the methods. The analog method considers the transducer signal as ordinary analog signal, at the same time the timer method treats the transducer signal as train of digital pulses. A separate methodology formulated for the signal acquisition. Digital signal processing using Hilbert transform and phase demodulation for the analog method. And counter based method for the timer method.
4. Analysis
4.1 Theoretical torsional vibration analysis
The equation of motion for a system can be found by many methods. Applying the Newton’s law for a torsional (rotational) vibratory system gives
(5.1)
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Before developing a solution of the above equation, it is useful to consider first some important simplified cases. The major case is considering damping, = 0 and external force = 0. This gives the following equation, known as equation of free vibration.
+ (5.2)Where, I and k are mass moment of inertia and torsional stiffness respectively.
The system (Figure 5.1) parameters required for the analysis are listed as below:
System Parameters:Mass Moment of Inertia J in kgm^2Motor Inertia + Fan Inertia J1=0.07 + 0.1117Coupling hub – motor side J2= 0.018395Coupling hub – compressor side J3= 0.018395Crank shaft + connecting rod assembly J4= 0.0425+0.005853
Stiffness in Nm/rad Motor input shaft stiffness Kt1= 344090Coupling stiffness Kt2= 3610 Compressor/Top block input stiffness Kt3= 276195
Compressor speed n= 980 rpmPower P= 14 HP
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For any torsional vibration analysis, it is mandatory to form a physical model
of the system. The model must have all the required for further calculations. The physical model (Figure 5.2) of the system is illustrated as below:
Fig. 5.2 Physical model of the system
J1 – Motor Kt1 – Motor shaft torsional stiffness J2 – Coupling motor side Kt2 – Coupling torsional stiffness J3 – Coupling compressor side Kt3 – Compressor torsional stiffness J4 – Compressor
Firstly the equation of motion for the system has to be formed, to proceed further to carry free vibration. The number of rotating masses or mass inertias in the system decides the number of equations of motion. As seen from the mathematical model, for the four rotating masses, four equations of motion formed as below:
(5.3)
(5.4)
(5.5)
(5.6)
The above equations are solved by standard Eigen value approach. Solving the characteristic equation, one gets the Eigen values and Eigen vectors. Those are nothing but the torsional natural frequencies and mode shapes. Determination of natural frequencies and mode shapes are important to avoid resonance is essential in machine elements. The effects of damping on natural frequencies have been neglected. Rotating machines can develop excessive stresses in torsion, because of low torsional natural frequencies of the system involving flexible couplings.
The natural frequencies (Table. 5.1) and mode shapes (Table. 5.2) are as follows.
J1
J2 J3
J4
Kt1 Kt2 Kt3
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Table 5.1 Torsional Natural Frequencies
Table 5.2 Mode ShapesMode No.
ω, rad/sec
J1 J2 J3 J4
10 1.9359 1.9359 1.9359 1.9359
2266.5 -1.1219 -1.0798 3.3214 3.3632
34545.1 -0.5662 5.6104 3.7747 -1.4426
44581.5 -0.4204 4.2393 -5.0334 1.8819
An interference diagram (Figure 5.5) also known as Campbell diagram is prepared. It includes the details regarding operating speed(s), safety margin(s), torsional natural frequency (ies) and the exciting orders. Making note of interference points and safety margin (20%) is essential.
ωn1 0
ωn2 266.5 rad/sec
ωn3 4545.1 rad/sec
ωn4 4581.5 rad/sec
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Vibration which occurs under the influence of external force is called forced vibration. The external force known as excitation keeps the system vibrating. Rotating machines during operations have this kind of vibration. The excitation may be periodic, impulsive or random in nature. In this analysis periodic excitation is assumed. The excitation may be external or internal. The external being easily detected and is due to motion produced by one dynamic system to another. The torsional vibration of the crank shaft is of particular importance, because this form of vibration not captured by normal vibration monitoring methods. Many crank shafts have been broken on account of it. Such failures are unnecessary.
Since the torsional vibrations in the crank shaft are excited by the non-uniformities either in the driving torque or driven torque. As known, it is made up of two parts, one due to air pressure and the other due to inertia. The analysis can be carried out based on the, Torque – Θ curve, which is a system characteristic. It is a periodic curve. When the machine operated at half load or no load conditions, the curve gets it shape changed. Or, otherwise whenever there is a change in torque, there exist change in the curve shape. The fact that the torque is so irregular constitutes one of the inherent disadvantages of the reciprocating compressor as compared with dynamic compressor where the torque curve is fairly constant.
It is possible to break up the torque – Θ curve into its harmonic components. The break up is known as decomposition. Decomposition can be done by wavelet or fourier series method. The superposition of split harmonic periodic motions gives the earlier non-harmonic periodic motion. The mathematic theory shows that any periodic curve f(t), of frequency ω can be split up into a series of sine curves of frequencies ω, 2ω, 3ω, etc.
(5.9)Where A0, A1, A2 are fourier coefficients, ω is operating frequency and φ is the
phase component. The magnitude and phase of each order ‘n’ can be found as below:
then
Decomposed data are then added to get the restructured torque – theta curve and then the restructured curve superimposed on the original torque-theta curve to check the accuracy of the approximation. The excitation torque (magnitude) arrived from the decomposition is used for the formation of force vector. From characteristic equation the modal matrix {φ} is formed. Using the modal matrix, modal inertia and modal stiffness are calculated.
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Since, the excitation force (torque) is acting at the compressor end; the exciting torque function for nth order is formed as;
Next the modal displacement has to found out. To get modal displacement, the
equations of motion have to be uncoupled first, to get the local displacement {q}. The uncoupling of the equations of motion result in;
The effects and differences between local and global displacement function is illustrated (Figure 5.9) below.
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4.2 Experimental torsional vibration analysis
The success of experiment starts with correct selection of transducer. The end depends on the data processing technique. It is a very difficult task to select a suitable transducer among many. It lot depends on the application and accuracy of measurement concerned. Signal conditioning and processing further enhances the result. The measurement torsional vibration, which is the transformation of instantaneous time data, the processing should be very effective. The method should able to process real time data. Torsional vibration data are obtained by the following procedure:
1. Selection of transducer - It is not a simple parameter to analyze torsional vibration, because transducer requirements are stringent and shaft access may be limited. The most commonly used parameter for expressing torsional vibration is angular displacement, whose units are degrees, peak – to – peak value. Torsional vibration is the rapid fluctuation of angular velocity of shaft during one rotation. Details about the speed at each specified instant has to be known, to calculate torsional vibration. Considering instantaneous sensing, this requires a signal of many pulses per revolution (PPR), an incremental rotary encoder (1000 PPR) is used.2. Signal conditioning3. Selection of data acquisition method - Based on the mechanism of data acquisition, the measurement of angular speed can be categorized into two broad groups:
Timer/counter-based methods and ADC-based methods
The timer/counter-based methods treat the signal from an angular transducer as pulse train. The ADC-based methods treat an angular speed as an ordinary analog signal. The angular speed is then derived from the sampled data using an appropriate angular speed extraction technique.4. Performance parameters and analysis - A number of parameters like minimum measurable speed, maximum measurable speed, resolution and number of channels characterize the performance of torsional vibration measurement methods.5. Method signal processing - Signal acquisition is process of converting physical phenomena into data the computer can use. The transducer (incremental rotary encoder) converts the electrical signals into digital pulse train.
Motor Coupling Encoder
DC Supply
DAQ System
PC withLabView
Oscilloscope
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The output of the encoder in a direct ADC method provides the time-domain data. Time-domain plots must be used for all linear and reciprocating motion machinery. They are useful in the overall analysis of machine-trains to study changes in operating conditions. It is then important to extract the phase from the signal, to know about the modulating component. Only an analytic signal can express the details about the phase. A method called Hilbert Transform is performed to get the so called analytic signal. This possesses instantaneous amplitude and instantaneous phase. The time derivative of the unwrapped instantaneous phase gives the instantaneous frequency. Removal of operating frequency from it yields the modulating component in time-domain. This time-domain waveform can be Fourier analyzed in order to obtain the frequency demodulated spectrum.
Using the two counter large-range method the measurement principles are clearly stated as the conversion of physical data into angular displacement data. The measurement of instantaneous angular speed is realized in the following difference form:
where is angular displacement and is the corresponding time duration.
Steady state means, the operating conditions remain the same throughout the measurement period. The machine speed should not be time dependent. The acquiring of signal should begin only after the driving element (motor) attains its constant operating speed. The motor is a constant speed AC induction motor. Therefore the maximum speed of the motor has to be reached before the acquisition of signal. Steady state analysis comprises of no-load and load tests. A physical layout is as shown in the block diagram figure 6.9.
The actual experimental setup is shown as below in the figure 6.10.
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Due to noise and disturbance present in the lines, distorted pulses are generated. The peaks are not horizontal line, but for torsional vibration analysis the amplitude of the signal is not important. Only the phase contains the angular speed details. The data acquiring and processing using LabVIEW is displayed in figure 6.12.
The ball-valve is used to apply the load. When the opening in the valve reduces, the load increases. In the timer/counter-based method, counter input is used. Angular speed is the topic interest here. Therefore direct frequency measurement will reduce time and unnecessary calculations.
(a) encoder connected to the motor
(b) A proximity sensor to check start and stop
(c) The complete set-up with ball valve (Circled)
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4.2.1.2 Transient-state analysis
The shafting of the compressor, encounters rapid fluctuation of speed during start-up and start-down. Torsional vibration during this period is visible for the naked eye. This reveals the behavior of the system under transient conditions, importantly about the torsional resonance frequencies. Under normal working conditions, the rate of speed change from idle state to steady state operation is rapid. Such that, it seems that the system operates smoothly without sign of torsional vibration. The block diagram lay-out of transient state analysis is same. The difference between this lay-out and steady state lay-out is that the motor is operated through Variable Frequency Drive (VFD).
As the behaviour of the system is to be known, transient analysis is performed for no load condition. Consequently, counter method of an angular speed measurement is used. It provides good measurement accuracy over a wide speed range, from standstill to thousands of revolutions per minute. During the transient period the amplitudes of vibration are high due to torsional resonance.
4.2.3 Coupling selection and analysis Couplings are compared based on different working parameters. Since, the effect
of misalignment is considered; the Jaw type coupling is selected. It is capable of accommodating both the misalignments, parallel (radial) and angular. The jaw couplings are highly reliable for light, medium and heavy duty electrical motor and internal combustion power transmission applications. This coupling consists of three parts: driving siding hub, elastomer and driven side hub. The hubs are considered for inertias, and it is simple. Maximum attention is paid in the selection of elastomer. It has to be selected considering, the hardness, stability, and stiffness (radial and torsional). All these in turn are material properties. Torsional vibration must be controlled to prevent equipment failure.
From the manufactures catalogue, four standard elastomers (TPU material) with different hardness are selected for the analysis. It is not possible to perform analysis for the entire range of hardness. It is not needed and also time consuming. The four choices are made as per the industry requirements. It covers the hardness range, at the higher end. Elastomers are designed to accept a certain amount of misalignment; when this is exceeded, identifiable symptoms are usually recognizable. For understanding purpose, the force induced due to parallel misalignment is discussed. Table 7.4 gives the details of force and moment due to parallel offset for the elastomers selected.
Hardness Parallel Misalignment, e = 0.2 mmForce, N Moment, Nm
Sh 92 A 1880.00 0.376Sh 98 A 3520.00 0.704Sh 64 D 5179.20 1.036
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Sh 72 D 9140.00 1.828With the new data available the free vibration and un-damped forced vibration
results are shown in the table 7.5.Natural Frequency, rad/sec
Coupling
Existing Sh 92 A Sh 98 A Sh 64 D Sh 72 D
ωn1 0 0 0 0 0ωn2 266 243 331 398 522
ωn3 4545 13787 13829 13869 13953
ωn4 4581 15211 15260 15312 15446
The hardness of the elastomer and the inertia of the hubs influence the interference diagram. It is seen from the diagram (for Sh 92 A), that the system is free from any operating resonance. Moments due to the misalignments are less compared to actual exciting torque at the compressor end. Due to the damping, a certain quantity of the vibration energy is transferred into heat (PV) in the elastomer. The loss should be minimal in fraction to the input power. Damping power loss has to be calculated for two reasons: to know about the magnitude of the loss, and to check correct elastomer is selected. The power loss is calculated for each order and added according to the following formula:
=Vibratory Torque order, kNm i = order number n = speed, rpm and
=Dynamic torsional stiffness,
Taking n = 980 rpm, , i = 12, ψ = 0.8, and from decomposition (Static torque)
displacements for a horizontal Jeffcott rotor are considered. A transverse surface crack is assumed at the center of the span. Consider a mass-less elastic shaft of diameter D and length L with a disk of mass m mounted at mid-span. The transverse crack at the centre of span has a depth a. The eccentricity of the centre of the disk mass from the geometric centre of the disk is and represents the orientation of the eccentricity in the direction of the shaft rotation from the axis. The damping coefficient is c. (t) is the instantaneous angle of rotation and is the constant rotational speed. Considering direct stiffnesses k, k in the , directions, and cross coupled stiffnesses k, k, which come into play due to partial opening of the crack, the equations of motion can be expressed in the rotating co-ordinates as
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Citations to be referred by numbers and listed in the order of occurrence in text,
Table Caption 10 pt Bold, Title Case
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(1)
4. Results and Discussion
A program in MATLAB (version 5.3) is written to implement the numerical solution procedure for the above equations [1]. Stiffnesses are assumed to be constant for one degree of rotation (/180 radian), for which the integration of Eq. (1) is carried out with sufficiently small time step (t=0.000002666 seconds for runb=0.8) for accurate solution. Table 1 shows the degree of ration with the stiffness,
Table 1: Variation of stiffness with Crack Orientation
The response obtained at the end of one degree of rotation is stored and again used to re-evaluate new stiffness values to be used in equation of motion to get next set of displacements ( and ). Thus response is used to evaluate stiffnesses which in turn give next set of response.
Figure Caption 10 pt Bold
Pl follow the style of citation reference (JSV Style)
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Fig. 1. Frequency domain plot of the response of the cracked rotor during unstable crack propagation (runb=1.5, =00) a) horizontal and b) vertical direction.
4. Conclusions
The unbalance lateral vibration response of a Jeffcott rotor with a single centrally situated crack is analysed for unstable crack propagation. It has been observed that for the supercritical region the response does not show diverging rotor response and the frequency response also shows only a dominant running frequency component. However, for the subcritical speed range, the response shows diverging vibration amplitude and the frequency response indicates a noticeable bending natural frequency component. The unbalance orientation angle has substantial influence over the vibration response with the bending natural frequency component is more prominently observed when the unbalance is in opposite phase with the crack.
References
1. Darpe A.K., Gupta K. and Chawla A, (2003), Experimental Investigations of the Response of A Cracked Rotor to Axial Excitation, Journal of Sound and Vibration, vol. 260, pp. 265-286.
2. Whitten, K. W., Davis, R. E., Peck, M. L. and Stanley, G. G. (2004), General Chemistry, 7th edition, Brooks/Cole-Thomson Learning, pp152-153
