International Journal of Scientific & Engineering Research, Volume 5, Issue 7, July-2014 ISSN IJSER

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1 International Journal of Scientific & Engineering Research, Volue 5, Issue 7, July-4 ISSN Advanced Dynaics & Control Lab Dept. of Mech Engg. CET, Kerala, India Dept. of Mech Engg. SCT College Of Engineering Trivandru, Kerala, India Abstract Rotating achines can develop inertia effects and this inertia effects in rotating structures are usually caused by gyroscopic oent introduced by the precise otions of the vibrating rotor as it spins. As spin velocity increases, the gyroscopic oent acting on the rotor becoes critical. It can be analyzed to iprove the design and decrease the possibility of failure. This paper focuses on dynaic unbalance detection in rotating achinery by incorporating whirl, gyroscopic effects, ass effects and the effect of unbalance using the Finite Eleent Method (FEM). dynaics are iportant [3]. Any instability in the rotor can easily lead to disaster and end up to be very expensive. Oscillations can shorten the lifetie of the achine. Oscillations can also ake the environent around the achine intolerable with heavy vibrations and high sound so it is advisable to avoid or reduce oscillations. ) Gyroscopic effects. The perpendicular rotation or precession otion is Keywords Gyroscopic effects, Rotating achine unbalance, applied to the spinning rotor about its spin axis, a reaction Rotatory Inertia, Coriolis Force. oent appears and this effect is described as gyroscopic effect [5]. The interaction of the angular oentu of the I. INTRODUCTION rotating rotor and the wobbling otion introduce gyroscopic The study of the echanical signature or the vibration effect. The gyroscopic effect is only observed for odes which includes an angular (wobbling) coponent. The gyroscopic spectru of a rotating achine allows to identify operating effect iplies a change in the stiffness of the syste and hence probles even before they becoe dangerous [4].Any its frequency. A forward whirling increases the stiffness of a deviation of the signature fro its usual pattern provides a syptos of a proble that is developing and allows the syste while backward whirling decreases the stiffness of a required counter easures to be taken in tie. The objective is syste. A frequency ap, is a graphically way to illustrate the influence of the gyroscopic effect on natural frequencies. to study the rotor dynaic aspects of rotors and develop an understanding of critical speed, the effects of ass in a siple odel and Rotor dynaics analysis using FEM by ) Critical Speed. incorporating whirl, gyroscopic effects and the effect of When the excitation frequency equal to natural frequency. The unbalance. critical speed can be identified with both a odal analysis and a forced response analysis. Critical speed should always be Modal analysis using daped Eigen value solver is carried out avoided in any achines parts because otherwise significant in order to predict the dynaic behavior of the rotating syste vibrations ay occur. These vibrations are the result of a []. In odal analysis, the natural frequencies are evaluated in resonant condition [8]. order to decide the critical speeds and ode shapes of the rotor. The process is repeated for different rotational speeds. Study of 3) Capbell Diagra a siple rotating shaft (Jeffcott rotor) with a) centrally ounted disc, b) ultiple discs, c) varying the ass, d) for Capbell diagra is the graphical representation of varying distance in order to find the gyroscopic effects and the syste frequency Vs excitation frequency as a function of ass effects in the syste. To understand the gyroscopic rotational speed. The diagra has the rotational speed of the effects a syste with four degree of freedo is considered.i.e. rotor on the x-axis and the ode frequencies on the y-axis. The two lateral and two rotational otion. odes are plotted for different rotational speeds. The A. Theory frequencies are not constant over the rotational speed range The theory of dynaics acts as a tool in understanding the because ost of the odes will be increased or decreased with physics of a rotating syste and the theory of vibration serves higher rotational speed. Frequencies of two circular whirling as a powerful instruent to atheatically quantify the otion, one occurring in the sae direction of the spin otion behavior of a rotor. Stability probles are the reason why rotor known as Forward whirling, and one in the opposite direction is known as reverse whirling or Backward whirling. Torsional and 5

2 International Journal of Scientific & Engineering Research, Volue 5, Issue 7, July-4 ISSN longitudinal odes are constant over the rotational speed range because they are not affected by the gyroscopic effect, the bearings or the stator. Forces in rotating achine ay occur due to isalignents, a bended rotor, or due to a certain ibalance in the rotating disk. Machines are designed on the basis of anufacturing tolerances; there will always be sall ibalance. An ibalance causes lateral forces, if the frequency of the force is equal the rotor speed the force is said to be synchronous. This produces a straight line of positive slope in the Capbell diagra, and is referred to as the one-ties spin speed and labeled as X. The critical speeds are where the excitation line crosses any of the ode line. Capbell diagra shown in figure. Classification of unbalances for a short rigid rotor is shown in figure.. Fig.. Classification of unbalances for a short rigid rotor II. MATHEMATICAL MODEL Fig. An exaple for Capbell diagra for a rotor syste 4) Rotor Unbalance A. Equations of otion The otion of the disc is described with four degree of freedo.two lateral and two rotational otion {U}={x y },where U is the nodal vector of displaceents and rotations. ) Equation Of Motion -Lateral Modes The disc has a ass and the centre of gravity offset fro the shaft centre is defined an eccentricity as shown in figure 3. It is assued that the daping of the shaft is zero Unbalance is the ost coon rotor syste alfunction. Its priary sypto is X vibration, which, when excessive, can lead to fatigue of achine coponents. In extree cases, it can cause wear in bearings or internal rubs that can daage seals and degrade achine perforance. Unbalance can produce high rotor and casing vibration, and it can produce vibration in foundation and piping systes. X vibration can also contribute to stress cycling in rotors, which can lead to eventual fatigue failure [6]. There are two types of rotating unbalance: A. Static Unbalance All the unbalanced asses lie in a single plane resulting in an unbalance of single a radial force. Static unbalance can be detected by placing the shaft between two horizontal rails and allowing the shaft to naturally roll to the position at which the unbalance is below the shaft axis. Static unbalance is shown in figure.. B. Dynaic Unbalance This is when the unbalanced asses lie in ore than one plane. The static test will only detect the resultant force. The unbalance has to be detected by rotating the shaft and easuring the unbalance. The achine for carrying out this detection are called balancing achines and consist of spring ounted bearings that support the shaft. By obtaining the aplitude and relative phase it is possible to calculate the unbalance and correct for it. This is a proble of two degrees of freedo as transitional and angular otions take place. Fig.3 Rotating ass on a flexible shaft Where, a Intersection with the line through the points of support b Intersection with the axis of rotation c Centre of the rotating ass The distance bc is indicated as and the distance ab as r, where is the eccentricity and r is the deflection of the bea. The equation of otion for the ass center is derived fro Newton's second law d r kr () dt Where is the bending stiffness of the shaft 5

3 International Journal of Scientific & Engineering Research, Volue 5, Issue 7, July-4 ISSN l 743 kaa x y kaaa (6) By solving,, Fig.4 The position of the ass center expressed in the x-y coordinate syste The equation of otion in x-y coordinate syste, is given by, d x H cos(: t) k x x dt d x H sin(: t) k y y dt kk kl Id 3,4 Id (7) II d () Where, Ȧn is the Natural frequency of the rotor III. FINITE ELEMENT ANALSIS Equation becoes, x kx cos( t) y ky sin( t) (3) ) Equation Of Motion Motion- Ang Angular Moentu ggular Moe Id Id k3 k3 A. Case:Modal Analysis Without Spin This analysis is done without any rotation, the gyroscopic effects does not take place. Basic characteristic of a rotating achinery can be deterined by introducing b a siple rotor odel (Jeffcott rotor) shown in figure 5. The odel consists of a single rigid disc centrally ounted on a unifor flexible ass less shaft supported by two identical rigid bearings at each side. The ass of the disc is Kg, Specifications of the rotor syste is given in the table. (4) Where, I -Diaetric ass oent of inertia, d I - Polar oent of inertia, p 3) The Stiffness Matrix l kaa kaaa (5) Fig.5 siple rotor odel TABLE.. p y 4) Equation Off Motion For The Coplete Syste x ÿ Id Id Description of the syste Disc: steel x y Shaft :steel Inner Diaeter:.3 Length : Outer diaeter:.6 Diaeter :.3 Thickness:.5 Modulus (E):. N/ 3 Density: 78 kg/ 3 Density: 78 kg/ The eigen frequencies obtained fro the analysis is tabulated in the table. 5

4 International Journal of Scientific & Engineering Research, Volue 5, Issue 7, July-4 ISSN F=x spin TABLE. Natural Frequencies (Hz) Analytical Ansys (Matlab) BW AUG 4 8:55:6 Fig.8. First and second ode shape STEP= STEP= SUB = SUB =3 FREQ=6.496 FREQ=4.93 Since the odel undergoes several rotation speed gyroscopic ZDMX =.9369 X DMX Z=.946 X effects take place. The critical speeds are calculated fro the Capbell diagra at the intersection of excitation line with odal frequency lines. Fro the Capbell diagra we can see the first ode is independent of the rotational speed and does not separate. This is because the disc does not have any Fig.6 First and second ode shape wobbling. In the backward whirling the natural frequency is It is observed that the Ansys results are very uch coparable decreasing as the rotational speed increasing, In forward with the analytical results obtained by Matlab. The rotor is not whirling and the natural frequency is increasing as the rotating and the odes are a pure planar otion in the z-x plane rotational speed is increasing. This behavior is due to the [9]. In the first ode the centrally ounted disc translate influence of the gyroscopic effect which is caused by the vertically in z direction.in the second ode the vertical wobbling of the disc. The intersections of the various branches translation is zero and the oveent is pure wobbling ode of of the Capbell diagra with the y axis are the natural the shaft. First ode is cylindrical ode and the second ode frequencies at standstill of the syste. is conical ode. The first two ode shapes shown in figure 8. The first ode shape is characterized as a cylindrical ode. The second ode B. Case:Analysis of Rotating Shaft shape is called a conical ode. These otions are described In this analysis, several set of daped Eigen frequency analysis with four degrees of freedo, two lateral and two rotational. is perfored on the rotor odel between speed range to The ode shapes look siilar to the non rotating odes, but rp. The rotational velocity is increased ins steps of rp. involve a circular otion instead of planar otion, the circular Daped odal analysis is perfored and validated by otion is called whirling. Due to gyroscopic effect the coparing the result against corresponding analytical solution. frequency line splits into two. There are three intersections between the excitation line and the forward(fw)and backward TABLE 3. whirling(bw) curves, two at 39 rp and one at 877 rp, Eigen Frequencies and Critical speeds where the latter is a backward whirling. The first forward ode Critical speed,rpm (slope Natural which crosses the excitation line at 39 rp is critical speed. of excitation line = ) Frequencies(Hz) The intersection at backward whirl is not considered as critical Ansys Matlab Critical speed Whirl speed because backward whirling odes not give any excitation BW CAMPBELL DIAGRAM FW C. Case 3.Offset Disc STEP= SUB = DMX Z=.9367 X ELEMENTS STEP= AUG 4 SUB =4 8:43:5 DMX =.879 AUG 4 8:44:3 AUG 3 4 3:34: M Fig.9 Model with ass placed off center Disc with the sae ass is placed at a distance fro the centre as shown in the figure 9. Analysis is perfored on the odel between the speed range rp to rp (rp) 9 Fig.7 Capbell diagra 5

5 ELEMENTS M M AUG 3 4 3:4:7 STEP= SUB = RFRQ= ZIFRQ= X DMX =.49 STEP= SUB =4 RFRQ= IFRQ=7.65 DMX =.99 STEP= SUB =3 RFRQ= IFRQ=3.849 DMX = STEP= SUB = DMX =.49 AUG 9 4 :5:38 International Journal of Scientific & Engineering Research, Volue 5, Issue 7, July-4 F=x ISSN spin CAMPBELL DIAGRAM 745 AUG 3 4 F=x spin CAMPBELL DIAGRAM F=x spin CAMPBELL DIAGRAM AUG 3 4 3:3:4 :59: AUG 9 4 :: (rp) (rp) 9.8 Fig.3 Capbell diagra for rotor a) sae ass.b) different asses Fig. Capbell Diagra 8 7 (rp) 9 STEP= SUB = RFRQ= IFRQ=9.49 DMX =. AUG 3 4 STEP= 3:3:53 SUB =3 RFRQ= IFRQ=9.944 DMX =.7858 AUG 3 4 3:33: Fig. First and second ode shape TABLE.4. Eigen frequency and critical speed Natural Frequencies (Hz) Critical speed(rpm) Fro the Capbell diagra shown in figure, frequency split is observed even for the first ode. That is when the disc placed off centre, which results in the wobbling for both the first and second ode shape. In the second ode the gyroscopic effect is ore and as a result the frequency splits as forward and backward whirl. This is due to the fact that at zero speed, the odes of vibration are coposed of a bending and a rotation one. When the syste turns, these two odes still IV. RESULTS AND DISCUSSIONS exist, but each of the is separated into two, one forward and A. Non rotating Dynaics one backward. Fro the equation of otion it is the coupling due to the diaetric oent of inertia, that causes the splitting up of forward and backward whirl. When the disc is placed offset natural frequencies of the syste also changed. Fro the Capbell diagra shown in figure.8, the line of excitation intersect the frequency lines only in the first ode and only two critical speeds are observed one in forward whirl and other one in backward whirl. D. Case4:Shaft with Two Rotor ) Exploring Gyroscopic and Mass Effects Considering two odels, first odel with the sae ass at equal distance fro the center and the in the second odel, asses are replaced with two different asses Fig. Two Rotor at equal distance Fig.4 ode shapes Figure 3 shows two sets of natural frequencies versus speed and fro the natural frequencies, it can be see that the different ass odel increases the first and second ode frequencies (ass is at a point of large whirling otion, ass oents of inertia changed). The reduced ass oent of inertia version (different ass) does not change the third ode (ay be disk center of gravity has very little conical otion).the reduced ass oent of inertia increases the frequency of the second ode, and decreases the strength of the gyroscopic effect (disc center of gravity has substantial conical otion). When the achine is not spinning and the bearings have essentially no daping, in the odal test, we would find a set of natural frequencies/odes. At each frequency, the otion is planar. The ratio of bearing stiffness to shaft stiffness has a significant ipact on the ode-shapes. For the soft and interediate bearings, the shaft does not bend very uch in the lower two odes. As the bearing stiffness increases (or as shaft stiffness decreases), the aount of shaft bending increases. In the first ode, the disk translates without rocking. In the second ode, it rocks without translation. This general characteristic repeats as the frequency increases. If we oved the disk offcentre, we would find that the otion is a ix of translation and rocking. This characteristic will give rise to soe interesting behavior once the shaft starts rotating and is called gyroscopic effects B. Rotating Dynaics When shaft rotates the odes look like the non rotating odes, but involve circular otion rather than planar otion to see the effects of changing shaft speeds perfored the analysis fro 5

6 International Journal of Scientific & Engineering Research, Volue 5, Issue 7, July-4 ISSN non spinning to a high spin speed and follow the two frequencies associated with the conical ode. The frequencies of the conical odes do change over the speed range. The backward ode drops in frequency, while the forward ode increases. The explanation for this behavior is a gyroscopic effect that occurs whenever the ode shape has an angular (conical/rocking) coponent. If consider forward whirl, shaft speed increases, the gyroscopic effects essentially act like an increasingly stiff spring on the central disk for the rocking otion. Increasing stiffness acts to increase the natural frequency. For backward whirl, the effect is reversed. Increasing rotor spin speed acts to reduce the effective stiffness, thus reducing the natural frequency.also the gyroscopic ters are generally written as a skew-syetric atrix added to the daping atrix, the net result is a stiffening or softening effect of the shaft. In the case of the cylindrical odes, very little effect of the gyroscopic ters was noted, since the centre disk was whirling without any conical otion. Without the conical otion, the gyroscopic effects do not appear. In conical type otion near the bearings a slight effect was noted. The frequencies are affected by both the ass and diaetric ass oent of inertia. The ass has the greatest effect at points of large circular otion (anti-nodes), while the ass oent of inertia has the greatest effect at points of large rocking otion (nodes).changes in ass precisely at a node do not change the corresponding natural frequency, and changes in ass oent of inertia at points of no conical otion do not change the corresponding natural frequency. V. CONCLUSIONS The odes affected by the ass oent of inertia (the conical ode, for exaple), are strongly affected by changes in speed. Assuing the bearing characteristics do not change, the backward whirl ode will decrease in frequency with increasing shaft speed, while the forward ode frequency will increase. The extent to which this occurs is related to both the ode shape and the ratio of the polar ass oent of inertia to the diaetric ass oent of inertia. Thus, a achine with a big disk/fan blade will probably show strong speed dependent effects in at least soe odes. A fairly syetric achine will probably have soe odes that are relatively constant with shaft speed. A forward whirling increase the natural frequency and a backward whirling decreases the natural frequency. A forward whirling increases the stiffness of a syste while backward whirling decreases the stiffness of a syste. For a non rotating syste ( = ), the odes of vibration are coposed of a bending and a rotation one. When the syste turns, these two odes still exist, but each of the is separated into two, one forward and one backward. Acknowledgent This work was supported by Centre for Engineering Research and Developent, Trivandru, Kerala and All India Council for Technical Education [RPS grant No: 83/RID/RPS-4/-]. Reference [] Michael R. Hatch,"Vibration Siulation Using MATLAB and ANSS",,Print Media. [] Achuth Rao, "Rotordynaic Capabilities in ANSS Mechanical",ANSS,Inc. [3] Forsthoffer,"Fundaentals of Rotating Equipent",Forsthoffer's Rotating Equipent Hand Book Vol.,5. [4] Giancarlo Genta, Dynaics of Rotating Systes, Springer Science Business Media, Inc.5. [5] [6] F. C. Nelson,"Rotor Dynaics without Equations, International Journal of COMADEM, (3) 7, PP. -. [7] A. Farshidianfar, S. Soheili, "Effects of Rotary Inertia and Gyroscopic Moentu on the Flexural Vibration of Rotating Shafts Using Hybrid Modeling", Scientia Iranica, Vol. 6, No.,9, pp [8] Lyn M. Greenhill, Guillero A. Cornejo, "critical speeds resulting fro unbalance excitation of backward whirl odes", Design engineering technical conferences volue 3,84-,ASME 995. [9] Mili J. Hota and D.P. Vakharia. "A Review of Rotating Machinery Critical Speeds and Modes of Vibrations", International Review of Applied Engineering Research. ISSN Volue 4, Nuber 3 (4), pp [] František Trebuna Et.Al, "Nuerically coputed dynaics rotor using Ansys software. Modelling of echanical and echatronic systes ", Modeling Of Mechanical and Mechatronic Systes (). [] M. Varun Kuar, B. Ashiwini Kuar, "Model Analysis of Axially Syetric Linear Rotating Structures", International Journal of Engineering and Advanced Technology (IJEAT) Volue-, Issue-4, April 3. 5