Dynamics of Rotating Discs Mini Poject Repot Submitted by Subhajit Bhattachaya (0ME1041) Unde the guidance of Pof. Anivan Dasgupta Dept. of Mechanical Engineeing, IIT Khaagpu. Depatment of Mechanical Engineeing, Indian Institute of Technology, Khaagpu 7130.
Cetificate This is to cetify that thesis entitled Dynamics of Rotating Discs submitted by Subhajit Bhattachaya to Depatment of Mechanical Engineeing, IIT Khaagpu in patial fulfilment of Bachelo degee in Mechanical Engineeing, is a bona fide ecod of wok caied out unde my supevision and guidance. This fulfils the equiement as pe egulation of the institute and meets the standads of submission. Pof. Anivan Dasgupta, Dept. of Mechanical Engineeing, IIT Khaagpu. Date:
1 Intoduction Rotating discs and simila otating objects appea in vaious pactical poblems in engineeing applications. These include otating shafts, disk clutches, cams, tubine blades, etc. One such specific and athe ecent application is otating data stoage devices in computes. Such stoage devices, like had disks and compact disks, geneally have to undego exteme conditions of stesses at extemely high otation speeds. These speeds typically assume odes of few thousand otations pe minutes. On the othe hand with high pecision opeations in pogess and highly sensitive components being pesent nea the otating disc (e.g. the eading and witing heads), high amplitude of vibations of the disk cannot be toleated. The poblem takes citical tun when the fequency of otation of the disc matches with the natual fequencies of vibation of the disc. At these citical fequencies even the slightest of eccenticity in the disk o vibation matching the fequency of otation of the disk may cause esonance and uncontollable vibations in the disk. This situations need to be avoided. One of the appoaches to avoid such a situation is to design the disk in such a way so that the natual fequencies of vibation of the disk ae inceased consideably. Consequently the pemeable ange of angula otation of the disk will be much wide. But the mateial of the disk needs to be chosen suitably so that it can hold the data laye on its suface satisfactoily. This often pevents the choice of a mateial with high yield stength which could have pushed up the citical fequencies fo the disk. Hence an investigation into the poblem seeking altenative ways fo stengthening the disk without alteing much of its dimensions and mateial popeties and esult in an inceased citical fequency is highly desiable. The pesent poject wok deals with the poblem of inceasing the natual fequencies and hence the citical fequencies of the disk by inseting thin stiffenes into the disk. Intoduction of such stiffenes of highe stength and igidity though does not affect the popeties and pefomance of the disk easonably, accoding to the pesent analysis it is found that they have successfully inceased the natual fequencies of the disk, both in static as well as otating conditions. In the pesent wok some analytical teatment of the poblem along with some FEM simulation of modal vibation of otating disks with stiffenes has been made. The addition of adial stiffenes of vaious shapes showed satisfactoy impovement in the esults. Along with these some inteesting obsevations have been made egading the modal shapes on addition of stiffenes. Pevious Woks The poblem of vibations of disk is not a new one. It finds it place well in vaious text books dealing with vibation of stuctues. Timoshenko [1] has given a detailed analysis of vibation of plates in Catesian Coodinates. The esults have also been obtained in Pola Coodinates by suitable coodinate tansfomation. Howeve the analysis has been done fo the case when the mateial of the plate is homogeneous and is unifom thoughout the solution egion. That is, the values of density, Young s modulus and Poisson s atio emains constant at evey point in the solution domain. With these assumptions the govening diffeential equation fo deflection of a plate unde static load is in geneal given by, q w = (1) D whee, w = deflection in nomal diection, 3
3 Eh D =, with, E = Young s modulus, h = thickness of the plate, ν = Poisson s 1( 1 ν ) atio of the mateial of the plate, q = nomal load pe unit aea on the plate. and = which assumes the expessions = in Catesian coodinate x y 1 1 system and = in Cylindical Pola coodinates. A seies solution of the diffeential equation has also been povided in [1]. Howeve the simple fom of the equation (1) is adically distubed once it is assumed that the mateial popeties ae functions of space. In fact if stiffenes ae inseted into the disk, the mateial popeties can no longe be egaded constant. Unde such cicumstances the whole analysis needs to be epeated with E, ν and hence, D as functions of the space vaiables. Moeove in equation (1) no pe-stessed effects have been consideed that may be caused due to the centipetal foces that act on the diffeent pats of a otating disk. Hence fo an analysis of the pesent poblem the pe-stess effects due to otation need to be consideed. As the pesent poblem deals with cicula disk-like plates which ae otating about thei axis, it is desiable to obtain the equations in cylindical pola coodinates. Hence unlike in Timoshenko [1], whee the oiginal analysis in Catesian coodinates have been late tansfomed to cylindical pola coodinates, the pesent appoach to the poblem has been done in the cylindical pola coodinates fom the vey beginning. 3 The Pesent Analysis In the pesent analysis we deal with only the out of plane modes of vibation of the plate/disk. Hence we have only one displacement vaiable, w, which denotes the vetical displacement of a point on the disk fom its un-displaced position. Hee w is a function of, and t, whee & ae the space vaiables in cylindical pola coodinates and t is the time. As we ae inteested in finding out the natual modes of vibation, we assume that w is a simple hamonic function of time with the same fequency and phase but vaying amplitudes fo all the points on the disk. That is, iωnt w(,, t) = u(, ) e () whee, ω n is the fequency of the paticula natual mode of vibation. 3.1 Moments on an elemental potion of the disk We stat with the expession fo bending and twisting moments on an element of the disk. The expessions fo moments pe unit length have been given by Timoshenko [1] fo any othogonal coodinate system. We have extended the expession fo cylindical pola coodinates. The bending moments pe unit length ae given by, 1 ν M = D (3) ρ ρ 4
1 ν M = D (4) ρ ρ whee, ρ and ρ ae the adii of cuvatue along a adial line and tangent espectively and ae given by, 1 1 ρ = and = w ρ 1 w And the twisting moment pe unit length is given by, 1 w 1 w M = D( 1 ν ) (5) and, M = M Hee it may be noted that E, ν, and hence D ae functions of and. The following figue shows the moment vectos due to the above acting on an elemental potion of the disk. It may be noted hee that the notations used by Timoshenko fo M and M have been intechanged in the pesent analysis. 1 w 3. Shea stesses fig 1 Now, figue shows the diection of the shea stesses acting on the element which contibute to the moments along e and e diections. fig 5
As the moment of inetia of the element about any axis (e o e ) embedded on it is a diffeential of ode 4, the otation of the element about the axes can be neglected. Hence we conside equilibium of the bending & twisting moments and the moments due to the shea foces on the element. Consideing moment about e, ( M ) M d d d d M 1 M M τ z = M h d d M τ h d d z = 0 (6) And, consideing moment about e, ( M ) M d d d d M d d 1 M M τ z = M M h τ h d d z = 0 (7) 3.3 Components of Radial and Cicumfeential stesses due to otation of the disk If we conside the disk to be pe-stessed, thee will be nomal stesses along e and e. As the element has a cuvatue both along e and e diections, thee will be components of foces due to σ and σ along e z (figue 3). fig - 3 On pefoming a simple analysis, it can be shown that the components of the foces due to adial and cicumfeential stesses along -e z ae espectively given by, d Fσ = ( σd h) dφ cos( ψ ) = σd h cos( ψ ) (8) ρ d and, F σ = ( σ d h) dφ cos( ψ ) = σ d h cos( ψ ) (9) ρ 1 w 1 w whee, ψ = tan and ψ = tan 1. Fo a disk with i and o as intenal and extenal adii espectively and fixed at the inne cicumfeence (as in the pesent case) and otating with angula fequency ω, the adial and cicumfeential stesses ae given by, 3 ν i o σ = ρω i o 8 (11) 6
and, = 3 3 1 8 3 o i o i ν ν ρω ν σ (1) 3.4 The final equation of motion Hence, the net foce on the element along e z due to the τ z, τ z, σ and σ causes it to acceleate along e z. Hence, the final equation of motion is given by, ( ) ( ) σ σ τ τ ρ F F d d h d h d t w d d h z z = whee, ρ is the density of the mateial and is a function of and. This gives, = 1 1 1 1 1 1 ρ σ ρ σ τ τ τ ρ w w t w z z z (10) Now, putting in (10) the expession fo w in tems of u and ω n fom (), and pefoming all the calculations and simplifications using Mathematica 5.1 the following diffeential equation was obtained, = u n ω ρ 7
The notations used hee ae as follows: i i, o o, ν[, ] ν, ρ[, ] ρ, ed[,] ( and ξ p, q) 3 Eh D =, u[,] u, 1 ν ( 1 ) ( p q) ξ [, ] whee ξ is D, ρ, ν o u fo any p and q. p q The above esult was coss-checked by putting constant values of D, ρ and ν. It gave back the esults as in [1] fo disk with constant mateial popeties. 8
The non-tivial solutions to this patial diffeential equation in and with appopiate bounday conditions give the modal shapes of the otating disk with inhomogeneous mateial popeties like stiffenes, etc. And the coesponding ω n s gives the natual fequencies. 3.5 Possibilities of solution As it can be seen, the obtained diffeential equation is a petty huge one and is difficult to handle analytically without any suitable appoximations. Attempts wee made to educe the patial diffeential equation to odinay ones using sepaation of vaiable method. The substitution u(,) = u 1 ().u () was done, but without much simplification o sepaation of the vaiable and. Howeve thee ae possibilities of futhe investigation into the equation and solving it analytically using suitable methods. Howeve, as the pesent poblem deals mainly with adial stiffenes (figue 4), a possible simplification of the equation may be pefomed by assuming that the popeties like D, ρ and ν ae functions of only. fig - 4 Moeove if we assume the stiffenes to be vey thin and having dastically diffeent mateial popety values compaed to that of the disk itself, the popety functions D, ρ and ν may be appoximated by a Diac Delta function as follows: n πk ξ (, ) = ξ ξ δ, disk stiffne whee, ξ is any popety of the mateial integated ove length, n = numbe of equispaced stiffenes on the disk. It may be noted that the domain of in which all the analysis ae done is assumed to be [-π, π]. k = n n 4 Finite Element Analysis The above patial diffeential equation can be attempted to be solved using suitable numeical techniques. Howeve as a pat of the pesent wok, the numeical solutions have been pefomed using the FEM softwae Ansys. The desciption of the geomety, mateial popeties, bounday conditions, gid type used, meshing and mode extaction method used ae given below. All the values mentioned hee ae in SI unit system. 9
4.1 Geomety of the Disk The disk was basically a thin annula cylinde with intenal adius( i ) = 0.01, extenal adius( o ) = 0.051, thickness(h) = 0.001. The geomety and numbe of the stiffenes wee vaied and diffeent sets of esults wee obtained fo each of them. 4. Mateial Popeties The mateial of the disk is consideed to be a type of plastic polyme, and the stiffenes wee assumed to be made of steel. Hence the mateial popety values wee chosen accodingly. The mateial of the disk was chosen to have the following popeties: E = 40 10 9, ρ = 000, ν = 0.5 And the mateial of the stiffene was chosen to have the following popeties: E = 00 10 9, ρ = 7800, ν = 0.3 4.3 Bounday Conditions fig - 5 The bounday condition was set so as to ensue that the disk is clamped at it s inne cicumfeence. In ode to ensue that, the suface aea of the inne cylinde of the disk was declaed to have zeo displacement along all the thee degees of feedom. 4.4 Gid type, meshing and mode extaction technique Fo meshing the volume of the disk, the 0-nodes solid element SOLID95 povided in Ansys was chosen. The paticula choice was made because the SOLID95 element can toleate iegula shapes without much loss of accuacy and the elements have compatible displacement shapes and ae well suited to model cuved boundaies. Hence fo the pesent poblem dealing with thin cicula disk, this element was found to be most suitable. The meshing of both the disk and the stiffene volumes wee done using unstuctued gids. Fo the pupose of contolling the size of the elements Ansys s Smat Size tool was used. Fo the volumes of the disk, the size level was set to 7 and fo the stiffenes the size level was set to 6. Fo each case, fist a static analysis was pefomed with the pestessed effect on and with an angula velocity of the global coodinated about z-axis to account fo the otation of the disk. It was followed by a modal analysis with the peviously obtained pestess data. Fo the modal analysis, the method used fo extaction of the eigenvalues is Block Lanczos. The following section descibes the geomety, position and numbe of stiffenes used and the coesponding esults obtained in each case. 10
5 Results The standad mode shapes fo disk without stiffenes and clamped at the inne cicumfeence consists of nodal cicumfeences and nodal diametes. A mode shape with i nodal diametes and j nodal cicles is temed as mode (i, j). The following figues show some typical mode shapes (the lines epesent the nodes): (0, 0) (1, 0) (, 0) (0, 1) (1, 1) (, 1) fig 6 11
5.1 Disk with no stiffenes Modes obtained: Fist 0 modes wee extacted, and the modal shapes obtained wee the standad ones. A plot of the modal fequencies against the angula velocity of the disk is pefomed. The intesections of the staight lines with slopes, 3, etc with cuves coesponding to modes (1,0), (,0), etc give the citical fequencies. The following gaph shows the plot fo only the fist 5 modes: It was obseved that the slope 1 line almost became asymptotic to the mode (0,0) cuve. This is a esult expected fom the standad calculations fo disk with no stiffenes. 1
5. Disk with thee staight equispaced adial stiffenes Stiffene Geomety: The stiffenes ae simple thin ectangula paallelopipeds with length o - i = 0.041 and both height and thickness = h = 0.001. Modes obtained: The mode shapes obtained wee same as befoe, but fo all the modes (i, j) with i as multiple of 3, the fequencies of the othogonal modes got splitted. The splitted modes ae denoted by A fo the modes which have stiffenes on antinodes and B fo the modes which have stiffenes on modal diametes. fig 7 : A typical (3, 0) mode Again, the plot of modal fequencies against the angula velocity was made. 13
5.3 Disk with fou staight equispaced adial stiffenes Stiffene Geomety: Same as section 5.. Modes obtained: Mode shapes obtained wee same as befoe, but splitting of othogonal modes was obseved fo modes with nodal diametes multiples of 4. It may be obseved hee that till now thee has not been any significant change in the citical angula velocities because of addition of stiffenes to the disk. Hence an investigation by alteing the geomety of the stiffenes may be done to see if the citical angula velocities go up. The following sections show the esults obtained by alteing the stiffene geometies. 14
5.4 Disk with thee expanding (naowe nea the inne cicumfeence, wide nea the oute cicumfeence) equispaced adial stiffenes Stiffene Geomety: The stiffenes ae tapezoidal shaped thin blocks with width of 0.0004 at the inne cicumfeence and 0.00 at the oute cicumfeence. The thickness is unifom thought and is equal to h = 0.001. On stating the analysis with zeo angula velocity of the disk, it was found that the modal fequencies, and hence the citical speeds deceased consideably compaed to the staight stiffenes case. This as the undesied case, hence futhe continuation of analysis with this geomety of stiffene was discontinued. Howeve it was clea fom the above mentioned obsevation that an inceased mass concentation nea the oute cicumfeence is not desiable. Hence it may be inteesting to do some study with stiffenes having highe mass concentation nea the inne cicumfeence. 15
5.5 Disk with thee contacting (wide nea the inne cicumfeence, naowe nea the oute cicumfeence) equispaced adial stiffenes Stiffene Geomety: As in 5.4, the stiffenes ae the same tapezoidal shaped thin blocks, but they ae now placed in a evese oientation. That is, they have a width of 0.0004 at the oute cicumfeence and 0.00 at the inne cicumfeence. The thickness is unifom thought and is equal to h = 0.001. Modes obtained: Mode shapes obtained wee simila to 5., with splitted othogonal modes fo modes with nodal diametes multiples of 3. Howeve in this case, a few modes wee found to be slightly defomed fom the standad mode shapes. 16
5.6 Intemediate Conclusions fo poceeding with futhe modifications on the stiffene geomety Though not vey evident fom the pevious gaphs, thee had been a mino incease in the citical fequencies with contacting stiffenes when compaed with the pevious ones. A close compaative study of the fequencies of mode (1, 0) and its intesection with slope line may eveal the fact. The above gaph eveals: With addition of stiffenes, the lowest citical velocity has gone up slightly. By inceasing the numbe of stiffenes fom 3 to 4 not much diffeence id made on the citical velocities. By inceasing the mass concentation of the stiffenes nea the inne cicumfeence thee has been some incease in the citical velocity. Howeve, in all the above mentioned cases the value of the fist citical angula velocity lies within the value 1500 (± 50) ad/s. Hence nothing much has yet been achieved. Hence futhe investigation is equied Fom the above dawn conclusions it was logical to investigate the poblem with stiffenes having even highe mass concentation nea the inne cicumfeence. The following section deals with such a stiffene geomety, which is a modification on the contacting stiffene, and was found to give much bette esults. 17
5.7 Disk with thee aised, contacting (aised above the suface of the disk and wide nea the inne cicumfeence) equispaced adial stiffenes Stiffene Geomety: The stiffenes ae the simila to those of section 5.5, but they ae now also aised above the suface of the disk nea the inne cicumfeence and gadually slopes down to meet the disk suface at the oute cicumfeence. Hence, they ae a sot of tuncated pyamidal shaped stiffenes with the base of the pyamid at the inne cicumfeence, and apex at the oute cicumfeence. At the inne cicumfeence they have a width of 0.00 and thickness of 0.0051. And at the oute cicumfeence they have a width of 0.0004 and thickness of 0.001. Thus the geomety appeas something as shown below (figue not to the scale): fig 8 Modes obtained: In this case the mode shapes obtained wee athe vey inteesting. Apat fom a few standad highe mode shapes (like (4,0), (5,0), (6,0) and (3,1)), a few new types of modes wee obtained, some of which wee much defomed an asymmetic. It was inteesting to obseve that the standad modes with the lowe modal fequencies wee completely eplaced by new modes with much highe modal fequencies. Hence the fist few citical speeds of the disk wee expected to incease consideably. The esults obtained follows. 18
As only a few of the standad modes wee available, they ae shown in the following gaph: But it will be of geate inteest to make a study on the new mode shapes obtained with the pesent stiffene geomety. The following figues show some of those mode shapes: 19
fig 9 : Unusual mode shapes obtained As most of the standad (i, j) modes ae absent with the pesent stiffene geomety, we will tem the modes mode-1, mode-, etc. in ascending ode of thei modal fequencies. Fo the pupose of compaison with the othe stiffene geometies the modal fequency vs. angula velocity gaphs wee plotted fo the diffeent stiffenes fo mode-1, mode- and mode-3. 0
1
Fom the above gaphs it can easily be seen how the 3 contacting and aised stiffenes used in this section have inceased the modal fequencies fo the disk substantially. Howeve, as many of the standad modes wee absent, it is difficult to daw any immediate conclusions egading the citical speed of the disk. But as the modal fequencies wee found to incease consideably, one can logically expect the citical velocities to incease accodingly. Hence the aised contacting stiffene gave extemely desiable esults by inceasing the modal fequencies. Even if we keep some allowance in these esults in ode to account fo the numeical eos caused due to diffeence in meshing, the esults show a high potential fo the success of the stiffene geomety mentioned in 5.5. Howeve some of the mode shapes obtained in the simulation of 5.5 wee highly defomed and asymmetic. This may be because of numeical eos caused by uneven meshing, limitations of the mode extaction and solving methods used, etc. Futhe investigation is possible in ode to explain these anomalous modal shapes. 6 Conclusions The final conclusions that can be dawn fom the above analysis and esults: An analytical solution has been attempted in ode to account fo vaiation of mateial popeties within the disk, which is in fact the case fo disks with stiffenes. A diffeential equation has been successfully set up and coss-checked by putting constant values of mateial popeties to obtain the equation in [1]. Howeve a final solution could not be achieved at the pesent moment due to the complexity of the diffeential equation. Futhe studies on the obtained patial diffeential equation with appopiate appoximations may lead to a satisfactoy analytical solution. Using the FEM softwae Ansys, modal analysis of the otating disk with stiffenes of diffeent geometies wee pefomed. A gadual development of the stiffene geometies on the basis of conclusions dawn fom intemediate esults finally yielded a stiffene which could successfully push up the modal fequencies, and hence potentially incease the citical speeds of the disk. Futhe investigation into the poblem may esult in a successful analytical method fo dealing with such disks with stiffenes. Moeove vaiation in the dimensions and geomety of the obtained stiffene may yield bette and inteesting esults.
7 Refeences [1] S.Timoshenko and S.W.Kiege, Theoy of Plates and Shells, Pentice Hall. [] Ego Popov, (1973), Intoduction to Mechanics of Solids, Delhi, Pintice Hall. [3] Ewin Keyszig, Advanced Engineeing Mathematics, John Wiley & Sons Inc. 3