A ROTATING DISC IN CONSTANT PURE SHEAR BY S. KUMAR AND C. V. JOGA RAO

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1 A ROTATING DISC IN CONSTANT PURE SHEAR BY S. KUMAR AND C. V. JOGA RAO (Deprtment of Aeronuticl Engineering, Indin Institute of Science, Bnglore-3) Received April 25, 1954 SUMMARY The disc of constnt pure sher cn be obtined by tking 9=, positive constnt, throughout the disc. Substituting this in the eqution of equilibrium (in polr co-ordintes) for symmetricl stress distribution we get homogenous differentil eqution of the first order. On integrting this eqution we get the eqution of the thickness distribution for the required disc. Tking f, h s bsciss nd ordinte, h o nd s prmeter, we plot the forms of different profiles. INTRODUCTION Investigtions hve been mde by severl workers to find the stress distribution in rotting discs of different profiles nd to find profiles corresponding to given distribution of stresses. De Lvl' hs designed rotting disc of uniflirm strength. Biezeno nd Grmmel 2 hve given number of profiles of the rotting discs nd their corresponding stress distributions. Stresses in rotting disc of exponentil thickness profile hve recently been given by Lee. 3 The uthors4 hve recently investigted stresses round circulr hole in rotting discs of hyperbolic nd prbolic profiles. This pper presents new type of disc in which the resulting pure sher stress is constnt nerly everywhere in the disc. NOMENCLATURE 102 r, 0 = polr co-ordintes w = ngulr velocity of the disc p,. 6 =-- density of the mteril of the disc = rdil stress = tngentil stress Tre = shering stress in the r, 0 plne R = body force h = thickness of the disc

2 Rotting Disc in Constnt Pure Sher Ii, = thickness of the disc t the periphery = the extreme rdius of the disc 103 = 17 = S. prmeter = constnt stress ow H = ho I. x The equtions of equilibrium in polr co-ordintes re (71- ± I ir r -00 r 1 7 2T 4. ± 11 q + } r -S-ci N. r n =--0 - If the thickness of the disc is smll in comprison to its rdil dimensions, we cn neglect the vrition of the tngentil nd rdil stress over the thickness of the disc. Also, the stress distribution being symmetricl bout the xis of rottion, Tr is zero nd the stresses re not dependent on 0 but on r lone. Since 0.0 = the resulting stress is pure sher. If we consider n infinitesimlly smll rdil element, this resulting pure sher will bisect the directions of nd r. In cse of disc of vrible thickness the first of equtions (I) becomes (2) dr (hr r )--- h + pw 2r 2h -r- The second of equtions (1) vnishes ltogether. Now for pure sher there - should be two perpendiculrly cting uniform norml stresses, one compres sive nd the other tensile. So here we tke (3) 0, positive constnt Therefore, eqution (2) becomes o d- --1, -- h or 6 \o' to2r p 2w2r2 h== 0 \ dr ri

3 104 S. KUMAR AND C. V. JOGA RAO Integrting this we obtin log h= P (1)2 /62 c2 2 log r 4- log k (4) where log k is constnt of integrtion or h p Wirt k e" r 2 (5) Now k being the constnt of integrtion it cn be determined by the boundry conditions of the disc. Let us ssume tht t r =, h= h p bo k e" h t'=-10 Substituting the vlue of k from (6) in (5) we get (6) h 2h 0 r 2 P cute' et 2 (7) This eqution gives the thickness profile of the required rotting disc. To study the vrition of h with respect to r, we introduce convenient prmeters. Eqution (7) cn be written s Now putting we hve 1 h 2 e 2 no r 2 Pco 2 2 h o 2 H 1 ki-4,) u nd 0 --x 2) x 2 e " This is the generl eqution of the profile of the disc. To mke the disc finite in the rdil dimensions nd lso to fcilitte the process we choose 0 < X < I ==x i.e., hs been tken s the peripherl rdius of the disc. At the outer boundry of the disc r should be zero, but this condition violtes our hypothesis of r = 0 =04, constnt, However, t1i4 (8)

4 Rotting Disc in Constnt Pure Sher 105 condition my be pproximtely stisfied by plcing rim t the periphery of the disc. Corresponding to different vlues of the prmeter cr we get different shpes of the disc. Tble 1 shows seven sets of vlues of h r correspondho' TABLE 1 H for S 1, l FIG. I. Sr= Co - =1,2 nd 4. cr Non-dimensionl proftics of the disks of consult,. iding to

5 106 S. KUMAR AND C. V. JOGA RAO ing to seven vlues of - vrying from 1 to 10. The corresponding - profiles re plotted in Fig. I. We find from the curves in Fig. 1, by giving some suitble vlues to h nd, tht the shpes of the discs obtined from the vlues of in the vicinity of 4 my be suitble for prcticl use. Agin by vrying in (7) we find tht this eqution holds good for ny vlue of, though very high vlues of mke the disc only of theoreticl interest, e.g., if is mde finitely gret number, h hs to tend to zero to mintin the ide of finite nd constnt stress throughout the disc. The drwbck in this disc is tht ny prticulr form of the disc is most suitble only for one ngulr velocity. The uthors thnk Mr. S. M. Rmchndr for his vluble criticism of the pper nd Mr. Glileo Bniqued for his help in rithmeticl clcultions. REFERENCES I. Stodol, A... Stem nd Gs Turbines, 1927, Biezeno, C. B. nd Grmme!, R. Technische Dynmik, 1939, Julius Springer, Berlin. 3. Lee, T. C... fount. App. Mech., Sep. 1952, 19, No Kumr, S. nd Jog Ro, C. V. :. J. Ind. Inst. Sc!., April 1953, 32, No. 2.

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