THE INFLUENCE OF TEMPERATURE S CHANGE ALONG THE RADIUS ON THE MEMBRANE STRESSES FIELD AT ROTATING DISKS

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1 6 th Intenational Confeence Computational Mechanics and Vitual Engineeing COMEC Octobe 015, Başov, Romania HE INFLUENCE OF EMPERAURE S CHANGE ALONG HE RADIUS ON HE MEMBRANE SRESSES FIELD A ROAING DISKS Conf. D. Eng. Ioana Comanescu 1, Pof. D. Eng. Gheoghe N. Radu 1 1 ansilvania Univesity of Basov ROMANIA Abstact: he otation of a disk with constant angula velocity leads to a membane stesses field; taction stesses which can be solved with the equations below [4]: EA 1 EA 1 EB 1 EB g g whee - the constant values A and B ae detemined as a function depending ae imposed suppot condition, fo each case patly; E is the mateial s longitudinal modulus of elasticity; - is Poisson s atio; - is the specific mass of the disk s mateial; g In the pesent pape, fo the tempeatue s distibution along the disk adius, one uses the following law: 0 o b a 1 b = 1/3, ½, 1, 30, a and b being the inne and the oute adius of the disk, espectively. Keywods: Disk, eigenvalues poblems, membane stesses, citical stesses, stability. (19) 0 (0) his stesses field intoduces a geat influence upon the membane stesses field given by the change of tempeatue along the adius and one established that its implication on the appeaance of elastic stability loss at disks is vey impotant. he above mentioned influence is put on evidence on basis of the exposed theoy, meaning equations (4) and (5) fom pat I, which ae eigenvalues p oblems which lead to the detemination of citical membane stesses, whee the stability loss may occu and to the detemination of vibation eigenfequencies and eigenmodes, espectively. Based on the known solutions of the two mentioned poblems, on thei intedependence, one sets the poblem of finding the solution of the eigenvalues poblem given by equation (6) (1) K K G M u (1) 0 In pat I the paabolic dependence between the eigenvalues i and i was established and one puts in evidence that in the case of moe existing membane stesses fields which wok simultaneously, the elastic loss of stability at disks occus when the following equation is accomplished: 431

2 n n i 1 One assumes that the citical stesses given by the two studied membane stesses fields ae known (otational motion and tempeatue). One puts in evidence the linea dependence of the citical fields mentioned above, given by equation: R R 1 (3) c c whee epesents the eigenvalues due to the membane stesses field given by the change of tempeatue along the adius; R epesents the eigenvalues due to the membane stesses field given by the otational motion. If one makes in equation (3) the following eplacements: 0, If the eigenvalues R c c () c c 0, we obtain (4) R c c, than fom equation (4) yields: 1 (5) 0 0 c c One eaches to a vey impotant conclusion, that = 1 at the supeposition of the effects of the two membane stesses fields only if the following condition is accomplished: (6) c c getting finally the equation : c (7) c whee 0 epesents the change of tempeatue along the adius due to which the elastic stability loss by the disk banching off (buckling) occus, in the pesence of the two membane stesses fields. Fig. shows the change of citical tempeatues fo stability loss as a function of the disk adius atio, fo diffeent combinations of the numbe of nodal diametes and nodal cicles. Based on the conclusions yielded fom equation (3) one calculated c, c and 0, the esults being tabula pesented. a One mentions that in calculations we consideed moe atios, and also moe types of occuing b the loss of stability, which is put in pactice by: zeo o moe nodal cicles joined with zeo o moe modal diametes. 43

Fig. he solutions of the elastic loss of stability at disks subjected to diffeent combinations based on computational methods, put in evidence that the condition = 1 is accomplished not only fo zeo

3 Fig. he solutions of the elastic loss of stability at disks subjected to diffeent combinations based on computational methods, put in evidence that the condition = 1 is accomplished not only fo zeo nodal cicles and zeo nodal diametes but fo othe situations too. One enumeates some of them: i 0 i 1 i 1 i i n 1, n 0, n 1, n 1, n 0. he othe studied situations ae not fo pactical impotance because it incease vey much the tempeatue diffeence and these cases ae not met in pactice. By i and n ae denoted the numbe of nodal cicles and of nodal diametes, espectively. In the tables met in the pesent pape one pesents the citical tempeatues of stability loss, and also the intedependence of the membane stesses fields. abel 3.1. able 3. Plate on mandel - he table contains the values on c - = 30 (coefficient fom the tempeatue s law of distibution) - = 10C - Disk on mandel - Numbe of nodal cicles: C = 0 - = 1 (coefficient fom the tempeatue s law of distibution) - = 10C; n = 1000 ot/min able Disk on mandel - Numbe of nodal cicles: C = 1 - = 1 (coefficient fom the tempeatue s law of distibution) - - = 10C; n = 1000 ot/min able Disk on mandel - Numbe of nodal cicles: C = - = 1 (coefficient fom the tempeatue s law of distibution - = 10C; n = 1000 ot/min - able 3.1. able

4 - able 3.3. able 3.4. Based on the exposed theoy and on the FEM study the way of stability loss fo diffeent combinations of nodal diametes and nodal cicles is shown in Fig. 3. One establishes that the condition = 1 is accomplished and so the solutions ae in concodance with the theoy. 434

Fig. 3 Fig.4 Fig.5. REFERENCES [1]. S. D. Ponomaiov, s. a., Calculul de ezistenta in constuctia de masini, vol. I, II, III, Editua ehnica, Bucuesti, 1964. []. S. P. imoshenko, S.

5 Fig. 3 Fig.4 Fig.5. REFERENCES [1]. S. D. Ponomaiov, s. a., Calculul de ezistenta in constuctia de masini, vol. I, II, III, Editua ehnica, Bucuesti, []. S. P. imoshenko, S. Woinowsky Kige, eoia placilo plane si cube, Editua ehnica, Bucuesti, [3]. Gh. N. Radu, Contibutii la studiul incovoieii si al stabilitatii elastice al discuilo subtii in miscae de otatie, in egim temic axial simetic. eza de doctoat,

6 [4]. Gh. N. Radu, Rezistenta mateialelo si elemente de teoia elasticitatii, vol.1, vol., Editua Univesitatii ansilvania din Basov, 00. [5]. J. ani, Elastic Instability of an Annula Plate unde Unifom Compession and Lateal Pessue, Jounal of App. Mech., sept

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