COUPLED MODELS OF ROLLING, SLIDING AND WHIRLING FRICTION

Transcription

1 ENOC 008 Saint Petesbug Russia June 30-July COUPLED MODELS OF ROLLING SLIDING AND WHIRLING FRICTION Alexey Kieenkov Ins ti tu te fo P ob le ms in Me ch an ic s Ru ss ia n Ac ad em y of Sc ie nc es Ru ss ia ki eenk@ipmnet.u Abstact It is poposed the essentially new combined models of fiction of ubbed igid solids unde conditions of combined kinematics when besides the sliding and whiling thee is a motion of olling. A coelation between fiction of olling and sliding is modelling on the base expeimental investigations fom the tye and ailway industy. In coespondence with these esults the main influence of the olling on the foce state in the aea of contact consists in the asymmety of the diagam of the distibution of the nomal contact stesses. This asymmety is well descibed by the linea function with one coefficient that depends on the diection of motion and velocity of olling and it leads to appeaance of nonzeo lateal component of the fiction foce. Unde the poposed model of fiction ae undestudied the inteelations between fiction foce components toques and velocities. The model involves the eplacement of exact integal expessions fo the net vecto and toque of the dy fiction foces fomed with the assumption that Coulomb's fiction law is valid at each point of the contact aea by appopiate Pade appoximations. This appoach substantially simplifies the combined dy fiction modeling making the calculation of double integals ove the contact aea unnecessay. Unlike available models the model based on the Pade appoximations enables one to account adequately fo the elationship between foce and kinematical chaacteistics ove the entie ange of angula and linea velocities. The appoximate model peseves all popeties of the model based on the exact integal expessions and coectly descibes the behaviou of the net vecto and toque of the fiction foces and thei fist deivatives at zeo and infinity. Moeove one does not have even to calculate the integals to detemine the coefficients of the Pade appoximation. The coesponded coefficients can be identified fom expeiments. Consequently the models based on Pade appoximations may be consideed as eological models of combined dy fiction. Key wods Combined fiction of olling sliding and whiling.. Intoduction Thee ae many woks in the scientific liteatue devoted to the dy fiction classification of that at the dependence on the aims of investigations can be found at [Zhuavlev V.Ph. Kieenkov A.A. 005]. At the most of these publications authos ae using the Coulomb model of dy fiction supposed that the fiction foce at the point of contact is diect opposite the elative velocities of sliding and it is not depend on the module of velocity. Howeve thee ae many expeimental facts about the violation of this law at case when the ubbed bodies ae paticipated simultaneously in the tanslational whiling and olling motions. Following the expeimental esults fom the tye manufactoy in the wok [Svendenius J. 003] was established the empiic dependence of the distibution of nomal contact stesses at aea of contact fom the velocity of olling. At the coesponded this dependence the influence of whiling is shifting of the symmetic fom of contact stesses distibution in the diection of olling. This shift is good appoximated by the line function with one coefficient depended on the diection and value of the olling velocity. Asymmety at distibution of the nomal contact stesses at the case of cicle aeas of contact cause the appeaance of the component of the fiction foce diected on nomal to the tajectoy of motion.. Combined model of fiction of olling and sliding Constuction of combined model of fiction of olling and sliding is pefomed at he supposition the validities of the Coulomb law at the diffeential fom fo the small element of aea ds inside of spot of contact in coespondence with the diffeentials of the net vecto df and toque dm of the fiction foces elatively C the cente of contact cicle ae defined by the fomulas V V df f ds dmc f ds V V V ( v y x ) whee f - coefficient of fiction x ( y) - adius vecto of the elementay squae inside of spot of contact (fig.) - distibution of nomal contact stesses ()

2 v - linea velocity of sliding and - angle velocity of whiling of contact spot cente. stesses on the value and diection of olling velocity: k (x y ) (x y ) (x y y x ) R (4) Figue 3. The typical behavio of function (4) fo diffeent values of the olling coefficient k at the supposition that at the absence of olling distibution of the nomal contact stesses (solid line) is descibed by Hetz low Figue. Asymmety at the symmetic distibution of nomal contact stesses ( x y) aisen at the non zeo velocity of olling in the ectangula coodinate system {xoy} axis x of which is diected alone the velocity of sliding (fig.) is descibed by the following dependence: (x y ) x y ( x y ) k k k 0 пр и 0 () R whee R - adius of contact cicle - axis of ectan- gula coodinate system diected pependiculaly to the instantaneous velocity of olling (fig. ) and k - dimensionless coefficient the sign of that is de- 3N x y (x y) R R R (5) is pesented at fig.3 by the dash lines. Integation of the expessions () on the spot contact taking in account the fomula (4) gives the exact integal model of combined fiction of sliding and olling ˆ y yr ˆ that in dimensionless vaiables: x xr (xˆˆ y ) ˆˆ (x yˆ) N R in supposition that distibution of contact stess at the absence of olling has cental symmety ( x y) ( ) has in pola coodinate system with oigin at the cente of contact cicle x cos y sin [0] [0 ] following fom (v u sin )( ) k x sin F fn u v uv sin pendent on the diection of motion F fnk y u 3( )cos u v uv sin (6) (u v sin )() k x sin M C frn u v uv sin whee F and F ae the components of the fiction foce diected coespondently on the tangent and nomal to the tajectoy of motion and M C is the toque of whiling espectively the cente of cicle aea diected pependiculaly to plane of whiling. Tansition at the model (6) fom the consideation of the connection of the fiction of olling and sliding in tem of pojection x and y of the velocity of olling Figue. Connection of the coodinate systems {xoy} and { O } ae given by otate tansfom on the angle [0 ] that is defined fom the values of pojec- tions x y of the instantaneous velocity of olling on the axis x and y (fig. ): xcos ysin xsin y cos cos y sin x x y (3) Substitution of expessions (3) to the fomula () gives dependence of distibution of the nomal contact to the its absolute value and to the angle between diection of olling and sliding gives the equivalent fom of this model (v u sin )( ) k sin sin F fn u v uvsin u3( )cos cos F fnk u v uvsin (7) (u vsin ) ( ) k sin sin M C frn u v uv sin One of the distinguish featue of model (6)-(7) is appeaance of none zeo component of fiction foce nomally diected to the tajectoy of motion. At the pes-

3 ence of combined motion of olling and sliding the net vecto of fiction foces is not opposite diected to the vecto of sliding velocity. At supposition that the distibution of the contact (x y ) is play ole of density the violation at its stesses cental symmety defined by the fomula (4) leads to shift of the gavity cente of contact cicle espectively the geometic cente in the diection of whiling (along axe (fig.)) on value s the pojections of which to axes x and y ae defined by the fomulas: s x sk y sk cos s y sk x sk sin (8) s R ( ) 3d 0 The shift of the cente of gavity of contact spot defined by fomulas (8) leads to appeaance of toque of olling M paallelly diected to the plane of sliding the pojections of that on the diections of the tangent M M (u muv) k M u v x M C 0 v muv u u v M (u muv) k M u v sin 0 u muv v u v F (v auv ) k Fu x F 0 v auv u u v F (v auv ) k Fu sin 0 v auv u u v k F (u buv ) y k F (u buv )cos F v buv u v buv u F u v MC a m F0 v v 0 M0 u u 0 y M cos M M x M sin (9) M sk N Thus the net toque of fiction foces at ectangula coodinate system one axis of that is diected on the tangent of tajectoy of motion is M M M M C (0) Expessions (6)-(7) fo toque M C and foce components F F as function of u v have seveal significant popeties detailed investigated in [Kieenkov A.A. 008]. These popeties allow simplifying the fiction modeling with the aid of eplacing of the exact integal models (6)-(7) by the appoximate models based on the Pade appoximations of coesponded ode. This appoach pemits to escape the integation ove the spot of contact. In coesponded with esults of the wok [ Kieenkov A.A. (008)] the combined model fiction olling and sliding of the fist ode based on the patial-linea Pade appoximation has fom: M u k M u v x M 0 u k M u v sin MC 0 u mv u mv F v k F u x F0 v k F usin F 0 v au v au u y k F u cos F k F u bu u bv F0 F u 0 F F v MC m M0 u u 0 v 0 M0 M C v 0 () M u M C u 0 u F v F a F0 v v 0 b k F u u 0 The model of the fist ode () is sufficient fo the dynamics investigation but fo moe pecise qualitative analysis the model of the second ode is equied. This model not only good appoximates the exact integal models (6)-(7) but conseves all thei popeties such as behavio of these functions and thei fist deivatives at zeo and infinity. v F b k F u u 0 The compaison of the integal model (solid line) and models of the fist () (dash-dot line) and the second () (dash line) fo the Hetz distibution of contact stesses (5) as function of paamete k v u is pesented on the fig.4: and nomal M to the tajectoy of motion ae defined by expession: M M () Figue 4.

4 Models ()-() of combined fiction of olling and sliding based on the Pade appoximations can be consideed as eological models because thee ae no equied in solving of eal poblems to calculate the double integals defined the coefficients of Pade appoximations. These coefficients can be defined fom the expeiments. 3. Hetz case If the distibution of nomal contact stesses obeys the Hetz law (5) then with the aid of the tansfe of the oigin of the coodinate system to the instantaneous cente of the velocities O (fig.5) to possibly obtain the pecise equations of model in the elementay functions. of the dimensionless adius-vecto q R is found fom the conditions of the intesection of pola ay with the cicle of the contact aea q [q q ] : q k cos k sin q k cos k sin (4) The distibution of contact stesses which is obeyed the Hetz law (3) in the intoduced vaiables takes the fom 3N (q ) q kq cos( ) k R k y q sin x (k q cos ) (5) The substitution of expession (5) into fomula (3) taking into account of fomulas (4) and location of the cente of instantaneous velocities defines tangential and nomal foce components of fiction foce as the piecewise-continuous functions of the paamete k which ae smooth at joint point: * q k x q ( k q cos ) cos dqd k * q F q k x q ( k q cos ) cos dqd k 0 q * q (q )q (sin ) dqd k y * q F k (6) q (q )q (sin ) dqd k 0 q 3N q kq cos k Integals (6) ae calculated in the quadatues [Kieenkov A.A. 008]. The esult of integation epesents the tangential and nomal component of the fiction foce as function of two paametes k [0 ) and k [ ]. The whiling toque it is calculated on the basis equality [Zhuavlev V.Ph. 998]: M C M h hf whee M h is the main toque of fiction foces elative to the instantaneous cente of the velocities: Figue 5. Nomal and tangential components of the net vecto of fiction foces in the pola coodinate system {O } with the oigin in the instantaneous cente of velocities (Fig. 5) being distant behind the geometic cente of the contact aea to the value h v kr in the diection of nomal to the velocity of sliding speed v ae defined by fomulas: F fr ( q ) q cos dqd G F fr (q )q sin dqd (3) G Integation limits in fomula (3) depend on the aangement of the instantaneous cente of velocities. If the instantaneous cente of velocities is located inside the contact aea k then pola angle [0 ] while if out of the aea of contact that [ * * ] sin * R h k. Inteval of the vaiation * q k x q (k q cos ) dqd k *q M h R q (7) k x q (k q cos ) dqd k 0 q Integals (.) ae also calculated in the elementay functions. Thus the tansfe of the oigin of the coodinate system to the instantaneous cente of velocities makes it possible to constuct in the Hetz case the pecise coupled model of the olling and sliding fiction epesented in the elementay functions. Howeve the obtained esult is too lengthy and is inconvenient. In ode to use it in the dynamics poblems it is necessay to build at the beginning the appopiate Pade appoximations. Consequently even if it is possible to accuately integate the equations of model the most effective appoach is the using of developed above models based on diect constuction of the Pade s appoximations.

5 Refeences Kieenkov A.A. (008). Coupled Models of Sliding and Rolling Fiction in the solids dynamics on the ubbed plane. Izvest. RAS. MTT. 008 No. 3 pp Svendenius J. (003). Tie models fo use in baking application. LICENTIATE THESIS (ISRN LUTFD/TFRT--33--SE). Depatment of Automatic Contol Lund Institute of Technology Box 8 SE 00 Lund Sweden. 95p Zhuavlev V.Ph. (998). The model of dy fiction in the poblem of the olling of igid bodies. Jounal of Applied Mathematics and Mechanics. Vol.6. No.5. pp Zhuavlev V.Ph. Kieenkov A.A. (005). Pade expansions in the two-dimensional model of coulomb fiction. Mechanics of Solids. Vol.40 No.. pp. 0.

6