Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 46, 2, pp. 367-381, Warsaw 2008 EFFECT OF ANGLE OF THREAD INCLINATION AND EQUIVALENT RADIUS ON ENERGY DISSIPATION IN SCREW JOINTS Wojciech Kaczmarek Warsaw Military University of Technology, Instytute of Mechatronics System, Warsaw, Poland e-mail: Wojciech.Kaczmarek@wat.edu.pl The paper presents the problem of structural friction appearing in a screw joint with frictional and elastic-frictional effects between its ele- ments. Two mathematical models of the screw joint taking into acount Lame’s problem are analysed. Key words: energy dissipation, structural friction, screw joint, thread, equivalent radius, Lame’s problem 1. Introduction In thepaper, the influenceof theangleof thread inclination ina screw jointand the effect of equivalent radius on energy dissipation in mathematical models representing screw joints are presented. Vibrations bring about formation of variable stress in machine elements. They lead to distortions in bearings, joints, etc. Vibrations of machines and vehicles have harmful influence on the human organism. Vibration of gases andmechanical elements are a source of noise. The largest intensity of vibration appears in resonance states. Therefore, selection of natural vibrations of a given system is very important.Escape from a resonance zone does not solve the whole problem, however. In many cases, designers use special dampers of vibrations. Admittedly, they raise costs and cause additional problemswith exploitation. Thus, it seems useful to continue works on suppression of vibrations bynaturalways, such as structural friction. Damping of vibration results from mechanical energy dissipation which is associated with motion of mechanical assemblies. Large vibrations can cause 368 W. Kaczmarek defective operation of equipment (reduced accuracy of performance of ele- ments). Vibrations of a system can result in jam of mechanical elements or it can disturb their operation. This can disconnection of joints (threads). In the paper, the problemof energy dissipation is presented on an example of structural friction. When small forces are exerted to shift two elements pressed one against the other, elastic deformation of the small contact surfaces follows.On these areas, very small slips appear.When the friction forceswork, dissipation of energy appears. Inmachines and systems, there aremany joints that are motionless by definition. Yet, in consequence of the deformations in joint bodies, dissipation of energy appears in them. The resistance caused by friction in motionless joints related to elasticity of the connected bodies is called structural friction (Giergiel, 1969, 1971; Osiński, 1986, 1997; Skup and Kaczmarek, 2005). The problem of energy dissipation in screw joints is not new and was presented by e.g. byKalinin et al. (1960, 1961), Panovko and Strakhov (1959). Anewapproach to the problem is investigation of elastic-frictional effectswith Lame’s problem taken into consideration. Fig. 1. Physical model of the considered screw joint 2. Mathematical models of screw joints Thepurposeof this paper is to showthe influenceof geometrical parameters on the value of natural dissipation of energy in a screw joint. The paper presents twomodels of the screw joint. In the firstmodel (Fig.2), it has been assumed that the interaction between the elements of the screw joint is frictional, and in the second the interaction is elastic and frictional (Fig.3). Notations on Fig.2 and Fig.3 r0 – equivalent radius rz,rw – external and internal radii of the surface of segment rz0 – external radius of the screw joint Effect of angle of thread inclination... 369 Fig. 2. Model 1 with frictional effects between its elements – segment of the screw joint: (a) axial loading, (b) axial releasing, (c) dislocation the screw joint segment Fig. 3. Model 2 with elastic and frictional effects between its elements – segment of the screw joint: (a) axial loading, (b) axial releasing p – unit pressure ∆x – height of the segment µ – friction coefficient σx1,σx2 – stresses β – angle of thread inclination of the screw joint u – dislocation of the segment ∆ – clearance between cooperating elements Ps – force of the elastic resistance. In the first place, a sector of the screw joint has been analysed (Fig.2 and Fig.3). Thewhole screw joint can be considered as a jointwith a large number of such sectors placed in the nut (each sector consists of one coil of thread). 370 W. Kaczmarek One cycle of loading consist of four stages: • stage one, the process of loading – the elements of the system get relo- cated (segment 1 – Fig.4) • stage two, the process of unloading – the friction forces change the sign and the elements of the system do not get relocated (segment 2 – Fig.4) • stage three, the process of unloading – the elements of the system get relocated (segment 3 – Fig.4) • stage four, the process of loading – the friction forces change the sign and the elements of the systemdonot get relocated (segment 4 –Fig.4). Fig. 4. Hysteresis loop of the screw joint During one cycle of loading, the energy dissipation in the system ismeasu- red by the field of a hysteresis loop in a triangle or quadrangle form (Fig.4). In the considerations, the following assumptions have beenmade: 1. The distribution of the pressure per unit area between cooperating sur- faces of the screw joint is even. 2. The joined elements can be characterized by a constant coefficient of friction for any value of the pressure per unit area. 3. Friction forces on the faying surface of mating elements are subject to Coulomb’s law. 4. Properties of the material are subject to Hooke’s law. 5. Friction is fully developed in the slip zone and amounts to zero outside of it. 6. Changes in the force P are smooth,which justifies the omission of inertia forces. Effect of angle of thread inclination... 371 7. Assumption of plane sections (transverse sections are plane and do not deform under stress) holds. 8. In the model, the role of internal friction is not taken into account. The conditions for the equilibrium of forces in the sector of the screw joint are (Kalinin et al., 1960) ∆σ1xF1 = ∆σ2xF2 (2.1) where: F1, F2 are the fields of the cross-sectional area of the bolt and nut sectors. 2.1. Model 1 The equation of the equilibrium of forces working in the system (in the direction of the x-axis, Fig.2) is for the screw sector −σ1xπr 2 0 −µp2πr0cosβ ∆x 2cosβ −p2πr0 sinβ ∆x 2cosβ +(σ1x+∆σ1x)πr 2 0 =0 (2.2) The equation of the equilibrium of forces operating in the system in the radial direction (y-axis, Fig.2) for the screw sector is −σ1r+µpsinβ −pcosβ =0 (2.3) The equation of the equilibrium of forces working in the system (in the direction of the x-axis, Fig.2) is for the nut sector −σ2xπ(r 2 z0−r 2 0)+µp2πr0cosβ ∆x 2cosβ +p2πr0 sinβ ∆x 2cosβ + (2.4) +(σ2x+∆σ2x)π(r 2 z0−r 2 0)= 0 In accordance with Lame’s problem, the equations of the radial and hoop stresses in the system (Fig.5) for the nut sector can be expressed as σ2r = pcosβ(1−µtanβ)r20 (r20 −r 2 z0) ( 1− r2z0 r2 ) (2.5) σ2t = pcosβ(1−µtanβ)r20 (r20 −r 2 z0) ( 1+ r2z0 r2 ) The radial dislocation up (in accordance with Lame’s problem, Fig.5c) is equal up ∣ ∣ ∣ (r=r0) =− ∆σ1x ∆x r20 cosβ(1−µtanβ) E2(µ+tanβ)(r 2 0 −r 2 z0) [(1−ν2)r 2 0 +(1+ν2)r 2 z0] (2.6) 372 W. Kaczmarek Fig. 5. Stresses and dislocations in the nut sector It has been assumed that during the loading the sectors do not displace axially, and amargin ∆ appears between them (the radius of the screw sector grows and the radius of the nut sector decreases – Fig.2c). Thedisplacement of thewhole screw joint has been found for the boundary conditions σ1x =        0 for x =0 αP πr20 for x = H (2.7) and can be described as u = αP πr20 [ w2− a0 2 w1+ w1 √ a20+4b0 2tanh ( H 2 √ a20+4b0 ) ] (2.8) where a0 = (w2−m4) w1 b0 = 1 E1w1 m4 = 2ν1r0cosβ(1−µtanβ) E1(µ+tanβ) Effect of angle of thread inclination... 373 w1 =− r20 cosβ(1−µtanβ)[(1−ν2)r 2 0 +(1+ν2)r 2 z0] E2 tanβ(µ+tanβ)(r 2 0 −r 2 z0) + + r20 cosβ(1−µtanβ)(1−ν1) E1 tanβ(µ+tanβ) w2 = ν1r0 E1 tanβ where H – height of the screw joint u – axial displacement P – axial force p – pressure per unit area µ – coefficient of friction Ei – Young’s modulus α – coefficient which changes from 0 to 1. Similarly, the dislocation of the extreme cross-section in Stage 3 has been determined u = αP πr20 [ w4− a1 2 w3+ w3 √ a21+4b1 2tanh ( H 2 √ a21+4b1 ) ] (2.9) where a1 = (w4−m5) w3 b1 = 1 E1w3 m5 = 2ν1r0cosβ(1+µtanβ) E1(tanβ −µ) w3 =− r20 cosβ(1+µtanβ)[(1−ν2)r 2 0 +(1+ν2)r 2 z0] E2 tanβ(tanβ −µ)(r 2 0 −r 2 z0) + + r20 cosβ(1+µtanβ)(1−ν1) E1 tanβ(tanβ −µ) w4 = ν1r0 E1 tanβ The energy dissipation for one cycle of stress (Fig.4) equals ψ = P2α21m6 2πr20 ( 1− m6 m7 ) − P2α23m7 2πr20 (m7 m6 −1 ) (2.10) 374 W. Kaczmarek where m6 = w2− a0 2 w1+ w1 √ a20+4b0 2tanh ( H 2 √ a20+4b0 ) m7 = w4− a1 2 w3+ w3 √ a21+4b1 2tanh ( H 2 √ a21+4b1 ) 2.2. Model 2 Thedislocation of the extreme cross-section and energy dissipation for one cycle of stress for Model 2 have been obtained similarly, and they have the following form u = αP πr20 [ w8− a0 2 w7+ w7 √ a20+4b0 2tanh ( H 2 √ a20+4b0 ) ] (2.11) u = αP πr20 [ w12− a1 2 w11+ w11 √ a21+4b1 2tanh ( H 2 √ a21+4b1 ) ] (2.12) where w9 =− r20 cosβ(1−µtanβ)[(1−ν2)r 2 0 +(1+ν2)r 2 z0] E2(r 2 0 −r 2 z0)(tanβ +µ) + + r20 cosβ(1−µtanβ)(1−ν1) E1(µ+tanβ) w10 = cS2[(1−ν2)r 2 0 +(1+ν2)r 2 z0] E2(r 2 0 −r 2 z0) (cosβ(1−µtanβ) π∆x(µ+tanβ) +r0 tanβ ) + + cS1cosβ(1−ν1)(1+tan 2β) E1π∆x(µ+tanβ) w13 =− r20 cosβ(1+µtanβ)[(1−ν2)r 2 0 +(1+ν2)r 2 z0] E2(r 2 0 −r 2 z0)(tanβ −µ) + + r20 cosβ(1+µtanβ)(1−ν1) E1(tanβ −µ) w14 = cS2[(1−ν2)r 2 0 +(1+ν2)r 2 z0] E2(r 2 0 −r 2 z0) (cosβ(1+µtanβ) π∆x(tanβ −µ) +r0 tanβ ) + + cS1cosβ(1−ν1)(1+tan 2β) E1π∆x(tanβ −µ) Effect of angle of thread inclination... 375 w7 = w9 tanβ +w10 w8 = ν1r0 E1(tanβ +w10) w11 = w13 tanβ +w14 w12 = ν1r0 E1(tanβ +w14) m8 = 2ν1r0cosβ(1−µtanβ) E1(µ+tanβ) m9 = 2ν1cS1cosβ(1+tan 2β) E1πr0s(tanβ +µ) m10 = 2ν1r0cosβ(1+µtanβ) E1(tanβ −µ) m11 = 2ν1cS1cosβ(1+tan 2β) E1πr0s(tanβ −µ) a0 = 1 w7 (w8−m8+m9w7) b0 = 1 w7 ( 1 E1 −m9w8 ) a1 = 1 w11 (w12−m10+m11w11) b1 = 1 w11 ( 1 E1 −m11w12 ) The energy dissipation for one cycle of stress (Fig.4) equals ψ = P2α21m12 2πr20 ( 1− m12 m13 ) − P2α23m13 2πr20 (m13 m12 −1 ) (2.13) where m12 = w8− a0 2 w7+ w7 √ a20+4b0 2tanh ( H 2 √ a20+4b0 ) m13 = w12− a1 2 w11+ w11 √ a21+4b1 2tanh ( H 2 √ a21+4b1 ) 3. Simulation results Simulations have been carried out with the help of a professional software (Mathematica 4.1). Themain purpose of the simulationswas to show the influence of the angle of thread inclination in the screw joint and of the equivalent radius on energy 376 W. Kaczmarek dissipation for the assumed frictional models of bolted joints. Table 1 shows parameters of the screw joint which were used in computer calculations. Table 1 Parameter [unit] Value Maximum stress σ [N/m2] 158.126 ·106 Maximum axial force P [N] 8000 Parameter α1 1 Parameter α3 0.125 Poisson’s ratio ν 0.32 Young’s modulus E [N/m2] 2.1 ·1011 Coefficient of friction µ 0.166 Angle of thread inclination of the screw joint β [◦] 45; 50; 55; 60 Outer diameter of the screw dz [m] 10 ·10 −3 Inner diameter of the screw dw [m] 8.026 ·10 −3 Equivalent diameter of the screw joint dz0 [m] 13 ·10 −3 Height of the screw joint H [m] 20.03 ·10−3 Pitch of thread s [m] 1.5 ·10−3 Equivalent radius r0 [m] 4.525 ·10 −3 The area of the hysteresis loop and themaximal values of displacement for different angles of thread inclination are presented in Table 2 and Table 3. Tabela 2 Angle Hysteresis area ψ [Nm] In percentages [%] β [◦] Model 1 Model 2 Model 1 Model 2 60 0.00268301 0.00223170 100 83.18 55 0.00337004 0.00274716 100 81.52 Angle Displacement u [mm] In percentages [%] β [◦] Model 1 Model 2 Model 1 Model 2 60 0.00214850 0.00187719 100 87.38 55 0.00285121 0.00247360 100 86.93 Simulations show (Fig.7) that the relations between the angle of thread inclination and the field of the hysteresis loop are not linear. Similarly is with the dependences between the angle of thread inclination and the displacement of the extreme cross-section. They are not linear too (Fig.6). Effect of angle of thread inclination... 377 Tabela 3 Model Hysteresis area ψ [Nm] In percentages [%] β =55◦ β =60◦ β =55◦ β =60◦ Model 1 0.00337004 0.00268301 100 79.61 Model 2 0.00274716 0.00223170 100 81.24 Model Displacement u [mm] In percentages [%] β =55◦ β =60◦ β =55◦ β =60◦ Model 1 0.00285121 0.00214850 100 73.35 Model 2 0.00247360 0.00187719 100 75.89 Fig. 6. Screw joint dislocation versus thread inclination angle Fig. 7. Energy losses versus thread inclination angle ψ(β) Moreover, one can easily notice that (for bothmodels Fig.6) it is very im- portant to remember that every screw joint should be self-locking and changes of the angle should be limited. Another problem is the equivalent radius r0. In frictional models of screw joints, its effect on the displacement of the extreme cross-section is not linear 378 W. Kaczmarek (Fig.8). Similarly, the influence of the equivalent radius on energy dissipation (Fig.9) non-linear too. Fig. 8. Dislocation of the screw joint versus equivalent radius r0 Fig. 9. Energy losses versus equivalent radius of a screw joint r0 The range of the equivalent radius is limited by the external and internal radius of a screw. Figures 8 and 9 show that for both mathematical models, the displacement and dissipation energy decrease at first and grow afterwards. All diagrams (Fig.6-9) show that the displacement as well as the area of the hysteresis loop are larger for the frictional than for the elasto-frictional model. 4. Conclusion The problem of energy dissipation in screw joints is not new and was pre- sented e.g. by Kalinin et al. (1960, 1961), Panovko and Strakhov (1959). A new approach to the investigation of the problem is the study elasto-frictional effects between cooperating elements (screw-nut) with Lame’s problem taken into consideration at the same time. Effect of angle of thread inclination... 379 Structural friction in screw joints can be used to damp vibrations in sys- tems subject to dynamic loads. Moreover, it is a natural way of energy dis- sipation. In the paper, basic laws of mechanics have been applied for the determination of the distribution of stresses and deformations at tension and compression. The assumptions are presented in Section 2, and the results of simulation in Section 3. The paper presents two mathematical models of screw joints, in which Lame’s problem is taken into consideration. Themain purpose is to show the influenceof theangle of thread inclination of a screw joint andof the equivalent radius on energy dissipation in frictional models of screw joints. Calculations have been conducted with the professional software Mathematica 4.1, which enabled quick analysis. The area of the hysteresis loop and the maximal va- lues of displacement for different angles of thread inclination are presented in Table 2 and Table 3. Simulations showed that (for bothmodelsFig.6)with a growth of the thre- ad inclination angle leads to decrement of both quantities. It can be concluded that in inch screw joints, in which the thread angle amounts to 55 degrees, the natural dissipation of energy is larger than in metric threads, in which this angle amounts to 60 degrees. One can formulate a theorem that the design of machines, devices and structures aimed to optimally exploit natural properties of vibration damping in screw joints, requires incorporation of screw threads with the smallest an- gle of thread inclination β (Fig.6) with all geometrical parameters and the property of self-locking taken into consideration. Mathematical models are created based on the concept of the equivalent radius.The value of the equivalent radius depends on the character of pressure per unit area. In the paper, it has been assumed that the distribution of the pressure per unit area between cooperating surfaces of the screw joint is even (2.1)-(2.13). This is not true. In real mechanical systems, the distribution of the pressure per unit area can have different shapes (triangular, parabolic, etc.), and this makes the value of the equivalent radius different too. Figures 8 and9 show (for bothmathematicalmodels) that if the equivalent radiusgrows, thedisplacementandenergydissipationdecreaseatfirstbut then grow as well. All figures (Fig.6-9) show that the value of the displacement as well as the hysteresis loop area are larger for the frictional model than for the elasto-frictional model of the joint. The obtained results confirm that research in this direction is purposeful and structural friction in screw joints can be used to damp vibrations in dynamic systems containing such joints. 380 W. Kaczmarek References 1. Giergiel J., 1969, Problemy tarcia konstrukcyjnegow nieruchomych połącze- niach, Zeszyty Naukowe AGH, 30 2. Giergiel J., 1971, Problemy tarcia konstrukcyjnegow dynamicemaszyn, Ze- szyty Naukowe AGH, 44 3. Kaczmarek W., 2001a, Badanie modelu połączenia gwintowego z czysto- tarciowym oddziaływaniem elementów,Biuletyn WAT, 592, 12, 127-145 4. Kaczmarek W., 2001b, Rozpraszanie energii w połączeniu gwintowym z czysto-tarciowym oddziaływaniem elementów,Biuletyn WAT, 588, 8, 89-107 5. Kaczmarek W., 2001c, Rozpraszanie energii w połączeniu gwintowym ze sprężysto-tarciowymoddziaływaniem elementów,Prace Instytutu Podstaw Bu- dowy Maszyn Politechniki Warszawskiej, 21, 83-101 6. Kaczmarek W., 2003a,Analiza czysto-tarciowegomodelu połączenia gwinto- wego,Biuletyn WAT,LII, 5/6, 123-139 7. Kaczmarek W., 2003b, Analysis of a bolted joint with elastic and frictional effects occurring between its elements, Machine Dynamics Problems, 27, 1 8. Kaczmarek W., 2003c, Badanie rozpraszania energii w modelu połączenia gwintowego za pomocąmaszynyMTS,Biuletyn WAT,LII, 5/6, 141-159 9. Kalinin N.G., Lebedev Yu.A., Lebedeva V.I., Panovko Ya.G., Stra- khovG.I., 1960,Konstrukcyonnoe dempfirovanie v nepodwizhnykh soedineniy- akh, Riga: AN Łotew. SRR 10. KalininN.G.,PanovkoJ.G.,LebedevV.J., 1961,Konstrukcyonnoe demp- firovanie v nepodwizhnykh soedineniyakh, Ryga 11. Osiński Z., 1982, Tarcie konstrukcyjne w utwierdzeniu belki przy czystym zginaniu,Prace Instytutu Podstaw Budowy Maszyn Politechniki Warszawskiej, Warszawa 12. Osiński Z., 1986,Tłumienie drgań mechanicznych, PWN,Warszawa 13. Osiński Z., 1997,Tłumienie drgań, PWN,Warszawa 14. Panovko J.G., Strakhow G.J., 1959, Konstrukcyonnoe dempfirowanie v rezbowykh soedeneniyakh, Izv. AN Łotev. SSR 15. Skup Z., Kaczmarek W., 2005, Analiza porównawcza badań teoretycznych i doświadczalnych rozpraszania energii w modelu połączenia gwintowego z uwzględnieniemtarcia konstrukcyjnego,XVKonferencja ”Metody i Środki Pro- jektowania Wspomaganego Komputerowo”, Kazimierz Dolny, 347-357 Effect of angle of thread inclination... 381 Wpływ kąta pochylenia gwintu i promienia ekwiwalentnego na rozpraszanie energii w połączeniu gwintowym Streszczenie W artykule przedstawiono problem tarcia konstrukcyjnegow aspekcie rozprasza- nia energii w połączeniu gwintowym. Rozpatrzono dwa modele matematyczne połą- czeń gwintowych: czysto- i sprężysto-tarciowyz uwzględnieniemtwierdzeniaLame’go. Manuscript received March 19, 2007; accepted for print November 12, 2007