Verification of cylinrical interference fits uner impact loas with LS-Dyna Prof. Dr.-Ing. elmut Behler, Jan Göbel, M.Eng. ochschule Mannheim, Paul-Wittsack-Straße 10, D-68163 Mannheim, Germany Steffen eute, M.Eng. Alpha Engineering Services Gmb, eßheimer Straße, D-677 Frankenthal, Germany Summary: Interference fits are a commonly use means to couple shafts an wheels for example. The usual imensioning is performe by a static verification. As long as the system geometry is not too complicate an the eformation is assume to be linear elastic, the interference pressure can easily be calculate with the more familiar solutions of the equations of elasticity. The maximum static contact forces can be calculate together with an assume coefficient of static friction. In orer to investigate whether a cylinrical interference fit provies sufficient stability against slip the real loas have to be known. owever in various applications this is not the case an the interference fit is subjecte to ynamic loas, especially to impact loas. We simulate a moel interference fit that is first axially mounte an later also axially loae. This is a typical case in hyraulic systems. Similar problems occur in gears, e.g. worm gears, especially if there are reverse torques as in many applications. The crucial number a esign engineer seeks is the safety against slip, S. A imensional analysis shows that S is epenent on the length l of the interference fit, its interference Z, the velocity v an the mass of the impacting boy m an the static friction coefficient µ. Altogether we fin: S ~ Z l µ v -1 m -0.5. Numerical experiments have shown that the easiest way is to vary the velocity of the impacting boy to fin the esign with minimum safety S = 1. The esire safety can then be achieve by simply changing the parameters. We investigate the influence of ifferent contact types, an fin the OSTS contact as optimal for the shaft-hub contact. The same way we consier the NTS contact as optimal for the shaft-impacting boy contact. The results also show that the forces ue to an impact are huge an that it is not possible to make an appropriate esign without a numerical or experimental analysis. Keywors: Interference fit, ynamic loa, impact loa, contact force, safety against slip
1 Introuction Interference fits are wiely use in mechanical engineering esign in orer to transmit torsional loas in the first place particularly in gears an frequently axial loas, e.g. in hyraulic systems. The general avantage of such joints is the avoiance of notches that cause a reuction in strength, especially when ynamic loas are applie. Also ifferent kins of material can easily be introuce, as for instance in worm gearings. The general mechanics is easy, the outer iameter of the inner part (shaft, inex S is larger than the inner iameter of the outer part (hub, inex the negative iameter ifference is calle interference Z. Figure 1 shows the situation. This interference geometry causes a surface pressure p that can bear huge torque or axial loas ue to static friction. This pressure also causes stresses in the shaft an the boy so that p an hence Z is limite, epening on material an geometry. Thus the imensioning of an interference fit is an optimization. If Z is too small, the result is slip an if it is too large, the result may be excessive stresses. v m S Impact mass, m l Figure 1: Interference fit with an impact mass Since for elastic behaviour p is linear in Z, the imensioning can be conucte accoring to DIN 7190 [1]. The unerlying formula (1 is an exact solution of the equations for the two-imensional interference fit []. Of course small three-imensional effects like notches or lubrication holes are usually isregare. A safety against slip S is easily compute since in the static or quasi-static case the loas are given. Moreover in [3] an optimal aspect ratio is introuce. In [] the interference fit uner plastic eformation is escribe. Z 1 p E 1 + Q 1 Q 1 + ES 1 + Q 1 Q ν S = + ν B I S In (1 the parameters Q represent the iameter ratios: Q S = QS ( = The maximum von-mises-stress is at the inner iameter of the shaft. Accoring to [1] an [] it is given by: σ v = p (3 1 Q S Frequently, e.g. in valves or other applications, the loas are ynamic or even of impact type. As with other impact problems the loa epens on the masses, velocities an the material properties an can not be easily calculate. Therefore we use LS-DYNA to simulate a simple impact on an interference fit to preict the safety S of the fit. There is little literature on the ynamic behaviour of interference fits (1
mostly the ynamic bening loas are consiere [4] an [5]. Würtz [6] escribes the mounting of such fits with impact forces as generally unclear an epenent on the energy that is applie to the fit. There are two ifferent ways of mounting an interference fit: raial an axial. In raial mounting the interference must temporarily be zero, for instance by subjecting the outer part to a higher temperature. In axial mounting two chamfers are necessary an the shaft is inserte with high axial loas. In axial mounting the surface roughness is smoothe by this process an thus the remaining interference is reuce by R, (4 gives a goo estimate, R z are the surface roughnesses of the two surfaces. owever the structure of the surfaces cannot be consiere in an analysis with LS-DYNA. In our moel Z is unerstoo as an effective interference after mounting. ( R 0. 8 R zs + R z (4 Theory an Dimensional Analysis N F µn Figure : Forces on the interference fit Figure inicates the parameters that etermine the safety S against slip. That means assuming Coulomb s friction: µn S = (5 F The resulting normal force is proportional to the pressure in the fit: N = π p l (6 With (1 the pressure can be represente by: Z p = (7 K K is a esign parameter that inclues material properties as well as iameter ratios. Thus we fin: µ π l Z S = (8 F K Therefore the main obstacle to etermine the safety against slip is the uncertainty in terms of F, which epens on mass an velocity, but also on material properties an geometry. To get a bit closer we consier the momentum uring the impact. As a reasonable guess we assume the force F(t to be sinusoial: ( ωt F( t = F0 sin (9 The frequency ω epens on the size of the impacting mass m an the stiffness c of the system an it is relate to t S. c π ω = ts = m ω (10
So it is a half-sine that is uner consieration uring t S. Therefore we obtain: m t S = π (11 c We investigate the impact equations for the case that the combine mass of shaft an hub is fixe, its velocity before an after the impact is zero as long as the interference fit is safe. Therefore for the impact number e we obtain e v' v = (1 In (1 v is the velocity of the impacting mass after the impact. Thus altogether for the force F: t S F' = F( t = m( v v = m( ev v = m(1 + e v (13 0 Obviously the worst-case-scenario is e = 1, meaning that the involve parts are infinitely har. That results in: F' = m v (14 Because of the half-sine the resulting force is: π F = t s ω F' = F' = c m F' = c m v Although these formulas reflect certain assumptions, we can, however, consier the safety against slip to be proportional to the involve parameters as follows: µ l Z S ~ (16 c m v K We note that K ~ E an c ~ E, but since both parameters also inclue geometry, (16 means we can either vary µ, l, Z or v linearly or the mass m in orer to examine the reliability of the fit. This is important because the variation of either parameter coul cause ifferent numerical obstacles, especially varying Z an µ can cause contact problems. The iea is that if an axial motion will occur in the simulation at a certain interference Z*, than one has to choose a Z that equals S-times Z*. Of course this can be performe using µ, v, m or l respectively as variation parameters. Thus the etermination of S is an iterative parameter stuy. The single parameter to be change is ϕ: µ l Z ϕ = (17 m v 3 Moel The moel we use for simulation is shown in figure 3. The mass m of the impacting cylinrical part is 0.075 kg an its esign velocity is 1 m/s. We assume axial mounting of the interference fit an also simulate this. After mounting there is a pause to make sure there is no kinetic energy left in the system before the impact. The whole process takes 5 millisecons [ms] the mounting takes 10 ms, the following pause is 5 ms, before the motion of the impact mass starts. Accoring to figure 3 the parameters in (1 are Q = 0.3 an Q S = 0.375 respectively. Comparing the von-mises-stress of (3 with an assume yiel stress of R e = 900 MPa the maximum pressure is 387 MPa an the maximum interference is 79µm. The analysis is performe with an interference of 54 µm however an a static friction coefficient of µ = 0.15. The irrelevant ynamic friction coefficient is set to 0.005, an we choose a ecay coefficient of 0.00008 therefore the mounting process can easily be performe. The interference fit is not valiate if an axial motion of the shaft occurs. The Young s mouli are E = 00 GPa (Steel an E S = 16 GPa (Vanais. (15
50 v = 1m/s 16 6 11 m = 0.076kg e.g. 30 Figure 3: Moel Geometry The contact between shaft an hub is a one-way-surface-to-surface contact, OSTS. The contact between the impacting mass an the shaft is a noe-to-surface contact, NTS. In the contact car we use the following options: ABC, SOFT = 1, SOFSCL = 0.1. In the ourglassing car we use IQ = 6 an QM = 1.0. Thus hourglassing is not a problem. owever, we performe a couple of simulations with fully integrate elements. Figure 4 shows the elements of the system. 4 Simulation an Results Of course it is reasonable not to alter the esigne geometry that is the length l. Therefore we have left static friction coefficient µ, the impact spee v or mass m or the interference Z to examine the safety against slip. Actually we experience contact problems in both using a very high static friction coefficient an a large interferences that cause numerical eformations of the shaft. To avoi this we vary the impact spee. The simulation with LS-DYNA is performe on the Sun Fire X 00 igh-performance-computing Cluster at Mannheim University of Applie Sciences. The cluster has 14 noes, each ual core (AMD Opteron with 16 GB RAM per noe an a spee of.6 Gz. The noes are connecte via Infiniban at 0 Gbit/s. The typical behaviour of such an interference fit uner impact loa is illustrate by the figures 5, 6 an 7. Figure 5 shows a valiate fit, the velocity of the impacting mass is 0.15 m/s. There is no slip at all. Figure 6 shows the moment of the impact of a faile fit, the velocity of the impacting mass is 1.0 m/s. There is also a notable elastic eformation of the shaft. Figure 7 shows the same fit, but 0.6 ms after the impact. The full slip then alreay happene an is 0.08 mm. Because of friction the shaft is at rest then. Figures 8 an 9 show a large slip brought on by an impact velocity of m/s. The slip in this case is 0.56 mm. Thus ifferent velocities give ifferent slips. The numerical experiments show that in the interference fit uner consieration slip will occur if the velocity of the impacting mass is larger than 0.58 m/s. Therefore we can also give a guess of the impact time t S. owever assuming a half-sine for F(t over t S. From (1, (14 an (15 we obtain: t S π F' = = F 0 m v µ p l (18 With our ata we obtain t S =.4 µs. Accoring to [7] the smallest time step of our moel will be automatically set to 0.086 µs, because the element size is 0.5 mm, so the element size can be consiere as sufficiently small. The CPU-time is about 000 s when using fully-integrate elements.
7th European LS-DYNA Conference Figure 4: Elements of the simulate system Figure 5: Interference fit without slip (valiate fit, simulate with v = 0.15 m/s
Figure 6: Interference fit with slip (faile fit, simulate with v = 1.0 m/s. The moment of impact. Figure 7: Interference fit with slip (faile fit, simulate with v = 1.0 m/s. 0.6 ms after the impact. The slip is 0.08 mm.
Figure 8: Interference fit with slip (faile fit, simulate with v =.0 m/s. The moment of impact. Figure 9: Interference fit with slip (faile fit, simulate with v =.0 m/s. ms after the impact. The slip is 0.56 mm.
5 Conclusions The interference fit uner impact loas can be simulate with LS-DYNA. In an iterative way it is possible to etermine safety against slip. Because of possible obstacles in changing contact parameters ivarying the velocity is recommene. The obtaine results shoul be valiate by a set of experiments. Because of the vast number of parameters it seems to be hazarous to suggest a esign formula that can be use to compute S as easy as in the case of a pure static loa. 6 Literature [1] ---, DIN 7190 Pressverbäne, Berechnungsgrunlagen un Gestaltungsregeln, 001 [] Kollmann, F.G.: Welle-Nabe-Verbinungen, Berlin, eielberg, New York, 1984 [3] Castagnetti, D.; Dragoni, E.: Optimal aspect ratio of interference fits for maximum loa transfer capacity, J. Strain Analysis, (40005, 177 184 [4] Leiich, E.: Neue Aspekte bei er Auslegung ynamisch beanspruchter Preßverbinungen, VDI-Berichte 1384, 1998, 03 5 [5] Gropp,.; Klose, D.: Grunlegene Ergebnisse experimenteller Untersuchungen zum Übertragungsverhalten ynamisch belasteter Preßverbinungen, VDI-Berichte 1384, 1998, 175 188 [6] Würtz, G.: Montage von Pressverbinungen mit Inustrierobotern, Schriftenreihe IPA-IAO Forschung un Praxis, Berlin, eielberg, New York, 199 [7] Weimar, K.: LS-DYNA User s Guie, Rev. 1.19, CADFEM Gmb, 001 Appenix: Notation Z Interference p Pressure Fit iameter Outer iameter of the hub S Inner iameter of the shaft E,S Young s moulus of hub an shaft Q,S Diameter ratio σ V Von-Mises-stress R Reuction of interference R z Roughness S Safety against slip N Normal force µ Static friction coefficient l Length K Parameter F, F 0 Loa F Time integral of F t Time t S Duration of the impact m Mass c Stiffness ω Natural frequency v, v Velocity e Impact number R e Yiel stress ϕ Similarity parameter