Force Characteristics of an Engaged Involute Gear Tooth

Transcription

1 Proceedings o the World Congress on Engineering 00 Vol III WCE 00, June 0 - July, 00, London, U.K. Force Characteristics o an Engaged Involute Gear Tooth Nesrin Akman, Member IAENG Abstract - An involute gear tooth under repeated pulse loading with varying magnitude as a result o a contact orce is analysed. The Euler beam equation allows the delection at the centre o mass o the gear tooth, modelled as a stubby cantilever, to be calculated. The relationship between the involute angle, aial coordinate and time determines the harmonic unction in time to represent the delection. An equivalent system subjected to a harmonic orce representing the component o the contact orce causing the delection is devised. This system comprises a mass matching that o the gear tooth, suspended rom a weightless spring and three parallel dampers. The author, ater establishing the validity o the model by correctly determining the characteristics o the spring orce, hypothesises that one o the dampers acts in a viscous ashion, that the second reacts linearly with the position o the contact orce along the gear tooth and that the third, as a harmonic unction o time directly proportional to the applied orce. Ten gear teeth with involute circle radius varying rom 6 mm to 5 mm are modelled and the damping orce calculated using the author s hypothesis. The damping orce is ound to dier to a maimum 0.4 percent o that obtained using the calculated delection, velocity, acceleration and the contact orce. Inde Terms Bending moment, damping orce constant, Euler beam equation, involute gear tooth. I. INTRODUCTION The characterisation o the damping orces in a vibrating structure has been actively pursued in structural dynamics [-4, 9]. The most common approach is to use viscous damping'', where the instantaneous generalized velocities are hypothesized to be the only relevant variables that aect damping orces. Many workers [,, 9, 4] have proposed ways o identiying, rom eperimental results, damping matrices in linear and multiple-degrees-o-reedom systems. Although these have led to a high degree o conidence in the perormance o the matri, they ail to address the undamental question o whether the model is indeed correct. Certainly, viscous representation is not the sole model, since all representations with non-negative energy dissipation unctional orm potential candidates. In contrast to the orces governing inertia and stiness, there are questions on the nature o the variables to be included in Manuscript received February, 000. Nesrin Akman is with the AKM Engineering and Education, 9 Pack Street, Jamboree Heights, QLD 4074, Australia (phone: ; e- mail: nesrin.akman@akm.com.au). epressions describing damping. It is generally recognised [- 4] that variables other than instantaneous generalized velocities make a signiicant contribution, urther complicating the model. The eistence o high damping orces in structural or mechanical elements leads to additional problems. Here, irst order perturbation methods are no longer appropriate. The author utilises previous research in the structural analyses and studies o the characteristics o gear teeth with respect to spring and damping properties. These are then used to conirm the eatures o the spring orce and establish potential variables that represent the damping orce on an involute gear tooth when modelled as a cantilever. Despite the act that gear teeth are short and stubby, they still possess elasticity; thus, the delection o teeth is one o the causes o the transmission error (TE). Case hardened and ground gears are precisely ormed, with proile errors o below 4 µm and cumulative pitch errors within 0 µm [5]. Hence, the tooth delection contributes signiicantly to the overall relative deviation rom smooth running at the mesh point. Loaded nontruly conjugate tooth proiles ail to uniormly transmit angular motion, thus adding to spacing or relative angular placement o driving teeth relative to driven ones. As a consequence, teeth do not engage smoothly and result in high dynamic loads. Then, periodic rather than transient eects determine the maimum load. om the mid-nineties onwards [6, 0- ], a ruitless search has continued or a modiication o proile to eliminate TE. This does not mean that certain modiications which encounter the displacement o gear tooth as a result o bending do not minimize transmission errors. These modiications depend on the magnitude o the contact orce and any modiication should be minimised. However, any proile adaptation which avoids edge contact at the beginning o mesh is highly desirable. During the engagement o teeth, energy is dissipated as the lubricant is epelled, adding to the sliding riction. The gear and the pinion, during the engagement, momentarily undergo a pure rolling action as the zone o contact coincides with the pitch point. In all other positions, however, the meshing action is a combination o rolling and sliding. Since rolling resistance is considerably smaller than the latter, its contribution to the total tooth riction is usually neglected. In contrast with the total rolling speed o the gears, the relative sliding velocity [7, ] varies with the meshing position as well as rom one tooth to the other. Vaishya and Singh [] studied non-linearity and parametric eects in gear dynamics and compared the results o both linear and non-linear time varying systems. They demonstrated that non-linearity in riction eerted its ISBN: (revised on 5 November 00) WCE 00 ISSN: (Print); ISSN: (Online)

2 Proceedings o the World Congress on Engineering 00 Vol III WCE 00, June 0 - July, 00, London, U.K. maimum inluence at primary resonance but even then did not signiicantly contribute to the TE. Kahraman and Vijayakar [8] highlighted the problems encountered by inite element and semi-analytical deormation treatments when applied to contact zones in planetary gear systems. These are etremely small (typically two orders o magnitude below the working depths o the teeth), and traverse its surace. Their approach uses a inite element model only to compute stresses and relative deormation or points that are away rom the contact zones and employs semi-analytical techniques to do the same in the contact zone. The near ield semi-analytical, and the ar ield, inite element solutions, are equated at a matching surace. Their model is signiicantly more diicult to program, but once implemented, provides comparable resolutions to that o a highly reined inite element mesh. Baud and Vele [5] used an etended inite element model. It accounts or leural torsional aial couplings, mesh stiness that varies non-linearly with time and departures rom the ideal geometry o a tooth. They conclude that the stiness o the bearing and bending o the shat must be included in any analysis. Wink and Serpa [5] used three procedures to calculate errors in static transmission o loaded pairs o helical gear pair. They used an incremental procedure with its gradual or iterative application, solving or the distribution o the ull load and calculating TE. They reduced the problem via a pseudo intererence method and solved it by separate procedures; one by linear programming, the other by a direct matri solver based on Cholesky actorisation. The later procedure was shown to be highly eicient in solving load distribution problems, encouraging its use in gear strength models that predict contact and bending. II. INVOLUTE GEAR TOOTH The tooth proile is created using an involute o circle with radius r. In this application, the value o is limited to r r requiring ϕ to be in the range o. to π radians. The overall length o the tooth is r. To avoid the negative values o, r is added to the value o, hence moving the origin to the let by r. Similarly, the y value is adjusted by subtracting a constant rom it to obtain the thickness o the tooth at the pitch to be equal to π P, where P is the diametral pitch. The parametric equation or such curve is = r(+ cosϕ + ϕ sinϕ) y = r ( sinϕ - ϕ cosϕ) y pitch π P Plotting y as a unction o, Fig., shows the tooth proile. When the mating gears engage, the contact orce perpendicular to the common tangent to the mating suraces at the point o contact shown in Fig., is responsible or the bending moment. () A. Bending Moment Figure : Gear tooth proile The bending moment M at (Fig. ) due to the y- component ( F y = Fcosϕ, ) o the contact orce F at = may be written as Substituting or, M cos M and ( ) i F j = F y in the equation or ( cos ϕ ϕ sin ϕ cos ϕ ϕ sin ϕ ) y M gives = ϕ + () During the engagement, the position o the contact orce varies rom = r ( ϕ =.) at the tip o the tooth to = c ( ϕ =.06 ), the beginning o the clearance illet; the range o horizontal values being the working length o the tooth. The position o the orce,, to result a maimum moment at a given value o varies or 0.46r and remains as the tip or larger values o. The bending moment at the base, as the orce travels along the tooth is shown in Fig. ; the maimum magnitude occuring at =.894r. I the delection o the tooth due to the contact orce is δ, then EI d δ = M d = cosϕ r 0 -r y c ( cosϕ + ϕ sin ϕ cos ϕ ϕsin ϕ) r Where E is the Young s modulus and I is the second moment o inertia. F r () ISBN: (revised on 5 November 00) WCE 00 ISSN: (Print); ISSN: (Online)

3 Proceedings o the World Congress on Engineering 00 Vol III WCE 00, June 0 - July, 00, London, U.K /r o ϕ and ϕ, the corresponding values o and can be calculated. For.4r, the maimum delection occurs when the orce is at the tip o the tooth and or <. 4 the position o the orce is slightly away rom the tip as shown in Fig.. Integrating again gives the delection Figure : The bending moment at the base o the tooth ( 0) moves along the tooth. B. Delection = as the orce Integrating the both sides o the above equation and putting d δ = 0 at = 0 results d EI dδ = cosϕ d ϕ ϕ cosϕsin ϕ + cos ϕ + + ϕ cos ϕ] [ cosϕ ( cosϕ + ϕsin ϕ) + ϕ sin ϕ ( cosϕ+ ϕsin ϕ) + + ϕ cosϕ sin ϕ + cos d δ Putting = 0 or a given ϕ gives the position o the orce d when the maimum delection occurs. Using the values M /r ϕ Figure : The position o the contact orce or maimum delection along. /r (4 ) EI δ = ϕ ϕ cos ϕ sin ϕ + + cos ϕ ϕ ϕ cos ϕ + ϕ cos ϕ sin ϕ + + cos ϕ 4 4 ϕ + ϕ sin ϕ ϕ cos ϕ sin ϕ + + cos ϕ ϕ sin ϕ ϕ cos ϕ + ϕ cos ϕ sin ϕ + + ϕ cos ϕ cos ϕ ϕ cos ϕ + ϕ cos ϕ sin ϕ + 4 sin ϕ + 7 cos ϕ + cos ϕ cos ϕ 6 cos ϕ 6ϕ sin ϕ) cos ϕ ϕ cos ϕ sin ϕ sin ϕ + cos ϕ + cos ϕ cos cos sin 9 + ϕ ϕ ϕ ϕ + + ϕ cos ϕ ϕ cos ϕ sin ϕ + sin ϕ 40 cos ϕ + ϕ cos ϕ 4ϕ sin ϕ 9 ϕ sin ϕ + + ϕ sin ϕ ] + cos ϕ ( cos ϕ + ϕ sin ϕ) ( cos ϕ + ϕ sin ϕ) + cos ϕ ( cos ϕ + ϕ sin ϕ) ( ϕ sin ϕ + ϕ sin 7 4 ϕ cos ϕ The delection at the tip, as the contact orce travels away rom it, must be etrapolated since the bending moment at the tip is zero and Eq. 5 is not applicable. The tooth is subject to the contact orce during each cycle or a time period o T N, where T = π ω is the period, N the number o the teeth and ω the angular velocity o the gear. The contact orce travels rom = r to 0.577r working length o the tooth- during the time period o 0 to τ = π ωn. To simpliy the dependence between the position o the orce and time t, a linear relationship between and t, as given in Eq. 6 below, is assumed. r ( cos ϕ + ϕ sin ϕ ) c c c = Nωt + (6) π + ( 5) ISBN: (revised on 5 November 00) WCE 00 ISSN: (Print); ISSN: (Online)

4 Proceedings o the World Congress on Engineering 00 Vol III WCE 00, June 0 - July, 00, London, U.K. The delection at = c as a unction o time, together with its cosine approimation is given in Fig. 4. The divergence between the two values can be accounted or, by the linear approimation o the horizontal distance covered in a given time. Calculus derived delections demonstrate its asymptotic approach to zero as the orce moves to the base o the tooth ( = 0) which is not the case or the trigonometric approimation. spring III. MODELLING parallel dampers The delection at the mass centre, as the contact orce moves along the tooth, is calculated in terms o the involute angle using Eq. 5. The relationships between the involute angle, aial coordinate, and time t, then allows the delection to be epressed as a harmonic unction o time. An, equivalent system Fig. 5, subjected to a harmonic orce, is devised to model the orces acting on the system. It comprises a mass matching that o the gear tooth, suspended rom a weightless spring and three parallel dampers. The author hypothesises that one o the dampers acts in a viscous ashion, that the second reacts linearly with the position o the contact orce along the gear tooth and that the third, as a harmonic unction o time, directly proportional to the applied orce. Ten involute gear teeth with involute circle radius varying rom 6 mm to 5 mm are modelled and the damping orce calculated using these parallel dampers is ound to dier to a maimum 0.4 percent o that obtained using the calculated delection, velocity, acceleration and the contact orce. The spring orce constant is ound when the system is at rest. Figure 5: The equivalent system To simulate the dependence o the delection on the bending moment, the vertical component o the contact orce is multiplied with a scaling unction, = t, in the τ equivalent system. This allows the orce to be maimum when the delection is greatest and be zero at the end o engagement, t = τ. The equation o motion is mass m & δ + cδ & + kδ = F e (7a) Nωt where δ is the displacement Now i we deine δ as δ = F δ, then δ & = Fδ & and & δ = F & δ EI δ Calculated values Cos approimation Substituting in Eq. 7a and dividing each term by the magnitude o the contact orce F, gives F m& δ + cδ & + kδ = e F (7b) The ratio o the equivalent and the contact orces, e = Fe F, can be written as a periodic unction o time. e = cos(bt + d) (8) Figure 4: Delection at = c as a unction o time, t. The delection at = c rather than at the tip is shown since the delection at = c is ound directly rom Eq. 5 as a unction o ϕ and using Eq.s and 6 epressed as a unction o time whereas the latter is ound through etrapolation. ϕc ϕ r where b =, d = ϕ r t t c r and ϕc, t c and ϕr, t r are the involute angles and time at the clearance and the tip o the tooth respectively. The departure o the actual and trigonometric approimations o the ratios e = Fe F and F y F is accounted or, by the non-linear relationship between and ISBN: (revised on 5 November 00) WCE 00 ISSN: (Print); ISSN: (Online)

5 Proceedings o the World Congress on Engineering 00 Vol III WCE 00, June 0 - July, 00, London, U.K. time t, although the trigonometci approimation closely approaches the actual one. A. Spring Force Constant The spring orce constant is ound when the tooth is at rest. It varies along the tooth as an inverse power unction, being ininitely large at the base o the tooth. It increases linearly as the overall height o the tooth increases with increasing base circle radius and as a power unction with an inde 0.5 with the second moment o inertia (Fig. 6), conirming the validity o the model. B. Damping Force Constant The variables in the equivalent system are taken as the ratio o the equivalent and the contact orces and ratio o the position o the contact orce on the gear tooth to the involute circle radius. The second ratio is needed to take into account that displacement results rom the bending moment o the tooth. The damping orce is assumed to be directly proportional to the instantaneous velocity and calculated using the orce equilibrium in the equivalent system. Multi variable regression analysis is perormed on all ten systems representing the gear teeth with base circle radius ranging rom 6 mm to 5 mm (overall length mm to 0 mm) to determine the dependency o the damping orce constant on these variables. The constants o the regression equation obtained are given below in Table. The adjusted R in each case is The irst constant in the analysis is given as the intercept and represents the viscous damping. The second and third constants are associated with the other two dampers in the parallel dashpot. Net, or each o the ten systems, the damping orce constant at each time interval is divided by the ourth root o the second moment o inertia. The multiple regression analysis is repeated or each o the ten equivelant systems and the Table : THE MULTI VARIABLE REGRESSION ANALYSIS CONSTANTS FOR THE PARALLEL DAMPERS. r viscous position o e magnitude o e Table : THE MULTI VARIABLE REGRESSION CONSTANTS FOR THE RATIO OF THE DAMPING FORCE CONSTANT AND THE FOURTH ROOT OF THE SECOND MOMENT OF INERTIA. R viscous position o e magnitude o e E+05.00E+07.4E+08 identical constants obtained are given in Table. The adjusted R is again The multi regression equation or all cases is c = 9.E E E + 08e (9) I Where c = damping orce constant I = second moment o inertia = the ratio o the position o the orce to the base circle radius Force/F k 0.6 Spring Force.0E+0.6E+0.E+0 8.0E+09 k= 9E+ I 0.5 R² = 4.0E E E-08.6E-07.4E-07 k Power (k) I Damping Force Equivalent Force t Figure 6: Spring orce constant as a unction o second moment o inertia. Figure 7: Equivalent, spring and damping orces as a unction o time. ISBN: (revised on 5 November 00) WCE 00 ISSN: (Print); ISSN: (Online)

6 Proceedings o the World Congress on Engineering 00 Vol III WCE 00, June 0 - July, 00, London, U.K. The damping orce then is written as 0.5 ( 9.E E E 08 ) I v d = + e (0) Where d = Fd F is the ratio o the damping orce to the contact orce and v is the velocity. The relevant magnitudes o the spring, damping and equivalent orces are shown in Fig. 7 above. It can be clearly seen that the damping orce is much smaller than that o spring orce. IV. CONCLUSIONS There is a particular allure and challenge in determining the variables governing damping orces. Many researchers deine kernel unctions to it eperimental measurements to their model o damping to obtain damping matrices. These unctions ensure non-negative energy dissipation unctions. However, they ail to identiy the undamental variables that govern damping. Here, the Euler beam equation is used to predict the bending moment and delection o the gear tooth, which is modelled as a two dimensional stubby cantilever. It shows that during the engagement o the tooth, the maimum bending moment ails to occur when the contact orce is at the tip without the addendum or tip modiication. Additionally, the model demonstrates that the maimum delection depends on the position o the contact orce and corresponds to the contact orce being at the tip when measured at two thirds o its height rom the base. An analysis o ten gear teeth with involute o circle radius varying rom 6 mm to 5 mm attests that the delection o the mass centre is close to being a harmonic unction o time. The author derives an equivalent system that comprises o a mass matching that o the gear tooth and suspended rom a weightless spring and three parallel dampers. This is shown to be subject to a harmonic orce derived rom the vertical component o the contact orce and can adequately be epressed as a unction o time. Multiple variable regression analysis o this system, perormed on the ratio o the damping orce coeicient and the ourth root o the second moment o inertia, produces a unique set o constants. The irst constant termed as the intercept corresponds to the viscous damping. The second and third relate to the position o the contact orce on the tooth and the magnitude o the equivalent orce correspond to the remaining two dampers. The close it ( by <0.4% ) to the resultant damping orce rom that obtained using the calculated delection, velocity, acceleration and the equivalent orce airm the legitimacy o the variables chosen. The spring orce constant has been shown to vary along the tooth as an inverse power unction, being ininite at its base. It s linear relationship on the overall height o the tooth, together with its dependence the second moment o inertia as a power unction with an inde 0.5 is demonstrated, conirming the validity o the model. Finally, the damping orce has been shown to be much smaller in relation to the spring orce and may be neglected in the initial analysis. The indings are applicable to studies o gear sets such as planetary ones. The mesh stiness can be written as a nonlinear unction o time considering the contact ratio and the stiness o each tooth at the point o engagement together with bearing stiness. Similarly energy dissipated due to damping may be included in the study using the unique set o constants obtained rom the multiple regression analysis. REFERENCES [] S. Adhikari, J. Woodhouse, Identiication o Damping: Part, Viscous Damping, Journal o Sound and Vibration, 4() (00) 4-6. [] S. Adhikari, J. Woodhouse, Identiication o Damping: Part, Non- Viscous Damping, Journal o Sound and Vibration, 4() (00) [] S. Adhikari, Y. Lei, M. I. iswell, Dynamics o Non-viscously Damped Distributed Parameter Systems, 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conerence, Austin, Teas. (005). [4] S. Adhikari, On the Quantiication o Damping Model Uncertainty, Journal o Sound and Vibration, 06 (007) 5 7. [5] S. Baud, P. Vele, Static and Dynamic Tooth Loading in Spur and Helical Geared Systems-Eperiments and Model Validation, Journal o Mechanical Design, 4() (00) [6] G. Bonori, B. Barbieri, F. Pellicano, Optimum Proile Modiications o Spur Gears by Means o Genetic Algorithms, Journal o Sound and Vibration, (008) [7] E. Buckingham, Analytical Mechanics o Gears, General Publishing Co. Ltd., Toronto, 949, pp [8] A. Kahraman, S. Vijayakar, Eect o Internal Gear Fleibility on the Quasi-Static Behavior o a Planetary Gear Set, Transactions o the ASME, (00) [9] N. Larbi, J. Lardies, Eperimental Modal Analysis o a Structure Ecited by a Random Force, Mechanical Systems and Signal Processing, 4() (000) 8-9. [0] S. Li, Eects o machining errors, assembly errors and tooth modiications on loading capacity, load- sharing ratio and transmission error o a pair o spur gears, Mechanism and Machine Theory 4 (007) [] W. Steeds, Involute Gears, Longmans, Green and Co. London, New York, Toronto, 948, pp. -4. [] D. P. Townsend, Dudley s Gear Handbook, Mc Graw Hill, Inc. New York, 99, pp [] M. Vaishya, R. Singh, Sliding iction-induced Non-Linearity and Parametric Eects in Gear Dynamics, Journal o Sound and Vibration, 48(4) (00) [4] C. Vallee, S. Y. Stepanov, S. Charles, Evaluation O The Determinant O Identiication Equations For A Linear Model O A Mechanical Vibratory System, Journal o Applied Mathematics and Mechanics, 69 (005) [5] C. H. Wink, A. L. Serpa, Perormance Assessment o Solution Methods or Load Distribution Problem o Gear Teeth, Mechanism and Machine Theory, 4() (008) ISBN: (revised on 5 November 00) WCE 00 ISSN: (Print); ISSN: (Online)