Impact of Toothed Gear Nonlinearities on Servomechanism Performance

Transcription

1 Proceedings of the th WSEAS International onference on AUTOMATI OTROL, MODELLIG & SIMULATIO Ipact of Toothed Gear onlinearities on Servoechanis Perforance Slobodan OBRADOVI Milan TUBA Dana SIMIA Faculty of oputer Science Faculty of oputer Science Departent of oputer Science Megatrend University Belgrade Megatrend University Belgrade Lucian Blaga University of Sibiu Bulevar uetnosti 9 Bulevar uetnosti dr. I. Ratiu str. SERBIA SERBIA ROMAIA sobradovic@ptt.rs tubailan@ptt.rs d_siian@yahoo.co Abstract: - Due to insufficient stiffness of gear transission and presence of backlashes, oscillations and reduced positioning accuracy ay occur in autoatic control systes. This paper gives an analysis of the ipact echanis of gear backlashes, as well as the ways of reducing or eliinating their influence on the syste perforance. Introduction of divided gears can eliinate backlash introduced effects but can also decrease the resonance axiu frequency and ove it near the cut-off frequency or even in the pass-band of the syste aking it unstable. Exaples and calculations are presented which show how to eliinate these probles. Key-Words: - Autoatic control syste, Geared servo syste, Gear backlash Introduction Many servo systes, especially these that anipulate large asses like tank turret, use a gear train structure to generate high torque and reduce angular speed. The selection of servo drive (electroechanical or hydraulic type) can be affect the disturbance transfer fro its source to the syste. Soe drive types can inherently dup the disturbance (even if the control loop is open) while others just accept the whole disturbance and leave the control loop to deal with it. Moreover, the quality of the selected servo drive has also a significant influence on the stability of control (accuracy of target tracking in tanks for exaple), which is analyzed in literature [], [] and is not a subject of this paper. This paper analyzes the gear as a part of the servo syste. Reducing gears are devices whose basic role is to perfor echanical adaptation of the load oent and the otor driving oent. Gear transission is one of the basic fors of echanical equipent which has the broadest application in accopanying systes, systes of gyro stabilization, regulators and other autoatic control systes. Different types of gears are applied and the ost widespread are reducing gears with spur wheels and bevel wheels, while other types are applied ore rarely and only in special cases. There are few nonlinear anoalies in geared servo systes (for exaple hindsight axis drive in tanks - turret by the aziuth and gun s tube by elevation): inertial variations, static and coulob friction and backlash are the ost iportant. These anoalies cause a lot of probles in stabilization, especially during the target tracking when it is necessary to switch the direction of the rear side axes. In such situations tracking velocities are generally very low, and the effects of friction and backlash are very significant [3], [4], [5]. Besides, reducing gears present a considerable source of possible oscillations (instability) in autoatic control systes due to the presence of backlashes and static and coulob friction and also because reducing gears (chassis and all obile part represent an elastic eleent. Ipact of Gearing Transission onlinearities on Autoatic ontrol Systes The presence of backlashes introduces, when the boosting coefficient in the syste is increased, the stability reduction and increased tranquilizing tie and even the undaped oscillations [6], [7], [8], [9]. onlinearities of the backlash type increase aplitude and phase deforations which lead to degradation of quality indicators. Backlashes occur due to lateral floats in cogged couples and due to the phenoenon of shaft and gear deforation. These elastic deforations are ore pronounced when changing direction of rotation. ISS: ISB:

2 Proceedings of the th WSEAS International onference on AUTOMATI OTROL, MODELLIG & SIMULATIO Moreover, backlashes are a highly nonlinear effect that can disturb, even jeopardize operation of linear control syste, especially in the low speed region where the torque, as a rule, changes the sign frequently to track the low speed reference (effects of friction are doinant). This is particularly dangerous in systes in which the controlled object is enclosed by feedback loop (this is the case in gyroscopic stabilization syste. Every tie the target tracking direction changes, rate gyroscope detects that the stabilization object is not oving, due to backlashes and static friction. The regulator as a response increases the control signal. When the syste overcoes static friction, angular velocity of rotation becoes significantly higher than necessary, so the regulator decreases the control signal abruptly. Angular velocity is also abruptly decreased to the necessary level or even changes sign. This leads to the appearance of stepwise inial velocities (Fig. ). This proble is even ore coplicated in stabilization systes with electrohydraulic activating devices. aely, besides static friction of the reducing gear, these systes require taking into the account static friction of the hydro-pup rotating control block, pup dead zone and leaking within the hydro-pup - hydro-otor syste (hydrostatic gearing). For very sall angles of pup block deflection, there is a dead zone of several iliradians. onlinearity is very pronounced in the vicinity of zero in that case. Besides that, hydro-pup block deflection is sall for low angular velocities, so the useful flow rates are of the sae order of agnitude as the leakage due to the anufacturing iperfections. For sall pup deflection angles the working pressure is very low, so the driving oent is practically zero. This is the additional cause for the iniu electrootor velocities to be irregular (stepwise), as can be seen in Fig.. The first channel represents the coand signal. Hydro-otor angular velocity depends on the block deflection angle of the pup with variable working volue (second channel). Tachoeter located on the hydro-otor axis easures angular velocity of the incoing reducing gear shaft. Angular velocity of the stabilization object is easured by rate gyroscope. Each tie the torque sign is changed, backlashes cause the sudden increase of the easured velocity and the linear controller responds with instantaneous decreasing of the coanded torque. This disturbance spoils the quality of control and can even result in oscillations. That is why controllers should be designed to keep the nubers of torque sign-changes as low as possible. Figure. Influence of static friction on iniu velocities The size of lateral floats depends on the selected quality and accuracy of axle distance of coupled gears and it is directly proportional to the size of the backlash. The ipact of reducing gear stiffness and quality of connection between reducing gears and load in the echanical part of the syste will be analyzed by eans of load otion equation in elastics systes, in the otor-reducing gear-load syste: J d θ θ dθ = M ( t) + r ( θ ) F dt () dt Equivalent equation of otion ay be written and reduced to otor shaft as follows: ( J where: J d θ r d + ) = M ( t) ( ) F θ dt θ θ () dt θ r ( θ ) [] - stiffness oent; r [/rad] - reducing gear stiffness constant; F [/(rad/] - speed friction load constant; F [/(rad/] - speed friction otor constant; θ [rad] - otor rotating angle; θ [rad] - load rotating angle. Laplace Transfors of the Equations () and () are: ISS: ISB:

3 Proceedings of the th WSEAS International onference on AUTOMATI OTROL, MODELLIG & SIMULATIO M ( θ( ( T s + ξ T s+ ) θ( = +, (3) r ( T s r M ( + ξ T s+ ) θ( = + θ (, (4) νj F T = T = [s]; ξ ; = r J where: J ; F J ξ = ; ν = +. νj r J o r ow a structural scheatic diagra of the otor starting syste by eans of elastic reducing gear can be drawn (Fig. ). Presence of the positive feedback contour is clearly observed on the scheatic diagra. When constructing frequency characteristics of this contour on the basis of Equations (3) and (4), it is clear that resonance axius will appear at frequencies ω =/T and ω =/T. They will be ore noticeable if r is larger since the daping coefficients ξ and ξ will be significantly reduced. Figure 3. Equivalent syste with gear transission elasticity at the output shaft Equations that describe dynaics of this syste are: J s θi + ( θi θ ) r = J s θ + F sθ + θ ) = M r ( θ i By solving this syste for θ i we get the transfer function: Figure. Scheatic diagra of elastic driving device Because of the large gear ratio i r in real systes, ξ in Equation (4) is greater than. That yields two real poles in the syste transfer function: one in the very low frequencies area (ideal integrator), the other near the cut-off frequency. Although it is considerably saller than the lateral floats, the ipact of elastic deforations of gears on backlash is also present. It is greatest in the inverse transission (change of direction) when there is an increased angle of shaft swiveling. This odel can in practice be substituted by a siplified version where the total torsion deforation of the gear transission is reduced to equivalent elasticity of the gears input shaft, or even better to the elastic deforation of the output shaft. Equivalent syste is depicted in the Fig. 3. θi( Gr ( = = M( F J J s + + F Gr ( = F s (+ T (+ ξt where:, + T s ) J+ J JJ aj [ ]; F T s T [ s]; = ξ F r ( J+ J ) r J, J J J 3 s+ s + s r r F aj T ' J J = ; T ; a. ' = = F r T J+ J When servo systes with feedback are designed (Fig. 4) this axiu has to be considered since it can cause instability of the syste, particularly in the area of sall gears stiffness r. If the syste is ipleented with pure integrator: /s (dotted line), then there is no pole at the frequency /T (this pole is the consequence of the ter: K/(T s+)). ISS: ISB:

4 Proceedings of the th WSEAS International onference on AUTOMATI OTROL, MODELLIG & SIMULATIO Figure 4. Bode diagra of the stabilization servo syste When the stiffness of the gears is sall, for exaple when a spring is inserted to eliinate the backlash of the shared gears, the frequency ω approaches the pass-band of the syste and there will be no redundant aplification stability (Fig. 5). Also, increase of the load oent of inertia will further decrease the redundant aplification. obile objects. By adjusting the entioned oent, this servo syste disturbance (being, as a rule, a stochastic value), becoes considerably ore definite. A solution, based on the principle of a copy lathe applied in practice is given in Fig. 6. Liited degree of freedo of oveent in X direction, in parallel with the conjugate axis action, is introduced to the whole reducing gear (that is, its casing). The running sall cylinder on the internal side of the following gear (rolling path), enables a very accurate radial clearance in cogging (tooting) and prevents serrations, and the radial force which is always present in coupling, prevents over cogging. Self-aligning coupling enables very high-quality coupling by eans of which, aongst other things, additional deforations are eliinated in case of bad coupling of insufficiently stiff reducing gear. Figure 6. Self-aligning gear with an object Figure 5. Shared gears reduce syste stiffness and stability redundancy 3 Ways of Eliinating or Reducing the Ipact of Backlashes and Stiffness on Syste Perforance Increasing the stiffness coefficient. The ipact of the gear elasticity reduction ay be copletely eliinated, or at least reduced, by increasing the stiffness coefficient r. Fro Equations (3) and (4) it is obvious that the resonance axiu will be increased, but it will in turn ove to the area of higher frequencies. Increase of the stiffness coefficient r is achieved by increasing the ass, and therefore increasing the size or corresponding constructive solution in the sense of reducing the gear casing which is often liited by the fitting-in conditions and peritted weight. This priarily refers to servo systes which are expected to provide high-quality onitoring of Introduction of divided gears and reducing gears. Ipact of lateral floats on backlashes is reoved by introduction of divided gears and reducing gears. aely, this coupled gear has one divided (two-part) gear, whose halves are braced in coupling with the other gear, by eans of one (torsion) or ore extensive (copressive) springs. Often, only the outlet reducing gear is derived as divided, for in its coupling, the ipact of lateral floats on the quality of servo syste onitoring is the greatest. This ethod decreases the resonance axiu frequency and oves it near the cut-off frequency or even in the pass-band of the syste aking it unstable. Table gives the easured values of the transfer function G( for one gyro stabilization syste. f[hz] L[dB] ϕ[ ] f[hz] L[dB] ϕ[ ]., ISS: ISB:

5 Proceedings of the th WSEAS International onference on AUTOMATI OTROL, MODELLIG & SIMULATIO Table. Elastic gear transfer function odule and phase The odule L is in decibels and the phase φ is in degrees. Transfer function is defined in Equation (4) as the ratio of the ain feedback (fro gyroscope) and the local feedback (fro tachoeter): U gyro ( G( = = U hp ( ( s + ) ω In the experient, the otor was loaded with the real load (guns tube). Tachoeter was connected directly to the otor shaft, before the gear. Gyroscope was connected directly to the guns tube. Fig. 7 gives the transfer function G( as a diagra of the odule L and phase φ, in a real syste of gyro stabilization, recorded as entioned signal ratio as a function of frequency f in Hz. The odule diagra is interpolated using cubic spline interpolation and the phase diagra is fitted using sixth order polynoial least square approxiation. Figure 7. Transfer function odule and phase diagra Using standard procedure in MATLAB one can approxiate this transfer function with the third order transfer function, Equation (5). orresponding diagras are given in Fig. 8. The values of coefficients K and ξ, and the second order pole frequency ω are set directly fro the diagra in Fig. 7, but to set the first-order pole frequency ω, another experient was necessary. The experient is described in the following paragraph. U gyro( K G( = = Utacho ( ξ s s s ( + )( + + ) ω ωn ωn.6 G ( = * s * s +.45s + where: K =.6, ξ =.76, ω = 4 rad / s, ω = 44 rad / s. (5) ISS: ISB:

6 Proceedings of the th WSEAS International onference on AUTOMATI OTROL, MODELLIG & SIMULATIO 4 onclusion Static friction and backlash cause a lot of probles in stabilization, especially during the target tracking with very low velocities when it is necessary to switch the direction of the rear side axes. In autoatic control systes, which are required to provide high-quality onitoring, regulation or stabilization, application of reducing gears with divided gears or self-aligning coupling considerably reduces the negative effects occurring due to insufficient reducing gear stiffness, as well as presence of lateral floats in the cogged transission. Acknowledgent: This research is supported by Ministry of Science, Republic of Serbia, Grant 447. Figure 8. Bode diagra of transfer function Equation (5) The first order pole is set to ω = 4 rad/s by eans of an experiental easureent of the transfer function of real, loaded hydro-otor. The transfer function of the otor is defined as a ratio U tacho /U hp where U tacho is the voltage fro tachoeter connected to the otor shaft, and U hp is the voltage fro the gauge which easures the pup-block angle (proportional to fluid flow). Fig. 7 (easured value and Fig. 8 (calculated transfer function) show a high resonant peak in the transfer function around the frequency ω. In servo syste design where the load is covered by the ain feedback loop, this peak ust be taken into account since it ay cause syste instability, especially in the area of decreased stiffness of reducing gear r. In that case the frequency ω approaches the passband of the syste and there is no redundant stability. It ay also be observed that the increase of the load inertia oent will cause even greater reduction of excessive boosting (strengthening). In order to avoid possible consequences (reduced operation accuracy and breakdown of the reducing gear), frequency ω should be at least 5 ties larger than the syste cut-off frequency. Hardening technologies. The ipact of elastic deforations of the gears is significantly reduced by teeth and shaft surfaces hardening technologies, such as: ceenting, nitriding and siilar. References: [] Mikerov AG: Brushless Dc Torque Motors Quality Level Indexes for Servo Drive Applications, Eurocon 9: International IEEE onference Devoted to the 5 Anniversary of Alexander S. Popov, Vol. -4, pp , 9 [] Driscoll S, Huggins JD, Book WJ: Electric Motors oupled to Hydraulic Motors as Actuators for Hydraulic Hardware-in-the-Loop Siulation, Proceedings of the ASME Fluid Power Systes and Technology Division, Vol., pp. -, 5 [3] Sun J, Yao YA: An Independent Active Torque Balancer Using a Servo-ontrolled Differential Gear Train, Transactions of The anadian Society For Mechanical Engineering, Vol. 33, Issue, pp , 9 [4] Ruderan M, Hoffann F, Bertra T: Identification and opensation of Stick-Slip Friction in Haronic-Drive Gear Transission, Proceedings of ISMA 8: International ISS: ISB:

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