Industrial Brushless Servomotors Episode 8 pot

Motor and load dynamics 135 The reflected inertia of the load mass should be matched with the inertia which already exists on the motor side of the screw threads, and so J~ = Jm + Jsw where Jsw is the inertia of the screw. Combining the last two equations and including the effect of force F gives the optimum screw pitch as / 1m +]~w where Az u n Cp xm dzx is the optimum screw pitch when the load follows any trapezoidal velocity profile and is subject to an opposing force. ~ i~i , ~,~ ~:. ~i!!~i :~ = ~z~ ~- 9 i~i i~ Figure 4.23 Two-axis, pick-and-place handling machine. (Photo courtesy of Hauser division of Parker Hannifin) Example 4.3 The sinusoidal motor in Table 4.1 is to be used to drive a ball screw. The load velocity is to follow the profile shown in Figure 4.24. The system constants are 136 Industrial Brushless Servomotors 4.5 Jm = 0.00022 kgm 2 Jsw - 0.00003 kgm 2 x - 0.025 m t? = 0.120 s F- 1000N m=5kg pql+p2-1 The profile constant is Cp = [1 - 0.5(p, +p2)] 2" In this case Pl - 20/120 and p2 -60/120, giving Cp load force factor is ~8 (1000x0"1221 2 A2 "- 0.025 x 5 - 737 - 18. The The optimum screw pitch is 2zr /0.00022 + 0.00003 dA V 5q-1 + 737 = 8.5 mm In this example, the optimum pitch is dictated mainly by the effect of the load force. In the hypothetical case of zero load force, the optimum screw pitch given by the calculation would be do - 44 mm! I I I 0.02 0.06 t 0.12 Figure 4.24 Load velocity profile for example 4.3 Motor and load dynamics 137 4.6 Torsional resonance In Sections 4.4 and 4.5 the mechanical connection between the motor and load is assumed to be inelastic, leaving the increase in system inertia as the only mechanical effect of an added load. In practice some flexibility in the connection is unavoidable, and an error may develop in the position of the load relative to that of the hub of the rotor of the motor as torsional forces come into play. When the error becomes oscillatory, the condition is known as torsional resonance. The problem can also arise in the section of shaft between the hub and the sensor, but we will assume throughout that any such effects have been eliminated through the design of the motor and sensor. Under these circumstances, the error between the sensor and the load can be assumed to be the same as the error between the hub and the load. Figure 4.25 shows a rotor of inertia Jm connected to a rotating load through a shaft which is subject to twist. Following the approach used in Section 4.4 for the totally rigid shaft shows that the poles of the transfer function of motor speed response are given by JL[$4TeTm -~- S3Tm -]- S 2] -1- C -1 [S2TeTM Jr- STM -q- 1] 0 (1) where TM Tm(Jm-~-JL)/Jm. If JL = 0, TM = Tm and the equation reduces to the form already derived for an unloaded motor: S 2 TeTm -1"- STm -[ 1 - 0 C is the compliance of the mechanical connection, or error factor for the angle between the motor position sensor and the driven load, normally expressed in microradians/Nm. The compliance varies widely according to the length and diameter of the drive shaft and the types of transmission between the motor and the load. Typical values are in the range 10-100 #rad/Nm. Industrial Brushless Servomotors 4.6 138 Solution of expression (1) gives the theoretical locations of the poles for the case where the damping effects of eddy currents and friction are ignored. There are two pairs of poles, one pair at low frequency and the other at the frequency of potential torsional resonance. I I %=1 I I I I I ! r,q ! Figure 4.25 Shaft compliance Jm I I I I I I I I hJ C ~trad/Nm \ \ \ Effect of compliance at a fixed load inertia The resonant frequency is affected by the ratio of the motor to load inertias, and also by the value of compliance. We start by looking at the way the resonant frequency varies as the compliance is changed, for a case where the inertias are approximately equal. Example 4.4 A rotating load is connected to the shaft of a brushless servomotor. The motor and load inertias are approximately equal. Torsional resonance frequency values are required for a wide range of shaft compliance. The system constants are Jm = 0.00215 kgm 2 JL = 0.00200 kgm 2 Motor and load dynamics 139 Te = 5.0 ms Tm = 2.6 ms TM 5.0ms Inserting the numerical values in equation (1) above and dividing through by JLTeTm gives S 4 + 200s 3 + (962C -1 + 77 • 103)s 2 + 192 • 103C-Is + 38 • 106C -1 = 0 Figure 4.26 shows how the poles move as C is varied from infinity to 10 #rad/Nm. The physical interpretation of infinite compliance is of course that the load is disconnected from the motor, at which point the last expression is reduced to S 4 + 200s 3 + 77 • 103S 2 = 0 The motor and disconnected load therefore have four poles, two showing the normal response (already dealt with in Section 4.3) of an unloaded motor to a step voltage input. The other two poles remain at the origin as long as the load stays unconnected. The arrows show the shift in position of the four poles as the compliance is reduced from infinity, or in other words as the stiffness of the transmission is increased from zero. As the compliance is reduced, the motor-load poles move from the position (at the origin) for a disconnected transmission towards the positions P1, P2 for a totally rigid connection. The poles at positions P3, P4 for the normal response of the unloaded motor rise in frequency but become increasingly oscillatory as the compliance falls, taking up relatively undamped positions close to the boundary between stable and unstable operation of the system. In practice it is found that the lower the frequency of such lightly damped responses, the more likely it becomes for the frequency to be excited by the system in general and for system instability to occur. 140 Industrial Brushless Servomotors 4.6 10 C i~rad/Nm 20 ,f Pa 100 C -~i~ P1 P2~ P4X V Figure 4.26 Pole loci as compliance is reduced j= 1600 1200 8OO 400 100 50 50 100 = 400 800 1200 1600 j= ~ Resonant frequency predictions and tests The low frequency poles in Figure 4.26 are relatively well damped and are in any case eliminated through the design of the control system. The other pair travel towards infinity as the compliance falls towards zero. In the present case the resonant frequency is predicted to lie between approximately 1550 and 1100 Hz for compliance values from 10 to 20 #rad/Nm. The motor and load of Example 4.4 were connected together. The compliance of the length of shaft between the front end Motor and load dynamics 141 of the hub of the rotor and the load was calculated to be C- 14.4 #rad/Nm. The resonant frequency was excited by striking the load, and measured by recording the stator emf produced by the resulting oscillations of the rotor. The result is shown in Figure 4.27(a). The low frequency envelope is due to the slow rotation of the rotor after the shaft has been struck. The resonant frequency is close to 1305 Hz, and this compares well with the value of 1300 Hz predicted by equation (1) at the compliance of 14.4 #rad/Nm. As the resonant frequency rises, it becomes less likely to be excited through a well-designed drive system. The resonant frequency rises as the compliance falls, and so the main conclusion is that the compliance should be as low as possible for maximum system stability. Damping Expression (1) automatically includes the damping effects of the i2R loss generated in the stator as the rotor oscillates, but these are negligible. The pole loci in Figure 4.26 do not take account of damping due to frictional and eddy current losses. When expression (1) is modified to include the viscous damping due to eddy currents, the effect is predicted to be insignificant in the test motor. The test results do not include the effects of any i2R loss in the stator as measurements must be made with the winding on open-circuit. Damping of the motor under such test conditions is therefore the result of eddy current loss and losses at the bearings, with the bearing loss likely to be the greater part. The time constant of the decay in Figure 4.27(a) is approximately 33 ms. The time constant affecting the rate of decay of the oscillations is seen to be high, when the load is mainly inertial. The rate of decay is of course increased in practice when the driven load is subject to friction, and also when damping appears in a transmission mechanism such as a belt and pulley drive. As [...]... resistance is R' - R[1 + 0.00 385 (150 - 25)] - 1. 48 R where the figure of 0.00 385 is the temperature coefficient of resistance of copper for a temperature rise above 25~ At any speed, the speed-sensitive loss is therefore given by Psp - R 1" 48~ -~T(T2s - '2 where Tsoac is the rated torque given by the Soac curve The speed-sensitive loss may amount to 25-35% of the total ISO Industrial Brushless Servomoters 5.2... example, the magnets may become hotter than the stator core The figure for Rth must therefore be high enough to allow any part of 1 48 Industrial Brushless Servomotors 5.2 the motor to remain below the maximum normally 150~ temperature, Soac curves The thermal characteristic of a brushless motor is usually given as the boundary of the Safe Operation Area for Continuous operation, which takes the form shown... the loss at 4500 rpm as R 1.29 (3.72 3.22 Psp - 1. 48 ~-~T(T2s - T2soac) - 1. 48 0.422 ) - 37 W |S2 IndustrialBrushless Servomoters 5.3 The figure of 37 W applies to both motors, remembering that the thermally effective resistance for the sinusoidal form is R = 1.5 RLL 6000 Speed rpm 5000 M03 M06 1 t M09 L 1 4000 , 3000 1 2000 i ii ' 1/ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 TorqueNm Figure 5.2 Soac curves Torques... RthPloss(av) where Ploss(av) is the average value of the motor losses and Rth is the published value of the thermal resistance Rth in ~ A large ripple can obviously lead to overheating if the average 154 IndustrialBrushless Servomoters 5.4 loss is large enough by itself to raise the average winding temperature by 110~ At an ambient temperature of 40 ~ the maximum winding temperature of 150~ would be exceeded... Psp - R 1" 48~ -~T(T2s - '2 where Tsoac is the rated torque given by the Soac curve The speed-sensitive loss may amount to 25-35% of the total ISO Industrial Brushless Servomoters 5.2 power loss when the brushless motor runs on load at the centre of the speed range A restriction on the average operation period of the motor may be imposed at mid to high speeds, where a substantial speed-sensitive loss is... same order as the cost of the next larger motor Any saving in cost is clearly negligible in comparison to the cost of failure of the blower, followed by the motor 5.3 Steady-state rating The rating of a brushless motor in terms of its continuous, constant torque output may be assessed in the way described in Chapter 1 for the brushed machine The maximum continuous stall current Is is normally given in... difficult to analyse by the method used above for the simple case where the load is connected directly to the end of the motor shaft, and modelling using electrical circuit analogues can be a better approach [8] In general the resonant frequency falls as the compliance increases, and also as the moment of inertia of the load increases in relation to that of the motor Assuming as much as possible has been done... that the intermittent rating of the brushed motor is affected by the presence of a temperature ripple, which is most pronounced at the rotor winding The same effects occur at the stator winding of the brushless motor Figure 5.3 shows an intermittent output torque, applied every t' seconds over a time of tp The remainder of the diagram shows the associated power loss and the steady-state temperature... chapter show calculations of nominal torques and temperatures using nominal constants, and also give the results at the extreme tolerance of the constants 5.2 M o t o r heating Heat is generated in the brushless motor as a result of the fir loss in the stator winding and iron losses in the stator and rotor In addition, some heat arises from the friction between the bearing seal and the rotor shaft Rotor... steady state depends on the thermal time constant ~'th For a typical motor, ~'th is of the order of 35 minutes The published value is normally the overall time constant of the main mass, which for the brushless motor is taken to be the stator winding, stator iron and motor case The temperatures of these three motor components do not vary at the same rate The i2R heating energy passes from the winding . 4:1. 144 Industrial Brushless Servomotors 4.6 Resonant ~e~e~ 180 0 1600 1400 1200 1000 900 ! 7- - i i l 0 ! iiiiiiiiiiiii~ iiiiiiiiiiiiii i i i ~ i i ! i i i ! 2 4 6 8 10 Ju. allow any part of 1 48 Industrial Brushless Servomotors 5.2 the motor to remain below the maximum temperature, normally 150~ Soac curves The thermal characteristic of a brushless motor is. between the motor and the load. Typical values are in the range 10-100 #rad/Nm. Industrial Brushless Servomotors 4.6 1 38 Solution of expression (1) gives the theoretical locations of the poles