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12 Machine ConsiderationsMachine Feed Drive Considerations for Resolver Feedback The usual configuration for resolver position feedback is to have the resolvergeared to the servo drive mo

12 Machine Considerations

Machine Feed Drive Considerations for Resolver Feedback

The usual configuration for resolver position feedback is to have the resolvergeared to the servo drive motor Computer-aided design programs forhydraulic drive sizing are based on the fact that the hydraulic resonance isthe predominant resonance in the servo loop With electric drives themechanical time constant is not of prime consideration since it can becompensated for with the drive compensation These are two importantdynamic considerations with electric and hydraulic servo drives usingresolver feedback at the servo motor Since the position feedback resolver isclose-coupled to the drive servo motor, all other mechanical resonances andnonlinearities are excluded from the position servo loop Recommendationsfor the required indexes of performance (I.P.) are identical with those given

in Section 9.2

Machine Feed Drive Considerations for Direct Machine Slide Position Feedback

The main reason that direct feedback could be used was the universal use ofthe ‘‘soft servo’’ with its low position-loop gain By definition a soft servohas a low position-loop gain of about 1 ipm/mil with no frequency breaksbelow the servo bandwidth The range of gains for the ‘‘soft servo’’ rangefrom 0.5 ipm/mil to 2 ipm/mil As long as no resonances are within six timesthe high end of this range (200 rad/sec or 32 Hz, Eq [9.2-27]), it should bepossible to close the position loop and be stable

As previously stated, if the feedback position transducer (resolver) islocated at the drive motor, any mechanical resonance in the machine will beoutside the servo loop The effects of these mechanical resonances on theclosed-position loop will be reflected load disturbances

However, if direct (e.g., Inductosyn-linear scales) position feedback is

to be used, the mechanical resonance in the mechanical drive componentswill be inside the position loop For this situation, some guidelines,recommendations, etc should be put forth to prevent a situation wheredirect feedback is used on a feed drive that will not be stable

REFLECTED INERTIAS FOR MACHINE DRIVES

As a constraint the lower limit on any resonance (hydraulic or mechanical)inside the velocity loop should not be less than 200 rad/sec (Eq [9.2-27]).Likewise, the bandwith ðocÞ of the closed velocity loop should not begreater than one-third the lowest resonance in the servo loop, which isusually the hydraulic resonance, oh, in hydraulic drives (Eq [9.2-23]).With the position loop, there may be a number of different mechanicalresonances These resonances are outside the velocity loop but inside theposition loop Considering the mechanical feed drive components, the one

of greatest concern is the drive screw axial resonance This resonance willprobably be the predominant low mechanical resonance in a ball-screw feeddrive

Typical values of ball screw stiffness are shown in the graph ofFigure 1

for the variables of screw diameter and length These values are based onconstant end bearing and ball nut stiffnesses In actual practice, these would

be varied according to the screw diameter InFigures 2to 5,the ball screwresonances are plotted for various screw diameters, lengths, and appliedload weights

Resonances occurring outside the velocity loop and inside the positionloop will contribute phase shift to the open-position loop frequency-

Fig 1 Ball screw stiffness.

Fig 2 Ball screw resonance

Fig 3 Ball screw resonance.

Fig 4 Ball screw resonance

response characteristics To maintain the desired 458 phase margin for theposition loop, the resonances (such as the ball screw) should be sufficientlyhigh relative to the position-loop velocity constant ðKvÞ that they will notreduce the phase margin It is recommended practice that the lowestresonance inside the position servo loop should be three times higher thanthe minimum hydraulic resonance (200 rad/sec) If this practice is followed,the reflected phase shift occurring at the velocity constant ðKvÞ from theposition loop resonance will not reduce the overall position-loop phasemargin In actual practice this is somewhat academic because in largemachinery mechanical resonances do occur above and below 200 rad/sec As

an index of performance (I.P.) it is recommended that the position loopresonance (usually the ball screw) should be at least six times the position-loop velocity constant ðKvÞ or greater than oh (Refer to Section 9.2,

Eq [9.2-22] to [9.2-28].)

A graph displaying this I.P can be drawn (Figure 6) showing therelation between position-loop velocity constant ðKvÞ and the lowestallowable resonance in the position servo loop For numerical controlsusing the ‘‘soft servo’’ technique, it can be determined from the graph inFigure 6 that resonances inside the position loop should, in general, be large

Fig 5 Ball screw resonance

enough (at least 200 rad/sec) such that the resonance will not have adetrimental effect on the drive.

For large machines where the velocity constantðKvÞ usually cannot belarger than 0.5 ipm/mil for reasons of stability, the lowest allowableresonance (from Figure 6) could be as low as 50 rad/sec For small machinesusing the ‘‘soft servo’’ with velocity constants of 2 ipm/mil, the lowestresonance in the position servo loop should not be less than 200 rad/sec.Therefore, it is difficult to make an across-the-board recommendation withchanging velocity constantsðKvÞ from machine to machine

One possible way to arrive at a recommendation is to assume that allmachines being built use position loop gains up to 2 ipm/mil, and therefore,machines with ball screw resonances under 200 rad/sec cannot use directfeedback Another possibility, which is more practical, is to list the machineswith the recommended maximum allowable position servo loop gains ðKvÞthat can be used with direct feedback Since all axes on a machine must havethe same position loop gain, the machine axis with the lowest resonance will

be the determining factor in how low the position loop gain or velocityconstantðKvÞ must be set

Other Resonance ConsiderationsAdditional mechanical resonances inside the position loop that may causestability problems should also be considered Reflected structural resonances

Fig 6 Lowest recommended resonance inside the position loop

due to large and sometimes limber machine structures may reflect theirresonances into the machine slide as the result of an antinode vibration.Likewise, very large and heavy workpieces placed on an otherwise stablemachine slide can cause stability problems when direct position feedback isused.

Additionally, mechanical power transmission devices, such as ‘‘woundup’’ gear trains, can be a source of stability problems when they are includedinside the position loop with direct slide position feedback The gearbox canhave two types of problems First, the amount of windup can vary causing achange in spring rate that can cause instability Second, any nonlinearitysuch as backlash that occurs as a result of lost windup will result in anunstable servo drive

Reflected Inertias for Machine Drives

An important parameter to consider in sizing a machine servo drive is thetotal inertia at the servo motor The definition of inertia is the property of abody, by virtue of which it offers resistance to any change in its motion.For an industrial machine slide the total reflected inertia to the servomotor is made up of the reflected belt and pulley inertia (or gearboxes), thereflected drive screw, and the reflected machine slide with its load weight.Knowing the total inertia at the servo drive motor is a requirement for anyform of servo analysis or drive sizing

The reflected inertia at the drive motor can be calculated as

Jreflected¼ Jload

ðD2=D1Þ2 (lb-in.-sec

Where D1¼ diameter of the motor pulley or gear (in.)

D2¼ diameter of the load pulley or gear (in.)

A very important consideration in applying a drive to a machine axis is thetorque to inertia ratio A mechanical equation for acceleration torque is:

Where JT ¼ total inertia at the motor shaft (lb-in.)

a¼ acceleration at motor shaft (rad=sec2)

Equation (12.2-1) can be rearranged as

be asked: Is there enough acceleration torque to meet these requirements?Quite often this becomes a critical drive-sizing problem providing enoughavailable current from the servo amplifier to accelerate the load required tomeet a productivity requirement?

For industrial servos using brushless DC drives there are two types ofmotors to be considered The brushless DC motor in general industrialapplications uses ceramic magnets For high-performance applications, lowinertia motors are used These motors use Neodymium-iron-boron (NdFeB)magnet material or Samarium cobalt (SmCo) magnet material Acomparison of torque to inertia ratiosðT/JÞ with various total inertia loads

is compared inFigure 7

At small inertia loads the torque to inertia ratios for low inertiamotors are significantly higher than standard ceramic motors, which is anindication that the low inertia motors are capable of higher performance(and acceleration) with small total inertia loads These curves also indicatethat the low inertia motors have a significant reduction in performance astotal load inertia increases The standard ceramic magnet-type motors haveless reduction in performance with added total load inertia The low inertiadrive is usually more expensive than the standard ceramic-magnet brushless

DC motor For larger total inertia loads it is not economically justifiable touse the low inertia type motor since performance will be compromised Ifhigh performance is a requirement, the reflected inertia load should bereduced with a ratio if possible

To further study the performance of brushless DC drives undervarious inertial loads, a transient step response test will be made for avelocity servo using a lag/lead compensation in the servo amplifier with again of 1000 volts/volt A more detailed analysis is presented inChapter 14

A 3-volt/1000 rpm tachometer feedback will be used The block diagram forthis servo drive is shown inFigure 8.A simple velocity servo was chosen forthis analysis to minimize complexity

A standard ceramic brushless DC motor and a low inertia brushless

DC motor will be used for this analysis The motor parameters are identified

as the following:

MOTOR STANDARD MOTOR LOW INERTIA MOTOR

Transient responses for a multiplicity of load inertias are shown inFigures 9

to 16 In this analysis it is assumed that each drive will be analyzed withidentical inertia loads as if a machine axis was being tested with a standardbrushless DC motor and then tested again with a low inertia motor

Load inertia Standard motor Low inertia motor(lb-in.-sec2) transient responses transient responses

to the motor armature can be tolerated For low-inertia brushless DCmotors it is recommended that the reflected load inertia match the motorinertia Since these motors are used for high-performance applications, it isadvisable not to reduce the servo bandwidth with inertia mismatch ratioslarger than one Reflected load inertias can be computed as follows:

Reflected Machine Slide Inertia

Jslide¼ W6ðL=NÞ260:0000656ðlb-in.-sec2Þ (12.2-7)

where: W ¼ total machine slide and load weight (lb)

L¼ ball screw lead (in./rev)

N¼ drive ratio

Fig 7 Torque to inertia ratio comparisons

Fig 8 Velocity servo block diagram.

Fig 9 Transient step response with a 1 6 inertia load for a standard magnet motor

ceramic-Reflected Drive Screw InertiaThe drive screw inertia (without threads) reflected to the servo motor can becalculated from Eq (12.2-1).

Jscrew¼ X1
½ðX2þ X3Þ6X46X5 (lb-in.-sec2) (12.2-8)where: DIA¼ screw diameter (in.)

LGTH¼ screw length (in.)Ball dia¼ diameter of the balls in the drive screw nut (in.)

Fig 10 Transient step response with a 1 6 load inertia for a low inertia motor

X4¼ 0:0046046½DIA=2
ð0:42446ball dia=2Þ

46LGTH67:2610
5

Pulley InertiasRatio¼ screw pulley diameter/motor pulley diameter

Woods pulleys, 3-inch pulley inertias (lb-in.-sec2)

No Pulley no Approx diameter Inertia

No Pulley no Approx diameter Inertia

No Pulley no Approx diameter Inertia

Woods pulleys, 2-inch pulley inertias (lb-in.-sec2)

No Pulley no Approx diameter Inertia

No Pulley no Approx diameter Inertia

No Pulley no Approx diameter Inertia

a rotary drive (Figure 19), and a rotary drive with a ring gear and pinion

user should be consistent throughout all calculations A rotary conversiontable is included asFigure 21

Fig 16 Transient step response with a 2 lb-in.-sec inertia load for a low inertiamotor

Fig 18 Reflected inertia for a rack drive.

Fig 17 Reflected inertia for a belt drive

N¼ ratioRadius¼ in:

Fig 19 Reflected inertia for a rotary drive

Fig 20 Reflected inertia for a rotary drive using a ring gear

12.3 DRIVE STIFFNESSMachine feed drives should have sufficient static stiffness to be insensitive toload disturbances In addition, a feed drive in a contouring numericalcontrol system must remain stationary or clamped when not in motion This

is called ‘‘servo clamping.’’ Also during the standstill period, the axis at restmust resist the load disturbances caused by the cutter reaction forces.The static stiffness of a feed servo drive is the equivalent springstiffness of the drive measured at the output The stiffness can be measured

at several different places, such as the drive motor shaft, at the drive input tothe ball screw, or at the machine slide When the stiffness is measured at therotary components the units of measurement are lb-in./rad When thestiffness is measured at the machine slide the unit of measurement is lb/in.Drive stiffness, measured as the machine slide displacement in inches caused

by a displacement force in pounds, increases as the square of the ratio andwith the fineness of the lead of the screw Therefore, in sizing a servo drive, it

is important to consider the benefits of using a drive ratio and small lead(in./rev)

Fig 21 Rotary inertia conversion table

Fig 22 Hydraulic servo-drive stiffness.

Fig 23 Electric servo-drive stiffness.

It is possible to have a stiffness measurement for each type of feedbackservomechanism For example, a velocity servo drive with tachometerfeedback would have a velocity stiffness with the units of lb-in./rad/sec Forthe purpose of this discussion the drive stiffness is derived for the case of apositioning servo drive with an internal tachometer loop.

To illustrate the engineering analysis of the static drive stiffness, twocases of direct slide positioning are considered: a hydraulic servo drive and

an electric servo drive Servo-drive stiffness equations for both hydraulicand electric drives are summarized inFigures 22and23.The derivation forthe stiffness of the hydraulic servo drive is followed by the electric servo-drive stiffness derivation

Hydraulic Servo-Drive Stiffness with Direct Feedback

A mechanical model of a machine feed slide is shown in Figure 24 It isassumed that the drive screw spring rate is represented by GT Themachine’s slide inertia reflected to the drive screw plus the inertia of thedrive screw is represented by JL The load forces acting on the machine slideare reflected to the drive screw as TL The friction forces of the machine slideare reflected to the drive screw and represented by BL The mechanical driveratio is represented by N

Fig 24 Machine slide free-body diagram

The equations for the mechanical model ofFigure 24are written in thefollowing steps:

y1¼ym

N T1¼ NTm T1¼ GTðy1
yoÞFor dynamic equilibrium we have:

Q1; Q2¼ hydraulic oil flow in and out (in:3=sec)

P1; P2¼ hydraulic pressure (psi)

KC¼ hydraulic valve pressure coefficient (in:3=sec
psi)

Kv¼ hydraulic valve gain (in:3=sec-in:)

Xv¼ hydraulic valve displacement (in.)

The valve flow is given by

Motor flow equations are developed as follows for each chamber:

Flow in
flow out ¼ theoretical flow þ compressibility

KLcp¼ cross port leakage (in:3=sec-psi)

KLex¼ hydraulic external leakage (in:3=sec-psi)

b¼ bulk modulus (16105lb=in:2)

V1; V2¼ hydraulic volume in and out (in:3)

fvðymÞ ¼ variation in each chamber volume (in:3)

Dm¼ hydraulic motor displacement (in:3=rad)

Vc¼ total trapped volume on one side of motor(that oil under compression); valve; and manifold (in:3)

s¼ Laplace operatorCombining Eq (12.3-17) to (12.3-21) results in

Q1
KLcpP1þ KLcpP2
KLexP1

¼ Dmsymþ Vc

b þfvymb

It is desired to combine the mechanical torque equations and the hydraulicflow equations to arrive at one system equation By rearranging Eq (12.3-39) in several steps, Eq (12.3-43) results

A second-order equation can be written as ðs2=o2

hÞ þ ð2d s=ohÞ þ 1.Therefore in Eq (12.3-39) oh¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s JmVc2b s

mþVcGT2bN2

G T

N yo

JmVc2b s3þ KLJms2þ D2

mþVc GT2bN 2

diagram can then be used to solve for the drive system stiffness FL=Xo.Using the substitutions for the block in Figure 26, drive systemequations can be written in Eq (12.3-44) to (12.3-52)

From Figure 26

Q¼
XoKfbKDGv
ymKTAsGvþ Xi (12.3-44)But Xi¼ 0, so

Fig 26 Hydraulic drive-servo block diagram.

Fig 27 Hydraulic drive-servo block diagram for stiffness

Combining Eq (12.3-44) to (12.3-46) yields

From Figure 26 the following simplifications are possible

Xo

DmNKfbKDK1Kv

KL

stiffness for direct feedback (12.3-61)

It can be shown that the velocity closed-loop equation from Eq (12.3-35) is

The velocity open-loop gain is

Substituting Eq (12.3-66) into (12.3-67) yields

FL

Xo

¼2pL

rev2 6rev

2

in:26

in:6rad66

1sec6

V

V6

sec lbin:3 in:2

¼lbin

Linear Electric Servo-Drive Stiffness with Resolver Feedback at the Drive Motor

The first step in deriving the drive system equations is to derive the DCelectric motor equations, which can be done with the equivalent electricalcircuit shown in Figure 28

The equation for the equivalent circuit of Figure 28 is

The torque equation forFigure 28is

the block diagram ofFigure 31for an output displacement on a torque loadinput while holding the reference input position yiat zero This solution will

be a linear solution considering all components as linear devices

Fig 29 Electric motor block diagram