Mechatronic Motion SystemModeling, Identification, & Analysis

• Objectives

– Understand compliance, friction, & backlash as it

relates to motion systems.

– Develop a control-oriented model of a motion

system with appropriate simplifying assumptions.

– Apply system identification techniques for the

parameters in the model.

– Analyze the model and run simulations using the

model to demonstrate its effectiveness.

Motor Rotor Inertia

with Coulomb &

Viscous Friction

Compliant

Shaft with

Inertia and

Viscous

Damping

Gearbox with

Backlash and

Gear Inertias

(neglect friction)

Complaint Shafts

and Coupling

with Inertias of

Shafts & Coupling

Pulleys 1 & 2

with Inertias and

No Belt Slip or

Backlash

Belt with Length-

Dependent

Compliance and

Damping; Belt

Inertia may be

included

Rigid Shaft

with Inertia

Rigid Load with

Coulomb & Viscous

Friction acting on it

Rigid Load Moving

with Belt by Coulomb

Friction or Attached

Motion System

• Shaft has infinite stiffness (rigid).

• Shaft has a stiffness represented by a spring constant that leads to a

resonance in the model.

• Shaft is represented by a PDE that leads to an infinite number of resonances.

Background on Compliance

• Key Questions– Where do the resonances and anti-resonances come from

and does the engineer have control over them?

– How do the physical system parameters affect the nature of

the resonances and anti-resonances?

– Is compliance truly a parasitic effect, the result of high-

speed behavior and lighter components, or an intentional

part of the design that can be modeled and dealt with well?

– When is a flexible coupling needed in a motion system

transmission? How stiff – transverse or torsional – does

this coupling need to be?

– Is there a difference between static and dynamic stiffness of

couplings?

• A goal for mechatronic motion

systems is high motion quality:

– high speed, high accuracy, high

precision, high resolution

– robustness to system changes

• Systems have time-varying and

position-varying characteristics:

– inertia, viscous friction, nonlinear

dry friction which leads to stick-

slip behavior, compliances from

flexible belts, couplings, and

shafts, and backlash from gears

• Understanding and modeling these

effects are essential for the

mechatronic design of the system.

• Resonance and anti-resonance are such mystifying

phenomena that even when we see a laboratory

demonstration we have seen many times before, like the

two-mass, three-spring system, we are still amazed by

what we see.

Colocated Bode Plot

1

K

M

3

3K

M

2

2K

M

M MK K K

M MK K K

M MK K K

fixed

undeflected

node

Mode ShapesM M

K K K

x2x1

F(t)

Frictionless Surface

1st

Natural Frequency

2nd

Natural Frequency

Anti-Resonance

3 2 1

• Resonance, and its destructive potential, always brings

to mind the Tacoma Narrows Bridge Disaster in 1940.

During a 42 mph-

wind the bridge,

designed for 120

mph winds, shed

vortices which

caused the bridge

to oscillate.

• Resonance in engineering mostly has a negative

connotation – something to be avoided. Of course,

without resonance we wouldn’t have radio,

television, music, or swings on playgrounds, but

mostly resonance brings to mind its dark side – it

can cause a bridge to collapse or a helicopter to fly

apart.

• Resonance requires three conditions:

– a system with a natural frequency

– a forcing function applied at the natural frequency

and in phase with velocity

– a lack of energy loss

M

K

F

B

+v

Resonance

Below Resonant FrequencyForce in phase

with Displacement

Above Resonant FrequencyForce 180° out of phase

with Displacement

At Resonant FrequencyForce 90° out of phase

with Displacement

At Resonant FrequencyForce in phase with Velocity

M

K

F

B

+v

Resonance

Hand-held Barcode Scanner

• Oscillating mirrors are used in hand-held barcode scanners

to reflect laser light out & collect reflected light from the

barcode. They use less power, have less moving mass, fit in

a smaller space, & survive shocks and drops better than

rotating polygon mirrors, which are used in fixed scanners.

Reducing the energy required to oscillate mirrors at the

required frequency & also producing wide oscillation angles

are important.

• To accomplish these objectives, the system, essentially a

torsional spring-mass-damper system, is driven into

resonance by a solenoid. The solenoid has two coils: one to

sense frequency of oscillation & another to drive the system.

The system has no inherent failure mechanism, as there are

no bearings and hence no friction. As the mechanical

stresses in the flexing member of the system are kept below

a threshold, fatigue failure is avoided. The result of this

design is extreme reliability.

Flexural

stiffness

Magnetic

plunger

Flexural

stiffness

Two coils:

(1) electromagnet to

oscillate mirror &

(2) coil to sense

frequency

Spring-Pendulum Dynamic System

Nonlinear Resonance

2

tkr F mgcos m r r

mgsin m r 2r

m

k

θℓ + r

r

• Then we have anti-resonance, an even more intriguing

phenomenon, which brings to mind tuned mass-damper

systems that quiet lively buildings excited by the wind or

earthquakes.

Taipei 101’s

730-Ton

Tuned Mass Damper

May 2005

• This is all very interesting, but what does this have to do

with mechatronic motion systems?

• There are no free lunches in design; there is always a

tradeoff. The best path to good design is to become

aware of these tradeoffs, assess the effects of these

tradeoffs through modeling and analysis, and then make

an intelligent choice based on what you need.

• Compliance is always present in real systems. It can be

parasitic and degrade motion, but it also can be used to

significantly enhance motion quality.

• The difficulty arises when it is not modeled effectively or

simply ignored.

• A goal for mechatronic motion systems is high motion

quality – high resolution, precision, accuracy, and speed

– as well as robustness to system changes.

• In an ideal world, machine components would be rigid,

machining and assembly imperfections or tolerances

would be non-existent, and there would be no friction or

backlash to overcome.

• Incorporating compliance into a system design can

significantly enhance motion quality and it can do so in

three ways.

• To eliminate friction and backlash in a load-bearing

situation, a designer might use a magnetic bearing or an

air bearing, where there is no contact. Both are very

complicated, high-maintenance systems. A flexure

bearing provides both load bearing and motion guidance,

albeit small motion, while eliminating friction and

backlash. It is designed to have an optimal distribution

of rigidity and compliance.

Example of a two-

axis flexure bearing

in a motion system

Dr. Shorya Awtar

U. Michigan

• In motion transmission, flexible couplings are used to

accommodate misalignments inherent to the design or

due to manufacturing and assembly tolerances, while

eliminating friction and backlash.

• Stiffness is a function of the misalignment in two ways:

– For a bearing selection, more misalignment requires

more transverse coupling compliance.

– More transverse compliance means more torsional

compliance.

– An ideal coupling has infinite torsional stiffness and

zero transverse stiffness.

– A universal joint has this, but with friction and

backlash!

• When fixing two components together, flexible clamps

provide similar benefits.

In-Plane

Clamping Mechanism

Flexible Clamps, the correct

substitute for set-screw-based

shaft connections

Flexure Clamps

• Designs in nature often exploit compliance, while man-

made designs often avoid compliance.

• As long as the compliance in the system design is

captured in the model, high-quality motion and

robustness can be achieved with the aid of compliance –

a friend!

• Compliance is a foe when it is not understood and

accounted for in the system design. Ignoring inherent

compliance or avoiding using compliance to one’s

advantage makes compliance a foe!

• Some Comments

– Accurate modeling of the dynamic behavior of a

mechanical system will result in a dynamic system of

higher order than you probably would want to use for

the design model.

– For example, consider a shaft that connects a drive

motor to a load. Possibilities include:

• Shaft has infinite stiffness (rigid)

• Shaft has a stiffness represented by a spring

constant that leads to a resonance in the model

• Shaft is represented by a Partial Differential

Equation that leads to an infinite number of

resonances

• Shaft has infinite stiffness (rigid).

• Shaft has a stiffness represented by a spring constant that leads to a

resonance in the model.

• Shaft is represented by a PDE that leads to an infinite number of resonances.

– In most situations, the frequencies of these

resonances will be orders of magnitude above the

operating bandwidth of the control system and there

will be enough natural damping present in the system

to prevent any trouble.

– In applications that require the system to have a

bandwidth that approaches the lowest resonance

frequency, difficulties can arise.

– A control system based on a design model that does

not account for the resonance may not provide

enough loop attenuation to prevent oscillation and

possible instability at or near the frequency of the

resonance.

– If the precise nature of the resonances are known,

they can be modeled and included in the design

model.

– However, in many applications the frequencies of the

poles (and neighboring zeros) of the resonances are

not known with precision or may shift during the

operation of the system. A small error in a resonance

frequency, damping, or distance between the pole

and zero might result in a compensator design that is

even worse than a compensator that ignores the

resonance phenomenon.

– Mechanical resonance is a pervasive problem in

servo systems usually caused by compliance of

power-transmission components.

– This compliance often reduces stability margins,

forcing gains down and reducing machine

performance.

– Servo performance is enhanced when control-law

gains are high; however, instability results when a

high-gain control law is applied to a compliantly-

coupled motor and load.

– Mechanical resonance needs only two inertias

coupled by compliant components to manifest itself.

– Machine designers specify transmission components

(e.g., couplings, gearboxes) to be rigid in an effort to

minimize mechanical compliance. Some compliance

is unavoidable. Also, cost and weight limitations force

designers to choose lighter-weight components than

would otherwise be desirable, leading to low

transmission rigidity.

Curing Resonance: Mechanical Cures

• Stiffen the Transmission

– This usually improves resonance problems.

– The key to stiffening a transmission is to improve

the loosest components in the transmission in an

effort to raise the total spring constant KS. This

has the effect of raising both the resonant and

anti-resonant frequencies and moving them away

from the frequencies where they cause harm.

– Some Suggestions:

• Use multiple belts, wide belts, or reinforce belts

• Shorten shafts;

use large-diameter shafts

4JG ( d / 32)GK

• Use stiffer gear boxes

• Use larger lead screws and stiffer ball nuts

• Use idlers to support belts that run long distances

• Reinforce the frame of a machine

• Oversize coupling components; be cautious here as

this will also add inertia, which may slow acceleration

– When stiffening a machine, start with the loosest components, as a

single loose component can single-handedly reduce the overall

spring constant significantly.

• Add Damping

– In practice, it is difficult to add damping between the

motor and load. Materials with large inherent damping

do not normally make good transmission components.

total

coupling gear box n

1K

1 1 1

K K K

– Steps used to stiffen a machine can actually make the

machine perform more poorly because they also

reduce damping.

– Sometimes the unexpected loss of damping can

cause resonance problems.

• Reduce Load-to-Motor Inertia Ratio

– Reducing the load-to-motor inertia ratio will improve

resonance problems. The smaller the JL/JM ratio, the

less compliance will affect the system.

– At low frequency, the system appears to have a

noncompliant inertia, JT = JL + JM.

– At high frequency, the load inertia is disconnected; the

system sees only JM, the motor inertia.

-100

-80

-60

-40

-20

0M

agnitu

de (

dB

)

102

103

-180

-135

-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

2

M

1

J s 2

M L

1

J J s

R

AR

Rigidly-Coupled

Motor + Load

Compliantly-Coupled

Motor + Load

anti-resonance

resonance

R AR always

– In a sense, compliance gives the system an apparent

inertia that varies with frequency. The smaller the

JL/JM ratio, the less variation in apparent inertia as

would be indicated by a smaller distance between the

two parallel lines representing the low-frequency and

high-frequency amplitude ratio vs. frequency plots.

– Reducing the load inertia is the best way to reduce

the ratio JL/JM; reduce the mass of the load or change

its dimensions.

– The reflected inertia (load inertia felt by the motor)

can also be reduced by increasing the gear ratio.

M L effective M L2

1N J J J Rigid Coupling

N

– Unfortunately, increasing N can reduce the top speed of

the application. Similar effects are realized by changing

lead screw pitch or pulley diameter ratios.

– Any steps taken to reduce the load inertia will usually help

the resonance problem; most machine designers work

hard to minimize load inertia for non-servo reasons, e.g.,

cost, peak acceleration, weight, structural stress.

– Increasing JM does help the resonance problem.

Unfortunately, raising motor inertia increases the total

inertia, which reduces total acceleration or requires more

torque and power from the drive to maintain the

acceleration. Increasing motor inertia therefore increases

the cost of both motor and drive. Despite the increase in

size and cost by increasing JM, it is commonly used

because it so effectively improves resonance problems.

– Common

Misconception:

JL/JM ratio is optimized

when the ratio is 1 or

inertias are equal or

matched.

– Based on a fixed JM

and JL, the gear ratio

N that maximizes the

power transferred from

motor to load is the

ratio that forces the

reflected load inertia

JL/N2 to be equal to

the rotor inertia JM.

eq L eqJ T

eq L 2 M

eq L M

J J N J

T T NT

L ML

L 2 M

T NT

J N J

Ld

0dN

2

L M LFor T 0 N J J

– This fact has little bearing on how motors and gear ratios

are selected in practice because the assumption that the

motor inertia is fixed is usually invalid; each time the gear

ratio increases, the required torque from the motor

decreases, allowing the use of a smaller motor.

– The primary reason that JM and JL should be matched is

to reduce resonance problems; actually, this is an

oversimplification.

– Larger JM improves resonance but increases cost. The

more responsive the control system, the smaller the JL/JM

ratio. JL/JM ratios of 3 to 5 are common in typical servo

applications. Highest bandwidth applications require that

the load inertia be no larger than about 70% of the motor

inertia. The JL/JM ratio also depends on the compliance

of the machine; stiffer machines will forgive large load

inertias.

Modeling of Compliance

Compliantly-Coupled Motor and Load

JM = rotor inertia of a motor

JL = driven-load inertia

KS = elasticity of coupling

BML = viscous damping of coupling

BM = viscous damping between ground and motor rotor

BL = viscous damping between ground and load inertia

T = electromagnetic torque applied to motor rotor

L

KS

BML

JL

JM M

BL

BM

T

• Comments

– KS, the elasticity of the coupling; it is often neglected in

low-power systems; modeling it in high-power systems is

essential.

– BML, the viscous damping of the coupling; it is usually

small, as transmission materials provide little damping.

– BM and BL can be neglected in the following analysis, as

they have a small effect on resonance. They are included

here for completeness.

– Coulomb friction has been neglected. The fixed value of

Coulomb friction has little impact on stability when the

motor is moving. At rest, the impact of stiction on

resonance is more complex. Sometimes stiction is

thought of as increasing the load inertia when the motor is

at rest. This accounts for the tendency of systems to

change resonance behavior when the motion stops.

M M L MM ML S M L M

L M L LL ML S M L L

T B B ( ) K ( ) J

B B ( ) K ( ) J

L

KS

BML

JL

JM M

BL

BM

T

Equationsof

Motion

Equationsof

Motion

M M L MM ML S M L M

L M L LL ML S M L L

T B B ( ) K ( ) J

B B ( ) K ( ) J

Laplace Transform of the Equations of Motion

2

M ML M S M ML S L

2

L ML L S L ML S M

J s B B s K s B s K s T s

J s B B s K s B s K s

2

MM ML M S ML S

2

LML S L ML L S

s T sJ s B B s K B s K

s 0B s K J s B B s K

MatLab / Simulink Block Diagram (BM = 0 and BL = 0)

ML M L S M L M M

ML M L S M L L L

T B ( ) K ( ) J

B ( ) K ( ) J

Transfer Functions

2

L ML L SM

ML SL

4 3

M L M L ML M L L M

2

M L S M L ML L M M L S

J s B B s Ks

T D s

B s Ks

T D s

D s J J s J J B J B J B s

J J K B B B B B s B B K s

Transfer Functions(BL = 0 and BM = 0)

2

L ML SM

22L MM L

ML S

L M

ML SL

22L MM L

ML S

L M

J s B s K1s

J JT J J ss B s K

J J

B s K1s

J JT J J ss B s K

J J

SAs K

or as s 0

M L 2

M L

1s s Rigid-Body Motion

J J s

Transfer Functions in Standard Form(BM = 0 and BL = 0)

2

AR

2

AR ARM

22 R

2

R R

L

22 R

2

R R

2 ssK 1

sT 2 ss

s 1

K s 1s

T 2 sss 1

M L

ML

S

S M L

R

M L

MLR

S M L

M L

SAR

L

MLAR

S L

1K

J J

B

K

K J J

J J

B

K J J2

J J

K

J

B

2 K J

Natural frequency of load

connected to ground through

the compliance

-150

-100

-50

0

50

Magnitu

de (

dB

)

101

102

103

104

-180

-135

-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

Sample Values:

2

L

2

M

S

ML

J 0.002 kg-m

J 0.002 kg-m

K 200 N-m/rad

B 0.01 N-m-s/rad

AR

R

AR

R

316 rad/s 50.3 Hz

447 rad/s 71.2 Hz

0.008

0.011

M sT

-100

-80

-60

-40

-20

0

Magnitu

de (

dB

)

102

103

-180

-135

-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

2

M

1

J s 2

M L

1

J J s

R

AR

Rigidly-Coupled

Motor + Load

Compliantly-Coupled

Motor + Load

anti-resonance

resonance

R AR always

M sT

2

L ML SM

22L MM L

ML S

L M

J s B s K1s

J JT J J ss B s K

J J

-100

-80

-60

-40

-20

0

Magnitu

de (

dB

)

102

103

-180

-135

-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

2

M

1

J s 2

M L

1

J J s

R

AR

Rigidly-Coupled

Motor + Load

Collocated System

anti-resonance

resonance

R AR always

L sT

-300

-200

-100

0

100

Magnitu

de (

dB

)

101

102

103

104

105

106

-360

-315

-270

-225

-180

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

R

resonance

ML SL

22L MM L

ML S

L M

B s K1s

J JT J J ss B s K

J J

Non-Collocated System

-300

-200

-100

0

100M

agnitu

de (

dB

)

101

102

103

104

105

106

-360

-270

-180

-90

0

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

Bode Plot Comparison L sT

M s

T

90º additional phase lag

-150

-100

-50

0

50

Magnitu

de (

dB

)

101

102

103

104

105

106

-180

-135

-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

L

2

M AR

2

AR AR

s 1s

2 ss1

in phase

out of phase

L

M

s

Note: There is no anti-resonant frequency in this transfer function.

Also, there is 90º more phase lag at high frequency.

Effect of JL / JM Ratio on Resonance and Anti-Resonance

-150

-100

-50

0

50

Magnitu

de (

dB

)

101

102

103

104

-180

-135

-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

L

M

J1

J

L

M

J5

J

L

M

J 1

J 5

5 1 0.2

M sT

ωR lower limit = 316 rad/sec

ωAR lower limit = 0

2

M

S

ML

J 0.002 kg-m

K 200 N-m/rad

B 0.01 N-m-s/rad

-100

-80

-60

-40

-20M

agnitu

de (

dB

)

102

103

-180

-135

-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

5 1 0.2

L

M

J

J

M sT

Effect of Varying

-100

-80

-60

-40

-20

0

20

Magnitu

de (

dB

)

102

103

-180

-135

-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

M sT

L

M

10J

, , 1100 1 , J

0.

100 10 1 0.1

ωR lower limit = 316 rad/sec

317.8 rad/sec

100

101

102

103

104

-180

-135

-90

-45

0

Phase (

deg)

-150

-100

-50

0

50

Magnitu

de (

dB

)

Bode Diagram

Frequency (rad/sec)

L

M

100J

, J

1000

1001000

M sT

ωR lower limit = 316 rad/sec

Effect of KS on Resonance and Anti-Resonance L

M

J1

J

-150

-100

-50

0

50

Magnitu

de (

dB

)

102

103

-180

-135

-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

S

N-mK 200

rad

S

N-mK 20

rad

S

N-mK 2000

rad

SK 2000

SK 200

SK 20

M sT

2

L ML SM

22L MM L

ML S

L M

2

M L

2

M

J s B s K1s

J JT J J ss B s K

J J

1 as 0

J J s

1 as

J s

Limiting Behavior:

S M L

R

M L

SAR

L

K J J

J J

K

J

SL AR R

M

KAs J , 0 and

J

• Observations

– For JM > 0, the anti-resonance frequency always

occurs before the resonance frequency.

– At a low JL/JM ratio, the resonance and anti-

resonance frequencies are close to each other at a

high frequency.

S M L

R

M L

SAR

L

K J J

J J

K

J

– As JL/JM increases, both the anti-resonance and

resonance frequency decrease, with the anti-

resonance frequency decreasing at a faster rate.

– At JL = JM, ωAR = 0.707ωR.

L M

S M L M LR S S

M L L M

SAR S

L L

J J1 1

K J J J JK K

J J J J

K 1K

J J

SL AR R

M

KAs J , 0 and

J

– For a given JL, to increase the resonance frequency,

either increase the shaft stiffness or decrease the

motor inertia.

– As KS increases, both ωR and ωAR increase.

– A general guideline to avoid instability problems is to

keep the desired closed-loop bandwidth well below

the resonance frequency and the ratio JL / JM less

than 5.

L M

S M L M LR S S

M L L M

J J1 1

K J J J JK K

J J J J

– Heavy-Load Approximation: JL > 5JM

M L L

M L MM

MM L

L

J J J

J J JJ

JJ J1

J

2

L ML SM

22L MM L

ML S

L M

2

L ML S

2 2

L M ML S

J s B s K1s

J JT J J ss B s K

J J

J s B s K1

J s J s B s K

-120

-100

-80

-60

-40

-20

0M

agnitu

de (

dB

)

101

102

103

104

-180

-135

-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

exact

approximate

Heavy-Load Approximation L

M

J5

J

Belt-Driven Load and Motor

M M L L

2

MM L M

L

R R

RJ J T

R

Rigid Belt Case:

2 2M M M LM M M M M M L L M L

2 2L L L ML L L L L M L M M L L

J B 2KR 2BR T 2KR R 2BR R

J B 2KR 2BR 2KR R 2BR R T

Equations of Motion

Laplace Transform of the Equations of Motion (TL= 0)

2 2 2

M M M M M M L M L L

2 2 2

L L L L L M L M L M

J s B 2BR s 2KR s 2BR R s 2KR R s T s

J s B 2BR s 2KR s 2BR R s 2KR R s

2 2 2

M M M M M L M L M

2 2 2LM L M L L L L L

J s B 2BR s 2KR 2BR R s 2KR R s T s

s 02BR R s 2KR R J s B 2BR s 2KR

2 2M M M LM M M M M M L L M L

2 2L L L ML L L L L M L M M L L

J B 2KR 2BR T 2KR R 2BR R

J B 2KR 2BR 2KR R 2BR R T

Transfer Functions

2 2 2

L L L LM

L M L M L

4 2 2 3

M L L M M L M L L M

2 2 2 2

L M M M L L M M L

2 2

L M M L

J s B 2BR s 2KRs

T D s

2BR R s 2KR Rs

T D s

D s J J s 2B J R J R J B J B s

2K J R J RL B B 2B B R B R s

2K B R B R s

Transfer Functions

(BL = 0 and BM = 0)

2 2 2

M L L L

4 2 2 3 2 2 2

M L L M M L L M M L

L M L M L

4 2 2 3 2 2 2

M L L M M L L M M L

J s 2BR s 2KRs

T J J s 2B J R J R s 2K J R J R s

2BR R s 2KR Rs

T J J s 2B J R J R s 2K J R J R s

2 2 2

M L L L

2 2 22L M M L M L

2 2

L M M L

L M L M L

2 2 22L M M L M L

2 2

L M M L

J s 2BR s 2KR1s

T J R J R s J Js 2Bs 2K

J R J R

2BR R s 2KR R1s

T J R J R s J Js 2Bs 2K

J R J R

Transfer Functions

(BL = 0 and BM = 0)

Transfer Functions in Standard Form

(BM = 0 and BL = 0)

2

AR1 2

AR ARM

22 R

2

R R

2L

22 R

2

R R

2 ssK 1

sT 2 ss

s 1

K s 1s

T 2 sss 1

M L2 2 2

M M L L

2

L1 2 2

M L L M

2 2

M L L M

R

M L

R

M L

2 2

M L L M

2

LAR

L

LAR

L

R RK

J R J R

R BK

J R J R K

2K J R J R

J J

B

2KJ J

J R J R

2KR

J

BR

2KJ

Modeling of Backlash

• In everyday language, the word backlash sounds as

undesirable as its meaning, i.e., a strong adverse reaction

or a violent backward movement. In engineering, the

situation is no different.

• Backlash, the excessive play between machine parts, as

often occurs in gears and flexible couplings, is highly

undesirable and usually exists with compliance. It gives

rise to inaccuracies in the position and velocity of a

machine, as well as to delays and oscillations.

• The model that is most beneficial for control design is the

least complex model that still retains sufficient accuracy to

capture the gross dynamic behavior of the system.

• The diagram shows the physical system under

investigation with the accompanying assumptions.

• In addition, we assume that collisions due to backlash

are sufficiently plastic to avoid bouncing.

• It is critical that the model capture the fact that the output

from the backlash element is a torque on the load inertia, not

a displacement of the load inertia. The model presented here

also captures the situation where the assumed-massless

compliant element has damping.

• The importance of this can be demonstrated as follows.

Imagine that you are compressing with your hand a massless

spring that possesses no internal damping. If you were to

suddenly move your hand away, the spring would stay in

contact as its response is instantaneous, since, being pure, it

has no mass or damping. But if the spring has damping and

you repeat the experiment, the spring’s response would not

be instantaneous and it would start to lose contact.

• The model shown in the block diagram, developed from the

system equations of motion, captures these essential

attributes and fosters insight.

M MM M M S

L LL L S L

shaft shaftS

J B T T

J B T T

T B K

Equations of Motion

c b

d M L

d d b bS S S

Define :

T T T (B K ) (B K )

ShaftTorque

d b b Sd

b d b b Sd

d b b Sd

Kmax 0, ( ) if (T 0)

B

K( ) if (T 0)

B

Kmin 0, ( ) if (T 0)

B