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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
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