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Forced response of a nonlinear translating brake band in the
presence of friction guides
Osman Taha Sen, Research Associate, [email protected], Department
of Mechanical Engineering, Istanbul Technical University, Istanbul,
34437, Turkey
Jason T. Dreyer, Research Scientist, [email protected],
Department of Mechanical and Aerospace Engineering, The Ohio State
University, Columbus, OH, 43210, USA
Rajendra Singh, Professor, [email protected], Department of
Mechanical and Aerospace Engineering, The Ohio State University,
Columbus, OH, 43210, USA
ABSTRACT The goal of this article is to investigate the response
of a nonlinear translating brake band system under different
external loading and constraint conditions. A friction bench
experiment is designed, built, and instrumented accordingly. In
this experiment, an actuation body (supported by the friction
guides) is pushed against a translating brake band over a
prescribed actuation cycle. Therefore, two friction regimes are
generated; one between the actuation body and brake band, and
another one between the actuation body and friction guide(s).
Locations of the friction guides and external load application
points are varied and all possible cases are experimentally and
computationally studied. First, the effect of the center of contact
force shift on the forced response is investigated, and conditions
that lead to this shift are examined. Second, a nonlinear
mathematical model is utilized to explain the relationship between
the center of contact force location and the forced system
response, as well as to observe certain trends. Finally, such
trends are confirmed by measurements, and a better understanding of
the effect of external load and constraint locations on a variation
in the friction force is obtained. Some of the findings are briefly
linked to the vehicle brake judder problem. Key words:
Friction-induced vibration, Experimental dynamics, Leading and
trailing edge dynamics in translating media, Brake judder,
Frictional guides 1. Introduction The effect of the center of
contact force location on the generation of brake squeal noise was
first studied by Spurr [1], where the so-called squeal mechanism,
“sprag-slip”, is first proposed. Spurr [1] illustrated this
mechanism with a rigid strut that was loaded against a rotating
surface with an angle α, and calculated the friction force (Ff) at
the interface as Ff = µbFN/(1−µbtan(α)), where µb and FN are the
friction coefficient at the contact interface and normal load,
respectively. This expression shows that Ff is boundless when the
condition α = cot−1(µb) is satisfied; hence the strut will lock and
will not permit any motion. As initiated by Spurr [1], the
sprag-slip concept has been subsequently studied using pin-on-disc
models [2-4] with experimental and analytical techniques, and the
system stability is related to the sprag angle, though other
mechanisms of the brake squeal problem have been extensively
studied [5, 6]. Fieldhouse and Steel [7] and Fieldhouse et al. [8]
expand the Spurr model with an additional friction regime, which
locates between the brake pad and the abutment (support) bracket.
Fieldhouse et al. [8] calculate the distance between the center of
contact force and the supported edge (ρ) with the following
expression: ρ = µbw + (1 ± µbµg)l/2, where µg, w and l are the
friction coefficient at the supported edge, thickness, and length
of the pad, respectively; the ± sign corresponds to the directional
behavior of the friction force at the supported edge. Note that
this expression is only valid for a pad that is supported at the
trailing edge. The above mentioned concept has yet to be applied to
the vehicle brake judder problem, which is a low frequency,
friction-induced forced vibration problem. The primary source to
this problem is assumed to be the geometric distortion of the brake
rotor at single or multiple orders of the excitation frequency that
is proportional to the vehicle speed [9-13]. Brake judder is
typically quantified in terms of the brake torque variation (T(t))
[9-12], though it is also observed as steering wheel nibble [13].
Disturbances generated at the source are usually amplified at the
path, e.g. the dynamic amplification in the angular displacement of
the vehicle steering wheel has been related to the rotor
imperfections and to the resonances of structural paths [13].
Additionally, experimental studies suggest that the frictional
torque amplitudes at higher speeds are influenced by the caliper
dynamics, though the available judder models do not seem to include
the actuation system [9-13]. Accordingly,
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the chief goal of this article is to experimentally and
numerically investigate the effect of actuation system dynamics on
the brake judder response by examining the dynamic interactions
between the brake pad and the caliper (including the guide pins).
Due to a large number of different caliper designs, it is extremely
difficult to construct an experiment to evaluate alternate designs
under controlled conditions. Therefore, investigations are carried
out with a simplified yet controlled translating brake band
experiment, and the effects of the frictional guide and normal load
application locations on the shift of the center of contact force
location are conceptually examined. Further, the effect of the
center of contact force location shift on the brake judder response
is explained with lower dimensional models. 2. Problem formulation
Schematics of the proposed physical system are shown in Fig. 1
where a friction material (simulates the brake pad) is pushed
against a translating band (simulates the brake rotor). In the
study, the friction material is supported either from leading
(superscript l) or trailing (superscript t) edge with a frictional
guide (simulates the guide pin). Therefore, two different caliper
designs known as “push” and “pull” type calipers in automotive
brake applications are examined [14]. The second frictional regime
that emerges at one of these edges is represented with Ffg, where
the preceding ± sign stands for the positive and negative velocity
conditions at the supported edge. In addition, the normal loads and
contact forces at the first frictional regime (between the friction
material and band) are given with FN
l , FNt ,
Fp
l and Fp
t , respectively.
Fig. 1 Illustration of the problem statement; (a) case A with
the frictional guide located at the trailing edge; (b) case B with
the frictional guide at the leading edge. Besides different
constraint location cases (cases A and B), the normal load ( FN
l and FNt ) application locations are also
varied as shown in Fig. 2: Close to the edges (1), center with a
narrow gap (2), leading edge (3), and trailing edge (4).
Accordingly, eight different cases are investigated in total (Fig.
2).
Fig. 2 Illustration of all configurations. Letters A and B
describe the different constraint location cases. Numbers 1 to 4
designate the different normal load configurations. Arrows
represent the forces acting on the friction material. Key:
, FNl and FN
t ; , Ff, , Ffg.
Band
Friction material
Friction regime #1
Friction material
Friction regime #2
Frictional guide
Direction of band travel
Ff Ff
FNl
FNt
FNl
FNt
Fpl
Fpt
Fpl
Fpt
ex ey ez
±Ffg ±Ffg
(a) (b)
Normal load locations
Frictional guide at the trailing edge
(A)
Frictional guide at the leading edge
(B) Point actuators closer to
the outer edges (1)
A1 B1
Point actuators close to the center
(2)
A2 B2
Point actuators at the leading edge
(3)
A3 B3
Point actuators at the trailing edge
(4)
A4 B4
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The main objectives of the current article are as follows: 1.
Calculate and compare the center of contact force locations for all
the configurations of Fig. 2 based on the static equilibria. 2.
Develop a simplified nonlinear model of the system of Fig. 1, and
obtain numerical solutions for all cases. 3. Design a simple
laboratory experiment to simulate the system of Fig. 1, perform
experiments, and validate the nonlinear model solutions. 3.
Calculation of center of contact force location The distance from
the center of contact force to the frictional guide (ρ) is
calculated for all cases by using force and moment equilibria,
which are derived from the free body diagram in Fig. 3. As seen in
the figure, the total friction force Ff = µbFp, where µb and Fp are
the friction coefficient and the total contact force at the
friction material/band interface, respectively. Obviously, Ff is
transmitted to the frictional constraint as a normal load, and thus
the friction force at this supported edge is Ffg = µgFf, where µg
is the friction coefficient at the friction material and guide
interface. Since the direction of Ffg changes for the positive and
negative velocity of the constrained edge, equilibrium equations
must be solved for these two sliding cases. Since –µbµgFp ≤ Ffg ≤
µbµgFp, the center of contact force location for the pure sticking
condition should be between these two sliding conditions. First,
the equations of force (Eq. 1) and moment (Eq. 2) equilibria (about
point O in Fig. 3) are given below for case A.
FN
l + FNt − Fp − µbµg Fp = 0 for x − 0.5l θ > 0 , (1a)
FN
l + FNt − Fp + µbµg Fp = 0 for x − 0.5l θ < 0 , (1b)
FN
l 0.5l + l l( ) + FNt 0.5l − l t( )− Fpρ + µbwFp = 0 . (2)
Fig. 3 Forces acting on the friction material for case A with
the frictional guide located at the trailing edge. The unknowns Fp
and ρ are solved with Eqs. (1) and (2), and ρ is obtained for the
two sliding cases as:
ρ+ = 1+ µbµg( ) l2 +
µbw1+ µbµg
+FN
l l l − FNt lt
FNl + FN
t
⎡
⎣⎢⎢
⎤
⎦⎥⎥
for x − 0.5l θ > 0 , (3a)
ρ− = 1− µbµg( ) l2 +
µbw1− µbµg
+FN
l l l − FNt lt
FNl + FN
t
⎡
⎣⎢⎢
⎤
⎦⎥⎥
for x − 0.5l θ < 0 , (3b)
where the superscripts + and – correspond to the positive and
negative sliding velocities, respectively. For the special case
of
FNl l l = FN
t lt , Eq. (3) simplifies to Fieldhouse et al.’s expression [8].
Similarly, ρ+ and ρ− for case B are obtained as:
ρ+ = 1+ µbµg( ) l2 −
µbw1+ µbµg
−FN
l l l − FNt lt
FNl + FN
t
⎡
⎣⎢⎢
⎤
⎦⎥⎥
for x + 0.5l θ > 0 , (4a)
ρ− = 1− µbµg( ) l2 −
µbw1− µbµg
−FN
l l l − FNt lt
FNl + FN
t
⎡
⎣⎢⎢
⎤
⎦⎥⎥
for x + 0.5l θ < 0 . (4b)
Based on Eqs. (3) and (4), the center of contact force locations
can be estimated for various conditions. These conditions and the
center of contact force locations are tabulated in Table 1 for both
positive and negative sliding velocity cases. As seen in the table,
µg is the key parameter that affects ρ+ and ρ−. Moreover ρ+ = ρ− as
µbµg = 0. For µb = 0, Ff = Ffg = 0; hence, both friction regimes
are inactive. However for µg = 0, Ff ≠ 0, but Ffg = 0, though ρ+ =
ρ− is still valid. This concludes that the friction regime at the
friction material and frictional guide interface has a significant
effect on the center of contact force location.
Ff
Ffgsgn x 0.5l( )
x
O
Fp
FNl
FNt
ex ey ez
w
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Table 1 Summary of the conditions and corresponding center of
contact force locations for both positive and negative sliding
conditions.
Case Positive Sliding Velocity Negative Sliding Velocity
Condition Center of contact force location Condition Center of
contact
force location
A1: (ll – lt = 0) Always ρ+ > 0.5l µg > 2w/l µg < 2w/l
ρ− < 0.5l ρ− > 0.5l
A2: (ll – lt = 0) Always ρ+ > 0.5l µg > 2w/l µg < 2w/l
ρ− < 0.5l ρ− > 0.5l
A3: (ll – lt > 0) Always ρ+ > 0.5l µg >
2µbw+ ll − l t
µb l + ll − l t( )
µg <
2µbw+ ll − l t
µb l + ll − l t( )
ρ− < 0.5l
ρ− > 0.5l
A4: (ll – lt < 0) µg > −
2µbw+ ll − l t
µb l + ll − l t( )
µg < −
2µbw+ ll − l t
µb l + ll − l t( )
ρ+ > 0.5l
ρ+ < 0.5l
µg >
2µbw+ ll − l t
µb l + ll − l t( )
µg <
2µbw+ ll − l t
µb l + ll − l t( )
ρ− < 0.5l
ρ− > 0.5l
B1: (ll – lt = 0) µg > 2w/l µg < 2w/l ρ+ > 0.5l ρ+ <
0.5l
Always ρ− < 0.5l
B2: (ll – lt = 0) µg > 2w/l µg < 2w/l ρ+ > 0.5l ρ+ <
0.5l
Always ρ− < 0.5l
B3: (ll – lt > 0) µg >
2µbw+ ll − l t
µb l − ll + l t( )
µg <
2µbw+ ll − l t
µb l − ll + l t( )
ρ+ > 0.5l
ρ+ < 0.5l
Always ρ− < 0.5l
B4: (ll – lt < 0) µg >
2µbw+ ll − l t
µb l − ll + l t( )
µg <
2µbw+ ll − l t
µb l − ll + l t( )
ρ+ > 0.5l
ρ+ < 0.5l
µg > −
2µbw+ ll − l t
µb l − ll + l t( )
µg < −
2µbw+ ll − l t
µb l − ll + l t( )
ρ− < 0.5l
ρ− > 0.5l
By using Eqs. (3) and (4), numerical values for ρ+ and ρ− are
calculated and listed in Table 2 in a normalized way, where the
normalization is done with: ρ = ρ 0.5l . Table 2 Normalized center
of contact force locations. Normalization is done by the geometric
center of the body (0.5l).
Normal load locations
Friction guide at trailing edge (Case A) Friction guide at
leading edge (Case B) ρ+ ρ− ρ+ ρ−
(1) 1.64 0.50 1.45 0.28 (2) 1.77 0.53 1.43 0.27 (3) 2.29 0.66
0.80 0.12 (4) 1.00 0.34 2.09 0.44
First, observe the following rank orders: ρA4 < ρ A1 < ρ
A2 < ρ A3 for case A and ρ
B3 < ρ B2 < ρ B1 < ρ B4 for case B. Second, it
is seen that ρ+ > ρ− for all cases. This means that the
motion in the +ex direction shifts ρ further away from the
supporting guide than the motion in the –ex direction. Third, for
symmetrically loaded cases (A1, A2, B1, B2), the motion of the band
in the +ey direction causes the center of contact forces to shift
closer to the leading edge. Therefore, ρ
A > ρ B for these cases. Fourth, for the asymmetrically
loaded cases (A3, A4, B3, B4), the center of contact forces moves
towards the loaded edge.
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4. Nonlinear mathematical model of the translating brake band
problem As a further investigation, a two degree of freedom
nonlinear model of the translating brake band problem is developed
and shown in Fig. 4. The model assumes only the translation (x) in
the ex direction and rotation (θ) about the ez direction. The
contact between the band and the friction material is modeled with
point contact models at two different locations, which are
described with linear springs (
kp
l and kp
t ) and viscous dampers ( cp
l and cp
t ).
Fig. 4 Two degree of freedom nonlinear mathematical model for
case A with the frictional guide located at the trailing edge.
Governing equations for case A (Fig. 4) are obtained as:
mx + cp
l + cpt( ) x + cpl lpl − cpt lpt( ) θ + kpl + kpt( )x + kpl lpl
− kpt lpt( )θ + Ffg sgn x − 0.5l θ( ) = FNl + FNt , (5)
I θ + cpl lp
l − cpt lp
t( ) x + cpl lpl( )2 + cpt lpt( )2⎛⎝ ⎞⎠ θ + kpl lpl − kpt lpt(
)x + kpl lpl( )2+ kp
t lpt( )2⎛⎝ ⎞⎠θ − Ff w− 0.5Ffgl sgn x − 0.5l θ( )
= FNl l l − FN
t lt, (6)
where m and I are the mass and the inertia of the friction
material. Distances ll, lt, lp
l and lp
t are between the center of mass and the leading edge point
actuator, trailing edge point actuator, leading edge point contact
location, and trailing edge point contact location, respectively.
The friction force Ff between the friction material and band is
calculated as:
Ff = µb cp
l x + lpl θ( ) + cpt x − lpt θ( ) + kpl x + lplθ( ) + kpt x −
lptθ( ){ } . (7)
Obviously, Ffg changes direction due to the direction of the
friction material velocity at the supported edge. Therefore, the
three valued “sgn” function is used to represent the directionality
of Ffg. However, for the numerical simulations “sgn” function is
approximated as sgn(z) = tanh(σz), where z is the argument of the
“sgn” function and σ is the regularizing factor [9, 10]. Similar to
case A, the governing equations for case B are derived as follows,
where Eq. (7) still applies:
mx + cp
l + cpt( ) x + cpl lpl − cpt lpt( ) θ + kpl + kpt( )x + kpl lpl
− kpt lpt( )θ + Ffg sgn x + 0.5l θ( ) = FNl + FNt , (8)
I θ + cpl lp
l − cpt lp
t( ) x + cpl lpl( )2 + cpt lpt( )2⎛⎝ ⎞⎠ θ + kpl lpl − kpt lpt(
)x + kpl lpl( )2+ kp
t lpt( )2⎛⎝ ⎞⎠θ − Ff w+ 0.5Ffgl sgn x + 0.5l θ( )
= FNl l l − FN
t lt. (9)
External loads FNl and FN
t are described as a combination of mean ( FNml and FNm
t ) and sinusoidal loads with constant
amplitudes ( FNal and FNa
t ) as given below.
FNl = FNm
l + FNal −1( )n
2n+1( )2sin 2n+1( )ωt( )
n=0
∞
∑ , (10a)
Ff
ll lt
l
w
Ffgsgn x 0.5l( )
x
lpt
lpl
FNl
FNt
kpl
cpl
kpt
cpt
ex ey ez
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FNt = FNm
t + FNat −1( )n
2n+1( )2sin 2n+1( )ωt( )
n=0
∞
∑ , (10b)
where the fundamental excitation frequency ω is assumed to be
20π rad/s (10 Hz) due to experimental limitations as explained in
section 5. Since it is aimed to explain the effect of center of
contact force location on the judder response, a periodic type
excitation is preferred. Note that the chief excitation in the
judder problem is the rotor surface distortion profiles. In the
translating brake band problem, the undulated surface effect is
generated with modulated normal forces as given with Eq. 10, where
the time-variant parts are essentially the representation of a
triangular wave. This waveform is chosen due to the experimental
limitations, and is explained in section 5. Moreover, the numerical
simulations are run with constant µb and µg values, and the
locations of point contacts are assumed to be at the edges of the
friction material, i.e.
lp
l = lpt = 0.5l .
Results of numerical solutions are given in Fig. (5) for cases
A1 and B1, in terms of displacements at the leading (xl = x +
0.5lθ) and trailing edges (xt = x – 0.5lθ). Observe that the
supported edge is almost stationary compared to the other edge for
both cases; supported edges move a total displacement of about 30nm
(xt for Case A1) and 40nm (xl for case B1). These small amounts of
motion at the supported edges arise due to the approximation of the
“sgn” function, and these edges should stick to the frictional
guide. This can be explained with the sticking condition, which
is
FN
l + FNt − Fc ≤ µbµg Fc . The term Fc
in this condition stands for the total contact force at the
friction material/band interface and is calculated with the
expression inside the brackets of Eq. (7). Rewrite this condition
after some algebraic manipulations:
FNl + FN
t
1+ µbµg≤ Fc ≤
FNl + FN
t
1− µbµg. (11)
The satisfaction of the condition given with Eq. (11) causes the
friction material to stick to the frictional guide. Since it is
always satisfied during numerical simulations, it is concluded that
the motion of friction material at the supported edge is due to
numerical approximation; hence a pure sticking condition cannot be
numerically achieved by approximating “sgn” function.
Fig. 5 Numerically calculated leading and trailing edge
displacements; (a) case A1; (b) case B1. Key: , xl; , xt. Since the
frictional material should not move at the supported edge, the two
degree of freedom nonlinear model can be simplified to a single
degree of freedom linear one, which does a pivoting motion about
the contact point at the frictional guide. Calculated mean (
Ffm ) and peak-to-peak (
Ffpp ) friction force values using this linear system are listed
in Table 3 for all cases. Note that the results are presented in a
normalized way where the normalization is done with the
corresponding value of case A1; i.e.
Ffm = Ffm Ffm
A1 and Ffpp = Ffpp Ffpp
A1 . First, observe that Ffm and
Ffpp show the same rank orders as in
ρ; Ff
A4 < FfA1 < Ff
A2 < FfA3 ,
Ff
B3 < FfB2 < Ff
B1 < FfB4 . This shows that as the center of contact force
location moves further
away from the supported edge, a higher Ff is obtained. Second,
it is seen that in all cases except (4), Ff
A > FfB for both mean
and peak-to-peak values. This can be explained with the
sprag-slip model of Spurr [1]. For cases A1 and A2, the center of
contact force locations are at the leading edge half of the
friction material (Table 2). Hence the total reaction force has a
lateral component, which is in −ey direction (opposite to the
direction of band motion). Therefore, the friction material digs in
to the band surface, and higher contact forces are generated.
However, for cases B1 and B2, the center of contact force locations
are at the trailing edge, and thus the directions of the lateral
component of the reaction force and band motion are
(a)
x l [
m]
t [s]
x t [
m]
(b)
x l [
m]
t [s]
x t [
m]
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both in +ey direction. For cases (3) and (4), normal load
locations dominate the response; i.e. the moment arm between the
external loads and frictional guide location is longer in cases A3
and B4. Hence, higher contact forces are generated in these cases
compared to A4 and B3. Third, observe that exact same values are
obtained for both normalized mean and peak-to-peak values. This is
expected since it is a linear system and the response should not
have any frequency dependency. Table 3 Predicted mean and
peak-to-peak friction force. Data is normalized with the
corresponding values of case A1.
Normal load locations Friction guide at trailing edge (A)
Friction guide at leading edge (B)
Ffm
Ffpp Ffm
Ffpp (1) 1.00 1.00 0.91 0.91 (2) 1.08 1.08 0.90 0.90 (3) 1.42
1.42 0.54 0.54 (4) 0.58 0.58 1.29 1.29
5. Translating friction band experiment In order to validate the
numerical solution of the nonlinear model, a translating friction
band experiment is designed and built as depicted in Fig. 6. In the
experiment, a custom-built stationary friction material is pushed
towards a translating friction band by two point actuators; hence
the band is compressed between the two surfaces of the actuation
system. On one side of the actuation system, Teflon is preferred
due to its low friction characteristics. Thus the friction force
generated at this side of the band is minimal. The following
sensors are utilized on the setup: a) Load cell to measure the
force (Ff(t)) that pulls the band; b) pressure transducer to
measure the pressure (pN(t)) at the air inlets of the point
actuators; c) linear potentiometer to measure the displacement of
the band (y(t)); and d) uniaxial piezoelectric accelerometers to
measure the accelerations of the custom-built friction material (
x
l and xt ) and actuation system ( xa
l and xat ) at the leading and trailing edges.
Fig. 6 Translating friction band experiment and the actuation
system including the sensors utilized. The experiment is
specifically designed to run all cases of Fig. 2. The design of the
custom-built friction material allows obtaining the friction regime
between the friction material and leading or trailing edge
frictional guides separately. In addition, point actuators can
slide in the slot located on the actuation system; hence the
different loading conditions of Fig. 2 can also be achieved. During
the tests, pN(t) is modulated with a solenoid valve, and thus a
triangular wave shaped normal force is obtained as defined earlier.
The modulation frequency is chosen as 10 Hz due to the physical
limitations of the solenoid valve. Experimentally, it is observed
that the band displacement (y) exhibits an almost constant velocity
behavior for all cases of Fig. 2. Therefore, the band has almost no
acceleration; the measured pulling force should be equal to the
friction force Ff. This measured friction force Ff and total normal
force FN ( FN = FN
l + FNt ) are displayed in Fig. 7, again for cases A1 and
B1.
Note that FN is calculated by multiplying the measured pN(t)
with the internal areas of the point actuators Al and At. Figures
show that both Ff and FN have triangular shaped waveforms with
almost constant mean and peak-to-peak values. In addition,
Teflon
Friction material
Air inlet
Leading edge guide
Trailing edge guide
Direction of travel
Point actuators
Brake band y(t) Ff(t)
Actuation system
xt (t) x
l (t)
pN(t)
xat (t)
xal (t)
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case B1 (Fig. 7(b)) shows less mean and peak-to-peak values
compared to case A1 (Fig. 7(a)), even for the almost same FN
values. This was explained with the sprag-slip concept in the
previous section.
Fig. 7 Measured normal (FN) and friction (Ff) forces; (a) case
A1; (b) case B1. Key: , FN; ,Ff. Measured normalized mean and
peak-to-peak values of Ff are tabulated in Table 4, and the
normalization is again done by the corresponding value of case A1,
i.e.
Ff = Ff Ff
A1 . Similar to the numerical simulations, measurements present
the same
rank orders for the peak-to-peak values where Ffpp
A4 < FfppA1 < Ffpp
A2 < FfppA3 and
Ffpp
B3 < FfppB2 < Ffpp
B1 < FfppB4 for cases A and B,
respectively. In terms of Ffm, the same rank ordering is still
valid for B cases. However, for A cases the orders of A1 and A4 are
switched. Table 4 Measured mean and peak-to-peak friction force.
Data is normalized with the corresponding values of case A1.
Friction guide at trailing edge (A) Friction guide at leading
edge (B)
Ffm Ffpp
Ffm Ffpp
(1) 1.00 1.00 0.88 0.67 (2) 1.04 1.07 0.81 0.63 (3) 1.19 1.38
0.56 0.56 (4) 1.03 0.78 1.05 1.16
Furthermore, the motion of the friction material on leading and
trailing edges is checked with the measured acceleration data
xl , x
t , xal and xa
t . The relative accelerations at the edges of the friction
material with respect to the actuation system are
calculated as xrel
l t( ) = xl t( )− xal t( ) and xrelt t( ) = xt t( )− xat t( ) .
Finally, calculated relative accelerations are transformed to
the
frequency domain, and the acceleration response ratio from
trailing edge to leading edge is calculated with:
atl ω( ) =xrel
t ω( ) xrell ω( )*xrel
l ω( ) xrell ω( )*, (12)
where * denotes the conjugate operator. Using Eq. (12), atl is
calculated at ω = 20π rad/s which is the fundamental excitation
frequency, and it is observed that the acceleration response ratios
at the unconstrained edge of the friction material are at least an
order of magnitude higher than the constrained edge, i.e. atl 1 for
cases (A) and (B), respectively. 6. Conclusion The current article
investigates the shift of contact force center location and its
effect on the system response due to different loading and
constraint combinations. It is successfully shown that the location
of the center of contact force alters the generated friction force
response; hence the judder response of the system changes. The main
contributions of this article are summarized as follows. First, the
location of the center of contact force is calculated with force
and moment equilibria, and the main factors that affect this
location are investigated. Second, a two degree of freedom
nonlinear mathematical model of the problem is developed, and
numerical solutions are obtained for 8 different cases where
specific trends are observed. Third, a simple yet controlled
translating friction band experiment is designed and built. The
same cases of numerical investigation are run on the setup, and
data is successfully collected. Similar trends are observed
experimentally; hence the numerical solutions are conceptually
validated.
0.04 0.06 0.08 0.1 0.12 0.14 0.16220
230
240
250
0.04 0.06 0.08 0.1 0.12 0.14 0.165
10
15
20(a)
F N [N
]
y [m]
F f [N
]
0.04 0.06 0.08 0.1 0.12 0.14 0.16220
230
240
250
0.04 0.06 0.08 0.1 0.12 0.14 0.165
10
15
20(b)
F N [N
]
y [m]
F f [N
]
-
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