caliper adapter calculus

8/13/2019 Caliper Adapter Calculus

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Brake caliper adapter calculus

Heiko Schroter

January 16, 2013

Contents

1 Dynamical and statical load 2

2 Force, moment equations 3

3 Theoretical maximum deceleration 4

4 Technically maximum braking acceleration 5

5 Limiting case of the theoretically and technically possible decelerationamax 65.1 Coefficient of adhesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

6 Maximum front braking force 7

7 Maximum braking force on the brake discs 8

8 Minimum thickness of a theoretical caliper adapter 98.1 Thickness of rectangular bar (principal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

8.2 Thickness of a theoretical adapter (tension calculus according to Mises) . . . . . . . . . . 98.3 FEM simulation of the theoretical adapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

9 Calculation of the adapter for the GS850 129.1 FEM Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

10 Forces at the hand brake lever and brake pistons 1410.1 B rake piston forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1410.2 H and lever forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1410.3 Stroke of the master cylinder shbz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1510.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

11 FEM simulation of the final draft adapter for Suzuki GS850 16

11.1 Matrial stress in N/mm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1711.2 Shifts inmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

12 Various adapters forms 20

13 Examples of adapters on motorcycles 21

14 Material values for aluminium materials. 22

15 Addendum (Grip strength) 24

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1 Dynamical and statical load

Figure 1: Forces, Dimensions

S = Center of gravityl = Wheelbase

m = Vehicle mass

g = Acceleration of gravity = 9,81

ms2

a = DecelerationFg = m*g

FN1 = Normal force frontFN2 = Normal force rear

Fdyn = m*a

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2 Force, moment equations

according to [2] and [1, S.41].

0 =FBr Fdyn (1)

0 =FN1+FN2 Fg =FN1+FN2 m g (2)

0 = l FN1+l2 Fg+hs Fdyn , moments around B (3)

=> FN1= Fg l2l

+Fdyn hsl

FN1= m g l2l

statical

+ m a hsl

dynamical

, force on front axle (4)

insert (4) in (1)

0 =

(Fg l2+FDyn hs)

l +FN2

Fg

=> FN2= Fg (Fg l2+FDyn hs)

l

=Fg (1 l2l)

l1l

hsl FDyn

=Fg l1l FDyn

hsl

FN2= m g l1l m a

hsl

(5)

During braking, the front axle is additionally loaded with the dynamic component m a hsl

and the

rear unloaded by the same amount.

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3 Theoretical maximum decelerationThe front axial force FN1 is maximum when the rear axle force FN2 0 (incipient rollover).

FN2= 0 =m g l1

l

m a hs

l

a= amaxTheor=g l1hs

(6)

and (6) inserted in (4)

FN1= m g (7)

i.e. at maximum decelaration amaxTheor the total vehicle mass lasts on the front wheel.

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4 Technically maximum braking acceleration

Figure 2: Stiction,= Coefficient of adhesion

FN2= 0 and (7) results in the braking acceleration:

FBr

= (FN1

+FN2

)m a = m g

a = g

a= amaxTech = g (8)

i.e. the deceleration is determined by the adhesion of the road.

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5 Limiting case of the theoretically and technically possibledeceleration amax

amaxTech = amaxTheor

g = g l1hs

= l1hs

(9)

i.e. if l1hs

the maximum usable adhesion is limited by the geometry of the bike. For example a dry

race track with Mu = 1.3 could only be exploited ifhs l11,3 .

5.1 Coefficient of adhesion

Table 1: Coefficient of adhesion for dry road surfaces:Type Value Sourcecountry road / highway 0,8 1.Racetracks 1,0 bis 1,4 1.Concrete, stone, granite 0 ,7 2.Asphalt 0,6 2.Blue basalt 0,55 2.Concrete 0,6 bis 0,9 3.Aspahlt 0,6 bis 0,8 3.Concrete bis 1,0 4.

Source:

1. Institute for Motorcycle Safety Association, Dr.-Ing.Achim Kuschewski

2. WABCO GmbH, 31028 Gronau

3. Schulz,H. (10%Schlupf)

4. Kirnich,G. (20% Schlupf)

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6 Maximum front braking force

with (9) two cases for the maximum possible braking force can be derived: FBrv = Braking force front

Limited by geometry (rollover)

FBrvmax = max FN1max = m g l1hs (10)

Limited by adhesion (slipping of the front wheel)

FBrv = FN1=

m g

l2l

+m a hsl

with (4) and (8)

FBrv =m g

l2

l +

2 hsl

(11)

Example with m = 500kg:

Bike l l1 l2 hs usable max

M1 1.5m 0,75m 0,75m 0,85m 0,88M2 1,5m 0,85m 0,65m 0.65m 1,3

Table 2: Comparison of possible brake forces

Figure 3: FBrs per Rotorx=adhesion coeffcient, y=braking force

The brake performance of the machine M1 is higher up to = 1.01 , although the center of mass ofM2 lies lower and back. i.e., the geometric possibility of the higher braking power of the lower and

further back center of mass can only be used at extremely handy dry road surfaces, such as concreteor race tracks.

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7 Maximum braking force on the brake discs

Figure 4: Force on the brake disk

FBr R= FBrs r R = radius wheel, r=radius brake rotor (Brs)

Braking forces at the front:

FBrsvmax= m g l1hs

Rvrv

, for

l1hs

with (10) (12)

FBrsv =m g

l2

l +

2 hsl

Rvrv

, for

(b) =6 My

b h2

Example:

FBrsv = 9392N, la= 50mm,ha = 75mm,zul(Al6082Rp0.2)= 250 Nmm2

=> bamin= 6 My(zul) h2

=6 FBrsv la

(zul) h2a= 2mm

8.2 Thickness of a theoretical adapter (tension calculus according to Mises)

Now the braking force is not directly applied on the adapter, but at the caliper (Fig. 7). The allowabletension in the adapter plate is derived from equivalent stress according to Mises [8] (only rough calculation,because the adapter is a parallelogram and not a rectangular plate.)

(v)= 2s+ 3 2b s= shear stress, b= bending stress, y-component = 0 (14)

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Figure 7: Caliper adapter

The brake system is loaded with impulses. Existing microscopic cracks shall not expand. Thereforethe equivalent stress shall not exceed the alternate bending strength of the material. The pulsed loadrequires a material which is not too brittle. Requested is a strong and ductile material.

(v) bW , bW= alternate bending strength in N/mm2

Because the adapter plate is not rectangular the following value is apllied for z:

zParallelogram 0, 7 ha

Fs = FBrsv sin() (shear stress)Fm = FBrsv cos() (bending stress)

bas = Fs

ha bW=

FBrsv sin()

ha bW(req thickness by Fs)

bam =12 0, 7 Fm (la+lz)

bW h2a

=8, 4 FBrsv cos() (la+lzange cos())

bW h2a(req thickness by Fm)

batot =

b2as + 3 b2am

(15)

Example:la = 40mm, ha = 75mm, lzange= 44mm, = 66

, = 11, FBrsv = 10000N, d(Al6082)= 95 Nmm2

batot =

10000N sin(66)

75mm 95 Nmm2

2+. . .

3

8, 4 10000N cos(66)

(40mm+ 44mm cos(11))

95 Nmm2

752mm2

2 12

=

1, 64mm2 + 84, 9mm2

batot = 9, 3mm

batot is the minimum thickness of the material which can stand all stresses and stays below bW. WithNdisks or adapters the thickness per adapter is: baAdapter =batot/N

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8.3 FEM simulation of the theoretical adapter

Figure 8: FEM Simulation ba = 9,3mmThe braking force FBr is applied perpendicular to the right eye. The fixed bearing of the adapteris the entire left edge. In the local lower region of the fixed bearing, the allowed tension value of2bW = 95N/mm is exceeded. The rest of the area stays below bW .

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9 Calculation of the adapter for the GS850

TOKICO Two-piston calipers shall be used.

Type HP weight top speed rotor

GS850 80 500kg 203km/h 275mmDL1000 98 470kg 208km/h 310mm

Table 3: technical data GS850,DL1000

Figure 9: Tokico Bremszange / Adapter Entwurf

Technical data Suzuki GS850:mmax = 478kg (calc value = 500kg), l = 1,5m, l1 = 0,75m, hs = 0,85m, Rv = 0,24m(19), rv =137,5mm (275mm) => rveff = (275/2 20) = 117, 5mm, ha = 73mm, la = 41,3mm, lcaliper =

44,9mm, = 9, = 70, l1hs = 0, 88.

Material chosen: Aluminium AL6082 =(d) = 95 Nmm2

FBrsvmax= m g Rv

rveff= 0, 88 500kg 9, 81

N

mm2

0, 24m

0, 1175m= 8888N

The braking force 1is distributed on to two rotors 8888N/2 = 4444Nand with (15) the minimumthickness of a single adapter is:

baAdapter =

4444N sin(70)

73mm 95 N

mm2

2+. . .

3

8, 4 4444N cos(70)

(41, 3mm+ 44, 9mm cos(9))

95 Nmm2

732mm2

2 12

=

0, 62mm2 + 3 2, 162mm2

baAdapter = 3, 79mm (16)

9.1 FEM Simulation

The FEM simulation (Fig. 10,11) of the adapters with a thickness according to (16) shows localareas where the stress value is a little bit higher than the maximum of 95 N/mm2. The leverageeffect of the caliper causes additional tension and bending moments in the Z-direction which arenot considered in the idealized estimate.

1

For comparison. The braking force of the DL1000 isFBrsvmaxDL1000 = 0, 9470kg 9, 81 Nmm2

0,24m

0,135m = 7377N. Wheeldiameter and wheelbase are identical. The height of the center of gravity assumed to be equal.

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Figure 10: Front-, Rearside

Figure 11: bolt fixing level, color table

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10 Forces at the hand brake lever and brake pistons

10.1 Brake piston forces

Figure 12: Master brake cylinder (HBZ), brake piston Scheme

FBrs = Fk 2 B (*2, because of 2 pads)

P = F

A =

FhbzAhbz

= FkAk n

(n = no of pistons per caliper)

Fhbz = FBrs Ahbz N

2 B Ak n =

FBrs d2hbz N

2 B d2k n (N = no of rotors)

adhesion coefficient B for brake pads according to ATE ca. 0,3-0,5. FBrs = braking force per rotor,see (13) and (12). dk = of brake piston, dhbz = of main braking cylinder.

10.2 Hand lever forces

Figure 13: Forces at the hand lever

Fhbz lhbz =FHand lh

FHand= lhbz

lh

d2hbz

d2k

FBrs N

2 B n (17)

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10.3 Stroke of the master cylinder shbz

With FkFhbz

= AkNnAhbz

= shbzsk

(Hydraulic Press)and an estimated air gap ofsk = 0.1 mm between rotor and the brake pads the hub of the master cylinderis:

shbz= d2k N n sk

d2hbz(18)

10.4 Example

Suzuki GS850m=500kg, N=2, FBrs = 4444N (Force per disk) , lhbz = 20mm, lh = 100mm, B = 0.4, dhbz = 16mm,P = brake tube pressure(bar).

GS850 n dk shbz FHand FHandkg FHBZ P

Original 1 38mm 1,13mm 394N 40kg 1970N 15,7bar

Tokico 2 30mm 1,41mm 316N 32kg 1580N 9,8bar

Table 4: Hand force and hub of the master cylinder

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11 FEM simulation of the final draft adapter for Suzuki GS850

FBrsvmax = 4509N, Material Al6082 (bW = 95 Nmm2

). The flanges are 10mm thick with 3,79mm needed

according to eq (16). The allowed stress level is adhered to. At the top fixing bolt a stress level of 136 Nmm2

can be observed. The other areas stay below v


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11.1 Matrial stress in N/mm2

Figure 16: Front-, Rearview

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Figure 17: Stud level, color table

11.2 Shifts inmm

Figure 18: X-direction

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Figure 19: Y-direction

Figure 20: Z-direction

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12 Various adapters forms

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13 Examples of adapters on motorcycles

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14 Material values for aluminium materials.

Selected data for some aluminium.

Material Rp0,2 bW Br Z S E Be

AW-2014 T6 380 100 7 1 5 4 4AW-2017 T3 240 8 2 5 5 4AW-5083 H111 120 110 10 2 2 4 4AW-6012 T6 260 80 8 2 5 5 2AW-6082 T6 250 95 12 2 2 3 2AW-7022 T6 370 110 6 1 5 5 3AW-7075 T6 480 160 5 2 5 5 3

Table 5: Material Properties (without guarantee)

Rp0,2 = offset yield strength [ Nmm2

], bW= alternate bending strength [ Nmm2

],Br=breaking strain [%], Z=machining, S=welding,E=anodising, Be=coating

1 very well

2 good

3 satisfactory

4 sufficient

5 not suitable

The corrosion protection is to be considered in the high-strength aluminum alloys. The alternatebending strength can be estimated from the tensile strength (b, Rm): A value of 0.2 . . . 0.25 bis allowable with alternating forces [7, S.103]

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References

[1] Projekt Antidive System, Markus Kriete, Max Kohler, Prof Dr.Ing. Uwe Reinert,WS07/08 , S.41,Hochschule Bremen

[2] Vittore Cossalter, Motorcycle Dynamics, 2006, ISBN-10: 1430308613, ISBN-13: 978-1430308614

[3] Vittore Cossalter, Roberto Lot , About the motorcycle braking,http://www.dinamoto.it/index.html

[4] z88aurora, Uni Bayreuth, Univ. Prof. Dr.-Ing. Frank Rieg,http://www.z88.uni-bayreuth.de/z88aurora/wasistz88.htm

[5] gmsh, a three-dimensional finite element mesh generator with built-in pre- and post-processing facil-ities, Christophe Geuzaine and Jean-Francois Remaclehttp://geuz.org/gmsh

[6] FreeCAD, A parametric 3D CAD modeler,http://sourceforge.net/apps/mediawiki/free-cad/index.php

[7] Feinwerkelemente, Horst Ringhardt,Carl Hanser Verlag Munchen, 1974

[8] http://de.wikipedia.org/wiki/Vergleichsspannung

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15 Addendum (Grip strength)

Standard value for grip strength (in kg):

Age Men Female

2024 years right 4655 right 2532Links 3847 left 2228

2529 years right 4455 right 2834left 4350 left 2329










75 years und above right 2030 right 1419left 1725 left 1317

Table 6: Grip strength in kg

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MALES FEMALES

Age (years) Weak Normal Strong Weak Normal Strong

10-11 22.4 < 11.8 11.8-21.6 >21.612-13 31.2 < 14.6 14.6-24.4 >24.414-15 44.3 < 15.5 15.5-27.3 >27.316-17 52.4 < 17.2 17.2-29.0 >29.018-19 55.5 < 19.2 19.2-31.0 >31.020-24 56.6 < 21.5 21.5-35.3 >35.325-29 57.5 < 25.6 25.6-41.4 >41.430-34 55.8 < 21.5 21.5-35.3 >35.335-39 55.6 < 20.3 20.3-34.1 >34.140-44 55.3 < 18.9 18.9-32.7 >32.7

45-49 54.5 < 18.6 18.6-32.4 >32.450-54 50.7 < 18.1 18.1-31.9 >31.955-59 48.5 < 17.7 17.7-31.5 >31.560-64 48.0 < 17.2 17.2-31.0 >31.065-69 44.0 < 15.4 15.4-27.2 >27.270-99 35.1 < 14.7 14.7-24.5 >24.5

Table 7: Grip Strength Ratings for Males and Females (in kg)source: Camry Electronic Hand Dynamometer Instruction manual

caliper adapter calculus

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