Notes_08_06
1 of 6
Coulomb Friction in Prismatic Joint
x
4
C
≠0
 F on 4  
4
X
Y
 FEXT F1on 4  0
X
F1Yon 4   FEXT
pseudo-code
if x  
F1Xon 4   D F1Yon 4 sign x 
else
X
F1Xon 4    FEXT
if F1Xon 4   S F1Yon 4

X
F1Xon 4   S F1Yon 4 sign  FEXT
end
end
ODE
X
m 4 x    FEXT
 F1Xon 4

C
Notes_08_06
2 of 6
Coulomb Friction in Revolute Joint
2
3
simple friction model

T2on3  D rBRG F2on3 sign  3   2

for  3   2  0
3
2
Notes_08_06
3 of 6
Free Vibration with Viscous Damping
spring-mass-damper with viscous damping has exponential envelope
mx  cx  kx  0
n 
k
 2 f n
m
c

c CR

c
2 m n
 d   n 1   2  2 f d
T  1/ f d
use x0 at a positive peak and xn at a positive peak that occurs n cycles later
log decrement

damping ratio

1  x0
ln 
n  x n

2
 
1 2

for exponential envelope

4   2
2
0.1
0.08
0.06
Position [m]
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
0
0.5
1
1.5
2
Time [sec]
2.5
3
3.5
4
Notes_08_06
4 of 6
Free Vibration with Coulomb Friction
spring-mass with Coulomb friction has linear envelope
mx  kx  f sign ( x )
use x0 at a positive peak and xn at a positive peak that occurs n cycles later
slope
s
x 0  x n 2 f n

nT
k
for linear envelope
friction force
f
 k s k(x 0  x n )

2 n
4n
rotary motion
s
0  n
nT
Tf 
 k  s k  ( 0   n )

2 n
4n
simple pendulum (small angle approx)
s
0  n
nT
Tf 
 m g a s m g a ( 0   n )

2 n
4n
0.1
0.08
0.06
Position [m]
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
0
0.5
1
1.5
2
Time [sec]
2.5
3
3.5
4
Notes_08_06
5 of 6
Pacejka Magic Formula
Pacejka, H.B., 2006, Tire and Vehicle Dynamics, 2nd edition, SAE International
y
values from curve
yP = peak value for y
xM = x value at peak
yA = asymptotic value for y
xM
yP
slope
yA
x
coefficients
B = stiffness factor
C = shape factor
D = peak value
E = curvature factor
y  D sin C a tan Bx  EBx  a tan Bx 
D = yP
slope 
D
BCD
xM / 2
y 
 2
C  1  1  a sin A 
D
 
B
slope
CD
E
B x M  tan  / 2C
B x M  a tan B x M 
for
C 1
Notes_08_06
6 of 6
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
% try_pacejka.m - test Pacejka magic formula
% HJSIII, 11.04.25
clear
% constants
% define curve
xm = 0.01;
yp = 0.9;
ya = 0.6;
% Pacejka coefficients
D = yp;
C = 1 + (1 - 2 * asin( ya/D ) / pi);
slope = yp / (xm/2);
B = slope /C /D;
E = (B*xm - tan(pi/2/C)) / (B*xm - atan(B*xm));
% plot
x = ( -1 : 0.001 : 1 )';
y = D*sin( C*atan( B*x - E*(B*x-atan(B*x)) ) );
figure( 1 )
clf
plot( x,y )
0
0.2
0.4
0.6
0.8
1