Page 11 - Exlar - EXTRAK Modular rodless belt-driven actuators

v v
F L
m T
i = d 1
J 2
d 2 J 2
i = d 1 m T
d 2 M M F L
P (screw pitch) M
M
J 1 n M
J M d 3
J 1
n M M J
M M
Motor speed (min ) n M = v · 6 · 10 4 n M = v · 6 · 10 4
-1
p · i ∏ d3 · i
6
Critical speed (min ) n k = 120 · 10 · d
-1
l 2
Load moment (Nm) M L = P · i F L M L = d 3 · i F L
2000 · ∏ 2000
( )
J T = m T P · 10 -6 J T = m T d 3 · 10 -6
2
2
Translatory mass moment of inertia (kgm ) 2 2 · ∏ ( )
2
Rotatory mass moment of inertia (kgm ) 2 J R = 7,7 · d · I · 10 -13
4
J = J M + J 1 + i (J R + J T + J 2 )
2
Total of reduced mass moments of inertia (kgm ) 2 (at gear reduction 2:1➝i = 0,5)
Acceleration torque resp. breaking M B = n M · J
moment M B = f(n M ) (Nm) 9,55 · t B
Acceleration torque resp. breaking M B = 4 · ∏ · S B · J M B = 4 · S B · J
moment M B = f(s B) (Nm) p · i · t B 2 d 3 · i · t B 2
t B = n M · J
Acceleration- / braking period t B = f(n m ) (s)
9,55 · M B
Acceleration- / braking period t B = f(s B ) (s) t B = 4 · ∏ · s B · J t B = 4 · s B · J
p · i · M B d 3 · i · M B
Resulting speed (rpm) after acceleration (min ) n M = 120 · s B n M = 120 · s B
-1
p · i · t B d 3 · ∏ · i · t B
Resulting distance of acceleration (mm) s B = n M · t B · p · i s B = n M · t B · d 3 · ∏ · i
120 120
Total of moments to override by the motor (Nm) M M = 1 (M L + M B
ŋ
Power output (W) P A = M M · n M
9,55
M eff = ∑ t B (M/M M ) + ∑ t L (M L /M M ) 2
2
Effective output torque of motor (Nm) = M M
∑ t B + ∑t L + t O
Key to the formulas:

d (mm) = screw diameter J R (kgm ) = rotatory mass moment of inertia m T (kg) = external load (linear moving mass)
2
d 1 (mm) = diameter driving wheel J T (kgm ) = translatory mass moment of inertia n k (min ) = critical speed for screw drive
2
-1
d 2 (mm) = diameter driven gear l (mm) = screw length n M (min ) = motor speed
-1
d 3 (mm) = diameter pinion or belt pulley M B (Nm) = acceleration torque resp. breaking p (mm) = screw pitch
F L (N) = feed power moment P A (W) = power output
i (–) = gear reduction M d (Nm) = motor – continuous torque s B (mm) = acceleration / brake path
J (kgm ) = mass moment of inertia (see motor spec.) t B (s) = acceleration / braking period
2
J 1 (kgm ) = mass moment of inertia driving M eff (Nm) = motor – effective output torque t L (s) = running time under load moment
2
wheel M L (Nm) = load moment t 0 (s) = stop period unloaded
2
J 2 (kgm ) = mass moment of inertia driven gear M M (Nm) = motor torque (see motor spec.) v (m/s) = rate of feed
J M (kgm ) = mass moment of inertia drive motor M max (Nm) = motor torque peak ŋ (–) = mechanical efficiency on motor shaft
2

EXTRAK Rodless Actuators 11