Page 155 - Mechatronics with Experiments

MECHANISMS FOR MOTION TRANSMISSION 141
It should be noted that m is in units of mass (not weight, weight = mass ⋅ g), and the J eff is
l
the mass moment of inertia. Therefore if the weight of the load is given, W ,
l
1
J = ⋅ (W ∕g) (3.58)
eff 2 l
(2 ⋅ p)
1
= ⋅ (W ∕g) (3.59)
l
N 2
ls
The lead-screw has a very large effective gear ratio effect. Notice that for p = 2, 5, 10, the
net gear ratio is N = 4 ,10 ,20 , respectively, and the inertial reflection of a translational
mass is a factor determined by the square of the effective gear ratio.
Let us determine the reflected torque at the input shaft due to a load force, F .The
l
work done by a load force during a incremental displacement is
Work = F ⋅ Δx (3.60)
l
The corresponding rotational displacement is
1
Δx = ⋅ Δ (3.61)
2 ⋅ p
Hence,
Work = F ⋅ Δx (3.62)
l
1
= F ⋅ ⋅ Δ (3.63)
l
2 ⋅ p
= T ⋅ Δ (3.64)
eff
The equivalent torque seen at the input shaft (T ) of the lead-screw due to the load force
eff
F at the nut is
l
1
T eff = ⋅ F l (3.65)
2 ⋅ p
1
= ⋅ F l (3.66)
N
Example Consider a ball-screw motion conversion mechanism with a pitch of p =
10 rev∕in. The mass of the table and workpiece is W = 1000 lb and the resistance force of
l
the load is F = 1000 lb. Determine the reflected rotary inertia and torque seen by a motor
l
at the input shaft of the ball-screw.
1 2
J eff = 2 ⋅ (1000∕386) lb∕(in∕s ) (3.67)
2
(2 ⋅ 10) rad ∕in 2
= 6.56 × 10 −3 lb ⋅ in ⋅ s 2 (3.68)
The torque that is reflected on the input shaft due to the load force F is
l
1
T eff = ⋅ F l (3.69)
2 ⋅ p
1
= ⋅ 1000 lb (3.70)
2 rad∕rev ⋅ 10 rev∕in
100
= lb ⋅ in (3.71)
2
= 15.91 lb ⋅ in (3.72)