Page 151 - Mechatronics with Experiments

MECHANISMS FOR MOTION TRANSMISSION 137
linear distance traveled by each gear at the contact point is same,

s = s (3.18)
1 2
Δ ⋅ r =Δ ⋅ r (3.19)
1 1 2 2
Δ 1
N = (3.20)
Δ 2
r 2
N = (3.21)
r 1
Since the pitch of each gear must be same, the number of teeth on each gear is proportional
to their radius,
̇ 1 N 2 r 2
N = = = (3.22)
̇ 2 N 1 r 1
where N and N are the number of gear teeth on each gear. It can be shown that for an
1
2
ideal gear box (100% power transmission efficiency),
P out = P in (3.23)
T out ⋅ ̇ out = T ⋅ ̇ in (3.24)
in
Hence,
N 2 ̇ in T out
N = = = (3.25)
N 1 ̇ out T in
The reflection of inertia and torque from the output shaft to the input shaft can be determined
by using the energy and work relationships. Let the rotary inertia of the load on the output
shaft be J and the load torque be T . Let us express the kinetic energy of the load
l l
1
KE = ⋅ J ⋅ ̇ 2 out (3.26)
l
2
1 2
̇
= ⋅ J ⋅ ( ∕N) (3.27)
l
in
2
1 1 2
̇
= ⋅ J ⋅ ⋅ ( ) (3.28)
in
l
2 N 2
1 2
̇
= ⋅ J in,eff ⋅ ( ) (3.29)
in
2
where the reflected inertia (inertia of the load seen by the input shaft) is
J l
J in,eff = (3.30)
N 2
Similarly, let us determine the effective load torque seen by the input shaft. The work done
by a load torque T over an output shaft displacement Δ out is
l
W = T ⋅ Δ out (3.31)
l
Δ
= T ⋅ in (3.32)
l
N
= T ⋅ Δ (3.33)
in,eff in
The effective reflective torque on the input shaft as a result of the load torque on the output
shaft is
T l
T in,eff = (3.34)
N