Modern Control Engineering

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Example Problems and Solutions 233

Solution.For this gear-train system, we can obtain the following equations: For shaft 1,

(5–63)

where is the torque developed by the motor and is the load torque on gear 1 due to the rest

of the gear train. For shaft 2,

(5–64)

where is the torque transmitted to gear 2 and is the load torque on gear 3 due to the rest of

the gear train. Since the work done by gear 1 is equal to that of gear 2,

or

If the gear ratio reduces the speed as well as magnifies the torque. For shaft 3,

(5–65)

where is the load torque and is the torque transmitted to gear 4. and are related by

andu 3 andu 1 are related by

Eliminating and from Equations (5–63), (5–64), and (5–65) yields

Eliminatingu 2 andu 3 from this last equation and writing the resulting equation in terms of u 1 and

its time derivatives, we obtain

(5–66)

Thus, the equivalent moment of inertia and viscous-friction coefficient of the gear train referred

to shaft 1 are given, respectively, by

Similarly, the equivalent moment of inertia and viscous-friction coefficient of the gear train referred

to the load shaft (shaft 3) are given, respectively, by

b 3 eq=b 3 +a

N 4

N 3

b

2

b 2 + a

N 2

N 1

b

2

a

N 4

N 3

b

2

b 1

J 3 eq=J 3 + a

N 4

N 3

b

2

J 2 +a

N 2

N 1

b

2

a

N 4

N 3

b

2

J 1

b 1 eq=b 1 +a

N 1

N 2

b

2

b 2 + a

N 1

N 2

b

2

a

N 3

N 4

b

2

b 3

J 1 eq=J 1 +a

N 1

N 2

b

2

J 2 +a

N 1

N 2

b

2

a

N 3

N 4

b

2

J 3

• cb 1 + a

N 1

N 2

b

2

b 2 + a

N 1

N 2

b

2

a

N 3

N 4

b

2

b 3 du

1 + a

N 1

N 2

ba

N 3

N 4

bTL=Tm

cJ 1 +a

N 1

N 2

b

2

J 2 +a

N 1

N 2

b

2

a

N 3

N 4

b

2

J 3 du

$

1

J 1 u

$

1 +b 1 u

1 +

N 1

N 2

AJ 2 u

$

2 +b 2 u

2 B+

N 1 N 3

N 2 N 4

AJ 3 u

$

3 +b 3 u

3 +TLB=Tm

T 1 ,T 2 ,T 3 , T 4

u 3 =u 2

N 3

N 4

=u 1

N 1

N 2

N 3

N 4

T 4 =T 3

N 4

N 3

TL T 4 T 3 T 4

J 3 u

$

3 +b 3 u

3 +TL=T 4

N 1 N 26 1,

T 2 =T 1

N 2

N 1

T 1 u 1 =T 2 u 2

T 2 T 3

J 2 u

$

2 +b 2 u

2 +T 3 =T 2

Tm T 1

J 1 u

$

1 +b 1 u

1 +T 1 =Tm