GATE Past Year Questions: Gears & Gear Trains | Theory of Machines (TOM) - Mechanical Engineering PDF Download

Q1: The Levai type-A train illustrated in the figure has gears with module m = 8 mm/ tooth. Gears 2 and 3 have 19 and 24 teeth respectively. Gear 2 is fixed and internal gear 4 rotates at 20rev/min counter-clockwise. The magnitude of angular velocity of the arm is ______ rev/min. (rounded off to 2 decimal places)   [GATE ME 2024]

Ans: 15 to 16
Data given
m8 mm
T₂ = 19
T₃ = 24
N₂ = 0
N₄ = 20 rpm
r₄ = r₂ + 2r₃

or, T₄ = T₂ × 2 T₃
i.e.,
T₄ = 19 + 2 × 24 = 67
As N₂ = 0 So,
$x+y=0...\left(1\right)$x + y = 0 ...(1)
N₄ = 20 = y−x × (19/67)
or, y − x × (19/67) = 20 ... (2) By (i) and (ii)
k = -15.58 rpm
= 15.58 clockwise
y = 15.58 counterclockwise 

Q1: Two meshing spur gears 1 and 2 with diametral pitch of 8 teeth per mm and an angular velocity ratio ∣ω₂∣/∣ω₁∣ = 1/4, have their centers 30 mm apart. The number of teeth on the driver (gear 1) is _______ . (Answer in integer)   [GATE ME 2023]
Ans: 95.999 to 96.001
Given
R + r = 30 ...(i)
where T = no. of teeth
or T₂ = 4T₁
Diametrical pitch = 8 teeth/mm = T/D = P_d
and m = D/T = 1/P_d
r = mT/2
As,
by equation (i) we have

As T₂ = 4T₁
So, 1/16 (T₁ + 4T₁) = 30
or σ₁ = (30×16)/5 = 96

Q1: A schematic of an epicyclic gear train is shown in the figure. The sun (gear 1) and planet (gear 2) are external, and the ring gear (gear 3) is internal. Gear 1, gear 3 and arm OP are pivoted to the ground at O. Gear 2 is carried on the arm OP via the pivot joint at P, and is in mesh with the other two gears. Gear 2 has 20 teeth and gear 3 has 80 teeth. If gear 1 is kept fixed at 0 rpm and gear 3 rotates at 900 rpm counter clockwise (ccw), the magnitude of angular velocity of arm OP is __________rpm (in integer).  [GATE ME 2022 SET-1]
Ans: 600 to 600
Speed of gear 1 N₁ = 0,
Speed of gear 3 N₃ = 900 rpm,
Speed of Arm N_arm = ?,
Teeth of gear 1 z₁ = ?
Teeth of gear 2 z₂ = 20
Teeth of gear 3 z₃ = 80
Teeth of gear 2
z₃ = z₁ + 2z₂
80 = z₁ + 2(20)
z₁ = 40z
N₁ = x + y = 0
⇒ x = -y
N₃ =
= y - (40/80)(x) = 900
⇒ y - (40/80)(-y) = 900
1.5y = 900
= 600 rpm

Q1: A power transmission mechanism consists of a belt drive and a gear train as shown in the figure.
Diameters of pulleys of belt drive and number of teeth (T) on the gears 2 to 7 are indicated in the figure. The speed and direction of rotation of gear 7, respectively, are  [GATE ME 2021 SET-2]
(a) 255.68 rpm; clockwise
(b) 255.68 rpm; anticlockwise
(c) 575.28 rpm; clockwise
(d) 575.28 rpm; anticlockwise
Ans: (a)
T₃ = 44
T₆ = 36
T₂ = 18
T₄ = 15

⇒ N₁ = (150/250) x 2500
N₁ = 15,00 = N₂

= 255.6818 = N₇ (Clockwise)
Gear 5 is idler.

Q1: The sun (S) and the planet (P) of an epicyclic gear train shown in the figure have identical number of teeth
If the sun (S) and the outer ring (R) gears are rotated in the same direction with angular speed ω_S and ω_R, respectively, then the angular speed of the arm AB is  [GATE ME 2020 SET-2]
(a)
(b)
(c)
(d)
Ans: (a)
r_s + 2r_p = r_R
⇒ T_s + 2T_p = T_R (T_P = T_S)
3T_P = T_R
⇒ 3T_P = T_R
y + x = ω_S
y − (x/3) = ω_R
Substract by,
4x/3 = (ω_{S -} ω_R)
x = 3/4 (ω_{S -} ω_R)
y = ω_S - x = ω_S - (3/4)(ω_{S -} ω_R)


Q2: A balanced rigid disc mounted on a rigid rotor has four identical point masses, each of 10 grams, attached to four points on the 100 mm radius circle shown in the figure.
The rotor is driven by a motor at uniform angular speed of 10 rad/s. If one of the masses gets detached then the magnitude of the resultant unbalance force on the rotor is ______ N. (round off to 2 decimal places). [GATE ME 2020 SET-1]
Ans: 0.09 to 0.11
ω = 10rad/s, r = 100mm = 0.1m
If one mass is detached then
Now, unbalance tone, F = mrω²
= (10/1000) x 0.1 x (10)² = (10×0.1×100)/1000
= 0.1 N