Gear Trains MCQ Quiz - Objective Question with Answer for Gear Trains - Download Free PDF

Explanation:

Gear train :

Given:

T₂ = 19, T₃ = 24, N₄ = 0, N₄ = 20 rpm (CCW)

As R₂ + 2R₃ = R₄

⇒ T₂ + 2T₃ = T₄

⇒ T₄ = 19 + 2 ×  24 = 67

N₂ = y + x ⇒ x + y = 0 ...(i)

and \(N_{4}=20 ~\mathrm{rpm} \)

\(\Rightarrow y-x \times \frac{19}{67}=20\) ...(ii)

On solving equations (i) and (ii);

x = -15.58 rpm = 15.58 rpm (CW)

y = 15.58 (CCW)

Explanation:

Gear Train:

Sometimes, two or more gears are made to mesh with each other to transmit power from one shaft to another. Such a combination is called a gear train.

They can be of many types:

• Simple Gear Train
• Compound Gear Train
• Reverted Gear Train
• Epicyclic Gear Train

Epicyclic gear train:

• When the gear train is having a relative motion of axes it is called the epicyclic gear train.
• The axis of at least one of the gear moves relative to the frame.
• An epicyclic gear train has at least one gear axis that moves relative to the frame.

Simple Gear Train:

• When every gear axes are fixed relative to the fixed frame and, there is only one gear on each shaft.
• The number of shafts may be any.

Compound Gear Train:

• When any shaft has more than one gear (first shaft or second shaft or intermediate shaft) i.e. two gear have the same angular velocity, the gear drive is called a compound gear train. 

Reverted Gear train:

• When the axes of the first gear (i.e. first driver) and the last gear (i.e. last driven or follower) are co-axial, then the gear train is known as a reverted gear train.
• The motion of the first gear and the last gear are alike. 
• The reverted gear trains are used in automotive transmissions, lathe back gears, industrial speed reducers, and clocks.

Explanation:

The net torque applied to the gear train must be zero

T1 + T2 + T3 = 0

The net kinetic energy dissipated by the gear train must be zero.

T1ω1 + T2ω2 + T3ω2 = 0

Here NA = 100 rpm (cw) ; NB = 250 rpm (cω); TA = 50 kN.m (cw) ; Nc = 0

TANA + TBNB + TCNC = 0

50 × 100 + TB × 250 = 0

\({T_B} = - \frac{{50 \times 100}}{{250}} = - 20\;kNm\)

TC = - TA – TB = -50 - (-20) = -30 kN.m

Explanation:

Speed ratio:

The speed ratio (velocity ratio) of a gear train is the ratio of the speed of the driver to the speed of the driven or follower.

\(Speed\;Ratio = \frac{{{N_1}}}{{{N_2}}} = \frac{{{T_2}}}{{{T_1}}}\)

Train value:

The ratio of the speed of the driven or follower to the speed of the driver is known as train value of the gear train.

\(Train\;value = \frac{{{N_2}}}{{{N_1}}} = \frac{{{T_1}}}{{{T_2}}}\)

\(Train\;value = \frac{{{1}}}{{{Speed\;Ratio}}}\)

From the above, we see that the train value is the reciprocal of the speed ratio.

Explanation:

Epicyclic Gear Train

• An epicyclic gear train (also known as a planetary gear train) is a complex gear system consisting of one or more outer gears (planet gears) revolving around a central gear (sun gear).
• It is called "epicyclic" because the planet gears also rotate around their own axes while revolving around the sun gear, similar to the motion of planets in the solar system.
• In an epicyclic gear train, the sun gear is usually fixed in the center, and the planet gears are mounted on a movable arm (carrier) that revolves around the sun gear.
• The outer ring with internal teeth (ring gear) may also be part of the system.
• The motion of the gears can be analyzed using various methods, including the tabular method and algebraic method, to determine the velocity ratios and directions of rotation.

Advantages:

• Compact design and high power density, making it suitable for applications with space constraints.
• Ability to achieve high gear ratios in a small volume.
• Uniform distribution of load among the planet gears, leading to improved efficiency and durability.

Disadvantages:

• Complex design and higher manufacturing costs compared to simple gear trains.
• More challenging to analyze and understand due to the compound motion of the gears.

Applications:

• Epicyclic gear trains are commonly used in automatic transmissions in automobiles, differential gears, and planetary gearboxes in various machinery and equipment.

Topper’s approach:

The follower turns = 18 × 16/8 = 36 times

Detailed Solution:

Let, the follower turns = x times

According to the question,

⇒ x × 8 = 16 × 18

⇒ x = 16 × 18/8

⇒ x = 36

Explanation:

Reverted gear train:

When the axes of the first gear and the last gear are co-axial, then the gear train is known as a reverted gear train. The reverted gear trains are used in automotive transmissions, lathe back gears, industrial speed reducers, and in clocks (where the minute and hour hand shafts are co-axial).

Advantages:

• The epicyclic gear trains are useful for transmitting high-velocity ratios with gears of moderate size in a comparatively lesser space.
• The epicyclic gear trains are used in the back gear of lathe, differential gears of the automobiles, hoists, pulley blocks, wrist-watches, etc.

Explanation:

Gear Train:

Sometimes, two or more gears are made to mesh with each other to transmit power from one shaft to another. Such a combination is called a gear train.

They can be of many types:

• Simple Gear Train
• Compound Gear Train
• Reverted Gear Train
• Epicyclic Gear Train

Epicyclic gear train:

• When the gear train is having a relative motion of axes it is called the epicyclic gear train.
• The axis of at least one of the gears also moves relative to the frame.

Simple Gear Train:

• When every gear axes are fixed relative to the fixed frame and, there is only one gear on each shaft.
• The number of shafts may be any.

Compound Gear Train:

• When any shaft has more than one gear (first shaft or second shaft or intermediate shaft) i.e. two gear have the same angular velocity, the gear drive is called a compound gear train. 

Reverted Gear train:

• When the axes of the first gear (i.e. first driver) and the last gear (i.e. last driven or follower) are co-axial, then the gear train is known as a reverted gear train.
• The motion of the first gear and the last gear is like.
• The reverted gear trains are used in automotive transmissions, lathe back gears, industrial speed reducers, and clocks.

Concept:

The speed ratio (or velocity ratio) of the gear train is the ratio of the speed of the driver to the speed of the driven or follower.

\(Speed\;ratio = \frac{{{\omega _{driver}}}}{{{\omega _{driven}}}} = \frac{{{\omega _1}}}{{{\omega _2}}} = \frac{{{N_1}}}{{{N_2}}} = \frac{{{T_2}}}{{{T_1}}} = \frac{1}{{Train\;Value}}\)

\(Train\;ratio = \frac{{{\omega _{drivev}}}}{{{\omega _{driver}}}} \)

Explanation:

Sometimes, two or more gears are made to mesh with each other to transmit power from one shaft to another. Such a combination is called a gear train.

When there is only one gear on each shaft, it is known as a simple gear train.

Since gear 1 drives gear 2, therefore gear 1 is called the driver and gear 2 is called the driven or follower.

The purpose of an idler gear used in gear trains is to have the required direction of rotation at the output shaft.

• It may be noted that when the number of intermediate gears (idler gear) is odd, the motion of both the gears (i.e. driver and driven or follower) is like.
• But if the number of intermediate gears is even, the motion of the driven or follower will be in the opposite direction of the driver.

Concept:

Velocity Ratio is:

\(Velocity~ratio = \frac{{Speed~of~driver}}{{Speed~of~follower}}\)

Train value is:

The reciprocal of velocity ratio is known as train value of the gear train.

\(Train~value= \frac{1}{velocity~ratio}\)

The velocity ratio of a simple gear train

When the driver and intermediate gear are in mesh

\(\frac{{{N_1}}}{{{N_2}}} = \frac{{{T_2}}}{{{T_1}}}\)

When the intermediate and followers are in mesh.

\(\frac{{{N_2}}}{{{N_3}}} = \frac{{{T_3}}}{{{T_2}}}\)

Multiplying both the equations,

\(\frac{{{N_1}}}{{{N_2}}} \times \frac{{{N_2}}}{{{N_3}}} = \frac{{{T_2}}}{{{T_1}}} \times \frac{{{T_3}}}{{{T_2}}}\)

\(\frac{{{N_3}}}{{{N_1}}} = \frac{{{T_1}}}{{{T_3}}}\)

\(Speed~ratio = \frac{{Speed~of~driver}}{{Speed~of~follower}} = \frac{{No.~of~teeth~on~follower}}{{No.~of~teeth~on~driver}}\)

Calculation:

Given:

T₁ = 40, T₂ = 20, T₃ = 80

\(Velocity~ratio=\frac{{T_3}}{{T_1}} = \frac{{80}}{{40}} = 2\)

\(Train~value = \frac{{1}}{{velocity~ratio}} = \frac{1}{2} = 0.5\)

Concept:

From Kutzbach's mobility criterion,

F = 3 (n – 1) – 2 j – h

n = Number of links

j = Number of binary joints

h = Number of higher pair

Calculation:

In the gear train system,

no. of links n = 4, no. of joints j = 3 (1 binary joint and 1 ternary joint by link 1, 2 and 3)

and no. of higher pair h = 1.

F = 3 (4 – 1) – 2× 3 – 1 = 2

A planetary gear train has 2 DOF and hence requires two inputs to get the desired output.

Explanation:

Path of contact: Contact of two teeth made where the addendum circle of the wheel meets the line of action and is broken where the addendum circle of the pinion meets the line of action

Let, r = pitch circle radius of pinion, R = pitch circle radius of the wheel, ra = addendum circle of radius of pinion, Ra = addendum circle radius of the wheel

Path of contact = Path of approach + Path of recess = \(\sqrt{R_a^2- R^2 cos^2 ~\phi}~+~\sqrt{r_a^2~-r_a^2cos^2~\phi }~-~(R~+~r)sin~\phi\)

Arc of contact: Arc of contact is the distance travelled by a point on either pitch circle of the two wheels during the period of contact of a pair of teeth.

\(Arc~of~contact = \frac{{path~of~contact}}{{\cos \left( ϕ \right)}}\)

Contact ratio: It is defined as the average number of tooth pairs in contact during one rotation.

\(Contact~ratio = \frac{{Arc~of~contact}}{{Circular~pitch}} \)

For continuous transmission of motion, at least one tooth of one wheel must be in contact with another tooth of the second wheel therefore contact ratio must always be greater than or equal to 1.

Concept:

The diameter of a gear is directly proportional to its teeth

d ∝ t 

where, t = no. of teeth, d = diameter of gear.

Calculation:

d_R = m × t_R = 2 × 15 = 30 mm

d_Q = 2 × d_R = 60 mm

\(\frac{{{d_P}}}{{{d_Q}}} = \frac{{{t_P}}}{{{t_Q}}} \Rightarrow {d_P} = \frac{{20}}{{40}} × 60 = 30~mm\)

\(\frac{{{d_R}}}{{{d_S}}} = \frac{{{t_R}}}{{{t_S}}} \Rightarrow {d_S} = dR × \frac{{{t_S}}}{{{t_R}}} = \frac{{30 × 20}}{{15}} = 40~mm\)

\(Centre\;distance = \frac{{{d_P}}}{2} + \frac{{{d_Q}}}{2} + \frac{{{d_R}}}{2} + \frac{{{d_S}}}{2}\)

Centre distance = 15 + 30 + 15 + 20 = 80 mm

Explanation:

Gear Train:

Two or more gears are made to mesh with each other to transmit power from one shaft to another. Such a combination is called a gear train.

When there is only one gear on each shaft, it is known as a simple gear train.

Since the gear 1 drives the gear 2, therefore gear 1 is called the driver and the gear 2 is called the driven or follower.

Idler Gear:

The purpose of an idler gear used in gear trains is to have the required direction of rotation at the output shaft.

• It is used for changing the direction.
• It does not change the velocity ratio or torque.
• It may be noted that when the number of intermediate gears (idler gear) is odd, the motion of both the gears (i.e. driver and driven or follower) is like.
• But if the number of intermediate gears is even, the motion of the driven or follower will be in the opposite direction of the driver.

The ratio for the given gear train is given by the following formula

\(\frac{{{{\rm{\omega }}_2}}}{{{{\rm{\omega }}_6}}} = \frac{{{{\rm{\omega }}_2}}}{{{{\rm{\omega }}_3}}}\frac{{{{\rm{\omega }}_3}}}{{{{\rm{\omega }}_5}}}\frac{{{{\rm{\omega }}_5}}}{{{{\rm{\omega }}_6}}} = \frac{{{{\rm{N}}_3}}}{{{{\rm{N}}_2}}}\frac{{{{\rm{N}}_5}}}{{{{\rm{N}}_4}}}\frac{{{{\rm{N}}_6}}}{{{{\rm{N}}_5}}} = \frac{{{{\rm{N}}_3}{{\rm{N}}_6}}}{{{{\rm{N}}_2}{{\rm{N}}_4}}}\)

∴ Only Gear(5) is the Idler gear, as the number of teeth on the wheel '5' does not appear in the velocity ratio.