Theory of Machines Test 2

Concept:

According to Grashof’s Law:

(1) S + L ≤ P + Q (Class I mechanism)

There will be at least one complete revolution between the two links.

• If shortest link is fixed: Double Crank Mechanism
• If any of the adjacent link of the shortest link if fixed: Crank rocker mechanism
• If the link opposite to shortest link is fixed: Double Rocker Mechanism

(2) S + L > P + Q (Class II mechanism)

• Only double rocker mechanism is possible.

Calculation:

For crank-crank mechanism, the condition to be satisfied are

(i) Shortest link must be fixed i.e. L_o is shortest link.

(ii) S + L ≤ P + Q

L_o + 300 ≤ 250 + 150

1. Dynamic-static analysis → D` Alembert‘s principal
2. Mobility (for plane mechanism) → Grubler’s criterion
3. Continuous relative rotation → Grashoff’s law
4. Velocity and acceleratio → Kennedy’s theorem

Concept:

T = Iα

ω₂ = ω₁ + αt

Calculation:

I = mk² = 400 × (0.2)² = 16 kg.m²

T = 400 N.m

T = Iα

\(\alpha = \frac{{400}}{{16}}\)

α = 25 rad/s²  

∴ ω₂ = ω₁ + αt

As ω₁ = 0, t = 10

∴ ω₂ = αt = 25 × 10

∴ ω₂ = 250 rad/s

\({\left( {KE} \right)_{t = 10}} = \frac{1}{2}I\omega _2^2\)

\({\left( {KE} \right)_{t = 10}} = \frac{1}{2}\left( {16} \right) \times {\left( {250} \right)^2}\)

∴ (KE) _(t=10) = 500 kJ

Concept:

Maximum arc of approach of pinion,

\({\rm{Max\;arc\;of\;approach}} = {\rm{}}\frac{{Max\;Path\;of\;contact}}{{Cos\;\emptyset }}\;\)

\({\rm{Max\;arc\;of\;approach}} = \frac{{{r_p}sin\emptyset }}{{cos\emptyset }}\)

 Max arc of approach = r_p tanθ

Where, r_p = radius of pinion

Given condition,

\({\rm{Arc\;of\;approach\;}} = {\rm{\;}}\frac{{\pi {d_p}}}{{{Z_p}}}\)

\({r_p}\tan \emptyset \ge \frac{{2\pi {r_p}}}{{{Z_p}}}\)

\({Z_p} \ge \frac{{2\pi }}{{tan\emptyset }}\)

For ∅ = 18°

\({Z_p} \ge \frac{{2\pi }}{{tan18^\circ }}\)

 Z_p ≥ 19.32

∴ Z_p = 20 

Concept:

\({\rm{Centre\;distance}} = \frac{{D + d}}{2}\)

D = Diameter of Gear, d = diameter of pinion

Calculation:

Given:

Z_G = 50 (Number of teeth on gear), Z_P = 13 (Number of teeth on pinion), m = 10 mm

We know

\(m = \frac{D}{Z}\; - - - \;\left( 1 \right)\)

Now,

For gear

D = mZ_G = 10 × 50 = 500 mm

For pinion

d= mZ_P = 10 × 13 = 130 mm

\({\rm{Centre\;distance}} = \frac{{D + d}}{2} = \frac{{500 + 130}}{2}\)

∴ Centre distance = 315 mm

Explanation:

Given:

F = ar + b

1500 = 200a + b

887.5 = 130a + b

⇒ a = 8.75 N/mm

⇒ b = - 250 N

Now,

For isochronous governor, b = 0

F = mrω² = ar

∴ a = mω²

8.75 × 1000 = 8ω²

ω = 33.07 rad/s

\(\therefore \omega = \frac{{2\pi N}}{{60}}\)

∴ N = 315.8 rpm

Kinematic pairs are classified under three headings namely, lower pair, higher pair and wrapping pair.

Lower Pair: 

A pair is said to be a lower pair when the connection between two elements is through the area of contact. Some of the types of Lower pair are:

• Revolute Pair
• Prismatic Pair
• Screw Pair
• Cylindrical Pair
• Planar Pair

Higher Pair: 

A pair is said to be higher pair when the connection between two elements has only a point or line of contact. Examples of higher pairs are:

• A point contact takes place when spheres rest on plane or curved surfaces (in case of ball bearings).
• Contact between teeth of a skew-helical gears.
• Contact made by roller bearings
• Contact between teeth of most of the gears.
• Contact between cam-follower
• Spherical Pair.

Wrapping Pairs:

In a higher pair, the contact between the two bodies has only a line contact or a point contact. Whereas in a wrapping pair, one body completely wraps over the other. The typical example is of a belt and a pulley or a chain and a sprocket where the belt completely wraps around the pulley or the chain completely wraps around the sp.

Mechanism: A mechanism is a set of machine elements or components or parts arranged in a specific order to produce a specified motion.

When one of the links of a kinematic chain is fixed, the chain is called a mechanism.

Mechanisms are of the following types:

Simple mechanism

• This is a mechanism which has four links.

Compound mechanism

• This is a mechanism which has more than four links.

Complex mechanism

• This is formed by the inclusion of ternary or higher-order floating link to a simple mechanism.

Planar mechanism

• This is formed when all the links of the mechanism lie in the same plane.

Spatial mechanism

• This is formed when all the links of the mechanism lie in the different plane.

Equivalent mechanism

• Turning pairs of plane mechanisms may be replaced by other types of pairs such as sliding pairs or cam pairs. The new mechanism thus obtained having the same number of degrees of freedom as the original mechanism is called the equivalent mechanism.

Explanation:

Given:

I_p = 20 kg – m²

\({\omega _s} = \frac{{2\pi \times 1000}}{{60}} = 104.67\;rad/sec\;\)

\({\omega _p} = \frac{V}{r} = \frac{{200 \times 1000}}{{3600 \times 150}} = 0.37\;rad/sec\)

Now,

Gyroscopic acceleration = ω_s × ω_p

∴ Gyroscopic acceleration = 104.67 × 0.37

∴ Gyroscopic acceleration = 38.7279 rad²/sec²

It should be lesser or equal, i.e. S + L ≤ P + Q (where S and L are length of shortest and longest link, respectively).

• If one of the links is fixed in a kinematic chain, it is called a mechanism.
• So, we can obtain as many mechanisms as the number of links (n inversions from a kinematic chain having ‘n’ number of links) in a kinematic chain.
• A slider-crank is a kinematic chain having four links so four inversions.
• It has one sliding pair and three turning pairs.
• Link 1 is a frame (fixed).
• Link 2 has rotary motion and is called a crank.
• Link 3 has got combined rotary and reciprocating motion and is called a connecting rod.
• Link 4 has reciprocating motion and is called a slider.
• This mechanism is used to convert rotary motion to reciprocating and vice versa.

Inversions of the slider-crank mechanism are obtained by fixing links 1, 2, 3, and 4.

• First inversion: This inversion is obtained when link 1 (ground body) is fixed.
  Application-  Reciprocating engine, reciprocating compressor, etc.
• Second inversion: This inversion is obtained when link 2 (crank) is fixed.
  Application- Whitworth quick returns mechanism, Rotary engine, etc.
• Third inversion: This inversion is obtained when link 3 (connecting rod) is fixed.
  Application - Slotted crank mechanism, Oscillatory engine, etc.
• Fourth inversion: This inversion is obtained when link 4 (slider) is fixed.
  Application- A hand pump, pendulum pump or Bull engine, etc.

Concept: 

Gyroscopic couple C = Iωωp

Calculation:

Given:

Radius of gyration (k) = 0.6 m, v = 100 km/hr, N = 1800 rpm, Radius (R) = 80 m, mass (m) = 8 tonnes 

Now,

\(\omega = \frac{{2\pi N}}{{60}} = \frac{{2\pi × 1800}}{{60}} = 188.5\;rad/s\)

Now,

Mass moment of inertia of rotor:

I = mk² = 8000 × (0.6)² = 2880 kg-m²

Angular velocity of precession:

\({\omega _p} = \frac{V}{R} = \frac{{100 × \frac{{1000}}{{3600}}}}{{80}} = \frac{{500}}{{1440}} = 0.3472\;rad/s\)

Gyroscopic couple (C) = Iωωp 

C = 2800 × 188.5 × 0.3472

C = 188500 Nm

∴ C = 188.5 kN-m

According to Grashoff’s Law for a four bar mechanism, the sum of shortest and longest link lengths should not be greater than the sum of the remaining two link length.

i.e. S + L ≤ P + Q

For Double crank mechanism Shortest link is fixed.Here shortest link is ‘P’.

Explanation:

T = 100 + 20 sinθ

\({T_{mean}} = \frac{1}{{2\pi }}\mathop \smallint \nolimits_0^{2\;\pi } (100 + 20\sin \theta )d\theta = 100N.m\)

Now,

Change in Torque (ΔT) = Torque developed by engine – Torque required by machine

Change in Torque (ΔT)  = 100 – (100 + 20 sinθ)

∴ Change in Torque (ΔT) = - 20 sinθ

Now,

To get (ΔT)_max,

\(\frac{d}{{d\theta }}( - 20\sin \theta ) = 0\)

⇒ - 20 cos θ = 0 ⇒ θ = 90 or 270

(ΔT)_max = -20 × sin 270

∴ (ΔT)_max = 20 N.m

Now,

(ΔT)_max = I α_max

⇒ 20 = 20 × α_max

∴ α_max = 1 rad/s² at 270° 

• A machine is a tool containing one or more parts that use energy to perform an intended action.
• A simple machine is a device that simply transforms the direction or magnitude of a force, but a large number of more complex machines exist.
• Examples include Levers, Screw Jack, Wheel and axle, Pulleys, Wedge, Inclined plane, etc.
• Spring stores and releases energy. it neither’s changed direction or magnitude of the force. To qualify as a simple machine, a device must exchange the magnitude of a Force.

Concept:

Sliding velocity is the velocity of one tooth relative to its mating tooth along the common tangent at the point of contact. It is given by

V_S = (ω₁ + ω₂) PC

Where PC is the distance between pitch point and the point of contact.

At the end of engagement, the distance between the pitch point and point of contact will be Path of recess.

\(PC = \sqrt {r_a^2 - {r^2}{{\cos }^2}\phi } - r\sin \phi \)

Calculation:

Given:

m = 5 mm, T = 60, G.R = 3, ω_P = 20 rad/s, ϕ = 20°, addendum = W = 1m = 5 mm,

From the given,

R = m T/2 = 5 × 60 /2= 300/2=150 mm;

G.R = T/t ⇒ t = 20

r = m t /2 = 5 × 20/2 = 100/2 =50 mm;

r_a = r + W = 50 + 5 = 55 mm;

ω_G = ω_P/G.R = 20/3 rad/s;

\(PC = \sqrt {{{55}^2} - {{50}^2}{{\cos }^2}20} - 50\sin 20 = 11.49\;mm\)

Now,

Sliding velocity = (20 + 6.67) × 11.49 = 306.7 mm/s

∴ Sliding velocity = 30.7 cm/s 

• When link 4 is moved upwards, it will push link 3 upwards but since link 3&4 are joined by a revolute joint, the link will have to rotate in anticlockwise direction to move upwards.
• Since link 2&3 are joined, link 2 will show the same motion as link 3. So link will rotate in counter clockwise direction.

• This is a case of rolling with sliding on a stationary curved surface, so the instantaneous centre of the body will lie on the common normal at the point of contact, it means line joining two extreme cases.

Important Points

• Instantaneous centre of a body rolling without sliding on a stationary curved surface will lie at the point of contact surface.
• Instantaneous centre of a body in case of pure sliding on a stationary curved surface at the point where it is centred at the curved surface.

When a point on one link is sliding along another rotating link, such as in quick return motion mechanism, then the coriolis component of the acceleration must be calculated. Quick return motion mechanism is used in shaping machines, slotting machines and in rotary internal combustion engines.