dynamics of reciprocating masses

1. Branch : Mechanical Engineering
Div: 6C-1
Name: Saahil Kshatriya
Enrolment Number: 150120119164

2. Content
 Introduction
 Condition for Completebalancing
 Primary Balancing of inline Multi-cylinder Engines
 Secondary Balancing of inline multi-cylinder Engines
 Balancing of four cylinder inline engines

3. Introduction
 Reciprocating parts of any machine is subjected to continuous
acceleration and retardation.
 Inertia acted opposite to the acceleration of part.
 This inertia force is disturbing or unbalanced dynamic force acting
on reciprocating part.
 Eliminate the unbalanced force by using suitable balancing masses.
 The balancing of reciprocating masses is divided into :
•
•
•
Balancing reciprocating massesin single cylinder engine
Balancing reciprocating massesin multi cylinder inline engine
Balancing reciprocating massesin V-engine

4. Balancingreciprocating masses in
Multi-cylinder inline engine
The multi-cylinder engine having the axes of all the engine of all the cylinders in the same plane and on
thesamesideof theaxis of thecrank shaft, areknown asinline engine.
In the multi-cylinder engines unbalanced force and couples are acting.
Themulti-cylinder engine have twoinner crank and twooutercrank.
Fortheinner crank angle =θ˚
Fortheoutercrank angle =180+ θ˚
Theunbalanced forces due toreciprocating massof each
Cylinder are ;[Primaryforce , Fp=mω 𝟐r cosθ ][secondaryforce= mω r{𝟐 cos2θ
𝒏
}

5. Condition for complete balancing
Primary balancing
 Primaryforces mustbe balanced.
∑mω2r cosθ= 0
 Primarycouple mustbe balanced.
∑mω2r l cosθ= 0
Secondary balancing
 Secondary forces mustbebalanced.
𝒏
∑ mω 𝟐r{cos 2θ}
 Secondary couple mustbe balanced.
𝒏
∑ mω 𝟐rl{cos2θ}

6. Analytical method for primary balancing
Forcompleteprimarybalancing, theanalytical solutionis;
∑ mr cosθ= 0
∑ mrl cosθ= 0
∑mr sinθ= 0
∑ mrl sinθ= 0
Analytical methodfor secondary balancing
∑ mrcos 2θ= 0 ∑ mrsin2θ=0
𝑛 𝑛
∑ mrlcos 2θ= 0 ∑ mrlsin2θ= 0
𝑛 𝑛

7. EXAMPLE : A four cylinder vertical engine has crank 150 mm long . The planes of rotation of first ,
secondandfourthcrankare400mm, 200mm and200mmrespectivelyfromthe thirdcrankandtheir
reciprocatingmasesare50kg,60kg and50 kg respectively.Findthemassof the thirdreciprocatingpart
and their relative angular position of the cranks in order that the engine may be in complete primary
balance.
Sol.:- Given , r1 = r2 = r3 = r4 = 150 mm = 0.15m ; m1 = 50 kg, m2 = 60 kg, m4 = 50 kg;
1 2 3 4
-ve R.P +ve
200 mm 200 mm
400mm
(a) Position of plane
In order to give the primary balance of the reciprocating parts of a multi-cylinder engine, the
problem may be treated as that of revolving masses
with the reciprocating masses transferred to their respective crank pins.

8. Plane Mass Radius Cent.
Force/ω 𝟐
Dist. From
plane 3
Couple/ω 𝟐
1 50 0.15 7.5 -0.4 -3
2 60 0.15 9 -0.2 -1.8
3( R.P) m3 0.15 0.15m3 0 0
4 50 0.15 7.5 0.2 1.5
a’
b’
o’
-1.81.5
-3
(b). Couple polygon
1
2
3
4
(c). Angular position of crank

9. 7.5
9
By measurement we find the angle between crank 2 and crank 1 is 160˚ in anticlockwise.
θ2 = 160˚
And angle between crank 4 and crank 1 is 26˚ in anticlockwise.
θ4 = 26˚
In order to find the mass of the third cylinder (m3) and its angular position, draw the force
polygon, to some suitable scale.
7.5
0.15m3 From the fig. , 0.15 m3 = 9 kg-m
m3 = 60 kg.
and θ3 = 227 ˚