Balancing of Reciprocating Mass MCQ Quiz - Objective Question with Answer for Balancing of Reciprocating Mass - Download Free PDF

Concept:

We analyze the kinematics of a Slider-Crank mechanism to determine the angular velocity of the connecting rod when the crank rotates at a given angular speed.

Given:

• Stroke length = (implies crank radius = )
• Connecting rod length =
• Ratio
• Crank angular velocity =
• Crank angle = (measured from inner dead center)

Step 1: Establish the geometric relationship

For a Slider-Crank mechanism, the displacement of the piston from inner dead center is:

Step 2: Differentiate to find velocity relationship

The angular velocity of the connecting rod () is found by differentiating the angular position of the connecting rod with respect to time.

The connecting rod angle relates to crank angle by:

Step 3: Derive angular velocity expression

Differentiating both sides with respect to time:

Thus:

Using :

Explanation:

Dynamically Equivalent System

• In mechanical systems, a dynamically equivalent system refers to a simplified system of masses that has the same dynamic behavior (i.e., same inertia properties) as the original complex body. This concept is especially useful in analyzing and designing mechanical systems because it allows engineers to replace a complicated body with a simpler, equivalent system without changing the dynamic response of the system.

Essential Condition:

• The essential condition of locating two masses such that the system becomes dynamically equivalent to the original body is given by the relationship between the distances of the two masses from the center of gravity of the body (I₁ and I₂) and the radius of gyration of the body (K_G). This condition ensures that the simplified system has the same moment of inertia as the original body.

The moment of inertia (I) of a body about an axis is a measure of the body's resistance to rotational motion about that axis. The radius of gyration (K_G) is a parameter that represents the distribution of the body's mass relative to the axis of rotation. It is defined such that:

I = M × K_G²

where M is the mass of the body.

Explanation:

In the case of reciprocating masses, Primary forces are partially balanced. Because in the reciprocating masses "the resultant forces" will be completely balanced but "the resultant couple" won't be balanced, that is why we say that reciprocating masses are only partially balanced.

Since reciprocating mass cannot balance fully, hence partial balancing of the reciprocating mass is done while revolving masses can be balanced fully.

Primary unbalanced Force 

Secondary unbalanced Force = 

Effect of Partial Balancing of Reciprocating Parts

Reciprocating parts are only partially balanced. Due to this partial balancing of the reciprocating parts, there is an unbalanced primary force along the line of stroke and also an unbalanced primary force perpendicular to the line of stroke. The effect of an unbalanced primary force along the line of stroke is to produce;

• Variation in tractive force along the line of stroke
• Swaying couple

Explanation:

Primary Reverse Cranks:

• The primary reverse cranks are imaginary cranks used to model the primary forces in the engine. They represent the effect of the reciprocating masses in a simplified way.
• Position: For a twin-cylinder V-engine with a 30º angle:
• The two primary reverse cranks are 180º apart, as they account for the dynamic forces in opposite directions due to reciprocation.
• They lie in the plane of the two cylinders, separated by the V-angle (30º).

Secondary Direct Cranks:

• The secondary direct cranks are used to analyze secondary forces that arise due to the second-order effects of reciprocating motion (like the acceleration of connecting rods).
• Position: The secondary direct cranks for a 30º V-engine are positioned as follows:
• They are aligned with the cylinder axes (direct representation of secondary forces).
• These forces are smaller in magnitude than the primary forces but are critical for smooth engine operation. 

Diagram Representation:

• Primary Reverse Cranks: Two cranks placed at an angle of 180º apart with a 30º angle separating them.
• Secondary Direct Cranks: Represented along the directions of the cylinder axes. 

In a 30º V-engine:

• Primary reverse cranks are positioned at 180º with a 30º separation.
• Secondary direct cranks align with the cylinder axes to model secondary forces.

Concept:

In a six-cylinder engine, the primary balancing of reciprocating masses is perfect. The primary unbalanced force can be calculated using the mass of the reciprocating part, the stroke, and the engine speed.

Calculation:

Given:

Mass of one reciprocating part,

Stroke,

Engine speed,

First, we calculate the crank radius :

Next, we calculate the angular velocity :

Now, we calculate the primary unbalanced force for one cylinder:

However, since the primary balancing of reciprocating masses is perfect in a six-cylinder engine, the unbalanced forces for all six cylinders cancel out.

The nearest value for the total primary unbalanced force is:

Explanation:

Primary and secondary Unbalance Forces of Reciprocating Masses.

Consider a reciprocating engine mechanism with

let m = mass of reciprocating part, l = length of the connecting rod, r = radius of the crank, θ = Angle of inclination of the crank with the line of stroke, ω = angular speed of the crank, n = ratio of the connecting rod to the crank radius = l / r.

The acceleration of the reciprocating parts is approximately given by the expression,

Inertia force due to reciprocating parts or force to accelerate the reciprocating parts,

F_I = F_R = Mass × Acceleration

The horizontal component of the force exerted on the crankshaft bearing (i.e. FBH) is equal and opposite to inertia force (F₁). This force is an unbalanced one and denoted by F_U.

∴ Total unbalanced force is:

Total unbalanced force is the sum of the primary and secondary unbalanced force.

The primary unbalanced force is: 

F_P = m × ω² × r × cosθ 

Secondary unbalanced force is:

• The primary unbalanced force is maximum when θ = 0° or θ = 180°. Thus, the primary force is maximum twice in one revolution of the crank.
• The maximum primary unbalanced force is given by F_p(max) = m × ω² ×  r
• The secondary unbalanced force is maximum, when θ = 0°, θ = 90°, θ = 180°, and θ = 360°.Thus, the secondary force is maximum for four angles in one revolution of the crank. 
• The maximum secondary unbalanced force is Fs(max) =  .
• Therefore, it is seen that the frequency of secondary unbalanced force is twice of a primary unbalanced force.
• However, the magnitude of the secondary unbalanced force is less (as n is generally 4 to 5) than primary unbalanced force.

Concept:

Swaying couple:

The unbalanced forces along the line of stroke for the two cylinders constitute a couple about the centerline YY between the cylinders. This couple has a swaying effect about a vertical axis and tends to sway the engine alternately in clockwise and anticlockwise directions. Hence the couple is known as a swaying couple.

a is distance between the centre lines of the two cylinders.

The formula for swaying couple:

It is maximum or minimum when θ = 45° or θ = 225°

The magnitude of swaying couples is directly proportional to the distance between the centrelines of the two cylinders. 

Explanation:

The unbalanced force due to reciprocating masses varies in magnitude but constant in direction while due to the revolving masses, the unbalanced force is constant in magnitude but varies in direction.

Unbalanced force:

 = Primary unbalance force.

 = Secondary unbalance force.

The maximum value of cos θ and cos 2θ is 1,

∴ maximum primary unbalance force (F_P) = mω²R

∴ maximum secondary unbalance force 

∴ 

Explnation:

The unbalanced force due to reciprocating masses varies in magnitude but constant in direction while due to the revolving masses, the unbalanced force is constant in magnitude but varies in direction.

Unbalanced force,

The expression (m. ω².r cos θ) is known as the primary unbalanced force and is called secondary unbalanced force.

So for complete balancing of reciprocating masses:

1. Primary and secondary forces must be balanced

2. Primary and secondary couples must be balanced

Concept:

From the diagram below,

By using Kennedy's theorem, I24 (Common I center of links 2 and 4) will lie on the line joining I21, I14, and I23, I24. The slider C can be considered as the link of infinite length rotating about point D on the ground. So I24 will coincide with I34. For links 2 and 4 (links of known and unknown variables),

⇒ ω2 × (I12 - I24) = ω4 × (I14 - I24)     

Where,

VS = ω4 (I14 - I24) = Velocity of the slider with respect to the ground  

So, the velocity of the slider,

⇒ Vs = ω2 × (I12 - I24) = ω2 × AC      ...(1) 

Calculation:

Given:

ω₂ = 10 rad/s, AB = 100 mm, BC = 160 mm

So, AC = √ (160² -100²) = 124.89 mm =  0.125 m 

Using equation (1),

⇒ Vs = ω₂ × (I₁₂ - I₂₄) = ω₂ × AB

⇒ V_S = 10 × 0.1

⇒ V_S = 1 m/sec

⇒ V_S = 1 m/sec

Alternate Method 

Calculation:

Given:

ω2 = 10 rad/s, AB = 100 mm, BC = 160 mm

Referring to the diagram below,

Where C is on the slider and D is on the fixed link.

Since points, A and D are on the fixed link so they have zero relative velocity. So, referring to the velocity triangle

,

Here, cd is the velocity of the slider which is clearly equal to the velocity of point b i.e. ab

cd = ab

⇒ Vs = ab = AB ω2 

⇒ VS = 10 × 0.100

⇒ VS = 1 m/s

Explanation:

The unbalanced force due to reciprocating masses varies in magnitude but constant in direction while due to the revolving masses, the unbalanced force is constant in magnitude but varies in direction.

Unbalanced force:

 = Primary unbalance force.

 = Secondary unbalance force.

The expression of the secondary unbalance force can also be written in the following way:

Given that the primary and secondary unbalance force is equal:

Thus, when the actual crank turned through an angle θ = ωt, the imaginary crank would have turned an angle of 2θ = 2ωt.

where  is known as the obliquity ratio.    

i.e. the effect of the secondary unbalance force is equivalent to an imaginary crank of length  rotating at double angular velocity i.e. twice the engine speed.

Concept:

x is the position of slider or piston from the bottom dead center

 ;  

For a high value of n

x = r (1 – cos θ)

v = rω sin θ

N is the rotation of crank in r.p.m

Calculation:

It is given that the position of the piston is 0.8 meters from the center.

Now θ can be found out from the expression

x = r (1 – cos θ)

0.2 = 1 (1 – cos θ)

θ = 36.86

Now, velocity

Explanation:

In the case of reciprocating masses, Primary forces are partially balanced. Because in the reciprocating masses "the resultant forces" will be completely balanced but "the resultant couple" won't be balanced, that is why we say that reciprocating masses are only partially balanced.

Since reciprocating mass cannot balance fully hence partial balancing of the reciprocating mass is done while revolving masses can be balanced fully.

Primary unbalanced Force

Secondary unbalanced Force =

Effect of Partial Balancing of Reciprocating Parts

Reciprocating parts are only partially balanced. Due to this partial balancing of the reciprocating parts, there is an unbalanced primary force along the line of stroke and also an unbalanced primary force perpendicular to the line of stroke. The effect of an unbalanced primary force along the line of stroke is to produce;

• Variation in tractive force along the line of stroke
• Swaying couple

Concept:

Direct and reverse crank method of analysis

• In this, all the forces exist in the same plane and hence no couple exists.
   

For the primary force is given by, Fp = mrω2cosθ which acts along the line of stroke, a force identical to this force is generated by two masses as follows:

1. A mass m/2, placed at the crankpin A and rotating at an angular velocity ω

    

   in the counter-clockwise direction.

2. A mass m/2, placed at the crankpin of an imaginary crank OA’ at the same angular position as the real crank but in the opposite direction of the line of stroke.It is assumed to rotate at an angular velocity ω in the clockwise direction (opposite).
    

​At any instant, the components of the centrifugal forces of these masses normal to the line of stroke will be equal and opposite.

The crank rotating in the direction of engine rotation is known as the direct crank and the imaginary crank rotating in the opposite direction is known as the reverse crank.

Now for balancing the secondary force, F_s = 

1. A mass m/2, placed at the end of the direct secondary crank of length 

    at an angle 2θ in  the counter-clockwise direction.

    

2. A mass m/2, placed at the end of the reverse secondary crank of length 

   ​​ at an angle -2θ in the clockwise direction.

​

Calculation:

Given:

The angle made by the crank θ = 30°  clockwise.

To completely balance the secondary forces a reverse crank at angle 2θ is applied in the anticlockwise direction.

Therefore, the reverse crank is at 60° in the anticlockwise direction.

Important Points

Secondary force balancing 

The conditions to be satisfied for the partial secondary force balancing are as follows:

• The imaginary crank length required is 
• The speed of imaginary crank is 2ω 
• Angle made by an imaginary secondary crank with inner dead center = 2θ
   

Additional Information

Reciprocating masses

• Reciprocating masses occur in internal combustion engines and steam engines.
• The reciprocating masses are due to the mass of the piston, piston pin, and part of the mass of the connecting rod considered as reciprocating.

The unbalanced force Fu, due to the reciprocating mass m, varies in magnitude but is constant in direction. It is given by,

Where primary force Fp =  which represents the inertia force of reciprocating mass having simple harmonic motion.

and secondary force Fs =  which represents the correction required to account for the obliquity of the connecting rod.

Explanation:

Direct and reverse crank method of analysis:

In this, all the forces exist in the same plane and hence no couple exists.

For the primary force is given by, F_p = mrω²cosθ which acts along the line of stroke, a force identical to this force is generated by two masses as follows:

1. A mass m/2, placed at the crankpin A and rotating at an angular velocity ω in the counter-clockwise direction.
2. A mass m/2, placed at the crankpin of an imaginary crank OA’ at the same angular position as the real crank but in the opposite direction of the line of stroke. It is assumed to rotate at an angular velocity ω in the clockwise direction (opposite).
    

At any instant, the components of the centrifugal forces of these masses normal to the line of stroke will be equal and opposite.

The crank rotating in the direction of engine rotation is known as the direct crank and the imaginary crank rotating in the opposite direction is known as the reverse crank.

Now for balancing the secondary force,   force identical to it can also be generated by two masses in a similar way as follows:

1. A mass m/2, placed at the end of the direct secondary crank of length  at an angle 2θ in the counter-clockwise direction. 
2. A mass m/2, placed at the end of the reverse secondary crank of length ​​ at an angle -2θ in the clockwise direction.