# Inertia force of a reciprocating masses

• hanson
In summary, the conversation discusses the derivation of the formula for the inertia force of reciprocating masses and introduces the concept of harmonic terms. It also mentions the use of an approximation and the confusion surrounding potential higher order terms in the equation. The equation includes a series expansion approximation to simplify the estimation of piston displacement and the mass used should include both rotating and reciprocating components. The most common use of the equation is for calculating bearing loads and rod load reversal angles.

#### hanson

hi all!
I am learning the derivation of the formula of the inertia force of reciprocating masses, which is a typical formula that I am sure all of you must know.
F=mrw^2{cosB+(cos2B)/n}
I know that the cosB term is called the 1st harmonic, and the cos2B term is the 2nd harmonic.
Also, I know that this is not an exact formula, since a approximation was made in the derivation. That, n>>l, right?
But I cannot see if without the assumption, what will be the relation?
I am told that there should be some higher order terms after mw^2{cosB+(cos2B)/n}. But I don't see what they are and how they are produced. Is that something like Taylor series is used in order to produce the higher order terms??
I am confused.

hanson said:
hi all!
I am learning the derivation of the formula of the inertia force of reciprocating masses, which is a typical formula that I am sure all of you must know.
F=mrw^2{cosB+(cos2B)/n}
I know that the cosB term is called the 1st harmonic, and the cos2B term is the 2nd harmonic.
Also, I know that this is not an exact formula, since a approximation was made in the derivation. That, n>>l, right?
But I cannot see if without the assumption, what will be the relation?
I am told that there should be some higher order terms after mw^2{cosB+(cos2B)/n}. But I don't see what they are and how they are produced. Is that something like Taylor series is used in order to produce the higher order terms??
I am confused.

According to "Internal Combustion Engines Applied Thermosciences" the equation you reference includes a series expansion approxmation [(1 - E) ^(1/2) is approxmately (1 - E/e)] to simplify the estimation of piston displacement as a function of crank angle.

The equation I have for force is:

F=maω^2 (cos⁡(β) + a/l cos(2β))

Where:
F is the instantaneous inertia force
m is the effecting rotating mass of the piston and connecting rod
a is the radius of the crankshaft
ω is the rotational velocity of the crankshaft in radians per unit time
β is the instantaneous crank angle
l is the length of the connecting rod

question- are you mixing rotating mass and reciprocating mass?
example only big end pf con rod is rotation
the piston end is reciprocating
will these not require different formulas to measure true inertia ?

There is a mix of the mass. The mass used in the equation should include the mass of all reciprocating components (piston, rings, and if employed, piston rod and cross-head) and a portion (estimated at 2/3) of the mass of the connecting rod. The inertia forces of the crankshaft should be evaluated separately as it is pure rotational and the inertia forces are dependent on the geometry of any counter weights. The most common use of the equation above is for calculating bearing loads and rod load reversal angles to ensure proper lubrication.

## 1. What is inertia force of a reciprocating mass?

Inertia force of a reciprocating mass refers to the force that opposes the motion of a reciprocating mass (a mass that moves back and forth in a straight line) due to its inertia, or resistance to change in motion.

## 2. How is inertia force of a reciprocating mass calculated?

Inertia force of a reciprocating mass can be calculated using the equation F = m x a, where F is the inertia force, m is the mass of the reciprocating object, and a is the acceleration of the object.

## 3. What factors affect the magnitude of inertia force of a reciprocating mass?

The magnitude of inertia force of a reciprocating mass is affected by the mass of the object, the speed and direction of its motion, and any external forces acting on the object.

## 4. How does inertia force of a reciprocating mass affect the operation of machines?

Inertia force of a reciprocating mass can cause vibrations and fluctuations in speed, which can affect the smooth operation of machines. It is important for engineers to consider and minimize these forces in the design of machines.

## 5. Can inertia force of a reciprocating mass be eliminated?

In some cases, inertia force of a reciprocating mass can be minimized or eliminated through the use of counterweights, dampers, or other engineering techniques. However, completely eliminating inertia force is not always possible and may not be desirable, as it can also provide stability and control in certain systems.