balancing of rotating masses

K.M.KumarME2302-Dynamics of MachineryUnit-IIUnit - II

ME2302-Dynamics of Machinery

K.M.KumarME2302-Dynamics of Machinery*Balancing of Rotating Masses whenever a certain mass is attached to a rotating shaft, it exerts some centrifugal force, whose effect is to bend the shaft and to produce vibrations in it. In order to prevent the effect of centrifugal force, another mass is attached to the opposite side of the shaft, at such a position so as to balance the effect of the centrifugal force of the first mass. The process of providing the second mass in order to counteract the effect of the centrifugal force of the first mass, is called balancing of rotating masses.The following cases are important from the subject point of view:1. Balancing of a single rotating mass by a single mass rotating in the same plane.2. Balancing of a single rotating mass by two masses rotating in different planes.3. Balancing of different masses rotating in the same plane.4. Balancing of different masses rotating in different planes.


Balancing of a Single Rotating Mass By a Single Mass Rotating in the Same PlaneK.M.KumarME2302-Dynamics of Machinery*


K.M.KumarME2302-Dynamics of Machinery*Balancing of a Single Rotating Mass By Two Masses Rotating in Different PlanesThe system in complete balance, two balancing masses are placed in two different planes, parallel to the plane of rotation of the disturbing mass, in such a way that they satisfy the following two conditions of equilibrium1. The net dynamic force acting on the shaft is equal to zero. This requires that the line of action of three centrifugal forces must be the same. In other words, the centre of the masses of the system must lie on the axis of rotation. This is the condition for static balancing.2. The net couple due to the dynamic forces acting on the shaft is equal to zero. In other words, the algebraic sum of the moments about any point in the plane must be zero. The conditions (1) and (2) together give dynamic balancing.


K.M.KumarME2302-Dynamics of Machinery*Balancing of a Single Rotating Mass By Two Masses Rotating in Different Planes


K.M.KumarME2302-Dynamics of Machinery*Balancing of Several Masses Rotating in the Same Plane


K.M.KumarME2302-Dynamics of Machinery*Balancing of Several Masses Rotating in the Same Plane1. Analytical method


K.M.KumarME2302-Dynamics of Machinery*Balancing of Several Masses Rotating in the Same Plane2. Graphical method


K.M.KumarME2302-Dynamics of Machinery*Balancing of Several Masses Rotating in Different PlanesThe following two conditions must be satisfied :1. The forces in the reference plane must balance, i.e. the resultant force must be zero.2. The couples about the reference plane must balance,i.e. the resultant couple must be zero.


K.M.KumarME2302-Dynamics of Machinery*Balancing of Several Masses Rotating in Different Planes


Balancing of Reciprocating MassesK.M.KumarME2302-Dynamics of Machinery*Several forces acting on the reciprocating parts of an engine. The resultant of all the forces acting on the body of the engine due to inertia forces only is known as unbalanced force or shaking force


Forces Acting in the I.C EngineK.M.KumarME2302-Dynamics of Machinery*


Primary and Secondary Unbalanced Forces of Reciprocating MassesK.M.KumarME2302-Dynamics of Machinery*


K.M.KumarME2302-Dynamics of Machinery*Primary and Secondary Unbalanced Forces of Reciprocating Masses


Partial Balancing of Unbalanced Primary Force in a Reciprocating EngineK.M.KumarME2302-Dynamics of Machinery*


K.M.KumarME2302-Dynamics of Machinery*

Unbalanced Primary Force in a Reciprocating EngineAs a compromise let a fraction c of the reciprocating masses is balanced, such that c.m.r = B.b


Partial Balancing of LocomotivesK.M.KumarME2302-Dynamics of Machinery*The locomotives, usually, have two cylinders with cranks placed at right angles to each other in order to have uniformity in turning moment diagram. The two cylinder locomotives may be classified as :1. Inside cylinder locomotives ; and 2. Outside cylinder locomotives.In the inside cylinder locomotives, the two cylinders are placed in between the planes of two driving wheels as shown in Fig. whereas in the outside cylinder locomotives, the two cylinders are placed outside the driving wheels, one on each side of the driving wheel, as shown in Fig. The locomotives may be(a) Single or uncoupled locomotives ; and (b) Coupled locomotives


K.M.KumarME2302-Dynamics of Machinery*Partial Balancing of LocomotivesA single or uncoupled locomotive is one, in which the effort is transmitted to one pair of the wheels only ; whereas in coupled locomotives, the driving wheels are connected to the leading and trailing wheel by an outside coupling rod.


Effect of Partial Balancing of Reciprocating Parts of Two Cylinder LocomotivesK.M.KumarME2302-Dynamics of Machinery*The effect of an unbalanced primary force along the line of stroke is to produce;Variation in tractive force along the line of stroke ; and Swaying couple.The effect of an unbalanced primary force perpendicular to the line of stroke is to produce variation in pressure on the rails, which results in hammering action on the rails. The maximum magnitude of the unbalanced force along the perpendicular to the line of stroke is known as a hammer blow.


Variation of Tractive ForceK.M.KumarME2302-Dynamics of Machinery*The resultant unbalanced force due to the two cylinders, along the line of stroke, is known as tractive force


Swaying CoupleK.M.KumarME2302-Dynamics of Machinery*The unbalanced forces along the line of stroke for the two cylinders constitute a couple about the centre line YY between the cylindersThis couple has swaying effect about a vertical axis, and tends to sway the engine alternately in clockwise and anticlockwise directions. Hence the couple is known as swaying couple.Let a = Distance between the centre lines of the two cylinders.


K.M.KumarME2302-Dynamics of Machinery*Swaying Couple


Hammer BlowK.M.KumarME2302-Dynamics of Machinery*


Problem No.1The following data refer to a two cylinder uncoupled locomotives, Rotating mass per cylinder = 300 kg, Reciprocating mass per cylinder = 330 kg, distance between wheels = 1.5 m, distance between cylinder centres (or) pitch of the cylinder = 600 mm, diameter of treads of driving wheels = 1.8 m, carnks radius = 325 mm, radius of centre of balance mass = 650 mm, locomotive speed = 60 km/hr, angle between cylinder cranks = 90, dead load on each wheel = 40 kN.determine (1) the balancing mass required in the planes of driving wheels if whole of the revolving and 2/3 of the reciprocating mass are to be balanced.(2) the swaying couple.(3) the variation in tractive force.(4) the maximum and minimum pressure on the rail and(5) the maximum speed of the locomotive without lifting the wheels from the rails.K.M.KumarME2302-Dynamics of Machinery*


Problem No.2The firing order of a six cylinder vertical 4-stroke engine is 1-4-2-6-3-5. the piston stroke is 80 mm and the length of the each connecting rod is 180 mm. the pitch distances between the cylinder centre lines are 80 mm, 80 mm, 120 mm, 80 mm, and 80 mm respectively. The reciprocating mass per cylinder is 1.2 kg and the engine speed is 2400 rpm. determine the out of balance primary and secondary forces and couples of the engine taking a plane midway between the cylinder 3 and 4 as the reference point.K.M.KumarME2302-Dynamics of Machinery*


K.M.KumarME2302-Dynamics of Machinery*
