Types of contrained motions

Types of constrained motion.

(i)                 Completely constrained motion.

(ii)                Incompletely constrained motion.

(iii)               Successfully constrained motion.

                              

4 Stroke I.C Engine using Turning moment Diagram

Single cylinder 4 – stroke I. C. Engine using Turning moment Diagram.

A turning moment diagram for a four stroke cycle internal combustion engine, we know that in a four stroke cycle internal combustion engine, there is one working stroke after a crank has turned through two revolution i.e.7200 .

Since the pressure inside the engine cylinder is less than the atmospheric pressure during suction stroke therefore a negative loop is formed. During the compression stroke, the work is done on gases, therefore a higher negative loop is obtained.

During the expansion or working stroke, the fuel burns and the gases expand, therefore a positive loop is obtained. In this stroke the work done is by the gases. During exhaust stroke, the work is done on the gases, therefore negative loop is formed. It may be noted that effect of inertia forces on the piston is taken is account.

Eddy current dynamometer

Construction and Working of Eddy current dynamometer

Sketch represents working principle of this transmission type dynamometer, to measure torque and hence power output of an engine.

 It consists of rotor disc made of steel or copper. The rotor shaft is supported in bearings and it is coupled to engine shaft.

 Stator is fitted with number of electromagnets and the stator cradles in the trunion bearings. When rotor rotates, it produces eddy currents in the stator due to magnetic flux by passage of field current in the electromagnets. These currents oppose the rotor motion, thus loading the engine.

 The torque is measured with the help of torque arm.

 This dynamometer requires some cooling arrangement since the eddy current generate heat.

 This dynamometer is compact and versatile; as it can measure high power output at all speeds.    These are used to test automobile and aircraft engines.

Elliptical trammel

Elliptical trammel- 

Since Elliptical trammel consist of two turning pairs and two sliding pairs, it is inversion of double slider crank chain.This instrument is used for drawing ellipses. This inversion is obtained by fixing a slotted plate (link 4) as shown in fig. It has got two right angled grooves cut into it. 

1-2  is    turning pair                       

2-3 is turning pair 1-4 is sliding pair                               

3-4 is sliding pair  

               

As the crank BC is rotated, any point on crank except midpoint of BC and point B and C will trace the ellipse. Midpoint of BC will trace a circle. The points B and C will move in straight line along the slot.

LAW OF GEARING

Law of Gearing  :-

Consider the portions of two gear teeth in mesh.

O1 and O2 are centre points.

Let K= point of contact T T = Common tangent at point of contact K

N’N’ = Common Normal at point of contact K 

O1M and O2N are perpendicular to Common Normal N’N’.

V1 and V2 =Velocities at point K w. r. t. gear 1 and 2 respectively If mating teeth to remain in contact while transmitting motion, components of velocities must be equal along N’N’. So, V1cos  = V2 COS   (ω1 x O1K) cos  = (ω2 x O2K) cos   From triangles O1MK and O2NK putting values of cos   and COS    ω1 X O1K X        =ω2 X O2K X     

 ω1 X O1M =ω2 X O2N       =               …………..(1) Since  O1MP and O2NP are similar triangles.

     

=     

          …………..(2)

From equations (1) and(2) , we get

    =     

 From this, it is proved that angular velocity ratio is inversely proportional to ratio of distance of fixed point ‘P’ ,which is pitch point. This gives constant angular velocity ratio.

In other words, the common normal at the point of contact between a pair of teeth must always pass through the pitch point for all positions of mating gears. This is the fundamental condition which must be satisfied while designing the profiles of teeth for gears.

This is Law of Gearing or Condition of correct gearing.

Balancing single rotating mass

Procedure of Balancing single rotating mass when disturbing mass in same plane

Fig. shows single rotating mass ‘m’ which is attached to a shaft rotating with angular velocity ‘ω’. Let ‘r’ = distance of centre of gravity of ‘m’ from axis of rotation of shaft

 Due to rotation of shaft, centrifugal force ‘mrω2 ‘acts  radially outwards due to inertia of mass. This force is called disturbing force which will produce bending moment on the shaft.  

   

Diagram:-

A balance mass mb is introduced in the plane of rotation of disturbing mass, such that, it neutralizes the effect of inertia force due to disturbing mass. 

Thus , the inertia forces of mass ‘m’ and mass ‘mb’ must be equal and opposite.       mrω2  = mbrbω2               mr  = mbrb.

Thus the balancing mass mb is used at convenient radius rb  .Generally, rb is considered as large as possible so that balance mass mb required is very small.  

Epicyclic Gear train

 Epicyclic  Gear train  :-    

In case of  Epicyclic  Gear train, the axis of shafts on which gears are mounted may have a relative motion between them, unlike other gear trains.

This gives advantage that, very high or low velocity ratio can be obtained compared to simple and compound gear trains; in the small space.

  

In above sketch, if gears A and B are rotating and arm RS is fixed, then it behaves like simple gear train.

 However, when Arm C rotates and gear A is fixed, then train becomes epicyclic. It is also known as planetary gear train.

Diagram:-

Applications-

Differential gears of the automobiles,

back gear of lathe,

hoists, pulley blocks