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

Klein's construction to determine velocity and acceleration of , single slider crank mechanism.

*Klein's construction to determine velocity and acceleration       

 of single slider crank mechanism.*

Let OC is the Crank & PC is the connecting Rod of a reciprocating steam Engine.

Let the crank makes an angle with the line of strokes PO ,And rotates with uniform angular velocity in rad/sec in a clock wise direction.

Diagram-

First of all draw OP perpendicular to OP,that’s intersect to line pc produce at M.

Then the TRAINGLE ‘’OCM ‘’ produced that’s known as kleins velocity diagram.

We have already discussed that the velocity diagram for given configuration is a triangle OCP as shown in Fig.

If this triangle is rotated through 90°, it will be a triangle oc1 p1, in which oc1 represents VCO  and is parallel to OC,

 op1 represents VPO velocity of P with respect to O or velocity of cross-head or piston P) and is perpendicular to OP, and c1p1 represents VPC and is parallel to CP.

A little consideration will show that the triangles oc1p1 and OCM are similar. Therefore, 

Klien’s acceleration diagram:

The Klien’s acceleration diagram is drawn as discussed below: 

1. First of all, draw a circle with C as centre and CM as radius.

2. Draw another circle with PC as diameter. Let this circle intersect the previous circle at K and L.

3. Join KL and produce it to intersect PO at N. Let KL intersect PC at Q. This forms the quadrilateral CQNO, which is known as Klien’s acceleration diagram.

We have already discussed that the acceleration diagram for the given configuration is as shown in Fig. We know that (i) o'c' represents CO  and is parallel to CO;

(ii) c'x represents PC and is parallel to CP or CQ;

 (iii) xp' represents PC at and is parallel to QN (because QN is perpendicular to CQ); and

(iv) o'p' represents PO and is parallel to PO or NO. A little consideration will show that the quadrilateral o'c'x p' is similar to quadrilateral CQNO .

Quick return Mechanism of Shaper .

*Crank and slotted lever quick return motion mechanism*

Or

*Quick return mechanism of shaper machine*

In the extreme positions, AP1 and AP2 are tangential to the circle and the cutting tool is at the end of the stroke.

The forward or cutting stroke occurs when the crank rotates from the position CB1 to CB2 (or through an angle β) in the clockwise direction.

The return stroke occurs when the crank rotates from the position CB2 to CB1 (or through angle α) in the clockwise direction. Since the crank has uniform angular speed, 

Diagram-

Oldhams coupling

Oldham’s coupling*

An Oldham’s coupling is used for connecting two parallel shafts whose axes are at a small distance apart.

The shafts are coupled in such a way that if one shaft rotates, the other shaft also rotates at the same speed.

This inversion is obtained by fixing the link 2, as shown in Fig. The shafts to be connected have two flanges (link 1 and link 3) rigidly fastened at their ends by forging. The link 1 and link 3 form turning pairs with link 2.

These flanges have diametrical slots cut in their inner faces, as shown in Fig. The intermediate piece (link 4) which is a circular disc, have two tongues T1 and T2 on each face at right angles to each other.

The tongues on the link 4 closely fit into the slots in the two flanges (link 1 and link 3). The link 4 can slide or reciprocate in the slots in the flanges.

Diagram :-

When the driving shaft A is rotated, the flange C (link 1) causes the intermediate piece (link 4) to rotate at the same angle through\ which the flange has rotated, and it further rotates the flange D (link 3) at the same angle and thus the shaft B rotates. Hence links 1, 3 and 4 have the same angular velocity at every instant. A little consideration will show that there is a sliding motion between the link 4 and each of the other links 1 and 3.

APPLICATIONS:-

1. Plastic center disc possesses power for voltaic isolation.

2. It performs as a type of fuse for machinery.

3. Exert a low reactive force that fortifies the support of bearings.

4. The driving and driven shafts gyrate at similar speed.

5. There is no backlash and torsion inelasticity.

6. It behaves as an electrical insulator.

Disadvantages :-

1. Comparatively, it harbors smaller angular misalignment.

2. It possesses lower peak torque and torsion elasticity.

3. Driven and driving shafts don’t move with same speed in a single rotation.

4. There is an angular asymmetrical.

5. On the maximum note of torque, it exerts an axial reactive load on the supportive bearings.

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