Two Speed Coal Crusher

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- Completely constrained motion: When the motion between a pair is limited to a definite direction irrespective of the direction of force applied then the motion is said to be a completely constrained motion. For example, the piston and cylinder (in a steam reciprocate) relative to the cylinder irrespective of the direction of motion of the crank, as shown in fig.

[pic 1]

Figure 2.1(a) Figure 2.1(b)

- Incompletely constrained motion: When the motion between a pair can take place in more than one direction then the motion is called an incompletely constrained motion. The change in the direction of impressed force may alter the direction of relative motion between the pair. A circular bar or shaft in a circular hole.

[pic 2]

Figure2.2(a) Figure2.2(b)

- Successfully constrained motion: When the motion between the elements, forming a pair is such that the constrained motion is not completed by itself, but by some other means, then the motion is said to be successfully constrained motion. Consider a shaft in a foot-step bearing as shown in Fig. 5.5. The shaft may rotate in a bearing or it may move upwards. This is a case of incompletely constrained motion. But if the load is placed on the shaft to prevent axial upward movement of the shaft, then the motion of the pair is said to be successfully constrained motion. The motion of an I.C. engine valve (these are kept on their seat by a spring) and the piston reciprocating inside an engine cylinder are also the examples of successfully motion.

2.7 Mechanism

When one of the links of a kinematic chain is fixed, the chain is known as mechanism. It may be used for transmitting or transforming motion e.g. engine indicators. Typewriter, etc.

A mechanism with four links is known as simple mechanism, and the machine with more than four links is known as compound mechanism. When a mechanism is required to transmit power or to do some particular type of work it then becomes a machine. In such cases, the various links or elements have to be designed to withstand the forces (both static and kinetic) safely. A little consideration will show that a mechanism may be regarded as a machine in which each part is reduced to the simplest form to transmit the required motion.

2.8 Number of Degrees of Freedom for plane Mechanisms

In the design or analysis of a mechanism, one of the most important concerns is the number of degrees of freedom (also called movability) of the mechanism. It is defined as the number of input parameters (usually pair variables) which must be independently controlled in order to bring the mechanism into useful engineering purpose. It is possible to determine the number of degrees of freedom of a mechanism directly from the number of links and the number and types of joints which in includes.

[pic 3]

Figure2.3

Consider a four bar chain, as shown in Fig 2.3 (a). A little consideration will show that only one variable such as 0 is needed to define the relative positions of all the links. In other words, we say that the number of degrees of freedom of a four bar chain is one. Now, let us consider a five bar chain, as shown in Fig 2.3(b). In this case two variables such as 01, and 02 are needed to define completely the relative positions of all the links. Thus, we say that the number of degrees of freedom is two.

In order to develop the relationship in general, consider two links AB and CD in a plane motion as shown in Fig 2.4(a).

[pic 4]

Figure2.4

The link AB with co-ordinate system OXY is taken as the reference link (or fixed link). The position of point P on the moving link CD can be compeltely specified by the three variables. i.e. the

The differential of an automobile requires that the angular velocity of two elements be fixed in order to know the velocity of the remaining elements. The differential mechanism is thus said to have two degrees of freedom. Many computing mechanism have two or more degrees of freedom. Co-ordinates of the point P denoted by x and y and the inclination 0 of the link CD with X-axis or link AB. In other words, we can say that each link of a mechanism has three degrees of freedom before it is connected to any otehr link. But when the link CD is connected to the link AB by a turning pair at A, as shown in Fig. 2.4 (b), the position of link CD is now determined by a single variable 0 and thus has one degree of freedom.

From above, we see that when a link is connected to a fixed link by a turning pair (i.e. lower pair), two degrees of freedom are destroyed. This may be clearly understood from Fig 2.5 in which the resulting four bar mechanism has one degree of freedom (i.e. n= 1)

[pic 5]

Figure2.5

Now let us consider a plane mechanism with/number of links. Since in a mechanism, one of the links is to be fixed, therefore the number of movable links will be (I - 1) and thus the total number of degrees of freedom will be 3 (I-1) before they are connected to any other link. In general, a mechanism with / number of links connected connected by j number of binary joints or lower pairs (i.e. single degree of freedom pairs) and h number of higher pairs (i.e. two degree if freedom pairs) then the number of degrees of freedom of mechanism is given by.

n = 3 (I - 1) - 2 j – h

This question is called Kutzbach criterion for this movability of a mechanism having plane motion.

If there are no two degree of freedom pairs (i.e. higher pairs). then h = 0. Substituting h = 0 in equation (i) we have

n = 3 (I - 1) - 2j

2.9 Movement of Inertia of a disk

Case 1st about an axes passing through center and 1 to plane o disk M= mass of disk R= Radius of disk Square area of disk = IIR2

Mass/area of disk = M

πR2

The whole disk can be consider to be made up of a very large no of rings whose radii varies from a to R: consider one such ring of radius X and thickness dx.

Hence, mass