Mobility/Degrees of freedom of mechanisms and Interpretation of mobility/Degrees of freedom equation

 

Mobility/Degrees of freedom of mechanisms and Interpretation of mobility/Degrees of freedom equation

Mobility/Degrees of freedom of mechanisms:

 The number of independent input parameters which must be controlled independently so that a mechanism fulfils its useful engineering purpose is called its degrees of freedom or mobility.

 Expression for degrees of freedom of a mechanism, consisting of lower pairs only is

                                           d.o.f of a mechanism, f = 3(n-1) -2l

the above equation is known as Grubler's equation, where n – number of mobile links

                                                                                              l – number of lower pairs

 Expression for degrees of freedom of a mechanism, consisting of lower pairs and higher pairs is

                                           d.o.f of a mechanism, f = 3(n-1) -2l - h

the above equation is known as modified Grubler's equation or Kutzbach's equation, where n – number of mobile links

                                                                                              l – number of lower pairs

                                                                                              h – number of higher pairs

 

Interpretation of mobility/Degrees of freedom equation:

 1. When the mobility/d.o.f of a mechanism is zero, no motion is possible by mechanism and mechanism becomes a structure.

2. When the mobility/d.o.f of a mechanism is -1, it results in an indeterminate structure. It means the linkage has an additional/redundant constraint, beyond what is necessary to keep the linkage in position.

3. When the mobility/d.o.f of a mechanism is +1, the mechanism can be driven by a single input motion to produce constrained output motion.

4. When the mobility/d.o.f of a mechanism is +2, the mechanism can be driven by two independent input motions to produce constrained output motion.

Principle of Virtual work - Beams

 Using the principle of virtual work, find the reactions Rb, Re and Rf at the roller supports of the compound beam shown in the below fig. If three equal vertical loads Q act as shown.

Principle of Virtual work - Beams

 A simply supported beam AB of span 5 m is loaded as shown in Fig. Using the principle of virtual work, find the reactions at A and B.

Principle of Virtual work - Beams

 An overhanging beam ABC of span 3 m is loaded as shown in Fig. Using the principle of virtual work, find the reactions at A and B.

Principle of Virtual work - Beams

 Principle of Virtual work - Beams 

Two beams AE and BD are supported on rollers at B and C as shown in Fig. Determine the reactions at the rollers B and C, using the method of virtual work.

Principle of Virtual work - Beams

Principle of Virtual work - Beams 

Using the principle of virtual work, find the reaction Rd for the system shown in fig. For any position of a vertical load P on the beam AC as defined by its distance x from A.

In whitworth quick return mechanism the displacement of forward stroke and backward stroke are equal or not?

 In whitworth quick return mechanism the displacement of forward stroke and backward stroke are equal or not?

In the below picture shown is a Whitworth quick return mechanism, L1, L2, L3 and L4 are lengths of various links.

The stroke of slider e can be written as

                                          Stroke, S = E2E1

                                                        = (L4 + L3) – (L4 – L3)

                                                        = 2xL3

 

From the above equation for stroke, it is evident that, stroke of the slider depends on the link length L3, which is of fixed length. Hence displacement of the slider in the forward stroke and in the reverse stroke is constant and equal to twice the length of the link 3.

 

Now let us see the angular displacement of link L2 during forward and reverse strokes.

Angular displacement of link 2 during forward stroke is Theta and during reverse stroke it is Beta. From the below picture it is clear that Theta is greater than Beta. So angular displacement of link 2 during forward stoke is greater than angular displacement in the reverse stroke.

How cylindrical cam is different from radial cam?

 How cylindrical cam is different from radial cam?

Radial cam: Below figure shows a radial cam (plate or disc cam) whose working surface is so shaped that the follower reciprocates or translates in a plane perpendicular to the axis of cam. These cams used to convert rotational motion to linear motion perpendicular to the axis of the cam.

Cylindrical cam: Below picture shows a cylindrical cam (barrel cam), in which the follower rides in a groove cut into the surface of a cylinder (Circumferential groove). These cams used to convert rotational motion to linear motion parallel to the axis of the cylindrical cam.

What is the double crank mechanism and double rocker mechanism?

 What is the double crank mechanism and double rocker mechanism?

First let us understand Grashoff’s law, Grashoff’s law states that for a planar four bar linkage, the sum of the shortest and longest link length must be less than or equal to the sum of the remaining two link lengths, if there is to be a continuous relative rotation between two members.

If S and L be the lengths of shortest and longest links respectively and P and Q be the remaining two link length, the shortest link will rotate continuously relative to the other three links, if and only if,

                                        S+L ≤ P+Q

If this inequality is not satisfied, the chain is called non-Grashoff chain in which none of the links can have complete revolution relative to other links.

For a Grashoff’s chain there are three distinct inversions possible, whereas a non-Grashoff’s chain has only one inversion, namely the Double rocker or Rocker-Rocker mechanism.

Inversions of Grashoff’s chain are 1. Double crank mechanism, 2. Double rocker mechanism and 3. Crank rocker mechanism.

1. Double crank mechanism: When the shortest link of a Grashoff’s chain is fixed, a double crank mechanism results, in which both the links connected to the frame rotate continuously.

2. Double rocker mechanism: When the link opposite to the shortest link is fixed, a double rocker mechanism results. None of the two links driver and driven connected to the frame can have complete revolution but the coupler link can have full revolution.

Can you make L/R in the slider crank mechanism less than unity?

 Can you make L/R in the slider crank mechanism less than unity?

In the below picture shown a Slider crank mechanism, having crank length R and coupler / Connecting rod length L.

In the below example, crank length of 50 units and coupler/connecting rod length of 40 units are taken, that is L/R ratio less than unity is selected. With this L/R ratio the crank can rotate only between angle +53° and -53°. If L/R ratio is less than unity then the crank can not make a complete rotation. So L/R ratio must always be greater than unity for the crank to have complete rotation. A slider crank mechanism with L/R ratio less than unity can not be made.

Is Scotch yoke mechanism a closed pair or open pair?

 Is Scotch yoke mechanism a closed pair or open pair?

Is Scotch yoke mechanism a closed pair or open pair? First let us see what is a closed kinematic pair and what is an open kinematic pair.

Closed kinematic pair:

Closed kinematic pairs are those pairs in which elements are held together mechanically.  All the lower pairs and a few higher pairs fall in the category of closed pairs.

Open kinematic pair:

Open kinematic pairs maintain relative positions only when there are some external means to prevent separation of contacting elements. Typical example of an open kinematic pair is a cam and follower, held in contact due to spring force.

In the Scotch yoke mechanism shown above, all the pairs are lower pairs. Hence a Scotch yoke mechanism is a Closed kinematic pair.

Principle of Virtual work - Beams

Principle of Virtual work - Beams 

Two beams AC and CF of spans 4 m and 6 m respectively are hinged at C and supported at A, D and F. The beams are loaded as shown in Fig. Using the principle of virtual work, find the reaction at D.

Now let us give a virtual displacement of delta c at C to find reaction at D.