Kinematic pairs according to the type of closure

 Kinematic pairs according to the type of closure:

According to the type of closure between the elements, kinematic pairs can be classified as 1. Self closed pair, 2. Force closed pair

1. Self closed pair: 

In a self closed pair, two elements of the pair are mechanically connected in such a way that only specified relative motion occurs

All lower pairs are self closed pairs

2. Force closed pair: 

In a force closed pair, two elements of the pair are not connected mechanically but are kept in contact by the action of external forces.

A cam and follower is an example of a force closed pair, as the cam and follower are kept in contact by the force exerted by spring

Kinematic link or element

 Kinematic link or Element 

Each part of a machine, which moves relative to other parts is called a kinematic link or element.
A kinematic link or element may consist of a single part or several parts rigidly connected together, so that they do not move relative to each other. For example, in a reciprocating steam engine, piston, piston rod and cross-head constitute one link ; connecting rod with big and small end bearings constitute a second link ; crank, crankshaft and flywheel a third link and the cylinder, engine frame and main bearings a fourth link.

A kinematic link need not necessarily be rigid but it must necessarily be resistant. A resistant body is one which transmits force with negligible deformation in the direction of force transmission. Based on this consideration, a spring which has no effect on the kinematics of a device and has significant deformation in the direction of applied force, is not treated as a link but only as a device to apply force. Springs are usually ignored during kinematic analysis and their force effects are introduced during dynamic analysis.

There are some machine parts which possess one – way rigidity. For instance, because of their resistance to deformation under tensile load, belts, ropes and chains are treated as links only when they are in tension. 

Similarly, liquids on account of their incompressibility can be treated as links only when transmitting compressive force. 

Differences between a structure and a machine

 Differences between a structure and a machine

Structure: 

A structure is an assemblage of a number of resistant bodies / members having no relative motion between them and meant for carrying loads having straining action. 

A railway bridge, a roof truss, machine frames etc., are some of the examples of a structure.

Machine:

A machine is a mechanism or a combination of mechanisms, apart from imparting definite motion to the bodies / members also transmit and modify available mechanical energy into desired work.

Internal combustion engine, Shaping machine etc. Are some of the examples of a machine

Differences between a structure and a machine are listed below.

1. The parts of a machine move relative to one another, whereas the members of a structure

do not move relative to one another.

2. A machine transforms the available energy into some useful work, whereas in a structure

no energy is transformed into useful work.

3. The links of a machine may transmit both power and motion, while the members of a

structure transmit forces only.

Mechanism, Machine and Theory of Machines

 Mechanism, Machine and Theory of Machines

Mechanism: 

If a number of bodies / parts are assembled in such a way that the motion of one body / part causes constrained and predictable motion of other bodies / parts is called a Mechanism. So a mechanism transmits and modifies motion.

 Machine: 

A machine is a mechanism or a combination of mechanisms, apart from imparting definite motion to the bodies / parts also transmits and modifies the available mechanical energy into desired work.

 Theory of Machines: 

The subject comprises study of kinematics and dynamics of machines is called Theory of Machines.

The study of relationship between the geometry of machine parts and their relative motion is called Kinematics.

The study of forces, which produce the relative motion between the parts is called Dynamics.

Dynamics further be divided into Statics and Kinetics. Statics deals with the forces that act on the machine parts which are assumed to be mass-less. Where as Kinetics deals with the inertia forces arising out of the combined effect of mass and motion of the parts.

Four bar mechanism - Coupler point analysis - Kinematic analysis

 Four bar mechanism - Coupler point analysis - Kinematic analysis

In the below picture shown a typical four bar mechanism (Crank rocker), L2 is crank length, L3 coupler length, L4 follower length and L1 is fixed link (frame) length. Angle between crank and horizontal axis is theta 2, Point E is a point on the coupler link. Length BE is e, angle is theta form BC.

In this blog I describe the procedure to derive equations to find position, velocity and acceleration of point E on the coupler.  

As a first step, four bar linkage ABCD is analyzed to find angles theta 3 and theta 4 in the below pictures.

Omega 3 is the angular velocity of coupler, omega 4 the angular velocity of follower link and alpha 3 is the angular acceleration of the coupler link. Equations are written below.

Check the below links for angular velocity ang angular acceleration equations.

https://www.kinematics-mechanisms.com/2022/05/four-bar-linkage-four-bar-mechanism_27.html

https://www.kinematics-mechanisms.com/2022/05/four-bar-linkage-four-bar-mechanism_65.html

To find velocity of the coupler point E, differentiate the position equations with respect to time once and follow the procedure shown in the below pictures.

To find acceleration of the coupler point E, differentiate velocity equations with respect to time once and follow the procedure shown in the below pictures.

After having derived the necessary equations, a problem is solved using Microsoft excel. Various dimensions are shown in the below picture.

Swiveling joint mechanism - Kinematic analysis (Position analysis)

 Swiveling joint mechanism - Kinematic analysis (Position analysis)

In the below picture shown is a swiveling joint mechanism. Lengths of various links are shown. In this blog equations derived to find position of the slider with respect to the crank angle theta 2.

As a first step, in the below pictures four bar mechanism ABCD is analyzed. Equations to find coupler angle theta 3 and follower angle theta 4.

In the below pictures described the procedure to find angle of link EFG (theta 11) and length L12.

In the below pictures shown the procedure to find angle of the link GH and position of the slider

After having derived the necessary equations, a problem is solved using Microsoft excel.

Email me if you want to have the excel spread sheet used in the above pictures. abu.adam1178@gmail.com

Circular disc cam with flat faced follower - Force analysis

 Circular disc cam with flat faced follower - Force analysis

In the below picture shown is a disc cam with a flat faced follower, Torque on the cam is T Nm (Clock wise) and force on the flat faced follower is F. Radius of the cam is R and radius of base circle is r. In this blog, a formula is derived to establish the relation between the above mentioned parameters for any given angle theta and a problem is also solved using Microsoft excel. (Note: friction between cam and follower is neglected.)

In the below picture, a problem is solved using Microsoft excel. In this problem R = 20 mm, r = 10 mm, T = 15 Nm and angle theta is taken from zero to 360 degrees at an interval of 15 degrees.

In the below picture, cam angle theta versus follower force F  is plotted. Theta along horizontal axis and Follower force F along vertical axis.

Inversions of a single slider crank mechanism

 Inversions of a single slider crank mechanism

Mechanism is a kinematic chain in which one link is fixed. By fixing the links of a kinematic chain one at a time, we get as many different mechanisms as the number of links in the chain. This method of obtaining different mechanisms by fixing different links of the same kinematic chain, is called as inversion of the mechanism.

In the process of inversion, the relative motions of the links of the mechanism produced remain unchanged.

Single slider crank mechanism: Single slider crank mechanism is the modification of the basic four bar chain. Single slider crank mechanism consists of four kinematic pairs, out of which one is sliding pair and rest three are turning pairs.

In the below picture shown is a typical single slider crank mechanism, in which link 1 and link 2 form one turning pair, link 2 and link 3 form second turning pair, link 3 and slider form a third turning pair and link 1 and slider form the sliding pair.

Inversions of Single slider crank chain: If different links of the single slider crank chain are fixed in turn four different mechanisms (called inversions) will be obtained.

First inversion: 

When link 1 is fixed, link 2 is made crank and link 4 is made slider, then the first inversion of a single slider crank chain is obtained.

This inversion is used in reciprocating engine and reciprocating compressor.

Second inversion: 

When link 2 is fixed, the second inversion of the single slider crank chain is obtained.

This inversion is used in Whitworth quick return mechanism and rotary engine.

Third inversion: 

When link 3 is fixed, the third inversion of single slider crank is obtained.

This inversion is used in oscillating cylinder engine and crank and slotted lever mechanism.

Fourth inversion: 

When link 4 is fixed, the fourth inversion of single slider crank chain is obtained.

This inversion is used in hand pump.

Saber saw displacement, velocity and acceleration analysis - Saber saw kinematic analysis

 Saber saw displacement, velocity and acceleration analysis

Saber saw kinematic analysis:

In the below picture shown is a saber saw, in which an offset slider crank mechanism is used to operate the saw blade. The crank wheel is driven by an electric motor, not shown in the below picture. Crank wheel radius is L2, coupler link length is L3, offset is e and crank angle is theta2.

The offset slider crank mechanism is redrawn upside down in the below picture. Angle between vertical and crank is theta2 and angle between vertical and coupler is theta3. Angular velocity of the crank is omega2 and angular acceleration of the crank is alpha2.

In the below pictures described the procedure to derive equations for blade displacement, blade velocity and blade acceleration.

In the below picture described the procedure to derive equation to find angular velocity and angular acceleration of the coupler link.

In the below picture, a problem is solved using Microsoft excel. In this problem crank length L2 is 12.7 mm, coupler length L3 = 44.45 mm, offset e = 25.4 mm, RPM of the crank is 900 and angular acceleration of the crank is zero.

In the below picture Crank angle theta2 is plotted along horizontal axis and along vertical axis displacement of saber saw blade.

In the below picture Crank angle theta2 is plotted along horizontal axis and along vertical axis velocity of saber saw blade.

In the below picture Crank angle theta2 is plotted along horizontal axis and along vertical axis acceleration of saber saw blade.

Slider crank force analysis - problem (Inline slider crank mechanism)

 Slider crank force analysis - problem

In this blog Inline slider crank mechanism static force analysis equations derived using analytical method. 

In the below picture shown is an inline slider crank mechanism, crank length is L2, coupler length is L3 and angle between crank and horizontal is theta 2. There is torque T is applied on the crank clockwise, due to this torque there will be a resting load F on the piston. 

In the below picture shown the forces and reactions on the mechanism.

In the below pictures described procedure to find relationship between applied torque T and resisting force F.

After having derived the equation, a problem is solved using Microsoft excel. In this problem, crank length L2 = 100 mm, coupler length L3 = 400 mm and a torque of 50 Nm is applied on the crank clockwise. For crank angle zero to 90 degrees force on piston F is calculated and plotted in the below pictures.