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.