Analytical approach for Kinematic Analytical approach for Kinematic Analysis of MechanismAnalysis of Mechanism

By

Shivaprasad.P

Introduction:Introduction:

� The analysis using graphical approach aresuitable for finding out velocity acceleration ofthe link mechanism in one or two position.

� The process of finding velocity and accelerationat various configuration by graphical methodat various configuration by graphical methodbecomes a tedious task .

� Hence possible alternative derivation ofanalytical expression using general parameters.

� These expressions can then be coded using anyprogramming language and executed.

BackgroundBackground� The principal goal of kinematic analysis is to determine the

accelerations of all the moving parts in the assembly.

� Dynamic forces are proportional to acceleration, from Newton's second law.

� We need to know the dynamic forces in order to calculate the stresses in the components.

� Thus the stresses in the materials must be kept well below � Thus the stresses in the materials must be kept well below allowable levels. To calculate the stresses, we need to know the static and dynamic forces on the parts.

� To calculate the dynamic forces, we need to know the accelerations.

� In order to calculate the accelerations, we must first find the positions of all the links or elements in the mechanism for each increment of input motion, and then differentiate the position equations versus time to find velocities, and then differentiate again to obtain the expressions for acceleration.

MethodsMethods

� Vector method.

� Absolute Cartesian method .

Vector method:Vector method:

� Description of Vector:

A vector r has two components in the (x,y) plane: rx and ry .

� It is important for the sake of consistencyand to formalize the analysis to adopt aconvention for the positive anglescharacterizing vector directions.characterizing vector directions.

� It is taken that the positive angle ‘α’ isalways directed counterclockwise and ismeasured from the positive direction ofthe x-axis.

Vector loop:Vector loop:

•If one starts from point O and continues through points A and B, one will come

back again to O.

•The system of vectors thus makes a loop.

•The mathematical representation of this requirement is that the sum of all

vectors in the loop must be equal to zero.

0coscoscos 332211 =Θ+Θ+Θ rrr

The above equation can be segregated in to two equations

0sinsinsin 332211 =Θ+Θ+Θ rrr

Example: 4 bar mechanismExample: 4 bar mechanism

� Vector loop equation :

�

� Rewriting the eqn :

� Since θ1 = 0

� Similarly,

Complex Numbers as vectorsComplex Numbers as vectors

Advantages over graphical methodAdvantages over graphical method

� If we choose to use the graphical approach to analysis,we will have to do an independent graphical solution foreach of the positions of interest.

� None of the information obtained graphically for thefirst position will be applicable to the second positionor to any others.or to any others.

� The analytical solution is derived for a particular mechanism, it can be quickly solved for all positions.(by coding)

� Reference:

� Robert.L.Norton, DESIGN OF MACHINERY: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, McGraw Hill Inc, 1999,Second Machines, McGraw Hill Inc, 1999,Second edition.

� Oleg Vinogradov, FUNDAMENTALS of KINEMATICS and DYNAMICS of MACHINES and MECHANISMS, CRC-Press, 2000,First Edition.