This document summarizes topics related to kinematics and theory of machines, including Grashof's law for 4-bar linkages, types of inversions (double crank and double rocker mechanisms), and Rayleigh's method for analyzing single degree of freedom vibrating systems. It provides examples and step-by-step explanations of applying Rayleigh's method to find the natural frequency of a spring-mass system undergoing simple harmonic motion. The document was submitted by a mechanical engineering student as part of a class assignment on these topics.

1. Dr B C Roy Engineering College -
Durgapur
Kinematics & Theory Of Machines, PC-ME503
Topic-A – Grashof’s 4R chain and inversions Criteria and Features
Topic-B – Application of Rayleigh’s method for formulate equation of
motion of a single DOF vibrating system
Shyamlal Mandy
University roll no – 12000721068
Sec – ME2
Department – Mechanical
Date Of Submission – 27/07/2022
CA - 1
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2. Content
•Grashof’s law
•Double Crank Mechanism
•Double Rocker Mechanism
•Rayleigh’s Method
•Application of Rayleigh’s Method
•Example Of Single DOF vibrating System
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3. Grashof’s Law states that for a four-bar linkage system,
if the sum of length of shortest and longest of a planar
quadrilateral linkage is less than or equal to the sum of
the remaining two links , then the shortest link can
rotate freely with respect to neighbouring link.
In a four bar chain there are four turning pairs and no
sliding pairs.
Let denote the smallest link of four bar linkage with S
and the longest link by L and the other two links by P
and Q.
The necessary condition to satisfy Grashof’s Law is :
S + L ≤ P + Q
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4. Different links one at a time this case produces three
mechanisms . These mechanisms are :-
i) Double Crank Mechanism
ii) Double Rocker Mechanism
iii) Crank and Rocker Mechanism
i) Double Crank Mechanism
:-
In double crank mechanism,
the shortest link which is S
is fixed or grounded. In this
mechanism, both the links
pivoted to the fixed link can
rotate 360 degrees.
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5. ii) Double Rocker Mechanism :-
In double rocker mechanism, the link opposite
to shortest link is fixed or grounded. In this
mechanism the shortest link can rotate 360
degrees. Shortest link is called coupler. Both
the links pivoted to the fixed links can
oscillate. These two links are called rockers.
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6. Rayleigh's method requires an assumed
displacement function. The method thus
reduces the dynamic system to a single-
degree-of-freedom system. Furthermore, the
assumed displacement function introduces
additional constraints which increase the
stiffness of the system. Thus, Rayleigh's
method yields an upper limit of the true
fundamental frequency.
Definition Rayleigh's method can be
summarized as
( KE)max =(PE)max = total energy of the
system (1)
where KE = kinetic energy
PE = potential energy
Rayleigh’s Method
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7. potential energy is also referred to as strain energy for
the case of certain systems, such as beams. Equation
(1) can only be satisfied if the system is vibrating at its
natural frequency.
When spring force (and not gravity) is the restoring
force causing oscillation, we can ignore gravity
effects and consider oscillations about the
equilibrium position. In these cases, the frequency of
oscillation is independent of g.
•Define an origin of coordinates for x at the
equilibrium position and assume x = Asin(wt).
Application of Rayleigh’s Method
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8. • Find expressions for the kinetic energy and the
elastic potential energy as a function of x and ˙x.
• Find Tmax , Vmin and Vmax .
• Use Tmax = Vmax − Vmin to find the frequency (A
should drop out of the equation)
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9. Reference
• Theory Of Machines, SS Ratten
• Kinematics & Theory of Machines, Sadhu Singh
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