This document discusses mechanical mechanisms and Grashof's law for calculating the degrees of freedom of linkages. It provides Grubler's equation for determining degrees of freedom as a function of the number of links (L), number of pivot joints (PL), and fixed joints (PH). An example applies this equation to a 4-bar linkage with 1 degree of freedom. The document also covers Grashof's criteria for classifying 4-bar linkages as single crank, crank-rocker, double rocker or others based on the relative lengths of the links. Various examples of 4-bar configurations are presented.

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MECHANICAL
MECHANISMS
(MAN 311) (MEC 360)
Dr. Ahmed Sobhi
“Sp 2023”
Lecture (2)
GRASHOF'S LAW
• Grubler’s Equation
𝑫𝑶𝑭 = 𝟑 × 𝑳 − 𝟏 − 𝟐 × 𝑷𝑳 − 𝑷𝑯
1. Identify the frame
2. Identify all other links
3. Identify joints
4. Identify any point of interest
5. Draw Kinematic diagram
6. Calculate the mobility
Degree of freedom (mobility)

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𝑫𝑶𝑭 = 𝟑 × 𝑳 − 𝟏 − 𝟐 × 𝑷𝑳 − 𝑷𝑯
𝑳 : 4
𝑷𝑳 : 4
𝑷𝑯 : 0
𝑫𝑶𝑭 = 𝟑 × 𝟑 − 𝟐 × 𝟒 − 𝟎
𝑫𝑶𝑭 = 𝟏
Degree of freedom (mobility)
Example
• Actuators and drivers
Linear motion that
usually generated
using hydraulic
power
Degree of freedom (mobility)

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𝑫𝑶𝑭 = 𝟑 × 𝑳 − 𝟏 − 𝟐 × 𝑷𝑳 − 𝑷𝑯
Example (3):
Degree of freedom (mobility)
𝑫𝑶𝑭 = 𝟑 × 𝑳 − 𝟏 − 𝟐 × 𝑷𝑳 − 𝑷𝑯
Example (3):
1. Identify the frame (1)
2. Identify all other links (2), (3) and (4)
3. Identify joints (A), (B), (C) and (D)
4. Identify any point of interest (X)
(1)
(2)
(3)
(4)
(A)
(B)
(D)
(C)
(X)
Degree of freedom (mobility)

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𝑫𝑶𝑭 = 𝟑 × 𝑳 − 𝟏 − 𝟐 × 𝑷𝑳 − 𝑷𝑯
Example (3): (1)
(2)
(3)
(4)
(A)
(B)
(D
(C)
𝑳 : 4
𝑷𝑳 : 4
𝑷𝑯 : 0
𝑫𝑶𝑭 = 𝟑 × 𝟑 − 𝟐 × 𝟒 − 𝟎 𝑫𝑶𝑭 = 𝟏
5. Draw Kinematic diagram
6. Calculate the mobility
(X)
Degree of freedom (mobility)
OTHER EXAMPLES

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The 4-Bar link Mechanism (single DOF)
There are many examples for 4-Bar link mechanism
Degree of freedom (mobility)
The 4-Bar link Mechanism (single DOF)
There are many examples for 4-Bar link mechanism
Degree of freedom (mobility)

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Draw the kinematic diagram and applied Grubler
equation to find the mobility.

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Grashof’s Criteria of 4-Bar link Mechanism
The following nomenclature is used to describe the length of
the four links.
s: length of the shortest link
l: length of the longest link
p: length of one of the intermediate length links
q: length of the other intermediate length links
Use Linkage Software to check all cases
of 4-Bar link mechanism
Grashof’s Criteria of 4-Bar link Mechanism

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Grashof ’s theory states that a four-bar mechanism has at
least one revolving (Crank) link if:
Conversely, the three non-fixed links will merely rock if:
Grashof’s Criteria of 4-Bar link Mechanism
Grashof’s Criteria of 4-Bar link Mechanism

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• Double Crank (Crank- Crank)
S
L
p
q
Grashof’s Criteria of 4-Bar link Mechanism
• Crank-Rocker
S
L
p
q
Grashof’s Criteria of 4-Bar link Mechanism

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• Double Rocker (Rocker-Rocker)
S
L
p
q
Grashof’s Criteria of 4-Bar link Mechanism
• Change Point
S
L
p
q
Grashof’s Criteria of 4-Bar link Mechanism

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• Change Point (Crank-Crank)
S
L
p
q
With momentum
Grashof’s Criteria of 4-Bar link Mechanism
• Triple Rocker
S
L
p
q
Grashof’s Criteria of 4-Bar link Mechanism

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Thanks for Attention