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Modified Gruebler's Equation for Kinematics Calculator

Mechanics and Machine Design, Equations and Calculators

Modified Gruebler's Equation for Kinematics Calculator

For a planar mechanism, the degree of freedom (mobility) is given by Gruebler's Equation:

In kinematics the number of mobile links is:

Gruebler's Equation - degrees of freedom (mobility)
F = 3(n – 1) – 2 f₁

Modified Gruebler's Equation - degrees of freedom (mobility)
F = 3(n – 1) – 2 f₁ - 1 f₂

where
f₁ = Total number of pin joints (some joints count as f = ½, 1, 2, or 3)
f₂ = number of roll-slide contact joints
n = Total number of links (including a fixed or single ground link)

For example,

Example 1: Slider-crank n = 4, f₁ = 4, F = 1

Figure 1 - 4 links (including ground), 4 joints

Example 2: 4-Bar linkage n = 4, f₁ = 4, F = 1

Figure 2 - 4 links, 4 joints

Source:

• Erdman, Arthur G., George N. Sandor, and Sridhar Kota, Mechanism Design, Vol. 1, 4th ed.,
  New York: Prentice Hall, Inc.:
• MIT, Mechanical Design Contest Presentation

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