--- title: "Degree of freedom and Gruebler's criterion" subject: "Theory of Mechanism and Machine I" chapter: 1 key: "semester_6/theory_of_mechanism_and_machine_i:degree-of-freedom-gruebler-mobility" source: https://www.bctengineer.com/q/semester_6/theory_of_mechanism_and_machine_i/degree-of-freedom-gruebler-mobility site: BCT Engineering --- # Degree of freedom and Gruebler's criterion Theory of Mechanism and Machine I · Chapter 1: Introduction - Asked in: year not given, 3+5 marks - Video: [Degree of Freedom examples using Grubler's Criteria](https://www.youtube.com/watch?v=p_YBWq0nHWc) — CLARI CONCEPTS - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/theory_of_mechanism_and_machine_i/degree-of-freedom-gruebler-mobility - Whole chapter: https://www.bctengineer.com/md/semester_6/theory_of_mechanism_and_machine_i/chapter-1 ## Question Define degree of freedom of a mechanism. State Gruebler's criterion for planar mechanisms. Find the mobility of (i) a planar mechanism having 7 links (including the frame), 8 turning pairs and 1 cam-follower higher pair, and (ii) a six-link planar chain having 7 turning pairs. ## Short answer Degree of freedom is the number of independent inputs needed to fix all link positions. Gruebler's criterion for planar mechanisms: F = 3(n-1) - 2j1 - j2. For (i) n=7, j1=8, j2=1: F = 3(6) - 16 - 1 = 1. For (ii) n=6, j1=7, j2=0: F = 3(5) - 14 = 1. Both have one DOF. ## Answer #### Degree of freedom The **degree of freedom (DOF)** or mobility of a mechanism is the number of independent input parameters (coordinates) that must be given to fix the position of every link with respect to the frame. A mechanism with $F = 1$ needs one input (for example one motor) to give a definite motion. #### Gruebler's criterion For a planar mechanism, a free link has 3 DOF. Fixing one link as the frame removes 3 DOF. Each lower pair (turning or sliding, 1 DOF) removes 2 DOF and each higher pair removes 1 DOF. So $$ F = 3(n-1) - 2j_1 - j_2 $$ where $n$ = total number of links (including the frame), $j_1$ = number of lower pairs (single-DOF), $j_2$ = number of higher pairs (two-DOF). - $F = 1$: constrained mechanism (definite motion). - $F \le 0$: structure (no motion). - $F \ge 2$: needs $F$ inputs, otherwise motion is indefinite. #### Numerical (i) Given $n = 7$, $j_1 = 8$, $j_2 = 1$. $$ F = 3(7-1) - 2(8) - 1 = 18 - 16 - 1 = 1 $$ #### Numerical (ii) Given $n = 6$, $j_1 = 7$, $j_2 = 0$. $$ F = 3(6-1) - 2(7) = 15 - 14 = 1 $$ This is the Watt or Stephenson six-bar chain, which has one DOF. **Answer: (i) $F = 1$, (ii) $F = 1$; both mechanisms have definite motion with one input.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does the degree of freedom of a mechanism tell us? A. The number of links in the mechanism B. The number of independent inputs needed to fix the position of every link relative to the frame C. The number of pairs in the mechanism D. The number of fixed links Answer: B. DOF is the number of independent input coordinates needed to fix all link positions with respect to the frame. A mechanism with F = 1 needs one input such as one motor. (Key point: Meaning of DOF) 2. Which expression is Gruebler's criterion for planar mechanisms (n links including frame, j1 lower pairs, j2 higher pairs)? A. F = 3n - 2j1 - j2 B. F = 3(n-1) - j1 - 2j2 C. F = 3(n-1) - 2j1 - j2 D. F = 2(n-1) - 3j1 - j2 Answer: C. A free planar link has 3 DOF and the frame removes 3, so 3(n-1). Each lower pair removes 2 DOF and each higher pair removes 1. (Key point: Gruebler's equation) 3. A planar mechanism has 7 links, 8 turning pairs and 1 cam-follower higher pair. What is its mobility? A. 0 B. 2 C. 3 D. 1 Answer: D. F = 3(7-1) - 2(8) - 1 = 18 - 16 - 1 = 1, so it has definite motion with one input. (Key point: Mobility of 7-link mechanism) 4. A six-link planar chain has 7 turning pairs. Its mobility is: A. 1 B. 2 C. 0 D. -1 Answer: A. F = 3(6-1) - 2(7) = 15 - 14 = 1, as for the Watt or Stephenson six-bar chain. 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