Question

A cuboidal part has to be accurately positioned first, arresting six degrees of freedom and then clamped in a fixture, to be used for machining. Locating pins in the form of cylinders with hemi-spherical tips are to be placed on the fixture for positioning. Four different configurations of locating pins are proposed as shown. Which one of the options given is correct? 

 

• A. Configuration P1 arrests 6 degrees of freedom, while Configurations P2 and P4 are over-constrained and Configuration P3 is under-constrained.
• B. Configuration P2 arrests 6 degrees of freedom, while Configurations P1 and P3 are over-constrained and Configuration P4 is under-constrained.
• C. Configuration P3 arrests 6 degrees of freedom, while Configurations P2 and P4 are over-constrained and Configuration P1 is under-constrained.
• D. Configuration P4 arrests 6 degrees of freedom, while Configurations P1 and P3 are over-constrained and Configuration P2 is under-constrained.

Hint:

For fixturing problems, always look for the 3-2-1 principle. Three locators define a plane, two define a line, and one defines a point. Any more than this on the respective faces usually leads to over-constraint. For example, four points on the primary plane (like P4) is a very common example of over-constraint.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
To uniquely and repeatably position a rigid body in space, all six of its degrees of freedom (DOF) must be constrained. The six DOF are three translations (along X, Y, Z axes) and three rotations (about X, Y, Z axes). The standard method for achieving this for a prismatic part is the 3-2-1 principle of location. - Under-constrained: Fewer than 6 DOF are arrested. The part can still move. - Fully constrained: Exactly 6 DOF are arrested. The part is stable and correctly located. - Over-constrained: More locators are used than necessary, leading to redundancy. This can cause the part to not seat properly or be deformed.
Step 2: Analyzing the Configurations:
- Configuration P1: Has 3 pins on the bottom face and 1 on each of two adjacent side faces. This totals 5 locating points. This configuration fails to arrest the rotation about the axis perpendicular to the bottom face (the Z-axis). It is under-constrained.
- Configuration P2: Has 3 pins on the bottom, 2 on one side, and 2 on the adjacent side (3-2-2 principle). This uses 7 locators. The second pin on the second side is redundant for locating the part along the X-axis and creates a conflict. This is over-constrained.
- Configuration P3: Has 3 pins on the bottom face, 2 pins on one side face, and 1 pin on an adjacent side face. This is the classic 3-2-1 principle. The 3 base pins constrain Z-translation and X/Y rotations. The 2 side pins constrain Y-translation and Z-rotation. The final pin constrains X-translation. This configuration correctly arrests all 6 DOF and is fully (or correctly) constrained.
- Configuration P4: Has 4 pins on the bottom face. Locating a plane on four points is a classic example of over-constraint. Due to manufacturing tolerances, the part will likely rest on only three of the four pins (a "rocking" condition), making its position unstable and non-repeatable.
Step 3: Evaluating the Options:
Based on our analysis:
- P1 is under-constrained.
- P2 is over-constrained.
- P3 arrests 6 DOF (fully constrained).
- P4 is over-constrained.
Let's check the options:
(A) Incorrect. Claims P1 arrests 6 DOF.
(B) Incorrect. Claims P2 arrests 6 DOF.
(C) Claims P3 arrests 6 DOF (Correct), P2 and P4 are over-constrained (Correct), and P1 is under-constrained (Correct). This statement is fully consistent with our analysis.
(D) Incorrect. Claims P4 arrests 6 DOF.
Step 4: Final Answer:
The correct statement is given in option (C).
Step 5: Why This is Correct:
Option (C) correctly identifies that Configuration P3 follows the 3-2-1 principle to properly arrest all 6 degrees of freedom. It also correctly classifies P1 as under-constrained due to an insufficient number of locators, and P2 and P4 as over-constrained due to redundant locators.