23-Mechatronics-A5 Mechanical Design, December 2019, Q1

Question 1 (20 marks: (a) 5, (b) 10, (c) 5)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Link census and Gruebler's equation

Given. The planar linkage below has three grounded pivots, one 4–way pin joint where four bars meet, one 3–way pin joint where three bars meet, and one pin–in–slot (higher pair) engagement, explicitly labelled on the source drawing.

Find. The number of binary (B), ternary (T), quaternary (Q) and pentagonal (P) links, and the mobility (DOF) by the Gruebler–Kutzbach equation.

Fig. 1(a) — digitised link graph. Every circle is a revolute pin; a plain junction of ≥3 lines with no shaded body is a multiple joint (several distinct binary links sharing one pin), while the hatched quadrilateral is one rigid (binary) plate. The labelled roller/capsule pair is the pin–in–slot half joint.

Approach. Count links by how many pin locations each rigid body owns, decompose every multiple joint into simple joints, then apply DOF = 3(n−1) − 2j₁ − j₂.

Classify each link. Ground carries three pivots (G₁ for link 2, G₂ for link 3, G₃ for the slotted rocker) → ground is ternary. The hub N₀ is a bare pin where four separate binary rods meet (link 2 from G₁, and three rods to P₁, P₁₀, P₁₁) — a 4–way multiple joint, not a link. Point P₁₁ is a 3–way multiple joint (rods to N₀, to ground G₂₂, and to the plate). The shaded plate is one rigid body with two functional pins (binary). Every other rod (N₀–P₁, P₁–slot pin, slot rocker G₃–slot, N₀–P₁₀, P₁₀–plate, N₀–P₁₁, P₁₁–G₂₂, P₁₁–plate) is a simple binary rod. $$n = 1\ (\text{ground, ternary}) + 10\ (\text{binary rods}) = 11 \text{ links}$$
Tabulate link types.

Type → B (binary) T (ternary) Q (quaternary) P (pentagonal)

Number → 10 1 (ground) 0 0

Check: $10+1+0+0 = 11 = n$. ✓
Count full joints $j_1$, resolving every multiple joint as (links meeting $-$ 1) simple pins. $$j_1 = \underbrace{3}_{N_0,\ 4\text{-way}} + \underbrace{1}_{G_1} + \underbrace{1}_{P_1} + \underbrace{1}_{G_3} + \underbrace{1}_{N_0P_{10}\,\text{pin}} + \underbrace{1}_{\text{plate top}} + \underbrace{2}_{P_{11},\ 3\text{-way}} + \underbrace{1}_{G_{22}} + \underbrace{1}_{\text{plate bottom}} = 12$$
Count the half joint $j_2$. One pin–in–slot engagement (a round pin sliding and rotating in a straight slot removes only one relative DOF): $j_2 = 1$.
Apply Gruebler–Kutzbach. $$\boxed{\text{DOF} = 3(n-1) - 2j_1 - j_2 = 3(10) - 2(12) - 1 = 30-24-1 = 5}$$

Type →	B (binary)	T (ternary)	Q (quaternary)	P (pentagonal)
Number →	10	1 (ground)	0	0

Quantity	Value
Links $n$ (B, T, Q, P)	11 (10, 1, 0, 0)
Full joints $j_1$	12
Half joints $j_2$	1 (pin-in-slot)
Degrees of freedom	$\boxed{\text{DOF}=5}$

The method (classify links by pin count, decompose multiple joints, apply Gruebler–Kutzbach with a half joint for the pin-in-slot) is the examinable content and is exact.

Topic: Mobility of planar linkages — Gruebler–Kutzbach equation.

Key relations: $\text{DOF}=3(n-1)-2j_1-j_2$; a pin shared by $k$ links is worth $(k-1)$ simple full joints; a full (lower) pair removes 2 DOF, a half (higher) pair such as a pin-in-slot or cam contact removes only 1.

Why this works: each unconnected rigid body in the plane has 3 DOF (2 translation + 1 rotation); the ground removes 3 immediately (folded into the $n-1$ term); every full joint then removes 2 of the remaining freedoms and every half joint removes only 1, because a half joint still permits one relative sliding/rolling freedom in addition to the pin's rotation.

Common pitfall: treating a point where three or more bars are pinned together as a single simple joint (undercounting $j_1$), or treating a rigid, shaded, multi-cornered plate as if each of its corners were an independent link.

Source: Norton, Design of Machinery, Ch. 2 (mobility, Gruebler's equation, multiple joints).

Question 1 of 10

Question 1 (20 marks: (a) 5, (b) 10, (c) 5)

(a) Link census and Gruebler's equation