Planar Mechanisms


Question 1
In the configuration of the planar four-bar mechanism at a certain instant as shown in the figure, the angular velocity of the 2 cm long link is \omega _2 =5rad/s. Given the dimensions as shown, the magnitude of the angular velocity \omega _4 of the 4 cm long link is given by _____ rad/s (round off to 2 decimal places).

A
1.25
B
0.75
C
2.25
D
3.75
GATE ME 2022 SET-2   Theory of Machine
Question 1 Explanation: 


Number of links = 4
Number of I-centers 4c_2=6 \omega _2=5 \; rad/sec
\omega _4=?
For
\begin{aligned} V_{I_{24}}&=I_{12}I_{24}\omega _{2}=I_{14}I_{24}\omega_4\\ &=2 \times 5 =8 \times \omega_4\\ \omega_4&=1.25 \; rad/sec \end{aligned}
Question 2
A planar four-bar linkage mechanism with 3 revolute kinematic pairs and 1 prismatic kinematic pair is shown in the figure, where AB\perp CE and FD\perp CE. The T-shaped link CDEF is constructed such that the slider B can cross the point D, and CE is sufficiently long. For the given lengths as shown, the mechanism is

A
a Grashof chain with links AG, AB, and CDEF completely rotatable about the ground link FG
B
a non-Grashof chain with all oscillating links
C
a Grashof chain with AB completely rotatable about the ground link FG, and oscillatory links AG and CDEF
D
on the border of Grashof and non-Grashof chains with uncertain configuration(s)
GATE ME 2022 SET-1   Theory of Machine
Question 2 Explanation: 
The given mechanism is

As we know sliding pair is a special. Case of turning pair with infinite lengths limit. So the equivalent diagram would be. Since two parallel lines meets at infinite point O, 2O_2 are same.

l_1=3 cm shortest line,
l_2=5cm
l_3=l_{3(\infty )}=L_x+3 , longest link
l_4=l_{4(\infty )}=L_x+1.5
For Grashoff's rule to satisfy
l_1+l_3 \leq l_2 +l_4
3+L_x+3 \leq 5+L_x+1.5
6 \leq 6.5
LHS is less than RHS.
Hence, Grashoff's rule is satisfied in this mechanism. Since shortest link is fixed. It will be a double crank mechanism.


Question 3
A rigid triangular body, PQR, with sides of equal length of 1 unit moves on a flat plane. At the instant shown, edge QR is parallel to the x-axis, and the body moves such that velocities of points P and R are V_P \; and \; V_R, in the x and y directions, respectively. The magnitude of the angular velocity of the body is
A
2V_R
B
2V_P
C
V_R/\sqrt{3}
D
V_P/\sqrt{3}
GATE ME 2019 SET-2   Theory of Machine
Question 3 Explanation: 
\begin{array}{l} \Rightarrow \mathrm{V}_{\mathrm{R}}=(\mathrm{IR}) \omega \\ \Rightarrow \omega=\frac{\mathrm{V}_{\mathrm{R}}}{(\mathrm{IR})} \\ \Rightarrow \omega \times \frac{\mathrm{V}_{\mathrm{R}}}{\frac{1}{2}} \\ \Rightarrow \omega=2 \mathrm{V}_{\mathrm{R}} \end{array}

Question 4
In a four bar planar mechanism shown in the figure, AB = 5 cm, AD = 4 cm and DC = 2 cm. In the configuration shown, both AB and DC are perpendicular to AD. The bar AB rotates with an angular velocity of 10 rad/s. The magnitude of angular velocity (in rad/s) of bar DC at this instant is
A
0
B
10
C
15
D
25
GATE ME 2019 SET-1   Theory of Machine
Question 4 Explanation: 
\begin{array}{l} \mathrm{AB}=5 \mathrm{cm} \\ \mathrm{AD}=4 \mathrm{cm} \\ \mathrm{DC}=2 \mathrm{cm} \\ \omega_{\mathrm{AB}}=10 \mathrm{rad} / \mathrm{s} \\ \because A \mathrm{B} \| \mathrm{DC} \\ \therefore A B \cdot \omega_{\mathrm{AB}}=\mathrm{DC} \cdot \omega_{\mathrm{DC}} \\ 5 \times 10=2 \times \omega_{\mathrm{DC}} \\ \omega_{\mathrm{DC}}=25 \mathrm{rad} / \mathrm{s} \end{array}
Question 5
In a slider-crank mechanism, the lengths of the crank and the connecting rod are 100mm and 160mm, respectively. The crank is rotating with an angular of 10 radian/s counter-clockwise. The magnitude of linear velocity (in m/s) of the piston at the instant corresponding to the configuration shown in the figure is_____.
A
2
B
1.5
C
1
D
0.5
GATE ME 2017 SET-2   Theory of Machine
Question 5 Explanation: 


After plotting I-centres

Here, I_{23} and I_{24} will come at same point
Applying angular Velocity Theorem at I_{24}
\begin{aligned} \therefore \quad \omega_{2}\left(I_{24} I_{12}\right) &=V_{4}=V_{B} \\ V_{B} &=\omega_{2}\left(I_{24} I_{12}\right)=10 \times 0.1 \\ \left(I_{24} I_{12}=100 \mathrm{mm}\right.&=0.1 \mathrm{m}=A B) \\ V_{B} &=1 \mathrm{m} / \mathrm{s} \end{aligned}


There are 5 questions to complete.

5 thoughts on “Planar Mechanisms”

  3. In question no 36, I think the answer should be B, bcoz cam and follower is the mechanism with DOF 1 having 3 links with 2 lower pairs & 1 higher pair.
    if in the problem they ask DOF 1 with lower pair only then minimum 4 links are required (from grablur’s eqn)

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