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Assignment 3

This assignment consists of four sets with specific questions that students must complete, with all students required to solve Question 3. The assignment involves plotting solution trajectories, determining frequency-amplitude relationships for various equations, and deriving governing equations for a pendulum system. The deadline for submission is February 19, 2025.

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Harsh Kumar
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19 views2 pages

Assignment 3

This assignment consists of four sets with specific questions that students must complete, with all students required to solve Question 3. The assignment involves plotting solution trajectories, determining frequency-amplitude relationships for various equations, and deriving governing equations for a pendulum system. The deadline for submission is February 19, 2025.

Uploaded by

Harsh Kumar
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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ASSIGNMENT 3 (50 marks)

Note:
• There shall be four sets for this assignment. Students shall be allotted ONLY ONE of
these sets based on their choice as per instructions from the course instructor.
• Questions 3 must be solved by all students.
• Submit in the forum specified by the course instructor.
• Deadline: 19 February 2025.

Question 1: For your respective set, plot the solution trajectories in the phase plane and
indicate the singular (fixed) points, their types and separatrices. [15 marks]

Set A: 𝑢̈ + 𝑢 − 𝑢3 = 0

Set B: 𝑢̈ − 𝑢 + 𝑢3 = 0

Set C: 𝑢̈ − 𝑢 − 𝑢3 = 0

Set D: 𝑢̈ + 𝑢 + 𝑢3 = 0

Question 2: Determine a two-term expansion for the frequency amplitude relationship


for systems governed by the following equations. [15 marks]

Set A: 𝑢̈ − 𝑢 + 𝑢3 = 0

Set B: 𝑢̈ + 𝜔02 𝑢 + 𝑢|𝑢| = 0

Set C: 𝑢̈ + 𝜔02 𝑢(1 + 𝑢2 )−1 = 0

Set D: 𝑢̈ + 𝜔02 𝑢 + 𝛼𝑢5 = 0


NONLINEAR VIBRATIONS (ME613)@IITG, JAN–MAY 2025,
ASSIGNMENT 3

Question 3: As shown in Fig. 1, a rigid rod is attached to an axle. The wheels roll without
slip as the pendulum swings back and forth. Only the ball at the end of the pendulum has
appreciable mass and it may be considered as a particle.
(a) Derive the governing equation for 𝜃 (5 marks)
(b) For small but finite motions, determine a two-term approximate frequency-
amplitude relationship. (15 marks)

Fig. 1

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