Assignment 1 - Indian Institute of Technology...
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K. R. Jayaprakash ES646: Elastodynamics and Vibrations Mechanical Engineering, IITGN 1 ASSIGNMENT 1 Term: Fall 2018 Date of posting: 10 September 2018, Submission deadline: 17:00, 24th September 2018 Instructions: i) Write all your assumptions (if any) clearly. ii) All questions are compulsory. iii) All questions carry equal marks. iv) All symbols have usual meaning unless otherwise stated. v) Assignments need to be submitted independently by each student. Repercussions of disobeying the institute academic honor code would be very serious. 1. Two uniform rigid bars each of mass and length are connected using a spring of stiffness as shown in Figure 3. Assume that the spring is unstretched when the bars are horizontal and consider all the hinges are frictionless. Derive the equations of motion for symmetric motion of the bars. Linearize the equation of motion about a stable equilibrium point and find the natural frequency for small oscillations. Discuss if the system can have asymmetric motion of the bars. Figure 1 2. A uniform disk of weight and radius rolls without sliding on a plane table. At its center it carries a hinge with a weightless pendulum of length and a concentrated weight at its end (Figure 2). Find the natural frequencies for motion in the plane of the paper. Figure 2
Transcript of Assignment 1 - Indian Institute of Technology...
K. R. Jayaprakash ES646: Elastodynamics and Vibrations Mechanical Engineering, IITGN
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ASSIGNMENT 1
Term: Fall 2018
Date of posting: 10 September 2018, Submission deadline: 17:00, 24th September 2018
Instructions:
i) Write all your assumptions (if any) clearly. ii) All questions are compulsory. iii) All questions carry equal marks. iv) All symbols have usual meaning unless otherwise stated. v) Assignments need to be submitted independently by each student. Repercussions of disobeying the institute academic honor code would be very serious. 1. Two uniform rigid bars each of mass 𝑚 and length 𝑙 are connected using a spring of stiffness 𝑘 as shown in Figure 3. Assume that the spring is unstretched when the bars are horizontal and consider all the hinges are frictionless. Derive the equations of motion for symmetric motion of the bars. Linearize the equation of motion about a stable equilibrium point and find the natural frequency for small oscillations. Discuss if the system can have asymmetric motion of the bars.
Figure 1 2.A uniform disk of weight 𝑊 and radius 𝑟 rolls without sliding on a plane table. At its center it carries a hinge with a weightless pendulum of length 𝑙 and a concentrated weight 𝑤 at its end (Figure 2). Find the natural frequencies for motion in the plane of the paper.
Figure 2
K. R. Jayaprakash ES646: Elastodynamics and Vibrations Mechanical Engineering, IITGN
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3. Consider the following system and derive the equations of motion and describe the kind of coupling in all the three cases. Find the generalized coordinate, which can lead to complete decoupling of the equations of motion considering 𝐿! = 2𝐿 3 , 𝑘! = 2𝑘!.
Figure 3a 3b 3c
4. Find the response of the system considering
𝑦 𝑡 = 𝐴 sin Ω𝑡 , 0 ≤ 𝑡 ≤ 𝜋 Ω0, 𝑡 > 𝜋 Ω
Figure 4 5. Consider the rods to be massless, all the joints to be frictionless, torsional stiffness 𝐾! to be linear
and the force 𝐹 to be always acting along 𝐴𝐵. Derive the EOM for the system shown in Figure 5.
Figure 5
𝑥(𝑡) 𝑦(𝑡)
K. R. Jayaprakash ES646: Elastodynamics and Vibrations Mechanical Engineering, IITGN
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6. Find the response 𝑥! 𝑡 , 𝑥! 𝑡 of the semidefinite system shown in Figure 6 assuming zero initial conditions.
Figure 6 7. A semidefinite system strikes a stopper as shown in Figure 7. Find the maximum force transmitted to the base of the stopper if the semidefinite system (a) doesn’t stick to the stopper after impact (b) does stick to the stopper after impact. Assume the velocity 𝑣! is constant and the springs are initially unstressed. Assume 𝑚! = 𝑚! and 𝑘! = 2𝑘.
Figure 7
