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Sheet 2 - nmks

nmks
Course

fliud mechanics (2222222)

16 Documents
Students shared 16 documents in this course
Academic year: 2018/2019
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جامعة كفر الشيخ

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Kafr Elsheikh University

Faculty of Engineering

Mech. Eng. Dept.

Working sheet and assignment No. ( 2 )

“Free Vibration of Single Degree of

Freedom Systems”

Third Year Mech. Eng. Section No. : ---------

Student name: ------------------- Student ID : ---------

Due date : Day: ------------ Date : / /

Marks

Put your Scan Code

Instructor : Dr. Maged Elhefnawey

1- An air-conditioning chiller unit weighing 2,000 lb is to be supported by four air springs (Figure. 1 ). Design the air springs such that the natural frequency of vibration of the unit lies between 5 rad/s and 10 rad/s.

Figure 1

3- Find the natural frequency of vibration of a spring-mass system arranged on an inclined plane, as shown in Figure 3.

Figure 3

4- A weight W is supported by three frictionless and massless pulleys and a spring of stiffness k, as shown in Figure 4. Find the natural frequency of vibration of weight W for small oscillations.

Figure 4

6- Figure 6 shows a small mass m restrained by four linearly elastic springs, each of which has an unstretched length /, and an angle of orientation of 45° with respect to the x-axis. Determine the equation of motion for small displacements of the mass in the x direction.

Figure 6

7- A mass m is attached to a cord that is under a tension T, as shown in Figure 7. Assuming that T remains unchanged when the mass is displaced normal to the cord, (a) write the differential equation of motion for small transverse vibrations and (b) find the natural frequency of vibration.

Figure 7

9- A helical spring of stiffness k is cut into two halves and a mass m is connected to the two halves as shown in Figure 9 (a). The natural time period of this system is found to be 0 s. If an identical spring is cut so that one part is one-fourth and the other part three-fourths of the original length, and the mass m is connected to the two parts as shown in Figure 9 (b), what would be the natural period of the system?

Figure 9

10- Derive the equation of motion of the system shown in Figure 10, using the following methods: (a) Newton's second law of motion, (b) D'Alembert's principle, (c) principle of virtual work, and (d) principle of conservation of energy.

Figure 10

12- Determine the displacement, velocity, and acceleration of the mass of a spring-mass system with k = 500 N/m, m = 2 kg, xo= 0 m, and ẋ= 5 m/s.

13- An automobile is found to have a natural frequency of 20 rad/s without passengers and 17 rad/s with passengers of mass 500 kg. Find the mass and stiffness of the automobile by treating it as a single-degree-of-freedom system.

15- A mass m is attached at the end of a bar of negligible mass and is made to vibrate in three different configurations, as indicated in Figure 12. Find the configuration corresponding to the highest natural frequency.

Figure 12

16- A uniform slender rod of mass m and length / is hinged at point A and is attached to four linear springs and one torsional spring, as shown in Figure 13. Find the natural frequency of the system if k = 2000 N/m, k = 1000 N-m/rad, m = 10 kg, and /= 5 m.

Figure 13

18- 2 A steel hollow cylindrical post is welded to a steel rectangular traffic sign as shown in Figure 15 with the following data: Dimensions: 1 = 2 m, ro= 0 m, ri= 0 m, b = 0 m, d = 0 m, t = 0 m; material properties: p (specific weight) = 76 kN/m³, E = 207 GPa, G = 79 GPa Find the natural frequencies of the system in transverse vibration in the yz- and xz-planes by considering the masses of both the post and the sign. Hint: Consider the post as a cantilever beam in transverse vibration in the appropriate plane.

Figure 15

19- Find the natural frequency of the traffic sign system described in Problem 18 in torsional vibration about the z-axis by considering the masses of both the post and the sign. Hint: The spring stiffness of the post in torsional vibration about the z-axis is given by Kt =(πG/2l) * (ro2-ri2). The mass moment of inertia of the sign about the z-axis is given by lo = (1/12) mo (d² + b²), where mo. is the mass of the sign.

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Sheet 2 - nmks

Course: fliud mechanics (2222222)

16 Documents
Students shared 16 documents in this course

University: جامعة كفر الشيخ

Was this document helpful?
Dr. Maged Elhefnawey
Kafr Elsheikh University
Faculty of Engineering
Mech. Eng. Dept.
Working sheet and assignment No. (2)
“Free Vibration of Single Degree of
Freedom Systems”
Third Year Mech. Eng. Section No. : ---------
Student name: ------------------- Student ID : ---------
Due date : Day: ------------ Date : / /
Marks
Put your Scan Code
Instructor : Dr. Maged Elhefnawey

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