Mechanical Vibrations Fall 2018

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Department of Mechanical and Materials Engineering

This is the EML 4220 Mechanical Vibrations Fall 2018 page

Here is the EML4220 Syllabus (updated 8/30/18)

We will meet in EC1113 on W from 10-1150am and in EC1113 on F 11-1150am

We will NOT have a TA. If you have any questions come to see me.

My office EC3442 and my telephone number is 305-348-3643; my email levyez@fiu.edu

First Assignment: READ Chapter 1 and Sections 2.1-2.5 in Rao’s book and do the following problems:

Here are problems 1-3, 1-7 and 1-8, 1-9 and 1-10

Here are problems 1-12, 1-13a and 1-13b,

Here are problems 1-29, 1-30, 1-35 and 1-35b

Here are problems 1-53, 1-54, and 1-59

Here is a videotape on energy methods and equivalent elements

Even though I will be giving you access to the solutions, you need to try to do the problems first so that you will know where you get stuck and then use the solutions to see you get past the problem. Just looking at the solutions does not help you understand the methodology in relation to the way you think about solving the problem. Just as with Dynamics, the only way to learn vibrations is to do as many problems as you can.

Here are problems 2-4, 2-6 and 2-7 Note that more comments about 2-7

Are found on the top of the page for problem 2-13

Here are problems 2-13 , 2-15. Note that the problem here is LIKE 2-13 in your book

But is not problem 2-13

Please Turn in on 9/5/18 Problems 1-9, 1-10, 2-15 for grading as a check on learning.

This material and all the linked materials provided, except where stated specifically, are copyrighted © Cesar Levy 2018 and is provided to the students of this course only. Use by any other individual without written consent of the author is forbidden.

These are more undamped SDOF problems for you to work on where the equivalent k or m must be obtained; or the frequency must be obtained using Rayleigh’s method or conservation of energy method.

Here are problems 2-17, 2-18, 2-19

Here are problems 2-26, 2-38

Here is the rest of 2-38, and problems 2-68, 2-79

Here is the rest of 2-79

Here is problem 2-69

Here is problem 2-33, problem 2-45, and problem 2-71

From here we begin Problems in free vibration with damping….

READ Chapter 2.6-2.8

Here is a videotape of materials on damped systems and log decrement.

Here is a videotape for coulomb friction

Here are problems 2.88 and 2.90

Here are problems 2.84, 2.92 and 2.95

Here are problems 2.93, 2.96 and the rest of 2.90

Here are problems 2.106, and 2.122

Here are problems 2.112, 2.113 and 2.115 part a

Here is the rest of 2.115 and 2.116 and the rest of 2.116

Here is problem 2.145

Some of these problem solutions for the previous problem sets are not activated as they are under copyright. Problems from the next set of problems will be revealed when you have attempted them and you ask them to be revealed.

Please do problems 2.17, 2.18, 2.19, 2.33, 2.88, 2.90, 2.106, 2.122 and they will be due on 9/28/18

First exam is announced for October 5 and will cover materials from Chapters 1 and 2 of Rao’s book.

Please start looking at the following problems for systems with forcing functions

Here are problems 3.1, 3.2, 3.8, 3.10

Problems 3.19 and 3.25 are given in the back of your book.

Here are problems 3.29 and 3.33part1, 3.33part2 and 3.32

Here is a videotape of forced vibration problems related to damped and undamped systems

On Lecture 11 we started discussing problems related to vibration of damped SDOF system with forcing functions, including beats, effects of adding damping to such sytems, resonance, how to find the magnification factor and where the maximum magnification factor is and how to calculate it.

Here is a video related to beats, undamped forced motion

On Lecture 12 we began discussing the topic of SDOF system with vibrating support and derived the equation of motion for it. The equation is similar to the forced vibration problem we began in Chapter 3 with Po =Y [k² +(cw_f)²]^1/2 and the sine function now including a lag angle, b, i.e., sin (w_f t - b).

On Lecture 13 we spoke of the relative motion of a SDOF system with oscillating support and showed how this case can be related to vibration measuring instruments such as vibrometers, velometers and accelerometers. We did problems 10.9 and 10.14 in Rao’s book. Please read chapters 10-10.4.

Other problems of Chapter 3 to be covered.

Here are problems 3.54, 3.58

Here are solutions to problem 3.53. Problems 3.26, 3.48, 3.51 and 3.59 solutions can be viewed privately.

These problems below deal with force transmitted (chapter 9) and with vibration measuring equipment (Chapter 10). Please go through them to understand the concepts.

Here are problems 9.27, 9.32 and 9.34

Here is the rest of 9.34pt2 and 9.35

Here are problems 10.11, 10.12, 10.15

Here is a video dealing with relative motion and its importance in accelerometer design. Relative Motion is explained in Chapter 3, but the applications are in Chapter 10.

A video dealing with problems related to this topic as well as vibration absorber, accelerometers, vibrometers can be found at the following link.

On Lecture 14 We had our first quiz

On Lecture 15 We spoke about the power removed from a system due to a damper and showed that it varied like the amplitude². We the related this to Q points (or half power points) as a way in which to use experimental data to determine damping ratio, ζ, in the case where the damping ratio is small.

We continued to speak about relative motion and force transmitted to the oscillating support. We reviewed the case of a single degree of freedom system with sinusoidal forcing function, namely

and showed that the Force transmitted to the oscillating support base was = -m where x(t) was the solution for the oscillating support. This led to

= -mw_f²X_ss,os)

where X_ss,os is the steady state amplitude for the oscillating support. We did problem 9.27.

Lecture 16 you were asked to read Chapter 9 on force transmitted and also asked to begin reading materials on the rotating unbalance case.

Lecture 17 we spoke about the rotating unbalance case as well as the force transmitted to the support due to the rotating unbalance

Here is a video related to rotating unbalance

Lecture 18 deals with solutions of the SDOF systems with a general sinusoidal forcing function and also in the case where the forcing function is NOT sinusoidal. This begins Chapter 4 material.

Please start reading chapter 4 which deals with non-periodic forcing functions of the type we discussed at the very end of the EML3222 classes (in Chapter 9 of Rowell’s book).

You are assigned problems 4.15, 4.16 and 4.23. Once you have completed them, you can view the solution below.

Lecture 19 and Lecture 20 will discuss displacements due to non-periodic forcing functions found in chapter 4. Once this material is covered, we will move on to two-degree of freedom systems found in Chapter 5 of Rao’s book.

Here are problems 4.15 and 4.16

Here is problem 4.23

Also for HW please do problem 4.12 and 4.13 (but let z=0.2 in this problem) using the methods we discussed in class Due by 11/2

Exam 2 is announced for Nov 7.

The exam will cover forced vibration of a single degree of system (chapter 3) both damped and undamped, beats, force transmitted to support, oscillating support, rotating unbalance, relative motion, accelerometers, vibrometers (chapter 9 and 10 materials). Most of the materials that I talked about comes from Chapter 3, 9, 10. We also talked about other materials, namely, Q factor, and situations when the frequency ratio (r=wf/wn) is not due to changes in wf but changes in k and m.

You will be allowed 5 8.5 x 11 inch sheets of formulas. One of the 5 pages may be the page with the 4 graphs that I gave you, if you wish to use that.

Lecture 21 and 22. 2 DOF systems

Chapter 5 begins two degree of freedom systems --linear motion, rotational systems, combined linear-rotational systems (like suspension type). Start reading Chapter 5, sections 1-5 and 7.

Here are problems that you should try: 5.1, 5.4, 5.5, 5.20, 5.21, 5.34. Only the first problem is worked out in its entirety for you to view. The others can be viewed privately.

Here is a video dealing with problems for vibration of 2DOF systems

Lecture 23 and 24. Application of 2DOF equations. We start discussing the case of the vibration absorber as an undamped method of reducing the amplitude of vibration of the main mass.

Please read chapter 9’s section on vibration absorber.

We had our second examination on Lesson 24.

Lectures 25 and 26 will cover the rest of 2-DOF problems and chapter 5, and, also will include the 3^rd exam.

We will have our third examination on Nov 21 (change from what I stated in class). The examination will cover the materials in chapter 4 (SDOF under general forcing function-both harmonic and non-harmonic), chapter 5 (2-DOF systems), Chapter 9 (vibration absorbers).

Here is a video dealing with problems for vibration of 2DOF systems

Lesson 27 and 28 will start discussing the Rayleigh and Dunkerley methods of determining natural frequencies. Chapter 7, Sections 1-3 begins solving 2 degree of freedom systems numerically by using energy methods.

Problems for Chapter 7: Please do Problems 5.1, 5.4, 5.55, 5.20, 5.21, 5.34 Using, Dunkerley and Using Rayleigh’s methods and compare to the values you obtained (see problems above). Practice on the problems 5.1, 5.4, 5.5, 5.20, 5.21, 5.34 by determining the M and K matrices and using values for the vector A

Please read Chapter 5, Section 6 and its application found in Chapter 9, Sect 11. This will be the last topic to be discussed in the course.

Please look at problems 9.54, 9.57 and 9.60 that deal with this last topic. Please note that these problems will not be uploaded but can be viewed privately if requested.