EML6223 Spring 2017

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Department of Mechanical and Materials Engineering

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This is the EML 6223 Vibration Analysis Course Spring 2017

Here is the EML6223 Syllabus which has been updated to include office hours.

We begin the review of what you have seen in your undergraduate vibrations course

READ Chapter 1 and Chapter 2 as part of the review. Do problems out of chapter1: 4, 7, 9, 10, 11, 16, 22, 50, 51, 55, 56, 88, 89. Turn in problems 1.7, 1.9, 1.22, 1.50, 1.56, 1.89 next Tuesday, January 24

READ Chapter 1 and Sections 2.1-2.5

This material and all the linked materials provided, except where stated specifically, are copyrighted © Cesar Levy 2017 and is provided to the students of this course only. Use by any other individual without written consent of the author is forbidden.

Even though I will give 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 how 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.

Problems out of chapter 2 will be assigned this week for you to turn in the following week.

Here are problems 1-4, 1-7 and like 1-9, 1-10 and like 1-11

Here are problems like 1-17, 1-23a and 1-23b,

Here are problems 1-48, 1-49, 1-55 and like 1-56

Here are problems 1-91, 1-85, and 1-88

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

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 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

Please do the following problems and turn them in by Feb 6, 2017: 2-38, 2-59(a), 2-121(a), 2-122(a), 2-123(a).

Please do the following problems and turn them in by Feb 13, 2017: 3.18, 3.19,3.25, 3.26—No solutions will be given for these…final solutions to 3.18 and 3.26 are given in the back of your book.

Please start looking at the following problems

Here are problems 3.1, 3.2, 3.8, 3.10

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

Other problems of Chapter 3 to be covered related to rotating unbalance and forced vibration.

Here are problems 3.54, 3.58

Here are problems 3.34 and 3.51 and 3.53

Here are problems 3.48 and 3.59

These problems deal with force transmitted and with vibration measuring equipment.

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

Feb 2 class: We covered the oscillating support case where both the support and the mass are moving. We discussed getting the equation of motion for the mass (see chapter 3) and the force transmitted to the oscillating support (see chapter 9). We used previous solutions we had found in the case F(t)=Po sin(wf t) to get the equation.

We then looked at the same equation but written in terms of the relative displacement, x-y (x is displacement of mass and y displacement of the support), and obtained the equation in terms of the motion of the mass as seen by an observer standing on the oscillating support (like you watching another car move relative to your moving car).

We showed how the solution found is relevant to creating a vibration measuring piece of equipment like a seismometer, a velometer and an accelerometer, and discussed the negatives and positives of each piece of equipment (see chapter 10).

Also went through the mechanics of solving problem 10.14 in Rao's book.

Feb 7 class: self induced motion and 2-DOF positive definite systems were discussed.

Feb 9 class: semi-definite systems, vibration absorbers, mixed coordinate 2-DOF systems (Chapter 5 in Rao) and damped 2-DOF systems are discussed.

Please start reading the topics related to continuous systems, namely the vibrations of strings, beams, membranes and plates starting in chapter 8.

We spoke about two degree of freedom (2DOF) systems relating to masses and springs; and relating to linear and rotational displacements (Chapter 5 of our book). We found the natural frequencies and eigenfunctions for such systems. We spoke about the vibration absorber case and also the semi-definite case (case where the lowest natural frequency is = 0).

Here are problems 5.1, 5.4, 5.5, 5.20, 5.21, 5.34

We looked at the forced vibration of 2DOF systems (Chapter 5 of our book). We showed how the 2DOF system moves the natural frequencies away from the original forcing frequency and how we can tune the system so that the main mass can stop oscillating.

We started speaking about the vibration of continuous systems (Chapter 8), such as strings and longitudinal vibration of bars with and without end masses, deriving the governing equation for these cases, defining the boundary conditions and how one can obtain the solutions using the method of separation of variables. We showed how the boundary conditions may be used to obtain the frequencies and the eigenfunctions.

We completed the work on bars with end masses and showed graphically how one can find the natural frequencies for different cases when mbar/M is less than, equal to and greater than 1. We derived and showed the similarity of torsional vibrations of bars.

We looked at torsional systems and showed how the equations were similar in nature to the axial vibrations of a bar with proper identification of masses, springs, etc.

We derived the governing equation of the Bernoulli-Euler beam, defined the boundary conditions and the assumptions made in the derivation of the equation. We looked at the inclusion of an axial force leading to the dynamic buckling problem and discussed the variation of the frequency of vibration as a function of the ratio of the axial force to the Euler buckling load.

We looked at the Timoshenko beam theory in which the rotatory inertia and shear effects are included. We looked at derivation of the equation when rotary inertia was kept but shear was neglected and vice versa.

We used the energy expressions to derive the variation in energy (total=strain+potential+work of outside forces/moments) to show how the governing equation and the appropriate BCs can be obtained. We discussed both natural and geometric BCs.

We have derived the membrane vibration equation to include the proper BCs. To include determination of the natural frequency and eigenfunction. We applied the equations to circular membranes, rectangular membranes and square membranes. We showed that for square membranes the nodal lines for the 12 and 21 cases can indeed rotate through eight cases depended on the period of vibration.

We began the topic of vibration of plates. We derived the kinematic equations, stress and moment resultants.

We continued the vibration of plates to include the determination of the boundary conditions needed for simply supported, fixed edge, free edge and edges connected to linear springs, rotational springs, etc. We use the Navier solution to solve the simply supported plate all around and discussed the case of the square plate and degenerate modes.

Please see the first 100 minutes of this video prior to attempting the final examination

The final exam was given