ME 7103: Theoretical Vibrations Course Notebook

Transcription

1 ME 7103: Theoretical Vibrations Course Notebook Instructor: Jeremy S. Daily, Ph.D., P.E. Fall 2009

2 1 Syllabus Instructor: Dr. Jeremy S. Daily Phone: Office: L173 KEP Office Hours: Monday from 10:00 am - noon and and Wednesday from 2:00 pm to 4:00 pm. Otherwise, drop in or schedule an appointment ( or phone) Classroom: L148 Days: Tuesday and Thursday Time: 11:00 12: Course Bulletin Description Multi-degree-of-freedom and continuous vibration systems. Introduction to the finite element method and approximation methods in vibration system analysis. Prerequisite: Knowledge ofonw degree of freedom systems or permission of instructor. 1.2 Topical Outline General Concepts in Mechanics Variational Principles and Energy Methods Single Degree of Freedom Systems Multiple Degree of Freedom Systems The Eigenvalue Problem Continuous Systems Numerical Methods (FEA and Time Integration) 2

3 1.3. DYNAMIC COURSE OUTLINE CHAPTER 1. SYLLABUS 1.3 Dynamic Course Outline The following is a dynamically updated schedule for the class. It is a shared Google calendar (Search for ME 7103 ) and can be integrated into your own personal calendaring system. While the calendar can be updated and changed as the course goes on, the schedule will remain fairly rigid during the semester. All changes and details concerning specific events and items on the schedule will be updated through the Google calendar for this course. The Google Calendar ID is: keoaomk6rlr0obvfhj4sgjp138@group.calendar.google.com 1.4 Course Policies Text Books Required: Principles and Techniques of Vibrations by Meirovich, Prentice Hall Reference: Engineering Vibrations by Daniel Inman, Prentice Hall Grading Procedures The following table gives the weights of the different aspects of the graded material for this class: Homeworks and Computer Projects: 25% Exam 1: 25% Exam 2: 25% Exam 3: 25% = A, = B, = C, = D, < 60 = F The instructor reserves the right to lower the minimum requirements for each letter grade Exam Policy Exams are open book and open notes; closed computer. 3

4 1.4. COURSE POLICIES CHAPTER 1. SYLLABUS Computer Usage Matlab and other specialty software will be used for labs and homework for data analysis and plotting. These programs are available on university machines Late Submission and Absences Late submission of homework will receive no score. Late computer projects will receive no score. Exams have mandatory attendance. Make-up exams will be offered only under very exceptional circumstances provided prior permission from the instructor is obtained. Neatness and clarity of presentation will be given due consideration while grading homework and exams Class Conduct Please do whatever necessary to maintain a friendly, pleasant and business-like environment so that it will be a positive learning experience for everyone. Please turn off all cell phone ringers or any other device that could spontaneously make noise Academic Misconduct All students are expected to practice and display a high level of personal and professional integrity. During examinations each student should conduct himself in a way that avoids even the appearance of cheating. Any homework or computer problem must be entirely the students own work. Consultation with other students is acceptable; however copying homework from one another will be considered academic misconduct. Any academic misconduct will be dealt with under the policies of the College of Engineering and Natural Sciences. This could mean a failing grade and/or dismissal. The policy of the University regarding withdrawals and incompletes will be strictly adhered to Center for Student Academic Support Students with disabilities should contact the Center for Student Academic Support to self-identify their needs in order to facilitate their rights under the Americans with Disabilities Act. The center for Student Academic Support is located in Lorton Hall, Room 210. All students are encouraged to familiarize themselves with and take advantage of services provided by the Center for Student Academic Support such as tutoring, academic counseling, and developing study skills. The Center for Student Academic Support provides confidential consultations to any student with academic concerns as well as to students with disabilities. 4

5 2 Analytical Dynamics 5

6 2.1. HOMEWORK PROBLEM SET 1 CHAPTER 2. ANALYTICAL DYNAMICS 2.1 Homework Problem Set 1 1. Given the following differential equation: mẍ + cẋ + kx = 0 when x(0) = x o and ẋ(0) = v o. Show every step in obtaining the solution of x(t) in the form x(t) = Ae ζωnt sin(ω d t + φ) (v o + ω n ζ x o ) where A = 2 + (x o ω d ) 2. In other words, determine the values of ω n, ζ, ω d, ω 2 d and φ in terms of m, c, and k. Hint: assume a form x(t) = e λt where λ is complex. Also, don t forget to use the product rule. 2. Derive the equations of motion for a simple pendulum using the conservation of energy. (See page 54 of the text for the derivation using Newton s Laws). 6

7 2.1. HOMEWORK PROBLEM SET 1 CHAPTER 2. ANALYTICAL DYNAMICS 3. Derive the equation of motion for the following system using a) the conservation of energy and b) Newton s laws of motion. c) Determine the equivalent spring constant for the 5-spring system when the individual value of the spring constants is 10 N/m. +x k 3 k 2 m k 1 k 4 k 5 7

8 2.2. VECTOR MECHANICS CHAPTER 2. ANALYTICAL DYNAMICS 2.2 Vector Mechanics Newton s Laws D Almebert s Principle 2.3 Energy Methods Principle of Least Action Generalized Coordinates Hamilton s Principle Lagrange Equations Kane s Equations 8

9 3 Single Degree of Freedom Systems 3.1 No Damping Spring Mass System Vector Formulation Energy Formulation Solution Tube within a Tube 3.2 Proportional Damping 3.3 Hysteretic Damping 3.4 Linearization Simple pendulum Rod Pendulump 3.5 Vibration Isolation 3.6 Base Excitation 9

10 4 Multiple Degree of Freedom Systems 4.1 Multibody Dynamics 10

11 5 Continuous Systems 5.1 Vibrations of Strings, Rods, and Shafts 5.2 Vibration of Beams 5.3 Rotordynamics 11

12 6 Computational Methods 6.1 Eigenanalysis Jacobi s Iteration QR Method 6.2 Finite Element Methods 12

13 7 Vibration Testing Sampling Theorems Instrumentation Time Analysis Frequency Analysis Frequency Response Functions Health Monitoring 13

14 8 Stability 14