**Phys280F part II. An Introduction to Vibrations and Waves**

Professor: Mark Jellinek (markj@physics.utoronto.ca , MP 516B)

Meeting times: M, W, Th (9-10 and 10-11; MP 202)

Office hours: Weds. 1 pm – 4 pm or by appointment.

Text: Vibrations and Waves, by A.P. French.

**A Course Outline **

**Weeks 1 – 2**. Harmonic motion and the design of mechanical systems

*Physical concepts*: The analysis of real physical systems, link between oscillations and waves, classification of problems in vibrations, kinematics, dynamics and energetics of undamped and damped simple harmonic motion. Introduction to general problem of response of linear systems to external forcing.

*Math topics*: Linear ordinary differential equations (ODEs), homogeneous equations, general solutions, complex numbers and the complex plane.

**Weeks 3 – 4**. Response to harmonic excitation.

*Physical concepts*: Steady-state phenomena, superposition of harmonics, resonance, energetics, power to driven oscillators.

*Math topics*: Nonhomogeneous ODEs, linear superposition, Fourier Series

**Weeks 5-6**. Coupled oscillators and waves (we will not get far into these topics with the time we have).

*Physical concepts*: Free oscillations due to coupled oscillators, normal modes. Introduction to waves, classification of waves, energy and momentum considerations.

*Math topics*: Coupled linear, constant coefficient ODEs, Linear algebra, eigenvalue problems, derivation and solution of the wave equation (a partial differential equation).

**Course Goals For Students**

1) Solve simple problems in vibrations. The student will acquire and apply necessary mathematical and physical skills to classify and solve simple given problems in vibrations.

2) Pose simple problems in vibrations. The student will develop a personalized approach for constructing a conceptual and technical understanding of the mechanics involved in straightforward vibrations problems.

3) Understand analog models of complex problems. The student will be able to reconstruct and understand existing mathematical representations of complicated problems in vibrations. The student will be able to discern and articulate verbally the strengths and weaknesses of such models.

4) Reduce complex (intractable) problems in vibrations into a number of simpler (tractable) component problems. The student will be able to construct a model of a complex system and understand the strengths and limitations of such a model system.

**Assessment (total is 40% of 280F)**

10% Self-graded Problem sets (PS#1due 11/3; PS#2 due 11/12; PS#3 due 11/24; PS#4 TBA)

15% Term test on Thursday 11/20/03.

15% Final Project due 12/3/03 (NOTE: Proposal is due Monday November 10)

**Problem Sets (4 out of 5)**

The goal of each problem set is to build understanding of concepts covered in class or covered in the textbook. Problem sets are to be turned in on the dates indicated above. Due dates are rigid and NO LATE WORK will be accepted. Solutions to the problem sets will be provided online on the Friday before the assignment is due. Your problem sets are to be SELF-GRADED and MUST INCLUDE COMMENTS. Problem sets submitted without your own assessment and remarks will be graded a zero.

Problem set 1

Problem set 2

Problem set 3

Problem set 4

Problem set 5

**Term Test (TBA)**

There will be one term test on XXX in MP125/126. The test will address basic concepts covered in class and in your homework. Three single-sided pages of HAND WRITTEN review notes and any sort of calculator will be permitted during the test.

**Final Project (Due date TBA)**

The final project is a report to be presented on a 3’ x 3’ (~1 m x 1 m) piece of white posterboard. Projects are due on TUESDAY, December 2 by 6 pm. On December 3 we will have a poster session where all of the projects from both sections of this class will be displayed and presented. The Poster session will be in MP110 from 6-10 pm.

Guidelines for assessment of your project

Examples of previous projects

The final project is to be a *team effort*. Teams are a minimu of 2 people and a maximum of 4 people. A 1-2 paragraph project **proposal from each group is due on Monday November 10**. The proposal should indicate the members of the group, the subject of the work and the question that is being addressed.

In detail most problems in physics and engineering are intractable. Consequently, determining ways to identify and pose simpler “analog problems” comprises a major part of research or design in these fields. The goal of the project is to give you an opportunity to learn about how a real problem involving vibrations or waves is addressed and ultimately solved. The topic you choose is entirely up to you. Some suggestions are provided below.

Some guidelines for your poster

Length and structure: 1 3’ x 3’ posterboard is all you get. It should include text and figures.

Your project should consider the following questions

1. What is the real problem?

2. What is the analog problem? (If appropriate) What are the design considerations?

3. How is the problem posed mathematically?

4. What is the strategy for solving the problem?

5. What are the meaning and significance of the results?

6. What are the strengths and limitations of the model (including the development of the model and the solution approach)?

7. How might the model be realistically improved?

**SOME suggestions for projects (you can choose your own!)**

The perfect slapshot

Melting of a crystalline solid

Dissolving a crystalline solid

Design of a road or mountain bike frame

Design of speed skates vs. figure skates or hockey skates

Radiative heat transfer

Earthquakes

Earthquakes triggered by earthquakes, changes in groundwater, or volcanic eruptions

Designing houses or skyscrapers in regions prone to large earthquakes

Heat capacity of solids and liquids

Tidally-induced volcanism on Jupiter’s moon Io

Variations in the length of day on Earth

Milankovitch cycles and climate change

Onset of convection in liquids

Onset of turbulence in fluids

Communication between growing fractures in stressed solids

Designing skis or snowboards for racing or for “off piste” conditions

Bridge design in windy environments

Electrical or thermal conductivity in metals

Sun spots

El Nino

Human heart

Nuclear magnetic resonance

Nanotechnology and drug delivery

Design of artificial cartilage for replacement joints in humans

Long distance running shoe vs. a trail running shoe

Thermal expansion of crystalline solids and liquids (at different pressures)