## Syllabus and Lecture Notes

**Course Goals: on completing EN0040, students will:**

- Be able to idealize a simple mechanical system or component as a collection of particles or rigid bodies, and to use Newtonian mechanics, with the aid of analytical or computational methods, to analyze forces and motion in the idealized system. Relates to ABET outcomes (a), (e), (k)
- Be familiar with the characteristics of vibrations in linear systems; and have the ability to analyze the free, damped, and forced vibrations of a 1 degree of freedom system. Relates to ABET outcome (a)
- Be able to design and conduct simple experiments to measure the dynamical properties of a mechanical system or components. Relates to ABET outcome (b)
- Be able to apply Newtonian mechanics to design a mechanical system to meet specified constraints, including: to function effectively in teams of 3-5 students; to communicate design specifications through clear and effective oral and written reports, to perform appropriate design calculations and optimization where appropriate, and to successfully manufacture and test a completed design Relates to ABET outcome (a), (b), (c), (g)

**Workload Expectation**

- Lectures: 23x80 mins = 31 hours
- Conference Sections 24x50 mins = 20 hrs
- Homework assignments: 8 at 8 hours each = 64 hrs
- Projects: 4 at 10 hours each
- Midterm exam: 80 mins in class, plus 10 hours preparation
- Final exam, 3 hours (as scheduled by Registrar) plus 15 hrs preparation

**TOTAL: **185 hours.

(Explanation: this represents a rough estimate of the time spent by an average student on the class. Your actual workload may be less or greater than the estimate (projects are open ended and the time commitment depends on how many candidate designs you develop and their complexity; we also drop the lowest homework grade so some people only submit 7 homeworks) . Federal regulations require 4 credit classes to exceed 180 hours)

**A note on downloading matlab codes: **if you download a code more than once, the second time the file name may be stored as something like matlab_file (2).m or matlab_file (3).m. If you run these, they will crash. That is because matlab is confused by the (2) and (3) in the file name. If you rename the file to remove it they will work.

**Notes from Lecture (9am Tu/Th). See also lecture capture on Canvas - go to Media Library**

- Jan 25: Organization slides from intro lecture. For the rest see lecture capture (due to copyright restrictions)
- Jan 30: Particles I: r-v-a relations, calculus review, staight line motion, Matlab to integrate accels data file
- Feb 1: Motion along curved paths: n-t and polar coords, Newtons laws, calculating F from F=ma
- Feb 6: Examples of F=ma and Mc=0; F=ma in n-t coordinates, deriving trajectory equations
- Feb 8: Deriving and solving differential equations for particlesI. Pump matlab, Suspension matlab
- Feb 13: Deriving and solving differential equations for particles II. Squirrel matlab Earthquake matlab
- Feb 15: Power-work-Kinetic energy relations for a single particle
- Feb 22: Conservative forces, potential energy, energy equations for a conservative system
- Feb 27: Linear impulse-momentum relations for particles; analyzing straight line collisions.
- March 1: 3D collisions, angular impulse-momentum relations for particles
- March 6: Vibrations of undamped 1-DOF systems
- March 13: Counting DOF; combining springs, using energy to derive EOM for conservative 1-DOF systems
- March 15: Finding natural frequencies for nonlinear systems; Analysis of vibrations in systems with damping
- March 20: Damped vibration examples; vibration of spring-mass system with harmonic external force
- March 22: Resonance in systems with many DOF; base excited and rotor excited spring-mass systems
- April 3: Rigid bodies: describing rotations
- April 5: Rigid bodies: rotation-spin-angular velocity relations; rigid body kinematics formulas and example
- April 10: Gears/pulleys/wheels, momentum and energy of a system of particles in terms of inertia matrix
- April 12: Calculating inertial propreties rigid bodies; 2D formulas for h and T; parallel axis theorem
- April 17: Using parallel axis theorem to compute I; 2D rigid body dynamics
- April 19: Examples of rigid body dynamics problems - slipping wheels; center of percussion for bat
- April 24: Solving rigid body problems with angular momentum; intro to 3D rigid body dynamics
- April 26: Solar Car: Power curves, Designing transmission to maximize speed, solar panels, DC motors.

**Notes from Conference Sections. **

- Jan 24: Vectors, Matrices, Loops, conditionals and plots in matlab m files Slides Matlab script
- Jan 29: Functions, solving differential equations. Slides Live Script ODE solution Matlab script
- Feb 5: Examples of straight line motion with variable a, n-t coords, polar coords
- Feb 12: Examples of solving particle problems with MATLAB. QP Filter script Pendulum script
- Feb 21: Matlab script illustrating optimizer
- Feb 26: Examples of power and energy methods
- March 9: Predator-Prey Project
- March 14: Free vibration of conservative 1DOF systems; counting # DOF and # vibration modes
- March 19: Vibration of nonlinear systems, examples of damped vibrations
- April 2: Forced Vibrations
- April 16: Rigid Body Kinematics
- April 23: 2D rigid body dynamics

**Detailed notes (electronic text)**

1. Brief introduction to the objectives and methods of dynamics

2. Review of forces and Moments (pdf version) (reading assignment - not covered in lectures2.3 Force couples, pure moments and Torques

3. Analyzing motion of systems of particles(pdf version)3.1 Equations of motion for a particle

3.2 Calculating forces required to cause prescribed motion of particles

3.3 Deriving and solving equations of motion for systems of particles

4. Conservation Laws for Particles(pdf version)4.1 Work, power, potential energy and kinetic energy relations

4.2 Linear impulse-momentum relations

5. Vibrations(pdf version)5.1 Features of vibrations and overview of issues in controlling vibrations

5.2 Free vibration of conservative single degree of freedom systems

5.3 Free vibration of damped single degree of freedom systems

5.4 Forced vibration of single degree of freedom systems

5.5 Solving differential equations for vibrating systems (for reference - not covered in lectures)

5.6 Introduction to vibration of multi-degree of freedom systems (advanced topic - not covered in lecture)

Summary of Solutions to EOM for vibration problems (pdf)

J

ava vibration demonstrationsThese will only run in Microsoft Internet Explorer - you can use the IE plugin to run them in Chrome or Firefox - and need to be added to the Site Exception List in your Java control panel to run. You can find additional instructions to set this up here.

Cut and paste the links below into the IE address bar in an Internet Explorer window

Free Vibration Simulator: http://www.brown.edu/Departments/Engineering/Courses/En4/java/free.html

Forced Vibration Simulator: http://www.brown.edu/Departments/Engineering/Courses/En4/java/forced.html

6. Analyzing motion of systems of rigid bodiespdf version - (see the html for the animations)6.1 Introduction to Rigid Body Motion

6.2 Describing Motion of a Rigid Body (rotation tensor; angular velocity and acceleration)

6.3 Analyzing Motion in Connected Rigid Bodies (mechanisms, rolling wheel, gears)

6.4 Linear Momentum, Angular Momentum and KE of rigid bodies (calculating COM and Inertia)

6.5 Rotational Forces: Review of moments exerted by forces and torques

6.6 Dynamics of rigid bodies (equations relating forces and moments to motion)

6.7 Summary of equations of motion for rigid bodies (long list of all important equations)

6.8 Examples of solutions to problems involving motion of rigid bodies