AME 30314 : Differential Equations, Vibrations and Controls I

Catalog Description:
First of a two-course sequence that introduces methods of differential-equation solution methods together with common engineering applications in vibration analysis and controls. Includes second-order, linear differential equations, partial differential equations feedback control and numerical solutions to systems of ordinary differential equations.

Prerequisites: MATH 20580

Textbooks:
Elementary Differential Equations and Boundary Value Prolems, Boyce and DiPrima, Wiley, 2001(required)
Mechanical Vibrations,Den Hartod, Dover , 1985 (optional but recommended).

Course Objectives:
After completing and passing this course a student will be able to classify a differential equation according to its type(linear or nonlinear, homogeneous or inhomogeneous, variable or constant coefficient, ordinary or partial etc) identify applicable solution techniques ( exponential solutions, and undetermined coefficients, variation of parameters, separation of variable, Laplace transforms, convolution integrals or numerical methods), use appropriate solution techniques to determine general solutions and solutions to initial value problems for ordinary differential equations, explain and prove the principle of superposition, linearize nonlinear ordinary differential equations about equlibria, identify domains in which solutions exist and are unique, write a computer program to solve an initial value problem for ordinary differential equatioins using various techniques(Euler’s method, the improved Euler’s method and fourth order Runge-Kutta) derive formulae for truncation error for different numerical solution methods, derive, linearize and solve the equations of motion for mechanical, electrical and electro-mechanical systems(both free vibration and force response), determine the natural frequency, damped natural frequency and damping ration for a second order system, determine an approximate solution for a mechanical system subjected to Colomb damping, derive and solve the equations of motion for a mechanical system subjected to harmonic base motion and solve the partial differential equations governing heat conduction, the wave equation and axial, torsional and transverse vibrations using separation of variable and Fourier series techniques.

Topics Covered:

Second-order differential equations (11classes)

• Introduction and review (1class)
• Linear equations, homogeneous equation, fundamental solution

Equations with constant coefficients (2classes)

• Inhmogeneous equations (2 classes)
• Nonlinear systems, linearization, phase plane (1 class)
• Stability of linearized nonlinear systems (1 class)
• Boundary value problems and Sturm-Liouville theory (4 classes)

Single degree of freedom vibrations (13 classes)

• Unforced damped and undamped vibrations (2 classes)
• Equivalent spring and damper systems (1 classes)
• Forced damped and undamped vibrations (2 classes)
• Energy methods (2 classes)
• System response and force transmission (2 classes)
• Transient response under harmonic and general forcing (1 class)
• Hysteresis, Coulomb friction and nonlinear vibrations(3 classes)

Partial differential equations (7 classes)

• Elliptic, parabolic and hyperbolic equations, separation of variables (1 class)
• Fourier series (2 classes)
• Even and Odd functions (1 class)
• Laplace’s equation (1 class)
• Heat equation (1 class)
• Wave equation (1 class)

Numerical methods for differential equations (7 classes)

• Euler method (1 class)
• Error analysis (1 class)
• Improved Euler method (1 class)
• Three-term Taylor series method (1 class)
• Runge-Kutta method (2 classes)
• Finite-difference method (1 class)

Schedule:
This course meets 3 times a week for 50 minutes each meeting, or twice a week for 75 minutes

Contribution to Professional Component:
Approximately 90% of this course is engineering science and 10% is engineering design.

Contribution to Program Learning Outcomes and Assessment:

Outcome Criterion	Topic	Pre-Knowledge	Direct Measure
a	Be able to solve first & second order differential equations	Calculus & 1st order equatioins	Homework & Exams
a	Be able to analyze solutions of 2nd order equations in the phase plane	None	Homework & Exams
a	Be able to solve 2nd order ordinary differential equations with specified boundary conditions	Calculus	Homework & Exams
a	Be able to solve partial differential equations using the method of seperation of variables including the wave equation, the heat equation and LaPlace's equation	None	Homework & Exams
e	Be able to solve single degree of freedom damped and undamped and forced and unforced vibration problems	Newton's Laws	Homework & Exams
a	Be able to express a periodic function as a Fourier series	Calculus	Homework & Exmas
k	Be able to use numerical techniques, such as Euler's Method, Runge-Kutta and Taylor series method, to write a computer program to solve ordinary diffential equations	Knowledge of computer programming	Homework assignments
k	Be able to use the finite difference method to write a computer program to solve partial differential equations	Knowledge of computer programming	Homework assignments
k	Understand the use of embedded microprocessors in engineering	Knowledge of computer programming	Homework assignments

Prepared by:
Bill Goodwine, August 28, 2006

 

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