AME 30315: Differential Equations, Vibrations and Controls II

Catalog Description:
Systems of nth-order differencial equations,mechanical vibrations, linear feedback s-plane controls analysis, frequency response analysis, partial differential equations.

Prerequisites: AME 30314

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

Course objectives:
After completing and passing this course a student will be represent nth order differential equations as a system of n first order differential equations, compute eigenvalues and eigenvectors and use them appropriately to determine the general solution to a system of ordinary differential equations including the cases of complex and repeated eigenvalues, solve a system of inhomogeneous first order ordinary differential equations using the methods of diagonalization, undetermined coefficients and variation of parameters, identify fundamental modes of vibration of mechanical systems, us Lagrange’s equations to determine the equations of motion for mechanical systems, explain the relationship between the eigenvalue problem and the corresponding frequency domain representation, compute the transfer function between a specified input and output for mechanical, electrical or electro-mechanical systems, list and explain the cause and effects of varying individual gains in a PID controller on system step response characteristics(rise time, overshoot, settling time, steady state tracking, etc.), sketch the root locus diagram for a given transfer function and use it to determine an appropriate gain value to meet or exceed controller design specifications, sketch Bode plots, interpret Bode plots for minimum phase systems with regard to stability for unity feedback, use root locus ad Bode plot techniques to design lean, lag, lead/lag and PID controllers.

Topics covered:

I. Feedback contol systemd (8 classes)

• First order plant response under PID control (1 class)
• Deifinition of Laplace transform (1 class)
• Solutions to initial value problems (1 class)
• Block diagrams and block diagram algebra (1 class)
• Step response versus pole locations (1 class)
• Time domain specifications for feedback problems (1 class)
• Effects of zeros and additional poles (1 class)
• Routh’s stability criterion (1 class)

II. Systems of first-order linear differential equations ( 4 classes)

• Linear independence, eigenvalues and eigenvectors (1 class)
• Homogeneous linear systems (1 class)
• Complex eigenvalues (1 class)
• Repeated eigenvalues (1 class)

III. Stability and frequency-domain analysis (16 classes)

• Basic pole placement in state space (2 classes)
• s-plane connection to state space representation (1 class)
• Root locus method (6 classes)
• Frequency domain methods (7 classes)

IV. Multiple degree of freedom vibrations (10 classes)

• Unforced vibration and fundamental modes (2 classes)
• Lagrangian dynamics (3 classes)
• Principal coordinators (2 classes)
• Forced vibrations, absorbers and force transmission (2 classes)
• Axial, torsional and transverse vibrations (1 class)

Schedule:
This course meets three times per week for 50 minutes or two times per 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 linear ordinary differential equations using Laplace Transforms	Ordinary first and second order equations	Homeworks and exams
(a)	Be able to represent a linear ordinary differential equation using block diagram algebra and to perform block diagram algebraic manipulations	None	Homeworks and exams
(a)	Be able to determine the responses of a linear input/output system to a step input referring only to the location in the complex plane of poles of transfer function	None	Homeworks and exams
(c)	Be able to design a feedback controller to meet, if possible, time domain specifications for a step input, including overshoot, rise time, settling time and steady state error specifications	None	Homeworks and exams
(a)	Be able to determine the stability of a transfer funtion using Routh's criterion	None	Homeworks and exams
(a)	Be able to compute the eigenvalues and eigenvectors of a matrix	Linear algebra	Homeworks and exams
(a)	Be able to solve systems of first order differential equations using real and complex eigenvalue and eigenvecotr computations, diagonalization and Jordan canonical form	Linear algebra, first and second order differential equations	Homeworks and exams
(e)	Be able to solve and analyze solutions for multiple degree of freedome vibration systems	Newton's Laws	Homeworks and exams
(e)	Be able to derive the equations of motion for mechanical systems using Lagrange's equations	Newton's Laws	Homeworks and exams
(c)	Be able to use the root locus design method to design feedback controllers	Ordinary differential equations	Homeworks and exams
(a,c)	Be able to use frequency domain methods(Bode plots) to analyze the stability of unity feedback systems and design feedback controllers	None	Homeworks and exams

Prepared by:
Bill Goodwine, August 28, 2006.

Direct comments, questions, and corrections to amedept@nd.edu