DOC PREVIEW
CSUN ME 501A - Review for Final Exam

This preview shows page 1 out of 3 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 3 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 3 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

ME 501A Final Review December 2, 20041Review for Final ExamReview for Final ExamLarry CarettoMechanical Engineering 501ABSeminar in Engineering AnalysisDecember 2, 2004Classroom call-in: 800.882.0125Technical problems: 800.600.99782Review for Final• Tuesday, December 7, 5:30-7:30 pm• Open book and notes, similar to two midterm exams• Will be cumulative, but will focus on material since second midterm– Questions about systems of ODEs involve ideas about matrix eigenvalues– Numerical analysis of ODEs can compare analytical and numerical solutions3Matrix Basics• Definition of a matrix and basic operations of addition, subtraction, and multiplication by a scalar or matrix• Properties of determinants and matrix inversion• Linear independence of vectors– Linearly dependent if Σaix(i)= 0 is possible with ainot zero– Rank of a matrix (LI rows = LI columns)4Systems of Equations• Solution to n equations, Ax = b– Unique if rank A = rank [A b] = n– Infinite if rank A = rank [A b] < n– None if rank A ≠ rank [A b]• Homogenous equations Ax = 0– Trivial solution (x = 0) if rank A = n– Check Det A; if Det A = 0, rank A < n– Use for matrix eigenvalues and eigenvectors5Eigenvectors and Eigenvalues• Ax = λx for square A (n x n)• In general, A has n eigenvalues, but some may be duplicate• If a matrix is Hermitian, it has n linearly independent, orthogonal eigenvectors– This is true even with duplicate eigenvalues– Real symmetric matrices are Hermitian6Likely Final Problems• Possibility, but low probability, of problems dealing only with matrices and systems of equations• May have problems with links to later materials such as – use of eigenvalues and eigenvectors in solutions of ODEs– numerical ODE eigenvalue problems– Thomas algorithm for BVPsME 501A Final Review December 2, 200427Ordinary Differential Equations• Definition of terms: order, linear, homogenous, general solution, particular solution• Look at linear ODEs with constant coefficients• First order linear ODEs have closed form integral solution• Higher order ODEs obtained as combination of general and particular solutions8Systems of ODEs• Any nth order ODE can be expressed as a system of n first-order ODEs (and vice versa)• Various approaches for solving systems– Combine into higher order– Eigenvalue approach– Laplace transforms• Must solve system to apply initial conditions9Phase Plane Analysis• Phase plots, critical points, and stability• Look at system of two linear homogenous, autonomous equations–dy/dt = Ay (no function of time)• Critical points and stability depend on matrix eigenvalues which depend on determinant properties• Describe node, center, saddle point and spiral10Numerical ODE Solutions• Basis is solution of initial-value problem (IVP), dy/dx = f(x,y) with y(0) = a• One step methods, Euler, Runge-Kutta• Multistep methods: Adams• Systems of ODEs handled by same algorithms as single ODEs– Must complete each substep for each ODE before moving to next one• Stiff systems require implicit algorithms11Boundary-value problems• Shoot-and-try is usually best approach• Finite-difference and finite-element approaches may be used– Application to ODEs is instructive of basic use of these methods for PDEs• Numerical eigenvalue problems12Likely Final Problems• Use an ODE algorithm, which you may or may not have seen before– Typically ask you to apply the algorithm for one or two steps– May be applied to a single equation or to a system of equations– Ask for analytical and numerical solution of the same differential equation– Combination of analytical, numerical and phase plane analysis on same ODEME 501A Final Review December 2, 2004313Sample Problem• The equation of motion for a pendulum of length, L, in a gravitational field with acceleration, g, is given by the following equation: d2θ/dt2+ (g/l) sin θ = 0• Use the following algorithm on the next chart to obtain a numerical solution for this equation with g/l = 0.1 s-2, starting at t = 0, h = 0.1, θ(0) = 0.1 and θ’(0) = 0.• Get solution for t = 0.2 (two steps)14Sample Problem II• Algorithm:yPn+1/2= yn+ hf(xn,yn)/2YCn+1= yn+ hf(xn+h/2, yPn+1/2)• In the algorithm the yPvalue is only used as an intermediate value•ODE: d2θ/dt2= -(g/l) sin θ= -0.1 sin θ• Initial conditions: θ(0) = 0.1; θ’(0) = 0• Use step size h = 0.1 for two steps15Sample Problem Solution• Split second order ODE into two first order ODEs–dθ/dt = y with θ(0) = 1– dy/dt = -(g/l)sin θ = -0.1sin θ with y(0) = 0• Derivatives at t = 0: dθ/dt = 0; dy/dt = -0.1sin 1 = -0.08415• Midpoint values θ = 1, y = -0.00421• Midpoint derivatives: dθ/dt = -0.00421; dy/dt = -0.0841516Sample Problem Solution• Midpoint derivatives: dθ/dt = -0.00421; dy/dt = -0.08415• h = .1 values: θ = 0.9996, y = -0.00841• h = .1 derivatives: dθ/dt = -0.008411; dy/dt = -0.08412• P values: θ = 0.9992, y = -0.01262• P derivatives: dθ/dt = -0.01262; dy/dt = -0.0841• h = .2: θ = 0.9983, y = -0.0168217Second Sample Problem• What is the solution to the ODE: d2θ/dt2=(g/l) sin θ, with boundary conditions: θ(0) = 0.1; θ(0.2) = 0• Use shoot-and-try with h = .1 or finite differences with h =


View Full Document
Download Review for Final Exam
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Review for Final Exam and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Review for Final Exam 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?