Determine the principal stresses at point A on the cross section of the beam at section a-a.
ENGR 232 Dynamic Engineering Systems Lecture 1 Introduction to Numerical Solutions with Differential Equations Dr. Michael Ryan Perspective Solutions to 1 Order ODEs • Analytically - Separation of variables– quantitative • Explicit solution Implicit Solution • Qualitative analysis ▯▯ • Sketch ▯▯, find EQ. Plot trajectories about EP • Direction field –Graphical technique to sketch solution curve versus time, grid spacing • Numerical Solution of DE • Euler’s methodRunge-Kutta etc. • Analog Computer - future lab(Simulink) Read Chapter1 and 18 of Schaum’s Outline ENGR 232 Spring14-15 2 Introduction • Differential equations: Mathematical equations for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders.dx ▯ at dt x ▯ 5x ▯ 6x ▯ u(t) ▯ ▯▯ = 0 1 ▯▯ + 0 ▯(▯) ▯▯ ▯▯ ▯2 ▯3 ▯▯ 1 (As opposed to algebraic equations▯▯▯+ ▯▯ + ▯ = ▯ ) • A differential equation that describes a physical process is often called a mathematical model • They play a prominent role in engineering, physics, economics, biology, and other disciplines. ENGR 232 Spring14-15 3
