Excel Method for Complex Dynamic Analysis

1. Excel Method for Complex Dynamic Analysis Boehm February, 2014

2. Example 2.5Acceleration Depends on Velocity a = g – cv2 v2 y2 = vdv (g – cv2) v1 x dx 1 = ln(x2-a/b) (a+bx2) 2c Best Integration formula I could find Good Luck proceeding to a Solution Falling Sandbag with Drag

3. Problem 2.1.28 Shows Integration Steps

4. Spreadsheet Method uses basic motion formulas l Velocity -y Direction l Time V2 = V1 + at = V1 + (V2 – V1)/Δt

5. Setting Up Formulas and Spreadsheet Problem Statement: Given: a = (g – cv2) where c = 6 x 10-4 ID Balloon’s Position g IRD y Formulas for Spreadsheet v=vi + (g-cvi2) x (delta t) Sandbag’s Position dy=vdt + (g-cv2)dt2/2 y=yi+dy A B D E F G H I J C 1 2 3 4 5

6. Excel Approach to Integrating a Solution Error = 1.5% Error = .065%

7. Reducing Delta Time Gives Better Accuracy

8. 20 a (m/s2) Problem 2.1.19 Non-Constant Acceleration 0 1 2 4 40 Time (sec) Velocity Acceleration 1 Time (sec) 2

9. Textbook Approach to Solution

10. Set up the First Three Rows, thenPull down rows until t = 2 seconds A B C D E F G 1 2 3

11. Acceleration is a function of Position F=ma = -kx a = -kx/m V2 = v1 + a1t X2 = x1 + v1t + a1t2/2

12. Spring Problem 3.1.37

13. Is a 1% Error Band Acceptable? • The vast majority of these analyses are used to determine design loads or performance requirements • Factors of 1.5 to 3 are then added for safety and robustness • Generally, analyses requiring complete accuracy, like Satellite re-entry rocket burns, are based on test results of the hardware involved. • MATLAB-type tools are used to refine the Spreadsheet method