1. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 1 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Engineering 25 Chp9 Tutorial: Prob 9.32 Solution

2. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 2 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Unit Summary  A summary of the Units of Measure in this problem: y(t) → meters dy/dt → meters/second (m/s) dy 2 /(dt) 2 → meter/second 2 (m/s 2 ) m → kg k → Newtons/meter (N/m) M → Newtons K, B → meters ω p, ω c → rads/sec (r/s)

3. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 3 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods The Physical Situation  A FrictionLESS mass-spring (m-k) system  SINSOIDAL Forcing Function, f(t) Pull/Push Magnitude, M = 10N (2.248 lbs)

4. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 4 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods The Math Model  Using Newton’s 2 nd Law (ΣF = ma) find the Mass- Spring System y(t) ODE  With 0 th & 1 st order I.C.’s  Find y(t) for ω p = 1, 5.1, 10 r/s

5. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 5 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

6. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 6 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

7. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 7 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

8. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 8 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

9. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 9 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

10. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 10 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

11. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 11 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods P9.32 Analytical Soln

12. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 12 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods P9.29 Analytical Plot

13. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 13 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Solve by ODE23

14. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 14 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Solve by ODE23

15. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 15 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods P9.32 Numerical Soln-a  The Function file for the ODE Solver Call

16. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 16 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods P9.32 Numerical Soln-a

17. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 17 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Anonymous Function  Quick Solution Entirely from Command Window Using an ANONYMOUS Function >> m = 3;k = 75; M = 10; wp = 1; >> dxdt = @(t,z) [z(2); (10*sin(wp*t) - k*z(1))/m] dxdt = @(t,z)[z(2);(10*sin(wp*t)-k*z(1))/m] >> [T, Y] = ode23(dxdt, [0,14], [0, 0]); >> plot(T,Y, 'LineWidth', 2), grid, legend('y(t)', 'dy/dt = slope')

18. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 18 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Anonymous Function  Plot ONLY(T) >> >> plot(T,Y(:,1), 'LineWidth', 2), grid, xlabel('t'), ylabel('y(t)')

19. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 19 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

20. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 20 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

21. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 21 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods P9.32 Numerical Soln-b  The Function file for the ODE Solver Call

22. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 22 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods P9.32 Numerical Soln-b

23. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 23 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Case-a: ω p = 1 rad/sec

24. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 24 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Case-b: ω p = 5.1 rad/sec

25. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 25 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Case-c: ω p = 10 rad/sec

26. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 26 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Case-b: 100 sec  Note the “beating” with a Period of about 63 sec

27. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 27 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods SimuLink Solution  ODE by Newton’s 2 nd Law  Solve for Highest Order Term  Find y by Double Integral

28. BMayer@ChabotCollege.edu ENGR-25_HW-01_Solution.ppt 28 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods SimuLink Model  P9_32_mk_1104.mdl  Note Changes in IC’s