1. Modeling the Vibrating Beam By: The Vibrations SAMSI/CRSC June 3, 2005 Nancy Rodriguez, Carl Slater, Troy Tingey, Genevieve-Yvonne Toutain

2. Outline  Problem statement  Statistics of parameters  Fitted model  Verify assumptions for Least Squares  Spring-mass model vs. Beam mode  Applications  Future Work  Conclusion  Questions/Comments

3. Problem Statement Develop a model that explains the vibrations of a horizontal beam caused by the application of a small voltage. IDEA Use the spring- mass model! Collect data to find parameters. GOAL

4. Solving Mass-Spring-Dashpot Model The rod’s initial position is y 0 The rod’s initial velocity is y o

5. Statistics of Parameters  Optimal parameters: C= 0.7893 K=1522.5657  Standard Errors: se(C)=0.01025 se(K)= 0.3999 Standard Errors are small hence we expect good confidence intervals.  Confidence Intervals: (-1.5892≤C≤-.7688) (-1521.76≤K≤-1521.7658)

6. Confidence Intervals  We are about 95% confident that the true value of C is between.8336 and.8786.  Also, we are 95% confident that the true value of K is between and 1523 and 1527.8.  The tighter the confidence intervals are the better fitted model.

7. Sources of Variability  Inadequacies of the Model Concept of mass Other parameters that must be taken into consideration.  Lab errors Human error Mechanical error Noise error

8. Fitted Model  The optimal parameters depend on the starting parameter values.  Even with our optimal values our model does not do a great job. The model does a fine job for the initial data. However, the model fails for the end of the data.  The model expects more dampening than the actual data exhibits.

9. C= 7.8930e-001 K=.5226e+003 C= 1.5 K= 100 Through the optimizer module we were able determine the optimal parameters. Note that the optimal value depends on the initial C and K values.

12. Least-Square Assumptions  Residuals are normally distributed: e i ~N(0,σ 2 )  Residuals are independent.  Residuals have constant variance.

13. Checks for constant variance!

14. Residuals vs. Fitted Values  To validate our statistical model we need to verify our assumptions.  One of the assumptions was that the errors has a constant variance.  The residual vs. fitted values do not exhibit a random pattern.  Hence, we cannot conclude that the variances are constant.

15. Checks for independence of residuals!

16. Residuals vs. Time  We use the residuals vs. time plot to verify the independence of the residuals.  The plot exhibits a pattern with decreasing residuals until approximately t= 2.8 s and then an increase in residuals.  Independent data would exhibit no pattern; hence, we can conclude that our residuals are dependent.

17. Residuals are beginning to deviate from the standard normal! Checks for normality of residuals!

18. QQplot of sample data vs. std normal  The QQplot allows us to check the normality assumptions.  From the plot we can see that some of the initial data and final data actually deviate from the standard normal.  This means that our residuals are not normal.

19. The Beam Model This model actually accounts for the second mode!!!

20. Applications  Modeling in general is used to simulate real life situations. Gives insight Saves money and time Provides ability to isolating variables  Applications of this model Bridge Airplane Diving Boards

21. Conclusion  We were able to determine the parameters that produced a decent model (based on the spring mass model).  We did a statistical analysis and determined that the assumptions for the Least Squares were violated.  We determined that the beam model was more accurate.

22. Future Work  Redevelop the beam model.  Perform data transformation.  Enhance data recording techniques.  Apply model to other oscillators.

23. Questions/Comments