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

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

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

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

Statistics of Parameters  Optimal parameters: C= K=  Standard Errors: se(C)= se(K)= Standard Errors are small hence we expect good confidence intervals.  Confidence Intervals: ( ≤C≤-.7688) ( ≤K≤ )

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

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

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.

C= e-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.

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

Checks for constant variance!

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.

Checks for independence of residuals!

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.

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

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.

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

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

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.

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

Questions/Comments