Modeling the Vibrating Beam

By: The VibrationsSAMSI/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

0)()()(

2

2

tKydt

tdyC

dt

tyd

0)0( yy

0)0( wy

The rod’s initial position is y0

The rod’s initial velocity is yo

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)

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.

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.

0 1 2 3 4 5 6-6

-4

-2

0

2

4

6

8x 10

-5

time(s)

disp

lace

men

t(m

)

Experimental Data

C= 7.8930e-001

K= .5226e+003

0 1 2 3 4 5 6-6

-4

-2

0

2

4

6

8x 10

-5

time(s)

disp

lace

men

t(m

)Experimental Data

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.

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-5

-4

-3

-2

-1

0

1

2

3

4

5

x 10-5

time(s)

disp

lace

men

t(m

)

Experimental Data

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6

-3

-2

-1

0

1

2

3x 10

-5

time(s)

disp

lace

men

t(m

)

Experimental Data

Least-Square Assumptions

Residuals are normally distributed: ei~N(0,σ2)

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.

-4 -3 -2 -1 0 1 2 3 4

-8

-6

-4

-2

0

2

4

6

8x 10

-5

Standard Normal Quantiles

Qua

ntile

s of

Inp

ut S

ampl

e

QQ Plot of Sample Data versus Standard Normal

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.

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7

-5

-4

-3

-2

-1

0

1

2

3

4

x 10-5

Time (s)

Dis

plac

emen

t (m

)

ModelData

The Beam Model

0 0.5 1 1.5 2 2.5 3 3.5-1

0

1x 10

-4

Time (s)

Dis

pla

cem

ent

(m)

ModelData

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