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
GOAL
IDEA
Develop a model
that explains the
vibrations of a
horizontal beam
caused by the
application of a
small voltage.
Use the springmass model!
Collect data to find
parameters.
Solving Mass-Spring-Dashpot Model
2
d y (t )
dy (t )

C

Ky
(
t
)

0
2
dt
dt
y (0)  y0
The rod’s initial position is y0
y(0)  w0
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.
-5
-5
Experimental Data
x 10
8
6
6
4
4
displacement(m)
displacement(m)
8
2
0
2
0
-2
-2
-4
-4
-6
0
1
2
3
time(s)
4
5
6
Experimental Data
x 10
-6
0
1
2
3
time(s)
4
5
C= 1.5
C= 7.8930e-001
K= 100
K= .5226e+003
6
Through the optimizer module we were able determine the
optimal parameters. Note that the optimal value depends on the
initial C and K values.
-5
Experimental Data
x 10
5
4
3
displacement(m)
2
1
0
-1
-2
-3
-4
-5
0.4
0.6
0.8
1
1.2
time(s)
1.4
1.6
1.8
2
-5
3
Experimental Data
x 10
2
displacement(m)
1
0
-1
-2
-3
4
4.2
4.4
4.6
4.8
time(s)
5
5.2
5.4
5.6
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.

-5
8
QQ Plot of Sample Data versus Standard Normal
x 10
Checks for normality of
residuals!
6
Quantiles of Input Sample
4
2
0
-2
Residuals are
beginning to deviate
from the standard
normal!
-4
-6
-8
-4
-3
-2
-1
0
1
Standard Normal Quantiles
2
3
4
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
-5
x 10
-4
1
x 10
Model
Data
Model
Data
4
3
2
Displacement (m)
Displacement (m)
1
0
0
-1
-2
-3
-4
-5
-1
0
0.5
1
1.5
2
Time (s)
2.5
3
3.5
2
2.1
2.2
2.3
Time (s)
2.4
2.5
This model actually accounts for the second mode!!!
2.6
2.7
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