Modeling the Vibrating Beam By: The Vibrations SAMSI/CRSC June 3, 2005 Nancy Rodriguez, Carl Slater,...
Transcript of Modeling the Vibrating Beam By: The Vibrations SAMSI/CRSC June 3, 2005 Nancy Rodriguez, Carl Slater,...
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.
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.
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.