Design and Dynamics
1145 The unknown parameters in X are determined using linear least squares technique
8
. The confidence level in using the parameters is determined by the condition number and
relative standard deviation RSD. With the intention of eliminating the delay in determining the velocities and accelerations, three points derivative method is used.
2
1 1
1 1
− −
+ −
− =
− −
+ +
i i
i i
i i
i i
i
t t
y y
t t
y y
y 7
2
1 1
1 1
− −
+ −
− =
− −
+ +
i i
i i
i i
i i
i
t t
y y
t t
y y
y 8
In order to compare the results from identified model to the experimental results, a fitting performance P
f
,
910
analysis of the results is performed.
100 1
×
−
− −
=
mx x
x i
f
y y
y y
P 9
Where y
i
is the identified model result, y
x
is the experimental result and y
mx
is the mean or average value of the experimental result. The error between model and experiment result
is determined by,
∑
=
=
n i
i T
i 1
ε ε
ε
10
where
ε
i
= | y
i
– y
x
|.
4. Results
The experiment is performed at different amplitude of displacement; Experiment I amplitude A
1yp1
= 1 mm, Experiment II amplitude A
2yp1
= 2 mm and Experiment III amplitude A
3yp1
= 3 mm. The time historical data of contact force f
1
, and displacements of y
p1
, y
p2
and y
p3
are shown by Figs. 6 to 8. Table 1 summarizes the experimental results. The frequency of the data is determined using Finite Fourier Transform. Tables 2 to 4 show the
result of pantograph parameters identification results for all experiments. Since the parameters identified in Table 2 gives the lowest condition number with comparatively low
RSD, these parameters are used in validation process. The same contact force data from experiment is used in the model and the displacements are compared. Figure 9 to 11 shows
the comparison of y
p1
, y
p2
and y
p3
between experiment and model. In order to calculate the fitting of the results between model and experiment, fitting performance analysis is
performed using Eq. 9. The fitting performances for all results are tabulated in Table 5. With averagely high fitting performances above 60 , and low mean errors below 1.0
mm the pantograph model with indentified parameters is considered validated.
Design and Dynamics
1146
Fig. 6 Contact force and displacements results of Experiment I
Fig. 7 Contact force and displacements results of Experiment II
Fig. 8 Contact force and displacements results of Experiment III
Design and Dynamics
1147
Table 1 Details of the experimental results Experiment Data Amplitude Offset
Frequency f
1
1.83 N 30.7862 N
y
p1
0.99 mm 54.3 mm
y
p2
0.99 mm 53.4 mm
I y
p 3
0.97 mm 51.2 mm
1.4219 Hz f
1
2.03 N 30.7913 N
y
p1
1.89 mm 54.3 mm
y
p2
1.88 mm 53.4 mm
II y
p 3
1.80 mm 51.1 mm
0.91017 Hz f
1
2.23 N 30.7682 N
y
p1
2.98 mm 53.9 mm
y
p2
2.96 mm 52.9 mm
III y
p 3
2.95 mm 50.9 mm
0.90335 Hz
Table 2 Identified parameters using results of Experimental I Parameters Value
RSD,
σ
Condition number c
1T
Nsm 12.31 2.8
c
2
Nsm 0.11 4.2
m
3
kg 10.38 15.6 c
3
Nsm 50.92 16.3
k
3
Nm 611.85 102.4 112.46
Table 3 Identified parameters using results of Experimental II Parameters Value
RSD,
σ
Condition number c
1T
Nsm 12.68 5.5
c
2
Nsm 0.31 7.4
m
3
kg 10.72 20.3 c
3
Nsm 51.92 41.6
k
3
Nm 624.17 155.4 448.00
Table 4 Identified parameters using results of Experimental III Parameters Value
RSD,
σ
Condition number c
1T
Nsm 13.01 8.4
c
2
Nsm 0.74 12.6
m
3
kg 10.88 29.2 c
3
Nsm 52.30 68.1
k
3
Nm 630.53 188.7 504.22
Design and Dynamics
1148
5. Active Pantograph Simulation
