Quantification of uncertainty by experiments without shunts . 116

6.3 Case II: Effect of spring element manufacturing and system assembly

Y_y/z,exp(Ω)(4.10) via envelopes of the amplitude and phase reponses in y- and z-direction. Figures 6.8 and 6.9 show that the used spring elements and assembly repetitions affect the short circuit resonance behavior in figure 6.8 and the dynamic transducer behavior in figure 6.9 in y- andz- direction.

0.1 100

|Gsc y|inm/s2 /V

100 300

−180

0

frequencyΩ/2πin Hz

argGsc yin◦

0.1 100

|Gsc z|inm/s2 /V

100 300

−180

0

frequencyΩ/2πin Hz

argGsc zin◦

a) b)

Figure 6.8:Case II: envelopes ofS×A=50exp. short circuited acceleration transfer functions, a)y-directionG^sc_y(Ω)( ), b)z-directionG_z^sc(Ω)( )

0.5 3

|Yy|inµF

100 300

−90

0

frequencyΩ/2πin Hz argYyin◦

0.5 3

|Yz|inµF

100 300

−90

0

frequencyΩ/2πin Hz argYzin◦

a) b)

Figure 6.9:Case II: envelopes ofS×A=50experimental capacitance transfer func-tions, a)y-directionY_y(Ω)( ), b)z-directionY_z(Ω)( )

Experimental characteristic quantities

Figure 6.10 shows theA=5experimental values of the short circuited resonance frequenciesω^sc_y/z,exp, the GEMCCsγy/z,expand static capacitancesC^s_y/z,expfor each of theS=10spring elements (a to c) and their respective mean values and standard deviations (d to f) calculated according to (2.5). In figure 6.10, the experimental values iny- andz-direction are shown in red and blue and slightly shifted for better visibility.

In figure 6.10a, the mean values of ω^sc_y/z,exp are generally decreasing. Together with the standard deviations of the assembly iterations, which are similar for all 10spring elements, it is assumed that the investigated spring elements show man-ufacturing variations. The presented order of spring elements does not represent the chronological order of conducted experiments and, therefore, a drift due to sys-tematic errors is excluded. The offset could be attributed to a higher experimental beam-column setup stiffness inz-direction, as already observed in section 5.1.1.

Forγy/z,exp and C^s_y/z,exp in figures 6.10b and c, no general upward or downward trend is observed. The variation of the GEMCCsγy/z,expin figure 6.10b are domi-nated by system assembly variations. The offset in figure 6.10c is also attributed to a higher experimental test setup stiffness inz-direction as well as to manufacturing differences in the used transducers.

As introduced in section 6.1.1, variations due to spring element manufacturing and system assembly in y- andz-direction in figures 6.10a to c are superposed to quantify probabilistic uncertainty in the beam-column dynamic behavior with-out shunts, as both, manufacturing and system assembly, affect the characteristic parameters.

In figure 6.11, the experimental values ofω^sc_y/z,exp,γy/z,expand C_y^s_/z,expfrom scat-ter plots in figures 6.10a to c are shown as histogram plots. The histograms are normalized to obtain probability densities with the sum of the bar areas equal to 1. Furthermore, the fitted normal distribution probability density functionsp_N(X) (2.4) are shown with normal distribution mean valuesµXand standard deviations σX summarized in table 6.8. According to probability plots shown in figure A.2 in the appendix, normal distributions can be assumed as the underlying distributions.

Figure 6.11 shows that the superposed spring element manufacturing and system assembly variations lead to significant variations in the characteristic quantities of the beam-column dynamic behavior without shunts. Furthermore, the relative variations of each quantity are similar in y- and z-direction, see table 6.8, with

γy/z,expvarying the most. Uncertainty in the vibration attenuation arising from the observed variations is discussed in section 6.3.3.

1 2 3 4 5 6 7 8 9 10 178

185 192

spring elements ωsc y/z/2πinHz

1 2 3 4 5 6 7 8 9 10 178

185 192

spring elements ωsc y/z/2πinHz

1 2 3 4 5 6 7 8 9 10 0.07

0.09 0.11

spring elements γy/z

1 2 3 4 5 6 7 8 9 10 0.07

0.09 0.11

spring elements γy/z

1 2 3 4 5 6 7 8 9 10 1.54

1.59 1.64

spring elements Cs y/zinµF

1 2 3 4 5 6 7 8 9 10 1.54

1.59 1.64

spring elements Cs y/zinµF

c) f)

b) e)

a) d)

Figure 6.10:Case II: exp. variations of short circuited frequenciesω^sc_y/z,exp, GEM-CCsγy/z,expand static capacitancesC^s_y/z,expforS=10membrane-like spring elements andA=5repetitions of assembly, a) to c) variations in y-direction (o) andz-direction (o), d) to f) mean valuesµX (•) and (•) and standard deviationsσXfor assembly iterations

178 185 192 0

0.3

ω^sc_y/2πin Hz p(ωsc y/2π)

178 185 192

0 0.3

ω^sc_z/2πin Hz p(ωsc z/2π)

0.07 0.09 0.11

0 135

γy

p(γy)

0.070 0.09 0.11

135

γz

p(γz)

1.54 1.59 1.64

0 85

C^s_yinµF

p(Cs y)

1.540 1.59 1.64

85

C_z^sinµF

p(Cs z)

Figure 6.11:Case II: exp. histograms and fitted normal pdfs p_N(X)of short cir-cuited frequencies ω^sc_y/z,exp, GEMCC γy/z,exp and static capacitances C^s_y/z,exp, left: y-direction ( ), right: z-direction ( ), values from figure 6.10

Table 6.8:Case II: exp. means valuesµX and standard deviationsσX of short cir-cuited frequenciesω^sc_y/z,exp/2π, GEMCCsγy/z,expand static capacitances C^s_y/z,expfrom normal pdfsp_N(X)in figure 6.11

absolute variation relative variation X µX σX unit σX/µXin %

ω^sc_y/2π 183.2 1.6 Hz 0.9

ω^sc_z/2π 185.0 1.4 Hz 0.8

γy 0.093 0.004 – 4.3

γz 0.093 0.004 – 4.3

C^s_y 1.598 0.006 µF 0.4

C_z^s 1.567 0.007 µF 0.4

6.3.2 Calibration of model without shunts considering uncertainty

The variations in the experimental beam-column system’s dynamic behavior with-out shunts as presented in section 6.3.1 are used to estimate normally distributed input parameter uncertainty for the lateral and rotational support stiffnessk_y,A/B, k_z,A/B,k_ϕ

y,A/B,k_ϕ

z,A/Band axial extension lengths l_ext,A/B of support A and B, sec-tion 6.1.1. The stiffness properties are assumed to vary independently in y- and z-direction but equally for support A and B. The axial extension lengths vary equally for support A and B iny- andz-direction.

Calibration

In total,N=S×A=50values of each parameterk_y,A/B,k_z,A/B,k_ϕ

y,A/B,k_ϕ

z,A/Band l_ext,A/Bare calibrated by solving the optimization problem (4.14)n=1, ...N times with

P_n= [k_y,A/B,k_z,A/B,k_ϕ

y,A/B,k_ϕ

z,A/B,l_ext,A/B]n (6.3) for the50experimental short circuited and the50capacitance transfer functions in figures 6.8 and 6.9. All other parameters are kept constant with the parameter values in table 5.3.

Figure 6.12 shows the50values ofk_y,A/B,k_z,A/B,k_ϕ

y,A/B,k_ϕ

z,A/Bandl_ext,A/Bfrom the calibration in normalized histograms. Additionally, the fitted probability density functions p_N(X)(2.4) are shown and their respective normal distribution mean valuesµX and standard deviationsσX summarized in table 6.9 represent proba-bilistic parameter uncertainty. The mean valuesµX ofk_y,A/B,k_z,A/B,k_ϕ

y,A/B,k_ϕ

z,A/B

andl_ext,A/Bin table 6.9 are the new mean support stiffness and geometry properties of beam-column system’s model in figure 3.3.

Table 6.9:Case II: parameter uncertainty of support A and B assumed from own experiments, num. meansµ_X,num and standard deviationsσ_X,num for normal pdfs from values in figure 6.12

property X µX σX unit

lateral support stiffness k_y,A/B 30.10·10⁶ 3.75·10⁶ N/m k_z,A/B 33.35·10⁶ 3.56·10⁶ N/m rotational stiffness k_ϕy,A/B 126.45 15.96 Nm/rad

k_ϕz,A/B 122.98 16.35 Nm/rad axial extension l_ext,A/B 6.70·10⁻³ 0.08·10⁻³ m

10 50 0

0.2

k_y,A/Bin N/mm

p(ky,A/B)

10 50

0 0.2

k_z,A/Bin N/mm

p(kz,A/B)

50 200

0 0.04

k_ϕ

y,A/Bin Nm/rad p(kϕy,A/B)

50 200

0 0.04

k_ϕ

z,A/Bin Nm/rad p(kϕz,A/B)

6.2 7.2

0 7

l_ext,A/Bin mm

p(lExt,A/B) e)

c) d)

a) b)

Figure 6.12:Case II: histograms ofS×A=50calibrated values and fitted normal pdfsp_N(X)

a) and b) lateral support stiffnessk_y,A/B( ),k_z,A/B( ) c) and d) rotational support stiffnessk_ϕ

y,A/B( ),k_ϕ

z,A/B( )

e) axial extension lengthsl_ext,A/B( ) Evaluation of calibration

To evaluate the assumed parameter uncertainty in table 6.9, the normally dis-tributed parameters are input for MONTE-CARLO-Simulation (MCS) to calculate nu-merical values of the short circuited frequenciesω^sc_y/z,num(3.50), GEMCCsγy/z,num

(3.53) and static capacitancesC^s_y/z,num (3.106), whose mean valuesµX and stan-dard deviationσX are then compared to experiments from table 6.8. In the MCS, ω^sc_y/z,num,γy/z,numandC^s_y/z,numare calculatedi=1, ...,Itimes withI=20000 sam-ples of independently and quasi-randomly varying parameter values in the set

P_i= [k_y,A/B,k_z,A/B,k_ϕy,A/B,k_ϕz,A/B,l_ext,A/B]i. (6.4)

Figure 6.13 shows the results of the MCS via normalized histograms and fitted probability density functions p_N(X) (2.4). The numerical normal distribution mean valuesµ_X,num and standard deviationsσ_X,num of the numerically calculated samples are summarized in table 6.10 and compared to the experimental mean val-uesµ_X,expand standard deviationsσ_X,expfrom table 6.10 by their relative deviation (4.16).

178 185 192

0 0.3

ω^sc_y/2πin Hz p(ωsc y/2π)

178 185 192

0 0.3

ω^sc_z/2πin Hz p(ωsc z/2π)

0.070 0.09 0.11

135

γy

p(γy)

0.070 0.09 0.11

135

γz

p(γz)

1.540 1.59 1.64

85

C^s_yinµF

p(Cs y)

1.540 1.59 1.64

85

C_z^sinµF

p(Cs z)

e) f)

c) d)

a) b)

Figure 6.13:Case II: num. histograms with I = 20000 MCS and fitted nor-mal pdfs p_N(X) of short circuited frequencies ω^sc_y/z,num, GEMCC γy/z,numand static capacitancesC^s_y/z,num, left: y-direction ( ), right:

z-direction ( )

The experimental and numerical mean valuesµX show a good agreement for all six parameters with a maximum relative deviation of1.1 %. In contrast to that, the numerical standard deviations σX,num underestimate the experimental stan-dard deviations σX,exp with a maximum relative deviation of 33.3 %. However, the deviations are acceptable as the assumed parameter uncertainty in table 6.9 adequately predicts uncertainty in the vibration attenuation as shown by the com-parison with experiments in section 6.3.4.

Table 6.10:Case II: normal pdf mean µX and standard deviationσX from exper-iments in table 6.10 and MONTE-CARLO-Simulation in figure 6.13, short circuited frequencies ω^sc_y/z/2π in Hz, GEMCCs γy/z and static capaci-tancesC^s_y/zinµF, rel. dev. in %

experimental numerical relative deviation X µX σX µX σX err(µX) err(σX) ω^scy/2π 183.2 1.6 183.0 1.4 0.1 14.3 ω^scz/2π 185.0 1.4 184.5 1.3 0.3 7.7

γy 0.093 0.004 0.093 0.003 −0.0 33.3

γz 0.093 0.004 0.094 0.003 −1.1 33.3

C^s_y 1.598 0.006 1.609 0.005 −0.7 20.0 C_z^s 1.567 0.007 1.578 0.005 −0.7 20.0

6.3.3 Prediction of uncertainty by model with shunts

Probabilistic uncertainty in the vibration attenuation with RL- and RLC-shunts due to spring element manufacturing and system assembly variations is predicted via MCS by using the assumed parameter uncertainty from table 6.9. To correctly predict uncertainty, the RL- and RLC-shunts are tuned with the new optimal shunt resistances in table 6.11, which optimally tune the shunts to the beam-column system with the new deterministic support stiffness and lateral extension length values represented by the mean valuesµXin table 6.9.

Table 6.11:Case II: reference RL- and RLC-shunt resistances for optimal vibration attenuation iny- andz-direction connected to transducersP_y,BandP_z,B

RL-shunt RLC-shunt

property symbol P_y,B P_z,B P_y,B P_z,B unit

resistance R^D 55.51 56.38 13.51 13.66 Ohm

R^L1 23.13 22.76 2.32 2.16 Ohm

R^N1 – – 1160.97 1138.31 Ohm

neg. cap. ratio δ – – −0.90 −0.90 –

inductance L 0.473 0.457 0.048 0.043 Henry

neg. capacitance Cⁿ – – −1.787 −1.753 µF

In the MCS, the numerical acceleration transfer functions with shuntsG^RL_y/z,num(Ω) (3.94) andG^RLC_y/z,num(Ω)(3.95) and their respective peak gainsGb^RL_y/z,num(3.96) and Gb^RLC_y/

z,num (3.98) are calculated I =20000 times with I samples of independently varying parameter values in the setP_i(6.4), while all other beam-column system and shunt parameters remain constant.

The variations of the acceleration transfer functions with RL-shunts G^RL_y/

z,num(Ω) and with RLC-shuntsG^RLC_y/z,num(Ω)obtained from MCS are shown in figures 6.14 and 6.15. The variations of the acceleration transfer function peak gains with RL-shunts Gb^RL_y/

z,num and with RLC-shuntsGb^RL_y/

z,num are shown in figure 6.16. The most likely peak gain valuesΣXand the maximum peak gain valuesQ⁹⁵_X, in this case obtained from GUMBELpdf fits, are used to quantify the peak gain variation, section 2.3.1.

Variation of numerical transfer functions

Figures 6.14 and 6.15 show the variations of I = 20000 samples of G^RL_y/z,num(Ω) andG^RLC_y/z,num(Ω)via envelopes of the amplitude and phase responses in y- and z-direction. Additionally, the respective optimally attenuated acceleration transfer functions, when uncertainty is disregarded, are shown with black lines.

0 12

|GRL y|inm/s2 /V

100 300

−180

0

frequencyΩ/2πin Hz

argGRL yin◦

0 12

|GRL z|inm/s2 /V

100 300

−180

0

frequencyΩ/2πin Hz

argGRL zin◦

a) b)

opt

Figure 6.14:Case II: num. acceleration transfer funct. with RL-shunts, a)G^RL_y,num(Ω): envelopes ( ) ofI=20000MCS, optimal ( ) b)G_z,num^RL (Ω): envelopes ( ) ofI=20000MCS, optimal ( )

0 12

|GRLC y|inm/s2 /V

100 300

−180

0

frequencyΩ/2πin Hz

argGRLC yin◦

0 12

|GRLC z|inm/s2 /V

100 300

−180

0

frequencyΩ/2πin Hz

argGRLC zin◦

a) b)

Figure 6.15:Case II: num. acceleration transfer funct. with RLC-shunts, a)G^RLC_y,num(Ω): envelopes ( ) ofI=20000MCS, optimal ( ) b)G_z,num^RLC (Ω): envelopes ( ) ofI=20000MCS, optimal ( )

Comparing the uncertain vibration attenuation, represented by envelopes, and op-timal vibration attenuation, represented by black lines, significant inclines in the vibration amplitudes are observed in the amplitude responses for vibration attenu-ation with RL- and RLC-shunts iny- andz-direction. To quantify uncertainty in the vibration attenuation, the variations of the peak gains of the varying acceleration transfer functions are analyzed in the next paragraph .

Variation of numerical peak gains

Figure 6.16 shows normalized histograms of the I =20000 numerical values of the peak gainsGb^RL_y/z,numandGb^RLC_y/z,numobtained from the MCS together with optimal peak gain valuesopt_X from figures 6.14 and 6.15 . All histograms are right-skewed with the mass of the pdfs concentrated on the left of the figure and can be approx-imated with GUMBELpdfsp_G(X)(2.6), as justified by the probability plots shown in figure A.1 in the appendix. The GUMBEL parametersΣX and∆X as well as the percentileQ⁹⁵_X are summarized in table 6.12.

The most likely peak gains for RL-shunts ΣG_b^RL_y/z and RLC-shunts ΣG_b^RLC_y/z in y- and z-direction from figure 6.16 are greater than their respective optimal peak gain valuesopt_X, see table 6.12. This is plausible as the optimal peak gain values rep-resent an optimum and deviations from the optimum normally result in reduced

0 9.25 15 0

2.2

Gb^RL_y in m/s²/V p(bGRL y)

0 8.51 15

0 2.2

Gb_z^RLin m/s²/V p(bGRL z)

0 2.82 15

0 2.2

Gb^RLC_y in m/s²/V p(bGRLC y)

0 2.74 15

0 2.2

Gb_z^RLCin m/s²/V p(bGRLC z)

c) d)

a) b)

Figure 6.16:Case II: num. histograms withI=20000MCS and fitted G^UMBELpdfs p_G(X)of peak gains with RL-shunts and RLC-shunts

a)Gb^RL_y,num( ), opt ( ) b)Gb^RL_z,num( ), opt ( ) c)Gb^RLC_y,num( ), opt ( ) d)Gb_z,num^RLC ( ), opt ( )

performance. Furthermore, some combinations of the used parameter uncertainty result in peak gain values smaller than the optimal peak gain values. This case is not further discussed as it does not lead to reduced vibration attenuation perfor-mance.

Uncertainty due to spring element manufacturing and system assembly, as repre-sented with the parameter uncertainty in table 6.9, has the same relative effect on vibration attenuation with RL- and RLC-shunts. This is seen by similar values of Q⁹⁵_X/optXfor all shunts in all directions. However, vibration attenuation with RLC-shunts still achieves higher vibration attenuation, as seen by most likely peak gains Σ_Gb^RLC

y/z and maximum peak gainsQ⁹⁵

Gb^RLC_y/z with RLC-shunts that are smaller than those with RL-shunts. All relative changes in table 6.12 show that uncertainty due to spring element manufacturing and system assembly variations effect the vibration attenuation with RL- and RLC-shunts similarly iny- andz-direction.

Im Dokument Evaluation of uncertainty in the vibration attenuation with shunted piezoelectric transducers integrated in a beam-column support (Seite 130-141)