94 S. Geier et al.
4. Forro L et al (2002) Electronic and mechanical properties of carbon nanotubes. In: Science
and application of nanotubes, Part III, 297–320
5. Fraysse J, Minett AI, Jaschinski O, Duesberg GS, Roth S (2002) Carbon nanotubes acting like
actuators. Carbon 40:1735–1739
6. Geier S, Mahrholz T, Wierach P, Sinapius M (2013) Carbon nanotubes array actuators. Smart
Mater Struct 22(9):094003
7. Grobert N (2006) Nanotubes—grow or go? Mater Today 9(10):64
8. Gupta S, Hughes M, Windle AH, Robertson J (2004) In situ Raman spectro-electrochemistry
study of single-wall carbon nanotubes mat. Diam Relat Mater 13:1314–1321
9. Iijima S (1991) Helical microtubules of graphitic Carbon. Nature 354:56–58
10. Kalbac M, Farhat H, Kavan L, Kong J, Sasaki K-I, Saito R, Dresselhaus M (2009) Electro-
chemical charging of individual single-walled carbon nanotubes. ACS Nano 3(8):2320–2328
11. Kreupl F (2012) Electronics: carbon nanotubes finally deliver. Nature 484:321–322
12. Mechrez G, Suckeveriene RY, Tchoudakov R, Kigly A, Segal E, Narkis M (2012) Structure and
properties of multi-walled carbon nanotube porous sheets with enhanced elongation. J Mater
Sci 47:6131–6140
13. Miller RL, Bradford WL, Peters NE (1988) Specific conductance: theoretical considerations
and application to analytical quality control. U.S. beological survey water-supply paper 2311,
116
14. Minett A, Fraysse J, Gang G, Kim G-T, Roth S (2002) Nanotube actuators for nanomechanics.
Curr Appl Phys 2:61–64
15. Mirfakhrai T (2009) Carbon nanotube yarn actuators. Doctoral thesis at the University of
British Columbia
16. Nightingale ER (1959) Phenomenological theory of ion solvation. Effective radii of hydrated
ions. J Phys Chem 63(9):1381–1387
17. Park JG, Smithyman J, Lin C-Y, Cooke AW, Kismarahardja A, Li S, Liang R, Brooks JS,
Zhang C, Wang B (2009) Effects of surfactants and alignment on the physical properties of
single-walled carbon nanotube buckypaper. J Appl Phys 106:104310
18. Riemenschneider J, Mahrholz T, Mosch J, Melcher J (2007) System response of nanotube
based actuators. Mech Adv Mater Struct 14:57–65
19. Riemenschneider J, Temmen H, Monner H-P (2007) CNT based actuators: experimental and
theoretical investigation of the in-plain strain generation. J Nanosci Nanotech 7(10):3359–3364
20. Sippel-Oakley JA (2005) Charge-induced actuation in carbon nanotubes and resistance changes
in carbon nanotube networks. Doctoral thesis at the University of Florida
21. Spinks G, Wallace GG, Fifield LS, Dalton LR, Mazzoldi A, De Rossi D, Khayrullin II, Baugh-
man RH (2002) Pneumatic carbon nanotube actuators. Adv Mater 14(23):1728–1732
22. Spinks G, Baughman RH, Dai L (2004) Carbon nanotube actuators: synthesis, properties and
performance. In: Bar-Cohen Y (ed) Electroactive polymer (EAP) actuators as artificial mus-
cles. SPIE—The international society for optical engineering, Bellingham, pp 261–295
23. Spitalsky Z, Aggelopoulos C, Tsoukleri G, Tsakiroglou C, Parthenios J, Stavroula G, Krontiras
C, Tasis D (2009) The effect of oxidation treatment on the properties of Multi-walled carbon
nanotube thin films. Mater Sci Eng B 165:135–138
24. Suppiger D, Busato S, Ermanni P, Motta M, Windle A (2009) Electromechanical actuation of
marcoscopic carbon nanotube structures: mats and sligned ribbons. Phys Chem Chem Phys
11:5180–5185
25. Vainio U, Schnoor TIW, Vayalil SK, Schulte K, Mller M, Lilleodden ETJ (2014) Orientation
distribution of vertcally aligned multiwalled carbon nanotubes. Phys. Chem. C 118:9507–9513
26. Wang D, Song P, Liu C, Wu W, Fan S (2008) Highly oriented carbon nanotube papers made
of aligned carbon nanotubes. Nanotechnology 19:075609
27. Whitten PG, Spinks GM, Wallace GG (2005) Mechanical properties of carbon nanotube paper
in ionic liquid and aqueous electrolytes. Carbon 43:1891–1896
28. Wittenburg J, Pestel E (2001) Festigkeitslehre, Ein Lehr- und Arbeistbuch, ed. 3. Springer,
Berlin, Heidelberg, New York
4 Experimental Investigations of Actuators Based . . . 95
29. Yu M-F, Lourie O, Dyer MJ, Moloni K, Kelly TF, Ruoff RF (2000) Strenght an breaking
mechanism of multiwalled carbon nanotubes under tensile load. Science 287:637–640
30. Yun Y-H, Shanov V, Tu Y, Schulz MJ, Yarmolenko S, Neralla S, Sankar J, Subramaniam S
(2006) A multi-wall carbon nanotube tower electrochemical actuator. Nano Lett 6(4):689–693
Chapter 5
Efficient Experimental Validation
of Stochastic Sensitivity Analyses of Smart
Systems
Steffen Ochs, Sushan Li, Christian Adams and Tobias Melz
Abstract A method for the efficient experimental validation of stochastic sensi-
tivity analyses is proposed and tested using a smart system for vibration reduction.
Stochastic analyses are needed to assess the reliability and robustness of smart
systems. A model-based design of experiments combines an experimental design
with the results of a previous numerical sensitivity analysis. To test this method, a
system of structural dynamics is used. Active suppression of disturbing vibrations
of a cantilever beam by means of active piezoelectric elements is considered. The
observed target variables are the level of vibration reduction at the beam’s end and
the fundamental frequency considering five uncertain system variables. Based on a
numerical model of the piezoelectric beam, a variance-based sensitivity analysis is
performed to determine each design variable’s impact on the target variables.
According to these numerical results, a model-based experimental design is
established and the experiments are conducted. In comparison to a fully five-factor
factorial experimental design, the model-based approach reduced the experimental
effort by 50%, without great loss of information.
5.1 Introduction
The development of smart systems becomes increasingly prominent, e.g., in the
field of vibration reduction. A smart structure system is characterized by a
structure-compliant integration of actuators and sensors based on multi-functional
materials such as piezoceramics. The resulting interactions between structural
components, sensors, actuators, and controllers hinder the analysis of the system
reliability. In order to analyze such interactions, statistical variation of the system
variables needs to be considered in a numerical simulation.
S. Ochs (✉) ⋅ S. Li ⋅ C. Adams ⋅ T. Melz 97
Research Group System Reliability and Machine Acoustics SzM, TU Darmstadt,
Magdalenenstr. 4, 64289 Darmstadt, Germany
e-mail: [email protected]
© Springer International Publishing Switzerland 2017
A.L. Araujo and C.A. Mota Soares (eds.), Smart Structures and Materials,
Computational Methods in Applied Sciences 43,
DOI 10.1007/978-3-319-44507-6_5
98 S. Ochs et al.
A number of research projects have been initiated to apply the sensitivity
analysis in performance and reliability studies of complex systems. Gunawan et al.
[1] introduced a concept of sensitivity analysis for stochastic processes. Coupé and
Gaag [2] described concepts of sensitivity analysis using Bayesian networks.
Beside that McCandless et al. [3] showed an application of sensitivity analysis
using Monte Carlo simulation for the risk analysis. Han et al. [4] used also Monte
Carlo simulations and variance-based sensitivity analyses to study the influences of
design variables on the scattering behavior of an active system. Thus, he showed
the usability of stochastic sensitivity analyses for smart systems. In addition, Li
et al. [5, 6] introduced the application of sensitivity analysis to calculate the effects
of design variables on the performance of a smart structure system with active
vibration control. Moreover, Li et al. [7] investigated the impact of uncertainties in
design variables on the quality of various control strategies of an active system.
However, a quantitative experimental validation of the numerical results is
difficult due to the random selection of simulation combinations in the stochastic
analysis. Only a small number of simulation combinations can be checked.
Therefore, the numerical results are often not validated experimentally. In addition,
the applicability of known experimental simulation techniques to validate stochastic
sensitivity analysis has only been explored to a small extend. Generally, the
methods of Design of Experiments (DoE) [8, 9] can be used. Li et al. [10] used DoE
to analyze the sensitivities of three design variables of an active framework
structure. However, the experimental design must be adapted to the investigated
system to keep the experimental effort low. In contrast to a non-optimal experi-
mental design, a model-based experimental design can be combined with an
expected model equation. Thus, the model-based design can be adapted to the
system behavior under investigation and enables an efficient validation of the
numerical results.
In this paper a method for generating an efficient experimental design for the
validation of a stochastic sensitivity analysis of smart systems is proposed. To test
this method, in Sect. 5.2 a variance-based numerical sensitivity analysis is pre-
sented. It was performed on a smart beam structure. Variance-based sensitivity
analyses serve as an example for such a stochastic simulation. Finally, in Sect. 5.3
the proposed method for the efficient experimental validation of the numerical
results based on the model-based experimental design is presented and tested.
5.2 Variance-Based Sensitivity Analysis of a Piezoelectric
Beam
5.2.1 Stochastic Sensitivity Analysis
The various methods that are used to explore the relationship between the input and
output variables of a system are part of sensitivity analysis. A distinction is made
5 Efficient Experimental Validation of Stochastic … 99
between local and global techniques [11]. The local sensitivity techniques utilize
the observation of a system’s localized output variables after making small changes
to its input variables. The major disadvantage of these techniques lies in the local
observation and its inability to identify interactions between design variables.
Global methods of sensitivity analysis, however, assume no limitations of the
considered values range of the design variables. The sensitivity is quantified using
the total values of the variable’s space. A corresponding approach relies on an
analysis of the variance seen at the observed output as a basis for assessing the
sensitivity [11]. Here, the scattering behavior of each design variable is determined
by assigning a density function. The respective influence of the design variable Xi
on the scattering behavior of the system is calculated and expressed by two sen-
sitivity indices. The direct impact of a design variable Xi is expressed by the main
effect
SMi = Var½EðY jXi Þ , ð5:1Þ
Var½Y
where the variance generated by Xi, represented by the variance of the conditional
expected value Var[E(Y | Xi)], is based on the total variance of the observed output
variable Var[Y]. The index i runs from one to k, the number of design variables.
The total effect
STi =1 − Var½EðYjX − iÞ ð5:2Þ
Var½Y
indicates the total influence of a design variable on the observed output and sum-
marizes all effects of Xi, where X–i represents all influencing design variables except
Xi. Variations that arise due to interactions between design variables are represented
by the difference between the total and the main effect.
The main effect of a variable can reach a value between 0 (no direct relationship)
and 1 (strong direct relationship). The total effect can be equal to or greater than the
main effect. Equality between the main and the total effect of Xi indicates no
interactions with X–i.
For the considered smart system both sensitivity indices are determined using a
Monte Carlo simulation with a sample size of N = 20000. The statistical estimators
∑rN= hi
bSMi N 1 1 1 fB, r ⋅ fAðBiÞ, r − fA, r
− 1 fAB,
= 2 ð5:3Þ
1 1 ∑r2N= 1 fAB, r − 1 ∑2r N= r
2N − 2N
∑Nr = hi
bSTi N 1 1 1 fA, r ⋅ fAðBiÞ, r − fB, r
− 1 fAB,
=1− 2 ð5:4Þ
1 1 ∑r2N= 1 fAB, r − 1 ∑r2N= r
2N − 2N
100 S. Ochs et al.
of Sobol‘ [12] and Jansen [13] combined with a sample strategy of Saltelli [14] are
used to calculate the indices from the results of the Monte Carlo simulation. The
index r runs from one to N, the number of simulations. fAB,r denotes the function
results of the sampling element, represented by the r-th row of the matrix AB,
which is defined as [A; B]. A and B are two independent sample matrices of size
N x k, and A(Bi) is a composite matrix that is identical to A except for the i-th column
which is replaced by the i-th column of B.
5.2.2 Mathematical Model of Piezoelectric Beam Dynamics
The investigated system is a cantilever beam with a flat collocated piezoelectric
sensor (S) and actuator (A) pair, as shown in Fig. 5.1. Its properties are summarized
in Table 5.1.
Piezoelectric beam systems are often used as an application example in current
research [15, 16].
The clamped beam is modeled as an EULER-BERNOULLI beam with a lateral load
F(t) close to the free end of the beam. A mathematical model governing the motion
of the piezoelectric cantilevered beam can be derived using Hamilton’s principle
and the assumed mode method [17, 18]. Only the eigenvalue problem of lateral
vibration in z-direction w(x, t) is used in this paper.
The lateral displacement in z-direction
n ð5:5Þ
wðx, tÞ = ΦðxÞqðtÞ = ∑ ΦiðxÞqiðtÞ
i=1
Fig. 5.1 Piezoelectric beam (view from top)
5 Efficient Experimental Validation of Stochastic … 101
Table 5.1 Characteristic data of the piezoelectric beam Unit
mm
Symbol Description Value mm
Length of beam 200 mm
lB Thickness of beam 3 kg/m3
hB Width of beam 40 GPA
bB Density of beam 2700 mm4
ρB Young’s modulus of beam 70 mm
EB Moment of inertia of beam 90 mm
IB Length of piezoelectric actuator 50 mm
lA Thickness of piezoelectric actuator 0.8 mm
hA Width of piezoelectric actuator 30 kg/m3
bA Position of piezoelectric actuator 15 GPA
aA Density of piezoelectric actuator 7800 mm4
ρA Young’s modulus of piezoelectric actuator 62.1 m/V
EA Moment of inertia of piezoelectric actuator 1.28 mm
IA Piezoelectric constant of actuator –1.8 × 10−10 mm
d31,A Length of piezoelectric sensor 10 mm
lS Thickness of piezoelectric sensor 0.5 mm
hS Width of piezoelectric sensor 10 kg/m3
bS Position of piezoelectric sensor 35 GPA
aS Density of piezoelectric sensor 7800 mm4
ρS Young’s modulus of piezoelectric sensor 66.7 m/V
ES Moment of inertia of piezoelectric sensor 0.104 mm
IS Piezoelectric constant of sensor –2.1 × 10−10 %
d31,S Position of lateral load 0.95 ⋅ lB
aF Structural damping ratio 1.5
ζ
is separated into the spatial solution Φ(x) and the temporal solution q(t). The overall
spatial solution is given by
ΦiðxÞ = sinhðβixÞ − sinðβixÞ − sinhðβilBÞ + sinðβilBÞ ðcoshðβixÞ − cosðβixÞÞ.
coshðβilBÞ + cosðβilBÞ
ð5:6Þ
The values for the product βilB emerge from the zero crossings of the charac-
teristic equation of a cantilever beam. For the first three eigenmodes they amount to
β1lB = 1.8751; β2lB = 4.6941; β3lB = 7.8548. The model is obtained by modal
truncation that only takes the beam’s first three modes of vibration into account.
The second-order differential equation in z-direction is given by
102 S. Ochs et al.
0 ZlB ZlB ZlB 1
@ρBhBbB Φ2i ðxÞdx + ρAhAbA Φi2ðxÞHAdx + ρShSbS Φ2i ðxÞHSdxA ⋅ qï ðtÞ . . .
0 0 0 0 1
ZlB ZlB ZlB
+ @EBIB Φ′i′ 2ðxÞdx + EAIA Φ′i′ 2ðxÞHAdx + ESIS Φ′i′ 2ðxÞHSdxA ⋅ qiðtÞ
00 0
1 ZlB
2
= ΦiðaFÞFðtÞ − ðhB + hAÞbAEAd31, AVAðtÞ ΦiðxÞHA′′ dx,
0
ð5:7Þ
where VA(t) is the voltage applied to the piezoelectric actuator.
Since the actuator and sensor are not attached over the entire length of the beam,
their positions need to be considered by means of Heaviside functions
HA = Hðx − aAÞ − Hðx − aA − lAÞ ð5:8Þ
and
HS = Hðx − aSÞ − Hðx − aS − lSÞ. ð5:9Þ
The left side of the equation of motion (5.7) is expanded to a speed-dependent
damping force. The damping is defined according to Rayleigh damping. The
coefficients α and β of the Rayleigh damping correspond to a damping ratio ζ of
1.5%, which is obtained from experimental studies. The coefficients are evaluated
by solving a pair of simultaneous equations, given by
α ! ω1− 1 ω1 ! − 1 ζ1 !
β ω2− 1 ω2 ζ2
=2 , ð5:10Þ
where ω1 and ω2 are the first two natural angular frequencies of the piezoelectric
beam and ζ1 = ζ2 = ζ. Finally, the second-order modal equation is converted into a
first-order state-space form to link the model with a controller.
5.2.3 Control Design
The vibration suppression method Positive Position Feedback (PPF) is imple-
mented to control the vibrations of the beam. PPF control was introduced by Goh
and Caughey [19]. It consists of a second-order compensator; thus, it is not sensitive
to spillover. Based on the fact that the position-proportional measurement is pos-
itively fed into the compensator and the signal from the compensator, magnified by
5 Efficient Experimental Validation of Stochastic … 103
Table 5.2 Characteristic Symbol Description Value Unit
data of the PPF compensator ωc 2π ⋅ 60
(based on [20]) and the charge ςc Compensator angular frequency 0.5 1/s
amplifier gc Compensator damping coefficient 0.9 –
Cf Feedback gain coefficient 1 s2
Capacity of charge amplifier nF
a gain, is positively fed back to the structure, the term ‘positive position’ is defined.
This property makes the PPF controller very suitable for collocated actuators and
sensors.
The advantage of PPF controller is that the damping of a specific frequency band
can be increased. However, one PPF controller can suppress only one mode at a
time. For this reason, only the vibration suppression of the beam’s first mode
(fundamental frequency) is presented in this paper. The charge generated in the
piezoelectric sensor due to his deformation is calculated according to Preumont [20]
QSðtÞ = ESd31, SðhB + hSÞbS Àw′ðx = aS + lS, tÞ − w′ðx = aS, Á ð5:11Þ
2 t.
A charge amplifier, which is connected to the piezoelectric sensor, converts the
charge at the input of the amplifier to a voltage at the output. The sensor’s voltage
VSðtÞ = QSðtÞ ð5:12Þ
Cf
is inputted into the PPF controller and VA(t) is the calculated output. Here, Cf
represents the capacitance of the charge amplifier. The transfer function in the
Laplace domain that describes the operation of the PPF compensator is
CðsÞ = VAðsÞ = s2 + gcω2c + ω2c , ð5:13Þ
VSðsÞ 2ςcωcs
where ωc is the compensator’s angular frequency, ςc is the compensator damping
coefficient, and gc is the feedback gain coefficient. All properties of the compensator
and the charge amplifier are summarized in Table 5.2.
5.2.4 Numerical Results of a Monte Carlo Simulation
Five design variables are selected for the stochastic analysis: the length of the beam
lB, the positions of the piezoelectric elements aA and aS, the position of the lateral
load aF, and the feedback gain coefficient gc. The scattering behavior of the design
variables is characterized by specified density functions; therefore, uniform
104 S. Ochs et al.
Table 5.3 Lower and upper limits of the uniform distributions
Symbol Description Lower limit Upper limit Unit
Length of beam 195 205 mm
X1 = lB Position of piezoelectric actuator 7 23 mm
X2 = aA Position of piezoelectric sensor 27 43 mm
X3 = aS Position of lateral load 0.85 ⋅ lB 0.95 ⋅ lB mm
X4 = aF Feedback gain coefficient 0.8 0.9 s2
X5 = gc
distributions between upper and lower limits are chosen. These limits are sum-
marized in Table 5.3.
It is expected that the variables X1, X2, X3, and X5 have a direct impact on the
system behavior because they affect the action of the PPF controller. The position of
the force X4 does not affect the control of the fundamental frequency; its influence
should not be detectable. Furthermore, the Monte Carlo simulation should show
whether interactions between the design variables affect the system behavior.
Figure 5.2 exemplifies the transfer function between velocity and load of the free
beam end for the passive system (without PPF control) and the active system (with
PPF control).
The implemented single mode PPF controller produces a significant reduction of
vibration at the tuned mode. The other modes are not affected. The level of reso-
nance amplitude reduction Y1 and the offset of the fundamental frequency Y2 are
considered as output variables of the system.
The sensitivity analysis is performed using the method presented in Sect. 5.2.1.
The sample for the corresponding Monte Carlo simulation is created using Sobol’
sequences [12]. The simulation was done with the aid of Mathworks MATLAB®.
Fig. 5.2 Schematic diagram of the transfer function between velocity and load of the free end of
the beam
5 Efficient Experimental Validation of Stochastic … 105
Fig. 5.3 Calculated main and
total effects of the design
variables to the outputs
The calculated main and total effects on the outputs Y1 and Y2 are shown in
Fig. 5.3. The individual columns stand for the respective main and total effect of a
design variable per output. The sum of all main effects of an output equals 1.
The feedback gain coefficient X5 has the highest impact on the amplitude
reduction Y1. The positions of the piezoelectric elements X2 and X3 also have an
influence, but not in the same magnitude as the gain coefficient. As expected, the
influence of the position of the lateral load X4 is zero, because it does not affect the
quality of the PPF control.
The analysis of the first resonance frequency offset Y2 shows a different
behavior. The length of the beam X1 has the highest impact. The influence of the
other variables is very low. The numerical analysis also demonstrates that no strong
interactions appear in both output variables. The validity of these results will now
be confirmed experimentally. The required approach is described in the next
section.
5.3 Experimental Validation of Stochastic Sensitivity
Analyses
5.3.1 Model-Based Experimental Design
In principle, the methods of DoE [8, 9] are suitable for the experimental validation
of the numerical sensitivity analysis because interactions can be detected with most
designs. According to the classical approach of DoE, either factorial or fractional
106 S. Ochs et al.
Table 5.4 Simulation combinations (SC) of the optimal experimental design
SC 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
X1 (–) (–) (–) (–) (–) (–) (–) (–) (+) (+) (+) (+) (+) (+) (+) (+)
X2 (–) (+) (–) (+) (–) (+) (–) (+) (–) (+) (–) (+) (–) (+) (–) (+)
X3 (+) (–) (–) (+) (–) (+) (+) (–) (–) (+) (+) (–) (+) (–) (–) (+)
X4 (–) (–) (–) (–) (+) (+) (+) (+) (–) (–) (–) (–) (+) (+) (+) (+)
X5 (–) (–) (+) (+) (–) (–) (+) (+) (–) (–) (+) (+) (–) (–) (+) (+)
factorial (non-optimal) designs can be used. Furthermore, a model-based design of
experiments, also called optimal design, combines an experimental design with the
results of a previous numerical sensitivity analysis. Thus, the number of necessary
experiments can be reduced, compared to the non-optimal design. Various statis-
tical criteria are available to optimize the experimental design. In this work, the
D-optimality criterion is used [21], which seeks to maximize the determinant of the
information matrix XTX of the design, where X is the matrix of the design vari-
ables. A D-optimal design is not generated with a fixed pattern, but constructed
iteratively so that the determinant of the information matrix is maximized. This
process is carried out with a coordinate exchange algorithm [22].
Under the assumption of a linear system behavior (theory of EULER-BERNOULLI)
the consideration of two levels per variable is acceptable. The upper and lower
limits of the numerical analysis (Table 5.3) denoted with (+) and (–) are used for
these two levels. A five-factor factorial experimental design with 25 = 32 simula-
tion combinations would be necessary. But the results of the numerical analysis in
Sect. 5.2.4 show that no strong interactions appear in both outputs. Hence, inter-
actions of higher order do not need to be considered; only second-order interaction
effects have to be observed. The generated D-optimal experimental design which
takes into account the results of the numerical analysis requires only 16 simulation
combinations (SC) as listed in Table 5.4. With this design, the main effects and the
second-order interactions can be analyzed to confirm the results of the numerical
analysis. Higher order interactions are not considered because they are not proba-
ble. In contrast to a five-factor factorial experimental design, the model-based
approach reduced the experimental effort by 50%, without great loss of information.
5.3.2 Experimental Setup
A test bench is set up to check the usability of the optimal experimental design for
the validation of numerical sensitivity indices. It is shown in Fig. 5.4.
Various beams consisting of aluminium and piezoelectric elements are fabricated
for the experimental studies, such as those described in the experimental design
(Tables 5.3 and 5.4). The piezoelectric materials used are PIC151 for the sensor and
PIC255 for the actuator. The piezoelectric materials are bonded to the beam with an
5 Efficient Experimental Validation of Stochastic … 107
Fig. 5.4 Experimental configuration (view from top)
adhesive film. The beam is clamped with two thick steel brackets bolted to a heavy
block of steel.
Real-time active vibration suppression is implemented using a dSPACE digital
control system. The block diagram of the PPF control system, built in Simulink, is
converted to C-code, which is then compiled and implemented on the dSPACE
hardware to achieve real-time simulation and control. The lateral load is simulated
by a force impulse with an electrodynamic shaker. The impact is recorded with a
force sensor and the velocity is measured with a laser vibrometer, whose laser beam
is directed at the end of the piezoelectric beam.
5.3.3 Experimental Validation
In this section, numerical (n) and experimental (e) response functions for the pas-
sive system (without PPF control) and the active system (with PPF control) are
compared, followed by an experimental validation of the numerical sensitivity
indices.
The challenge in the experimental validation of numerical results lies in ensuring
that boundary conditions remain identical. A stiff connection of the electrodynamic
shaker to the beam would affect the system behavior of the piezoelectric beam.
108 S. Ochs et al.
Therefore, the beam system is excited by a force impulse. The response of the smart
beam (SC 16) to a force impulse of 3 N at 0.1 s is shown in Fig. 5.5. The
numerically calculated system response is illustrated in the upper diagram. The
decay time of the active system is reduced by 85%, compared to the uncontrolled
system, due to the damping introduced by the PPF controller. The numerically
calculated system behavior is consistent with the real system behavior, as shown in
both diagrams of Fig. 5.5.
Also the experimental and the numerical values for the fundamental frequency
correlate well as shown in Fig. 5.6.
A quantitative agreement between the numerical and experimental absolute
values is not necessary for the following sensitivity analysis. The calculation of the
sensitivity indices is carried out by looking at the relative changes in the outcomes
between the simulation combinations. Thus, it is only necessary to ensure that the
numerical model has the same qualitative behavior as the experiment.
The experimental sensitivity indices are not calculated like the numerical indices
in Sect. 5.2.1 because the number of simulation combinations is insufficient. In a
two-level factorial design, the effect
Fig. 5.5 Impulse response
function (SC 16), with a load
F = 3 N, calculated
numerically (n) in the upper
diagram, measured
experimentally (e) in the
lower diagram
5 Efficient Experimental Validation of Stochastic … 109
Fig. 5.6 Impulse transfer
function between velocity and
load of beam (SC 16),
calculated numerically (n) in
the upper diagram, measured
experimentally (e) in the
lower diagram
EXi = Y Xi+ − Y Xi− ð5:14Þ
of a variable Xi is defined as: the change in response Y produced by a change in the
level of that variable, averaged over the levels of the other variables. Thus, the
effect of X1 is calculated as the average of the results of SC 9–16 minus the average
of the results of SC 1–8. The effect index
EbXi = jEXi j ð5:15Þ
∑i5= 1 jEXi j
is based on the sum of all absolute effects for comparison with the calculated main
effect SM of the numerical simulation. All calculated sensitivity indices (numerical
and experimental) of the design variables to the outputs Y1 and Y2 are shown in
Fig. 5.7.
For output Y1 the calculated indices closely match. The recognized influence of
the position of the load X4 contradicts the numerical results. This influence can also
be seen in the results for output Y2. Probably the applied load in the experimental
setup deviated from the load in the numerical simulation. The large influence of the
110 S. Ochs et al.
Fig. 5.7 Comparison of the
calculated main effects of the
design variables to the outputs
length of the beam X1 on the offset of the fundamental frequency Y2 cannot be
confirmed in the experiment. Instead, a much larger impact of the feedback gain
coefficient X5 is identified. However, it can be confirmed that the length of the beam
and the feedback gain coefficient have a greater impact than the positions of the
piezoelectric elements and the load.
In Sect. 5.2.4 the results of the numerical analysis show that no strong inter-
actions appear in both outputs; but second-order interactions are possible. To verify
these results, the effects of interactions Xij can be calculated similarly to Eq. (5.14)
using the experimental data. The results for Y1 and Y2 are shown in Fig. 5.8.
Stronger as expected, second-order interaction effects can be seen. The com-
parison with Fig. 5.7 shows that some interaction effects have a similar magnitude
to the design variables; thus, they cannot be ignored. The strongest interaction is
observed between the positions of piezoceramic actuator and sensor X23. In addi-
tion, interactions between the structure and the electronic components can be rec-
ognized, e.g., X12 and X13.
Higher order interactions cannot be clearly determined with the used experi-
mental design. Due to the reduced test effort, the higher order interactions effects
are superposed with the main and first order interaction effects [9]. To analyze the
higher order interactions as well, the experimental design has to be extended to a
full factorial design. This requires the testing of 16 additional simulation combi-
nations. However, the testing of the additional combinations and the evaluation of
the full factorial design demonstrated no significant higher order interactions. This
outcome was expected based on the numerical results. Thus, the proposed
D-optimal experimental design is the most efficient design to validate the results of
the stochastic analysis.
5 Efficient Experimental Validation of Stochastic … 111
Fig. 5.8 2nd order
interaction effects to the
output Y1 and Y2
5.4 Conclusions
Experiments are an essential method of scientific and technical work. However,
they are often expensive and time-consuming. Therefore, the number of attempts
should be kept as small as possible. The present paper has suggested an efficient
procedure for experimental validation of stochastic sensitivity analyses of smart
systems. The proposed method was performed on a smart piezoelectric beam
structure. The main effects of system variables on the performance of an active
system, as well as the interactions between system variables, were determined.
The authors show that the methods of DoE are generally suitable for the
experimental study of smart systems. However, not all experimental designs allow
an efficient validation of smart system behavior. In addition to the main effects
various interaction effects have to be analyzed, too. Those interaction effects arise
for example from the combination of mechanical and electronic components. The
authors demonstrated that interaction effects are partly more important than main
effects. Thus, one-factor-at-a-time designs are not suitable to validate the results of
stochastic sensitivity analyses. A full factorial design allows the validation of the
main and interaction effects; however, the test effort increases exponentially with
the number of the design variables. For this reason, the use of an optimized design
is proposed in the present paper.
Various statistical criteria are available to optimize the experimental design. In
this project, the D-optimality criterion is tested. An optimal design, also called
model-based design, combines an experimental design with the results of the
112 S. Ochs et al.
previous numerical sensitivity analysis. The results of numerical analysis of the
piezoelectric beam showed that the behavior of the smart system is influenced only
by the main and first order interaction effects. Therefore, higher order interactions
should not be included in the experiment. Thus, the number of necessary simulation
combinations could be reduced, by 50%, compared to the non-optimal design.
The conducted experimental study validated the results of the previous numer-
ical sensitivity analysis. Hence, the suitability of variance-based methods for the
numerical analysis of smart systems could be confirmed. In addition, a full factorial
design was carried out to check the results of the used model-based design.
However, no further interaction effects were identified. Finally, this study shows
that sensitivity analyses provide a useful tool for analyzing the performance
robustness of smart systems.
References
1. Gunawan R et al (2005) Sensitivity analysis of discrete stochastic systems. Biophys J
88:2530–2540
2. Coupé VMH, v. d Gaag LC (2002) Properties of sensitivity analysis of Bayesian belief
networks. Ann Math Artif Intell 36:323–356
3. McCandless LC, Gustafson P, Levy A (2006) Bayesian sensitivity analysis for unmeasured
confounding in observational studies. Stat Med 26:2331–2347
4. Han SO, Wolf K, Hanselka H (2009) Reliability analysis of a smart structure system
considering the dependence on parameter uncertainties. In: Proceedings of 2nd international
conference on uncertainty in structural dynamics
5. Li Y et al (2012) Approaches to sensitivity analysis for system reliability study of smart
structures for active vibration reduction. Int J Reliab Qual Saf Eng 19:1250025
6. Li Y et al (2013) Sensitivity analysis-assisted robust parameter design of an adaptive vibration
neutralizer. In: Proceedings of the 11th international conference on structural safety and
reliability ICOSSAR
7. Li S, Ochs S, Melz T (2015) Design of control concepts for a smart beam structure with
regard to sensitivity analysis of the system. In: Proceedings of the 7th ECCOMAS thematic
conference on smart structures and materials
8. Fischer RA (1935) The design of experiments. Oliver & Boyd, Edinburg
9. Montgomery D (2009) Design and analysis of experiments. Wiley, Chichester
10. Li Y et al (2012) Experimental sensitivity analysis robustness studies of a controlled system.
J Smart Mater Struct 21(6):064002
11. Saltelli A et al (2008) Global sensitivity analysis—The Primer. Wiley, Chichester
12. Sobol’ IM (1993) Sensitivity estimates for nonlinear mathematical models. Math Model
Comput Exp 1:407–414
13. Jansen MJW (1999) Analysis of variance designs for model output. Comput Phys Commun
117:35–43
14. Saltelli A et al (2010) Variance based sensitivity analysis of model output. Design and
estimator for the total sensitivity index. Comput Phys Commun 181(2):259–270
15. Zori´c ND et al (2012) Optimal vibration control of smart composite beams with optimal size
and location of piezoelectric sensing and actuation. J Intell Mater Syst Struct 24(4):499–526
16. Stavroulakis GE et al (2005) Design and robust optimal control of smart beams with
application on vibrations suppression. Adv Eng Softw 36(11–12):806–813
5 Efficient Experimental Validation of Stochastic … 113
17. Hong S et al (2006) Vibration control of beams using multiobjective state-feedback control.
Smart Mater Struct 15:157–163
18. Park CH (2003) Dynamics modelling of beams with shunted piezoelectric elements. J Sound
Vib 268:115–129
19. Goh CJ, Caughey TK (1985) On the stability problem caused by finite actuator dynamics in
the collocated control of large space structures. Int J Control 41:787–802
20. Preumont A (2002) Vibration control of active structures—an introduction. Kluwer Academic
Publishers
21. Walter E, Pronzato L (1985) Robust experiment design via stochastic approximation. Math
Biosci 75:103–120
22. Meyer RK, Nachtsheim CJ (1995) The coordinate-exchange algorithm for constructing exact
optimal experimental designs. Technometrics 37:60–69
Chapter 6
Design of Control Concepts for a Smart
Beam Structure with Sensitivity Analysis
of the System
Sushan Li, Steffen Ochs, Elena Slomski and Tobias Melz
Abstract A smart structure is a structure that can reduce a structural vibration by
means of the integration of one or more sensor, actuator, and controller. A sensor de-
tects a vibration in the structure and transfers a signal to a controller. The controller,
designed to compensate the structural vibration, then computes the desired control
signal and sends it to an actuator. A piezoelectric ceramic patch is often used, as
in the present study, as a sensor or an actuator in a smart structure. A smart struc-
ture with a properly designed controller can reduce the structural vibration without
changing the structure’s physical dimensions. Since a smart structure contains more
uncertainty factors, not least due to the additional interfaces in comparison to a pas-
sive structure, the smart structure should be well analyzed to ensure its reliability
and robustness. This paper focuses on how to set up a numerical model for a smart
beam structure, how to design its control concept, and how to investigate the con-
troller’s robustness by means of the Design of Experiments (DoE) method (In this
paper experiment refers to numerical simulation.). A full factorial design is used,
in which the parameters of the smart structure are either varied in their distributed
range or held constant, so that several structures are designed with slight variations.
This study aims at determining, which controller is the most robust by comparing the
performance for different structural variations of the smart structure. Only if a smart
structure is connected to a robust controller, its reliability can be analyzed, which is
the aim of further research.
S. Li (✉) ⋅ S. Ochs ⋅ E. Slomski ⋅ T. Melz 115
Research Group System Reliability and Machine Acoustics SzM, TU Darmstadt,
Magdalenenstr. 4, 64289 Darmstadt, Germany
e-mail: [email protected]
S. Ochs
e-mail: [email protected]
E. Slomski
e-mail: [email protected]
T. Melz
e-mail: [email protected]
© Springer International Publishing Switzerland 2017
A.L. Araujo and C.A. Mota Soares (eds.), Smart Structures and Materials,
Computational Methods in Applied Sciences 43,
DOI 10.1007/978-3-319-44507-6_6
116 S. Li et al.
6.1 Introduction
Nowadays, lightweight materials are widely used in many machines to reduce the
costs of production and power consumption. Unfortunately, this leads to another is-
sue: Under the same excitation, a structure made of lightweight materials tends to
be affected by stronger vibrations than a structure made of conventional materials.
This issue can be solved by using a smart structure [15]. With a properly designed
control concept through the integration of one or more sensor and actuator, a smart
structure can reduce the vibration of a lightweight structure without changing the
physical dimensions of the structure. But due to the integration of sensor, actuator,
and controller, the smart structure contains more uncertainty factors in comparison
to a passive structure. Therefore, it should be well analyzed to ensure its reliability
and robustness.
In this paper a smart beam structure (described in detail in Sect. 6.2) is used as
a reference structure to clarify how to design a control concept that reduces the vi-
bration of the beam structure and how to investigate the controller’s robustness ac-
cording to a full factorial design. At first, the parameters of the smart beam structure
are combined in predefined variations, so that structures with varied parameters are
established. Consequently, the control concept must be proven to be robust for the
reference structure and its structural variations.
The numerical model of a smart beam structure has two parts: the finite element
(FE) model of the structure and its control strategy. Karagülle et al. [8] explain a way
to build the numerical model of a smart beam structure and control system using the
FE software ANSYS. The displacement of the beam end in the time domain indicates
the performance of the control system subjected to an instantaneous excitation. Un-
fortunately, this modelling is inconvenient for a frequency response analysis, which
can directly represent the vibrational behavior of the beam over a wide frequency
range. MATLAB is also widely used for controller design [10, 19], but since the
FE model matrices are oversized in this study, they are almost impossible to be cal-
culated in the process of controller design. Rudnyi and Korvink [16] point out that
by using a model order reduction (MOR), the size of the structural matrices can be
reduced to a calculable size. The reduced matrices can then be used in the control
system. Hence, the modelling process used in this paper, to set up the numerical
model of the smart beam structure, starts by setting up an FE model in ANSYS.
Then the FE model matrices are exported to a commercial software called MOR
for ANSYS [16], where their orders are reduced. Based on this reduced model, the
controller used to compensate the beam’s vibration is designed in MATLAB. The
complete modelling process chain is shown in Fig. 6.1.
There are a lot of different control concepts that may be used to reduce a beam
structure’s vibration. The Linear Quadratic Regulator (LQR) and its extension, as
well as the Linear Quadratic Gaussian regulator (LQG), are frequently discussed as
controllers for a smart beam structure [17, 20]. Lead control (LC) is a popular active
damping control method, which can be used to compensate the beam structure’s vi-
bration [1]. Therefore, LQR and LC are considered in this study for further analysis.
6 Design of Control Concepts for a Smart . . . 117
Fig. 6.1 The process chain of the numerical model building
Fig. 6.2 The smart beam structure, consisting of an aluminum beam, one sensor, one actuator, and
a controller
In Sect. 6.2 the smart beam structure is introduced and the way to build its FE
model in the reduced state-space form is shown. Section 6.3 focuses on the design
process of the potential control concepts for the reference smart beam structure. In
Sect. 6.4, their performances are compared according to selected criteria. The robust-
ness of the control concepts is investigated by varying parameters of the smart beam
structure according to a full factorial design, and the results are finally discussed in
Sect. 6.5.
6.2 The Smart Beam Structure
In this study, the smart beam structure consists of an aluminum beam structure, one
sensor, one actuator, and a controller (Fig. 6.2).
This beam, with one clamped end and one free end, is assumed to be an Euler-
Bernoulli beam. Therefore, its deformation is based on the basic equation of struc-
tural dynamics [18]. A vertical harmonic force at the free beam end acts as an excita-
tion on the beam structure. Piezoelectric ceramic patches are widely used as sensors
or actuators in a smart structure [8, 13, 19]. In this study the piezoelectric ceramic
PIC151 is chosen for this smart beam structure because of its high permittivity, high
coupling factor, and high piezoelectric charge constant [14]. Two of these piezoelec-
tric ceramic patches are collocated at the top and the bottom of the beam and act as
an actuator and a sensor, respectively. The sensor detects the vibrations of the beam
and transfers a signal to the controller. The controller, designed to compensate the
118 S. Li et al.
Table 6.1 The dimensional and material’s data of the reference smart beam structure
Beam Actuator Sensor
Length L in mm 200 50 10
Width W in mm 40 30 10
Thickness T in mm 3 1 1
Density in kg/m3 2,700 7,800 7,800
Young’s modulus E in GPa 70 62.1 62.1
16.1 16.1
Elastic constant s11 in 10−12 m2∕N – –180 –180
Charge constant d31 in 10−12 C/N – 10 –
– 30
Position SA in mm –
Position SS in mm –
beam vibration, then computes the desired control signal and sends it to the actua-
tor. The geometric parameters and material properties of the reference smart beam
structure are listed in detail in Table 6.1.
6.2.1 Finite Element Model
The FE model of the smart beam structure is built using the software package AN-
SYS Workbench [6]. First, the mechanical structure of the aluminum beam and two
piezoelectric ceramic patches is set up in ANSYS. The contacts between the piezo-
electric ceramic patches and the beam are defined as ideally bonded [5]. The element
type for the beam is SOLID186,1 which is a three-dimensional structural SOLID ele-
ment [2]. The element type SOLID226,2 which is also a three-dimensional structural
SOLID element, but used for coupled field components, is selected for the piezoelec-
tric ceramic patches [2]. Because the beam and the piezoelectric ceramic patches are
all rectangular cuboid in shape, the elements are set to be cube. To determine a proper
size for the elements, some pretests are carried out. As there are no differences be-
tween the simulation results when the element is tested at sizes of 2 and 4 mm [12],
the size of the elements used for the simulations described in this paper is set to be
4 mm, as this considerably reduces the computational effort.
The structural damping is defined according to Rayleigh damping, which is a
mass- and stiffness-proportional damping given by
D = M + K , (6.1)
1SOLID186: standard element type in the software ANSYS.
2SOLID226: standard element type in the software ANSYS.
6 Design of Control Concepts for a Smart . . . 119
where D is the approximated structural damping matrix in Ns/m, M is the struc-
tural mass matrix in kg, K is the structural stiffness matrix in N/m, is the mass-
proportional damping coefficient in s−1, and is the stiffness-proportional damping
coefficient in s [3]. According to the ANSYS help system [3] and Cai et al. [4], the
mass damping represents the friction damping and in most case it can be assumed
to be zero. In [9] an FE model of a smart beam structure, which is similar to the
smart beam structure in this study, is set up in ANSYS, and the mass damping is
neglected. Therefore, the mass damping is considered to be = 0 s−1 in this study.
The overall structural damping ratio obtained from experimental studies is about
= 5 %, and the corresponding stiffness value is set to be = 10−5 s.
After building the numerical model, the structural matrices including M, K, B
(the input matrix), and C (the output matrix) can be extracted to describe the dynamic
behavior of the whole structure in the form of differential equations [3, 12]. The
dimensions of these matrices depend on the number of FE nodes. Therefore, the
dimensions vary according to the structural variations.
6.2.2 Model Order Reduction
The FE model of the reference smart beam structure has 4,735 nodes in total and
each node has six degrees of freedom, resulting in a matrix dimension of 28,410 ×
28,410. Hence, the structural matrices are oversized and cannot be calculated in the
process of controller design by the software MATLAB. To solve this issue, MOR is
carried out in this study using the software MOR for ANSYS [16], which is based on
the Krylov subspace method. The difficulty of MOR is to define expansion points to
ensure the accuracy of the MOR and the non-singularity of the reduced matrices. Ac-
cording to the results of the pretests, two expansion points are defined at (−10, −105)
and six dimensions are expanded at each point [12].
After reducing the structural matrices, the beam structure can be described by
Mrq̈ + Drq̇ + Krq = Bru (6.2)
y = Crq ,
where Mr, Dr, Kr, Br, Cr are the reduced matrices of M, D, K, B, and C, respectively
[16], q is the state vector with unknown degrees of freedom [11], u is the input vector,
and y is the output vector. The dimensions of the reduced structural matrices and
vectors are listed in Table 6.2. The input vector u is composed of the force at the beam
end u1 and the actuator voltage u2. The output vector y includes the displacement of
the beam end y1 and the sensor voltage y2.
To prove the accuracy of the MOR, the vibration behavior of the reduced beam
structure is compared to that of the non-reduced structure. As already mentioned,
the non-reduced matrices are oversized to be read in MATLAB, thus a harmonic
analysis in the frequency range from 1 to 1,300 Hz with 0.1 Hz solution intervals is
120 S. Li et al.
Table 6.2 The dimensions of the structural matrices and vectors after the MOR
Mr Dr Kr Br Cr q u y
2×1
Dimensions 12 × 12 12 × 12 12 × 12 12 × 2 2 × 12 12 × 1 2 × 1
Fig. 6.3 Comparison of the
Bode magnitude plots of the
reference smart beam
structure before and after
MOR (input an 1 N harmonic
force at the free beam end,
output the displacement of
the free beam end)
carried out in ANSYS. The results show the vibration behavior of the beam structure
under an 1 N harmonic force in the z-direction (grey curve in Fig. 6.3). The red curve
represents the result for the reduced beam structure. It is obvious that the two curves
fit well.
The differential equation with reduced structural matrices can also be converted
to the state-space form
ẋ = Assx + Bssu (6.3)
y = Cssx + Dssu
[] [ ]
q u1
with the state vector x= ]q̇ , the input vector u = u2 ,
[ [
y1 0 I ]
[y2 −Mr−1Kr −Mr−1Dr
the output vector y = , the state matrix Ass = ,
the input matrix Bss = ] []
0 = Cr 0 ,
Br , the output matrix Css
and the feed through matrix Dss = 0.
The control plant of the smart beam structure in state-space form is shown in
Fig. 6.4. The whole system is a multiple input and multiple output (MIMO) system.
A controller, which is the interface between the sensor voltage y2 and the actuator
voltage u2, is designed to compensate the displacement of the beam end y1 when the
structure is subject to the force u1.
6 Design of Control Concepts for a Smart . . . 121
Fig. 6.4 The smart beam structure’s control plant in state-space form
Fig. 6.5 Block diagram of an optimal LQR applied to the smart beam structure
6.3 Control Concepts
In general, control concepts are distinguished into two groups. The first group
consists of model-based controllers including the LQR. By designing the model-
based controllers, the control plant of the structure and the structural matrices
must be known. The second group consists of non-model-based controllers, e.g.,
LC, which require no information about the structure except its natural frequencies
[15]. The process of designing these two controllers and their technical parameters,
particularly for the reduction of vibration at the fundamental frequency f1 of the
smart beam structure, is explained in this Section.
6.3.1 Linear Quadratic Regulator
An LQR can be designed based on the state-space model according to the principle
of state feedback (Fig. 6.5). An optimal LQR assumes the knowledge of the state
vector x instead of the sensor voltage y2 [15].
An optimal LQR seeks a linear state feedback with constant gain
u = −Gx (6.4)
122 S. Li et al.
to ensure that the following quadratic cost function J is minimized
J = ∫ (xT Qx + uT Ru)dt , (6.5)
with suitably chosen matrices Q and R. For choosing Q and R, Q must be positive
semi-definite (Q ≥ 0), which means that some of the states may be irrelevant for
the design of the controller, while R must be positive definite (R > 0). Thus, this
expresses that every control has a cost. With suitably chosen matrices Q and R, the
matrix S in the Riccati equation
ATssS + SAss − (SBss)R−1(BTssS) + Q = 0 (6.6)
can be calculated, which means the optimal G in the LQR can also be defined ac-
cording to
G = R−1BsTsS . (6.7)
The structure matrices Ass and Bss are exported from ANSYS to MATLAB via
MOR for ANSYS. Since the smart beam structure has two inputs, the force u1 lo-
cated at the beam end and the actuator voltage u2, the matrix Bss has two columns.
According to the control plant in Fig. 6.4 and the block diagram in Fig. 6.5, the con-
troller computes the desired control signal based on the sensor signal. Therefore,
only the second input, the second column of the matrix Bss, is needed for the con-
troller design in Eqs. (6.6) and (6.7). Since the state feedback is only from one output
(y2) to one input (u2), the R in this study is not a matrix but rather a scalar R. And
since the actual output of the system is used as the control variable, Q is defined as
Q = 1C0rT4Cthr.e Multiple testing of the controller lead to the conclusion that by setting
R = LQR compensates the displacement at the end of the reference smart
beam structure better than other settings.
6.3.2 Lead Control
As already mentioned, the LC belongs to the non-model-based control concepts. A
structure with a collocated, dual actuator/sensor pair can be actively damped with an
LC [15]. The transfer function of the LC is
H(s) = g s + z (6.8)
s + p
with p ≫ z.
Figure 6.6 shows the block diagram of the LC. This controller produces a phase
lead in the frequency band between the zero z and the pole p with an amplification g.
6 Design of Control Concepts for a Smart . . . 123
Fig. 6.6 Block diagram of the LC applied to the smart beam structure
As a result, all modes z < i < p are actively damped. Therefore, the pole p must be
set to be larger than the zero z.
The MATLAB Simulink Control Design Toolbox is used to obtain the optimal
settings for the parameters g, z, and p. By repeatedly varying the pole or the zero
position in the toolbox and by comparing the simulation results from various set-
tings, it is determined that when the settings are g = 3.85, z = 6, and p = 19,664, the
vibration at the beam end is best compensated in comparison to other settings.
6.4 Comparison of Control Concepts Applied
to the Reference Smart Beam Structure
In this section, the performances of these two controllers are compared in the fre-
quency domain and time domain through inspection of their respective Bode magni-
tude plots and step responses, when the controllers are applied to the reference smart
beam structure.
6.4.1 Bode Magnitude Plot
The frequency response of the smart structure, with and without the controller, is il-
lustrated in the same Bode magnitude plot (Fig. 6.7). Since the controller is designed
to reduce the beam vibration when the structure is subject to an 1 N harmonic force
at the free beam end, even though the system is a MIMO system, the most important
subsystem is the relation between the displacement and the force at the beam end.
Therefore, only the magnitude plot for this subsystem is shown here. By comparing
the two curves, it can be directly determined whether the controller can compensate
the vibration at the beam end or not. In Fig. 6.7, it can be observed that the first peak
of the solid line is always sharper than that of the dashed line, regardless of the con-
troller. This means that both controllers perform well, and the displacement of the
beam end at the fundamental frequency can be compensated.
Two criteria in frequency domain are used to compare the performances of the
considered controllers. The first criterion is the vibration reduction percentage at the
fundamental frequency
124 S. Li et al.
(a) Compensation with LQR (b) Compensation with LC
Fig. 6.7 Bode magnitude plot of the reference smart beam structure (input an 1 N harmonic force
at the free beam end, output the displacement of the free beam end)
y1,1 = (y1,1,no − y1,1,c) × 100 % , (6.9)
y1,1,no
where y1,1,no is the displacement of the beam end y1 at the fundamental frequency f1
without a controller, and y1,1,c is the displacement of the beam end y1 at the funda-
mental frequency f1 with a controller (Fig. 6.7). The second criterion is the shift of
the fundamental frequency f1 (Fig. 6.7b), which indicates if the structure behavior
is altered by shifting the fundamental frequency after the controller is connected.
The smart beam structure connected with the LQR can reduce 92.3 % of the vibra-
tion (corresponding to 22.3 dB) at the fundamental frequency f1, while the reduction
percentage is 96.7 % (corresponding to 29.7 dB) when the beam structure is con-
nected with the LC. The LC, as an active damping controller, modifies the structural
vibration behavior. In this case, the fundamental frequency f1 of the reference smart
beam structure with LC is shifted by f1 = 5.3 Hz, but nevertheless it is not a critical
factor to assess the robustness of the controller.
6.4.2 Step Response
The step response of the smart beam structure with controller is investigated for
each control concept to ensure that the smart structure is stable when it is subject
to a step force u1 = 1 N at the free beam end. The settling time ts and the response
final value Ms of these two controllers are compared as the criteria in time domain.
The settling time ts represents how long it takes until the system is stable. In this
study, stable means the response value reaches and stays within a range of 10 % of
the response final value Ms. The response final value Ms indicates the accuracy of
the controller and it is approximated by calculating the mean response value after
6 Design of Control Concepts for a Smart . . . 125
10-4 10-4
0 0
-1 -1
-2 ts =0.20 s
10%
-3
Ms = - 0.30 mm
-4
-5
-6
0 0.1 0.2 0.3 0.4 0.5
time in s
(a) Compensation with LQR
displacement in m -2 ts =0.11 s
displacement in m
10%
-3
-4 Ms = - 0.30 mm
-5
-6
0 0.1 0.2 0.3 0.4 0.5
time in s
(b) Compensation with LC
Fig. 6.8 Step response of the reference smart beam structure, settling time ts and response final
value Ms
Table 6.3 Comparison of the controllers according to the previously determined criteria
LQR LC
y1,1 in % 92.3 96.7
f1 in Hz 0.4 5.3
ts in s 0.20 0.11
Ms in mm −0.30 −0.30
0.4 s in this study. The step response of the reference smart beam structure with both
controllers is illustrated in Fig. 6.8.
From Fig. 6.8 it can be seen that both controllers lead to a stable state, even if
the beam is subject to a step force. When comparing the settling time ts of both
controllers it can be found that the LC compensates the vibration faster than the LQR.
The structure with the LC needs only 0.11 s to settle the vibration within ±10 % of
the final value Ms, but the structure with the LQR needs 0.20 s, almost twice as long.
The response final values Ms of both structures are almost identical.
The compared data are listed in Table 6.3. f1 describes the offset of the funda-
mental frequency, but it is not a critical factor to assess the robustness of the con-
troller. Therefore, in the following robustness analysis of the controllers, only the
three remaining criteria are used to compare the performance of the two controllers.
126 S. Li et al.
Table 6.4 Full factorial design with 33 SCs LB TB SA
SC LB TB SA SC LB TB SA SC (−) (−) (+)
1 (−) (−) (−) 10 (−) (−) ( 0 ) 19 ( 0 ) (−) (+)
2 ( 0 ) (−) (−) 11 ( 0 ) (−) ( 0 ) 20 (+) (−) (+)
3 (+) (−) (−) 12 (+) (−) ( 0 ) 21 (−) ( 0 ) (+)
4 (−) ( 0 ) (−) 13 (−) ( 0 ) ( 0 ) 22 ( 0 ) ( 0 ) (+)
5 ( 0 ) ( 0 ) (−) 14 ( 0 ) ( 0 ) ( 0 ) 23 (+) ( 0 ) (+)
6 (+) ( 0 ) (−) 15 (+) ( 0 ) ( 0 ) 24 (−) (+) (+)
7 (−) (+) (−) 16 (−) (+) ( 0 ) 25 ( 0 ) (+) (+)
8 ( 0 ) (+) (−) 17 ( 0 ) (+) ( 0 ) 26 (+) (+) (+)
9 (+) (+) (−) 18 (+) (+) ( 0 ) 27
Table 6.5 The design factor values at each level (+)
(−) ( 0 ) 0.2050
0.0035
LB in m 0.1950 0.2000 0.0120
TB in m 0.0025 0.0030
SA in m 0.0080 0.0100
6.5 Robustness Analysis of the Controllers
The robustness analysis of both controllers is carried out according to a 3-level full
factorial simulation. Based on the analysis results, the more robust controller will be
finaly chosen for the smart beam structure.
6.5.1 The Full Factorial Simulation
In the 3-level full factorial simulation, the simulation is carried out by combining the
minimum (−), the midpoint (0), and the maximum (+) values of each varied parame-
ter. Unfortunately, if all the geometric parameters or material properties of the smart
beam structure are varied, the number of the simulation combinations (SCs) is too
large to be executed. Han [7] did a sensitivity analysis of a very similar smart beam
structure based on its analytical model. According to his results, the beam length
LB, its thickness TB, and the actuator position SA have more influence on the beam
vibration than the other parameters. Therefore, these three parameters are chosen
as design factors to investigate the controller robustness, while the other parameters
are held constant at the same values as for the reference smart beam structure shown
in Table 6.1. In this study, 33 = 27 SCs are simulated according to the full factorial
design (Table 6.4). Table 6.5 shows the three varied levels of the design factors.
6 Design of Control Concepts for a Smart . . . 127
100 100
90 90
80 80
y in %
70 70
1,1
y in %
1,1
60 60
50 50
(-) (0) (+) (-) (0) (+)
the varied level the varied level
(a) Compensation with LQR (b) Compensation with LC
Fig. 6.9 The vibration reduction percentage y1,1 at the fundamental frequency f1 at different
levels of LB, TB, SA (To ensure a good legibility of the error bars, the curves for the LB and SA are
slightly shifted to the left or to the right, which does not mean the levels of these two parameter are
shifted)
6.5.2 Results and Discussion
According to the full factorial design plan in Table 6.4, 27 smart beam structure
variations are simulated. Each structure is successively connected with the two con-
trollers without changing the controllers’ parameters. The influences of the three
parameters of the smart beam structure and of the controllers on the vibration are
compared in this subsection according to the three criteria y1,1, ts, and Ms.
From Table 6.4 it can be seen that there are nine SCs (SC 1, 4, 7, 10, 13, 16, 19,
22, and 25) that are set to the low level of LB. The average influence of LB at the low
level on the criterion y1,1 is the average y1,1 of these nine SCs. The maximum y1,1
shows the best case of these nine SCs, and the minimum y1,1 shows the worst case.
Each average y1,1 of the varied levels is plotted in Fig. 6.9. The error bars specify
the maximum and the minimum values at each level. Similarly, the influence of the
other parameters TB and SA at varying levels are calculated and plotted in Fig. 6.9.
From Fig. 6.9 it can be seen that the varying tendencies of the averages y1,1 from
a low to a high level are not linear. All y1,1 in Fig. 6.9 are positive, which indicates
that both controllers can compensate the beam vibration for all SCs (Table 6.4). Tak-
ing into account the error bars in Fig. 6.9a, it can be seen that when using the LQR
the y1,1 is in some cases smaller than 50%. Compared to this, when using the LC
the minimum y1,1 at each level of each parameter is still larger than 96%. The com-
parison of Fig. 6.9a, b indicates that the LC has a better compensation ratio with a
smaller variance than the LQR.
Likewise, the averages of the two other criteria, ts and Ms, of the SCs, calculated
at each level of the three beam parameters, are plotted in Figs. 6.10 and 6.11. The
error bars specify the maximum and the minimum values at each level.
128 S. Li et al.
According to Figs. 6.10 and 6.11, it is difficult to identify the varying tendencies of
ts and Ms according to changes in LB, TB, and SA from a low to a high level. Because
the variations in the error bars of ts and Ms at each level are huge in comparison of
the varied range of the averages from a low to high level. In general, regardless of the
parameters at which level or which controller is used, the maximum ts is shorter than
0.27 s and the maximum Ms is smaller than 0.56 mm, which means both controllers
stabilize the structure that is subjected to the step excitation, in a short time with a
minor displacement at the beam end. However, when comparing Fig. 6.10a, b, it can
be seen that the LC needs half the time to stabilize the step excitation compared to
the LQR.
Based on the numerical simulation results of these 27 SCs, it can be concluded
that the LC, with higher compensation speed, superior compensation ability, and
more stable performance, is more robust than the LQR.
(a) Compensation with LQR (b) Compensation with LC
Fig. 6.10 The step response’s settling time ts at different levels of LB, TB, SA (To ensure a good
legibility of the error bars, the curves for the LB and SA are slightly shifted to the left or to the right,
which does not mean the levels of these two parameter are shifted)
(a) Compensation with LQR (b) Compensation with LC
Fig. 6.11 The step response’s final value Ms at different levels of LB, TB, SA (To ensure a good
legibility of the error bars, the curves for the LB and SA are slightly shifted to the left or to the right,
which does not mean the levels of these two parameter are shifted)
6 Design of Control Concepts for a Smart . . . 129
6.6 Conclusion
This paper focuses on the numerical modelling process of a smart beam structure.
Based on the smart beam structure, two different controllers are designed in order to
compensate the beam vibration. The performances of both controllers are compared
in the frequency domain by means of a Bode magnitude plot, as well as in the time
domain by investigating the step response. A numerical simulation according to a full
factorial design is carried out to investigate the robustness of these two controllers by
varying three parameters of the beam structure at three levels. Both controllers are
found to be robust. By comparing the three criteria y1,1, ts, and Ms, the LC with its
higher compensation speed is found to have a superior compensation ability with a
smaller variance than the LQR. Therefore, it is concluded that the LC is more robust
than the LQR. All the results discussed in this paper are numerical simulation results,
and they need to be validated in experimental simulations. The LQR is designed
based on a complete state feedback, but the complete state vector cannot be measured
in experimental simulation. Hence, an additional observer is needed to estimate the
complete state vector, which makes the control plant more complex. In conclusion,
the LC performs better for the smart beam structure and it is recommended to be used
to compensate the vibration of the beam end. To examine the structure reliability
and robustness, a sensitivity analysis of the smart structure is planned for further
research.
List of Symbols
Parameters of the smart beam structure
LB beam length
WB beam width
TB beam thickness
LA actuator length
WA actuator width
TA actuator thickness
LS sensor length
WS sensor width
TS sensor thickness
B beam density
EB beam Young’s modulus
P piezoelectric ceramic density
s11,P piezoelectric ceramic elastic constant
d31,P piezoelectric ceramic charge constant
SA actuator position
SS sensor position
130 S. Li et al.
Parameters of the model before MOR
M structural mass matrix
K structural stiffness matrix
D approximated structural damping matrix
B input matrix
C output matrix
mass-proportional damping coefficient
stiffness-proportional damping coefficient
damping ratio
Parameters of the model after MOR
Mr reduced structural mass matrix
Kr reduced structural stiffness matrix
Dr reduced approximated structural damping matrix
Br reduced input matrix
Cr reduced output matrix
q state vector with unknown degree of freedom
q̇ derivation of state vector q with respect to time
q̈ 2nd derivation of state vector q with respect to time
u input vector composed of u1 and u2
u1 force at the beam end
u2 actuator voltage
y output vector composed of y1 and y2
y1 displacement of the beam end
y2 sensor voltage
x state vector for state-space form, composed of q̇ and q
ẋ derivation of state vector x with respect to time
Ass state matrix for state-space form
Bss input matrix for state-space form
Css output matrix for state-space form
Dss feed through matrix for state-space form
Parameters of the LQR
G gain matrix by state feedback
J cost function of LQR
Q interim matrix for calculation of Riccati equation
R/R interim matrix/coefficient for calculation of Riccati equation
S interim matrix for calculation of Riccati equation
Parameters of the LC
H transfer function of LC
g amplification of LC
z zero of LC
p pole of LC
i the i-th angular frequency
6 Design of Control Concepts for a Smart . . . 131
Characteristics and criteria to compare the controllers
f1 the fundamental frequency of the smart beam structure
y1,1,no displacement of the beam end at the f1 without controller
y1,1,c displacement of the beam end at the f1 with controller
Δy1,1 vibration reduction percentage at the fundamental frequency
Δf1
ts offset of the f1 with a controller
settling time for step response
Ms
step response’s final value
Acknowledgements The authors would like to thank the German Research Foundation (DFG)
for funding the project “Stochastische Simulationstechniken zur Bewertung der Sensitivität und
Zuverlässigkeit adaptronischer Struktursysteme (Eng.: Stochastic simulation techniques to evaluate
the sensitivity and reliability of smart structural systems)” (grant no.: HA 1634/35-1).
References
1. Al-Ashtari WK (2008) The deflection control of a thin cantilever beam by using a piezoelec-
tric actuator/sensor. In: The 1st regional conference of Eng. Sci. Nahrain university college of
engineering journal (NUCEJ) special Issue, vol 11, pp 43–51
2. ANSYS® Academic Research, Release 12.1: Help system: Element reference (2009)
3. ANSYS® Academic Research, Release 14.0: Help system: ANSYS Mechanical APDL theory
reference (2012)
4. Cai C et al (2002) Modeling of material damping properties in ansys. ANSYS 2002 users’
conference, Pittsburgh USA, April 22–24
5. De Faria AR, De Almeida SFM (1996) Modeling of actively damped beams with piezoelectric
actuators with finite stiffness bond. J Intell Mater Syst Struct 7(6):677–688
6. Gebhardt C (2014) Praxisbuch FEM mit ANSYS workbench. Hanser, Munich Germany
7. Han SO (2011) Varianzbasierte Sensitivitätsanalyse als Beitrag zur Bewertung der Zuver-
lässigkeit adaptronischer Struktursysteme (Eng.: Variance-based sensitivity analysis of smart
structural dynamical systems). Ph.D. thesis, TU Darmstadt
8. Karagülle H et al (2004) Analysis of active vibration control in smart structures by ANSYS.
Smart Mater Struct 13(4):661–667
9. Kostka P et al (2013) In situ integrity assessment of a smart structure based on the local material
damping. J Intell Mater Syst Struct 24(3):299–309
10. Kurch M et al (2009) A framework for numerical modeling and simulation of shunt damping
technology. In: The 16th international congress on sound and vibration (ICSV16), Krakow.
Poland, July 05–09:
11. Kurch M et al (2011) Model order reduction of systems for active vibration and noise control.
In: The 4th international conference on computational methods for coupled problems in science
and engineering, Kos Island Greece, June 20–22
12. Li S et al (2015) Numerical modeling of a smart structure with regard to sensitivity analysis
of the system. In: ANSYS conference and 33rd CADFEM users’ meeting, Bremen Germany,
June 24–26
13. Peng S et al (2012) Modeling of a micro-cantilevered piezo-actuator considering the buffer
layer and electrodes. J Micromech Microeng 22(6): 065005
14. Piezoelectric Materials: Website. http://piceramic.com/products/piezoelectric-materials.html.
Accessed 16 April 2016
132 S. Li et al.
15. Preumont A (2011) Vibration control of active structures: an introduction, 3rd edn. Springer,
Berlin Heidelberg Germany
16. Rudnyi EB, Korvink JG (2006) Model order reduction for large scale engineering models de-
veloped in ANSYS. Lect Notes Comput Sci 3732:349–356
17. Stavroulakis GE et al (2005) Design and robust optimal control of smart beams with application
on vibrations suppression. Adv Eng Softw 36(1112):806–813
18. Strømmen EN (2014) Structural dynamics. Springer International Publishing, Switzerland
19. Xu SX, Koko TS (2004) Finite element analysis and design of actively controlled piezoelectric
smart structures. Finite Elem Anal Des 40(3):241–262
20. Zorić ND et al (2012) Optimal vibration control of smart composite beams with optimal size
and location of piezoelectric sensing and actuation. J Intell Mater Syst Struct 4:499–526
Chapter 7
Adaptive Inductor for Vibration Damping
in Presence of Uncertainty
Bilal Mokrani, Ioan Burda and André Preumont
Abstract This paper considers the RL shunt damping of vibration with a piezoelec-
tric transducer of a structure with a variable natural frequency. The inductive shunt
damping is notorious for not being robust when the natural frequency of the elec-
trical circuit does not match the natural frequency of the structure. In the proposed
implementation, the shunted piezoelectric transducer is supplemented with a small
additional one (with open electrodes) measuring the mechanical extension of the
structure at the location of the transducer. The adaptation strategy uses the property
that, at resonance, the electric charge in the shunted transducer is in quadrature of
phase with the mechanical strain at the location of the transducer (i.e. the voltage in
the transducer with open electrodes). A Phase Shift to Voltage Converter, inspired
from the Phase Locked Loop technique (PLL), is built to evaluate the phase shift
between these two signals and to adapt the (synthetic) inductor L via a voltage con-
trolled resistor, involving a photoresistive optoisolator (photoresistor). The proposed
strategy is supported by simulations and experimental results.
7.1 Introduction
Several piezoelectric shunt damping techniques for structural damping have been
proposed during the last two decades. Typically, a piezoelectric transducer is attached
to the structure to convert the vibratory energy into electrical energy and an electrical
circuit is connected to its electrodes to dissipate the transformed energy.
B. Mokrani (✉) ⋅ A. Preumont 133
Active Structures Laboratory, Université Libre de Bruxelles,
Brussels, Belgium
e-mail: [email protected]
A. Preumont
e-mail: [email protected]
I. Burda
Babes Bolyai University, Cluj-Napoca, Romania
e-mail: [email protected]
© Springer International Publishing Switzerland 2017
A.L. Araujo and C.A. Mota Soares (eds.), Smart Structures and Materials,
Computational Methods in Applied Sciences 43,
DOI 10.1007/978-3-319-44507-6_7
134 B. Mokrani et al.
The simplest piezoelectric shunt involves a single resistor R tuned properly; it is
robust but inefficient. The linear RL shunt provides better performances; it involves
a resistor and an inductor and it works as a dynamic absorber [2, 3]: The piezoelec-
tric transducer, being an electrical capacitance, when it is connected to the inductor,
the resulting circuit is resonant. If the electrical frequency is tuned on the natural
frequency of the structure, then the mechanical vibration excites the electrical reso-
nance; the resistor is used to dissipate the transformed energy and it can be mounted
in series or in parallel with the inductor [4]. Damping several modes can be per-
formed by involving several RL branches [5].
Inductive shunt is notorious for requiring huge inductors and being very sensitive
to the variation of the electrical frequency with respect to the natural frequency of the
targeted mode. These problems are addressed by the Synchronized Switch Damping
on Inductor shunt, referred to as SSDI [13]. It is a nonlinear shunt involving a syn-
chronized switch and a very small inductor, and it offers performances similar to the
RL shunt [8]; however, it requires complex implementation and it is very sensitive to
the switching synchronization. Based on the same concept as the SSDI, several non-
linear shunt techniques have been proposed, where voltage sources are used instead
of the RL circuit (see e.g. [7]).
Other non-linear shunts, based on the same concept as the SSDI, involving syn-
chronized switches and voltage sources exist.
Another way to overcome the problems related to the RL shunt is to use active
synthetic inductors, called gyrators [11], and to adapt the value of the inductor when
the resonance frequency of the targeted mode changes (in such a way that the elec-
trical frequency matches the resonance frequency of the targeted mode). The adap-
tation can be based on two approaches: (i) the first approach consists of choosing
the inductor which minimizes the RMS value of the vibration level [6] (e.g. voltage
across the electrodes of the piezoelectric transducer); (ii) the second approach con-
sists of tuning the inductor based on the fact that at resonance, the electrical charge
Q flowing in the circuit and the strain at the location of the piezoelectric transducer
are in quadrature of phase, only if the electrical frequency matches the natural fre-
quency of the structure [9]. In this paper we consider an adaptive RL shunt based on
the second approach.
The use of the relative phase shift between the electrical charge and the strain (or
velocity), for the adaptation of the inductor, has been first proposed in [9, 10]. In their
work, the relative phase shift between the reference signals is estimated based on the
mean value of their product: when the signals are in phase the product is always
positive and inversely when they are in antiphase, it is zero only when the signals are
in quadrature of phase. The measurement system is performed using analog circuits
and the output is then used to control the resistance of the drain-source channel on a
Field Effect Transistor (FET).
In this paper we propose an adaptive RL shunt damping using a voltage controlled
synthetic inductor, based on a voltage controlled photoresistive optoisolator, referred
to as vactrol [1]. A phase shift to voltage converter circuit has been built to control the
adaptive inductors. The circuit, inspired from Phase Locked Loop technique (PLL),
utilizes a phase detector to measure in real time the relative phase between the voltage
7 Adaptive Inductor for Vibration Damping in Presence of Uncertainty 135
Fig. 7.1 One-dimensional
spring-mass system
equipped with a piezoelectric
linear transducer (the
inherent damping of the
system is neglected for the
sake of simplicity)
drop in the inductor and the voltage of a small additional piezoelectric transducer
(in phase with the local strain). The output signal is then used by a Digital Signal
Processor (DSP) to control the inductor.
The paper is organized in three main parts. The first part discusses the principle of
the linear RL shunt and the effect of parameters detuning on simple single degree of
freedom system. The second part discusses the adaptive shunt and the importance of
the relative phase between the strain and the electrical charge in the circuit. Finally,
the third part is devoted to the experimental implementation of the adaptive RL shunt
on a cantilever beam and to the description of the various circuits.
7.2 Linear RL Shunt
Consider the one-dimensional spring-mass system of Fig. 7.1 subjected to an exter-
nal disturbance d and equipped with a linear piezoelectric stack made of n slices
working according to the d33 operating mode.
The constitutive equations of the piezoelectric transducer1 are:
{ } [ ]{ }
V Ka 1∕Ka −nd33 Q
f = C(1 − k2) −nd33 C x (7.1)
where V is the electrical voltage between its electrodes, Q the electrical charge stored,
x its elongation and f the force applied at its tips; Ka is the stiffness of the transducer
in short-circuit (i.e. V = 0), C is the electrical capacitance when no force is applied
(i.e. f = 0), d33 is the piezoelectric constant, and k is the electro-mechanical coupling
factor, it measures the capability of the transducer of converting mechanical energy
into electrical energy and vice versa.
1In practice, the transducer is mounted with a prestressing structure to prevent the splitting of the
PZT slices under traction. This can be represented by a linear spring K1 mounted in parallel to
the piezoelectric stack. By considering the prestressing spring K1, one can find readily the same
form of the piezoelectric constitutive equations with the effective properties: Ka⋆ = Ka + K1, C⋆ =
C(1 − k2 + k2), d3⋆3 = d33 and k⋆2 = k2 ∕(1 − k2 + k2), where is the fraction of strain energy
in the transducer defined as: = Ka∕(K1 + Ka), see [12].
136 B. Mokrani et al.
The mass M is governed by:
Mẍ = −f + d (7.2)
After substituting the constitutive equations of the transducer, one obtains the gov-
erning equations of the whole electromechanical system of Fig. 7.1 as:
Mẍ + (1 Ka x = d + nd33Ka Q (7.3)
− k2) C(1 − k2)
V = 1 k2) Q − nd33Ka x (7.4)
C(1 − C(1 − k2)
by introducing the definition of the resonance frequency (with open-electrodes):
√
n = Ka ,
(1 − k2)M
and using the Laplace transform, one gets
x = 1 (s2 1d + nd33 n2 Q (7.5)
M + n2) C (s2 + n2)
When the piezoelectric transducer is shunted, in series, on a linear RL circuit, the
voltage V and the charge Q are related through the impedance of the shunt:
V = −LQ̈ − RQ̇ (7.6)
substituting in Eq. (7.4), one finds
Q + RC(1 − k2)Q̇ + LC(1 − k2)Q̈ = nd33Kax (7.7)
(7.8)
and, defining the electrical frequency and damping, (7.9)
e = √ 1 , and 2 e e = R
L
LC(1 − k2)
one gets (after using the Laplace transform):
Q = ( nd33Ka 2e ) x
s2 + 2 e es + 2e
Equations (7.5) and (7.9) describes the full dynamics of the system.
7 Adaptive Inductor for Vibration Damping in Presence of Uncertainty 137
7.2.1 Optimal Tuning
The transmissibility between the mass displacement x and the disturbance force d is
obtained by substituting2 Eq. (7.9) into Eq. (7.5):
x(s) = 1 s4 + 2 e es3 + s2 + 2 e es + e2 + e2 n2 (7.10)
d(s) M ( e2 + n2)s2 + 2 e e n2s
where √
n = Ka
M
is the resonance frequency of the system with short-circuited electrodes.
Equation (7.10) can be used as a metric for the tuning of the RL circuit compo-
nents. Depending on the frequency content of the excitation, the RL circuit can be
tuned in two ways [4] (other types of design are also possible):
∙ Equal peak design, where the circuit is tuned to minimize the H∞ norm, i.e. to
minimize the maximum of the frequency response of x( )∕d( ). This design is
usually referred to as the equal peak design because it results in a double peak with
equal amplitudes of the frequency response, Fig. 7.2. The optimal parameters are
obtained by tuning the electrical frequency e as:
∗e = √1 , or e∗ = 1, (7.11)
n 1 − k2 n
and the electrical damping e as
√ 3k
8
e∗ = (7.12)
∙ Maximum damping (stability) design: In this design, the RL circuit is tuned to
maximizes the damping of the targeted mode, and thus, the stability of the system,
Fig. 7.2c. The optimal tuning of the RL circuit is achieved when the targeted mode
and the electrical mode have the same frequency and damping; it is achieved by
tuning the electrical frequency e as
∗e n = 1, or e∗ = 1 (7.13)
n2 n 1 − k2
and the electrical damping as
e∗ = k (7.14)
2The definition of k2 has been used, where k2 = .n2d323Ka
C
138 B. Mokrani et al.
(a)
(b) (c)
Fig. 7.2 RL shunt tuning: a frequency response of the mass x∕d with open circuited electrodes,
with RL shunt tuned according to the equal peak design and with RL shunt tuned according to the
maximum damping design; b, c the root locus of the system when the resistor R varies from ∞
(open circuit) to 0 (short circuit), with the electrical frequency tuned, respectively, according to the
equal peak design and to the maximum damping design
Next, we will consider only the equal peak design and analyse the effect of the elec-
trical frequency and damping variation on the performance.
Finally, one should notice that for a multi degree of freedom system, the gener-
alized electromechanical coupling factor Ki, of a specific mode i, should be used
instead of k2. It is defined as:
i2 ,
Ki2 = 1 − i2
where i and i are, respectively, the resonance frequencies of mode i with open
electrodes and short-circuited electrodes.
7 Adaptive Inductor for Vibration Damping in Presence of Uncertainty 139
7.3 Robustness of RL Shunt
The three main parameters that may vary and affect the tuning of the RL shunt are
the inherent capacitance of the piezoelectric transducer C, the electromechanical
coupling factor k and the resonance frequency n; a variation of one of the these
parameters deviates the value of the electrical parameters from being optimal, and
thus, affects the performance of the RL shunt. In this section, we study the effect of
the electrical parameters detuning on the performance of the RL shunt.
7.3.1 Sensitivity to R
Figure 7.3 shows the influence of the deviation from the optimal tuning of the resistor
R, on the attenuation generated in the response of the structure (the H∞ norm of x∕d
is considered). An error of the tuning of ±30 % affects the performance by only
(a)
(b)
Fig. 7.3 Influence of the resistor tuni√ng R on the attenuation. The maximum attenuation (corre-
sponding to H∞∗ ) is obtained for e∗ = 3∕8k. a Frequency response x∕d for various values of R
(k = 0.1); b H∞ norm of x∕d (normalized with respect to H∞∗ ) as a function of the variation of R
140 B. Mokrani et al.
10 %, independently from k. The figure shows that the RL shunt is weakly sensitive
to mistuning of the resistor R.
7.3.2 Sensitivity to e
The performance of the RL shunt is very sensitive to the variability of the resonance
frequency of the targeted mode with respect to the electrical frequency e, and thus
to L; this sensitivity is illustrated in Fig. 7.4. The figure shows that an error on the
electrical frequency of ±10 % reduces the attenuation by a factor 2–4, for k vary-
ing from3 0.05–0.2. This high sensitivity justifies the need of an adaptive tuning of
the inductor. The sensitivity to the electrical tuning e tends to decrease when the
electromechanical coupling factor increases.
7.4 Adaptive RL Shunt
A simple way to recover the degradation of the performance of the RL shunt, due to
uncertainties on the electrical frequency, is to adapt the value of L so as to preserve
the optimal tuning. In this section, we show the effect of the electrical frequency
detuning on the relative phase shift between the electrical charge Q and the displace-
ment x. Then we consider an adaptive inductor L, based on the fact that, at resonance,
the relative phase shift between Q and x is ∕2 only if e = n (Eq. 7.11).
7.4.1 Adaptation Law
Consider again the system of Fig. 7.1 with the piezoelectric transducer shunted on
a RL circuit. If the disturbance force d is a band-limited white noise, then, from
Eq. (7.10), the response x is a narrow band random process of central frequency
equal to n; it can be considered as a sine wave with slowly varying amplitude and
frequency (around n). As a first approximation, it can be represented as:
x = A sin nt (7.15)
Therefore, according to Eq. (7.9), since x is nearly harmonic, the electrical charge Q
is also harmonic of the form:
Q = A sin( nt + ) (7.16)
3These values are representatives of typical generalized electromechanical coupling factors Ki met
in practice.
7 Adaptive Inductor for Vibration Damping in Presence of Uncertainty 141
(a)
(b)
Fig. 7.4 Influence of the electrical frequency tuning e on the attenuation. The maximum attenu-
(corresponding to H∞∗ ) obtained for e∗ = n.
ation (k = 0.1); b H∞ norm is x∕d (normalized with a Frequency response x∕d for various values
of e of respect to H∞∗ ) as a function of the variation
of e
where is the magnitude of Q∕x evaluated at s = j n, and is its phase given by:
()
2e
= arg(Q∕x) = arg nd33Ka − n2 + 2 e e nj + 2e (7.17)
From Eq. (7.17), one gets for:
∙ n ≫ e: ≃ − ,
∙ n ≪ e: ≃ 0,
∙ n = e, which corresponds to the optimal tuning of the equal peak design
(Eq. 7.11): ()
= arg − nd33Ka j = −
2 e 2
Therefore, by measuring the relative phase between x and Q, it is possible to detect
the direction of the electrical frequency detuning and to compensate it by adapting
the inductance L in such a way to keep = − ∕2.
142 B. Mokrani et al.
The relative phase between x and Q must be = − ∕2, so that Q is in antiphase
with the velocity ẋ , in order to produce a pure positive damping effect at resonance
where most of vibration energy is concentrated. Indeed, the RL shunt has a damping
effect only around the resonance, within the bandwidth where the relative phase,
between x and Q, −3 ∕4 < < − ∕4, and an amplification effect outside this
bandwidth. This explains the high sensitivity of the RL shunt to the frequency detun-
ing when the electromechanical coupling factor k is small, which is translated by a
small electrical damping e∗ and thus a quick variation of the relative phase around
the resonance frequency.
Finally, one should notice that the adaptation approach presented here assumes
that the structural response is dominated by a single mode. If this response includes
several modes, x and Q will not be pure harmonic signals and the measurement
of their relative phase shift (at resonance) becomes more difficult. This problem
requires a deeper study.
7.4.2 Measurement of Q and x
In practice, it is not easy to measure Q explicitly. However, if one considers the
measurement of the voltage of the inductor VL:
VL = −LQ̈ = −Ls2Q = L 2Q,
then, one obtains a signal VL always in phase with Q. The relationship between VL
and x is then obtained by multiplying Eq. (7.9) by −Ls2:
VL = nd33Ka s2 + −L 2e s2 2e (7.18)
x 2 e es + (7.19)
From this equation, the relative phase between x and VL is deduced as
()
L 2e n2
= arg(VL∕x) = arg nd33Ka − n2 + 2 e e nj + 2e
leading to the same relative phase as that of Eq. 7.17. Once again, when e = n:
()
nd33KaL n2
= arg − 2 e j = − (7.20)
2
The maximum performance of the RL shunt can be guaranteed by tuning the elec-
trical frequency in such a way to keep = − ∕2.
Finally, one should notice that the measurement of x (or the strain at the location
of the shunted transducer) is not a straightforward task. Indeed, when the transducer
has its electrodes open, its voltage is proportional to x, but when they are connected
7 Adaptive Inductor for Vibration Damping in Presence of Uncertainty 143
Fig. 7.5 Magnitude and phase of VL∕x, for various tuning of the electrical frequency e (k = 0.1,
corresponding to e = 6.1 %)
to an impedance Z (e.g. RL circuit), the voltage across the electrodes depends on the
shunt impedance. This fact led us to use a second transducer as a sensor to measure
the strain at the location of the shunted transducer. Obviously, an ideal situation
would be to use a single transducer for the damping and for the measurement of x.
A new circuit inspired from self sensing circuits is under investigation.
7.4.3 Sensitivity of
Consider now the frequency response of VL∕x shown in Fig. 7.5. The figure shows
the effect of the electrical frequency detuning on the magnitude and the relative phase
.
An optimal tuning of e corresponds to a phase shift of exactly − ∕2 at = n,
and the peak of magnitude (which corresponds to the frequency at which the RL
shunt is more efficient). For a deviation of ±5 % of e, the curves are right or left
shifted along the frequency axis and the relative phase at = n varies as a
function of the variation of the electrical frequency e. This demonstrates the high
sensitivity of to the variation of e and also justifies the high sensitivity of the
performance of the RL shunt to the detuning of e. Indeed, for a very small electrical
144 B. Mokrani et al.
Fig. 7.6 Magnitude and phase of VL∕x, for various tuning of the resistor R (or electrical damping
e)
damping e, which corresponds to a smaller electromechanical coupling factor k, this
sensitivity is much pronounced because the phase of VL∕x varies quickly around the
resonance.
Figure 7.6 shows the effect of the resistor detuning R on the magnitude and the
phase of VL∕x. The figures shows that the relative phase at = n is unchanged
for the various values of R. Once a gain, this justifies the low sensitivity of the RL
shunt performance to the tuning of the resistor R.
7.5 Experiments
This section describes the experimental procedure pursued to confirm the theoretical
aspects discussed previously and to implement an adaptive inductor in a linear RL
shunt.
7.5.1 Setup
The adaptive RL shunt has been implemented experimentally on a cantilever beam.
The experimental setup is schematized in Fig. 7.7. The structure consists of a lightly
damped cantilever aluminum beam on which 2 PZT patches are glued close to the
7 Adaptive Inductor for Vibration Damping in Presence of Uncertainty 145
Ground Clamp Damping
Electrode PZT
Cantilever Optional
Reference beam mass
PZT
Damping
PZT
Voltage controlled
synthetic inductor
Fig. 7.7 Experimental setup: Cantilever aluminium beam equipped with two PZT patches attached
with a conductive glue. The beam, being conductive, is used as a common electrical ground
Reference Desired phase
PZT shift 90 (5V)
Structure Phase Shift to
Voltage Converter
Damping
PZT (PSVC)
Phase shift
error
DSP
Fig. 7.8 Principle of the adaptive inductance control
clamp, with opposite polarization and in front of each other. An optional mass is
added at the tip of the beam using small magnets to allow the modification of the
resonance frequency during operation. The excitation consists of a force applied by
a voice-coil at the beam tip with a collocated measurement of the velocity using
a non-contact laser vibrometer. The excitation is a band-limited white noise in the
frequency range [8–180] Hz, to ensure the excitation of the first flexural mode only.
The implemented adaptive RL shunt is schematized in Fig. 7.8. One PZT patch is
used for the damping and it is shunted in series on a tunable resistor R and a voltage
controlled synthetic inductor L. The second PZT is used as a reference, since its
voltage Vref is proportional to the strain at its location, and thus, with analogy to the
single degree of freedom system, it represents the measurement of x. The relative
phase between VL and Vref is measured using a dedicated circuit, and compared to
the desired relative phase shift (− ∕2 for an equal peak design). Then, the relative
phase error is integrated4 using a dSpace DSP and the output is used as a command
of the voltage controlled synthetic inductor.
4We used a pure integral controller since we aim at cancelling the static error between the measured
and the reference phases.
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