VIBRATION SUPPRESSION OF ROTATING BEAMS THROUGH PIEZOELECTRIC SHUNT CIRCUITS by PRESTON POWELL JINWEI SHEN, COMMITTEE CHAIR WEIHUA SU STEVE SHEPARD HAO KANG A THESIS Submitted in partial fulfillment of the requirements for the degree of Master of Science in the Department of Aerospace Engineering and Mechanics in the Graduate School of The University of Alabama TUSCALOOSA, ALABAMA 2016
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VIBRATION SUPPRESSION OF ROTATING
BEAMS THROUGH PIEZOELECTRIC
SHUNT CIRCUITS
by
PRESTON POWELL
JINWEI SHEN, COMMITTEE CHAIRWEIHUA SU
STEVE SHEPARDHAO KANG
A THESIS
Submitted in partial fulfillment of the requirementsfor the degree of Master of Science
in the Department of Aerospace Engineering and Mechanicsin the Graduate School of
The University of Alabama
TUSCALOOSA, ALABAMA
2016
Copyright Preston Powell 2016ALL RIGHTS RESERVED
ABSTRACT
This thesis analytically investigates the feasibility of passive vibration damping of a rotat-
ing beam-like structure, such as a helicopter rotor, through the use of piezoelectric materials.
Piezoelectric materials are unique, in that, they produce an electrical charge under the pres-
ence of mechanical stresses. Conversely, they generate mechanical stresses under applied
electrical loads. When mounted to a structure undergoing bending stresses, such as a can-
tilever beam, there is an exchange of mechanical and electrical energy between the beam
and the piezoelectric material. This electrical energy can be used to power small electronics
such as onboard data transmitters. This energy can also be dissipated through electrical
shunt circuits rather than being harvested for external use. Electrical components in a
shunt circuit (resistors, capacitors, and inductors) release energy from the system as Joule
heat. Energy dissipation corresponds to a vibration damping effect in the electromechanical
system. Numerous configurations of electrical components and mechanical structures are ex-
plored. First, the Rayleigh-Ritz method of assumed modes is adopted for a rotating uniform
single degree-of-freedom cantilever beam. Both in-plane (lag) and out-of-plane (flap) bend-
ing directions are considered. The beam model is modified to include piezoelectric elements
and electrical shunt circuits. Two types of shunt circuits are considered: one with a single
resistive element and one with a resistor and inductor in series. Various resistances are used
in finding the frequency and impulse responses of the rotating beam with a shunt circuit.
The change in damping potential between resistors is evaluated for each electromechanical
ii
system. The effects of the number of modes assumed when modeling the beam are also
highlighted. Single-mode approximations are found to be helpful in understanding the foun-
dations of the physics in the beam/piezo systems. It is also determined that multiple-mode
approximations account for important electromechanical behavior that is neglected by the
single-mode formulations. The settling times of the impulse responses are used as the figure
of merit to assess energy dissipation in the systems. Successful vibration damping of rotating
cantilever beams is predicted through the piezoelectric shunt circuits.
iii
NOMENCLATURE
b width
βi roots of characteristic equation for mode shapes
c modulus of elasticity
Ci mode shape coefficients
Cp piezoelectric capacitance
D electric displacement
d31 piezoelectric constant
e piezoelectric coupling coefficient
E electric field
ε dielectric constant
f external force
floc location of applied force
FT centrifugal force
g31 voltage coefficient
I current
K stiffness matrix
k31 coupling coefficient
iv
L inductance
m mass per unit length
M mass matrix
N number of modes
Ω rotational speed
ωe electrical frequency
φ electric potential location
Ψi mode shapes
q electrical charge
R beam length, resistance
r spatial coordinate along axis of beam
ρ density
S strain
t thickness
T stress, kinetic energy
Θ electromechanical coupling matrix
U potential energy
V volume
v
v lag displacement
Vi temporal lag displacement coordinate
v voltage
w flap displacement
Wi temporal flap displacement coordinate
We electrical energy
W work
Superscripts
E parameter at constant electrical field
S parameter at constant strain
T parameter at constant stress
( )′ first derivative with respect to spatial coordinate
( )′′ second derivative with respect to spatial coordinate
˙( ) first derivative with respect to time
( ) second derivative with respect to time
Subscripts
b beam parameter
vi
p piezoelectric parameter
sh shunt parameter
v pertains to lag bending motion
w pertains to flap bending motion
vii
ACKNOWLEDGMENTS
I wish to thank my advisor and committee chair, Dr. Jinwei Shen, for providing the
opportunity to work on this project. His patience and guidance were essential in the comple-
tion of this thesis. His door was always open for my frequent visits and questions, which he
always welcomed even during his busy schedule. He challenged and motivated me to work
my hardest when I got stuck or lost focus.
I want to acknowledge the resources and experiences provided by the Army Research
Laboratory in Aberdeen Proving Ground, Maryland. I would like to thank Dr. Hao Kang
specifically for his mentorship during my time at ARL. His support and research interests led
to the development of my thesis topic. He served on my committee and also gave valuable
input throughout the course of my research.
I would like to show my gratitude towards the other committee members, Dr. Weihua Su
and Dr. Steve Shepard. I recognize that their participation in this work took time from their
daily schedules. Their contributions and advice were helpful and their friendly personalities
made this process significantly less intimidating.
Additionally, I have a sincere appreciation for The University of Alabama and the De-
partment of Aerospace Engineering & Mechanics. The faculty and staff have all contributed
to my education and, for that, I am truly indebted. I am grateful for all of my friends and
fellow graduate students who helped me relax at times when I was most stressed. Their
input and encouragement throughout the past two years was crucial in not only maintaining
viii
my focus and drive, but also reminding me to take it easy sometimes.
Finally, I want to thank my family for their constant support since the day I was born.
Mom, Dad, Lindsay, and Ashlyn, you each have had an immeasurable influence on my life.
Your being a role model and showing me the importance of hard work and education has
given me continual motivation and is the reason I am where I am today.
Figure 4.29: Lag Bending Frequency Responses of RL-Shunt System for Various Resistors
Figure 4.30 gives the impulse response for the system with a 5000 Ω resistor. The settling
time of 5.36 seconds is a vast improvement on the impulse response for the system tuned to
the first mode.
71
0 1 2 3 4 5 6 7−0.03
−0.02
−0.01
0
0.01
0.02
0.03Tip Displacement for Impulse Load
Time [s]
Dis
plac
emen
t [m
]
R = 5000 Ω
Figure 4.30: Lag Bending Impulse Response with RL-Shunt and 5000 Ω Resistor
The amplitude steadily decreases as both the first and second mode vibrations are damp-
ened. However, we learned from the flap bending behavior for multiple modes that the op-
timal resistor according to the second mode frequency response is not necessarily the best
choice for the overall damping of the beam. By inspection of Figure 4.29(b), we conclude
that the 35000 Ω resistor would provide the most energy dissipation. Although that value
corresponds to a slightly larger magnitude for the second mode frequency response, it may
lead to a lower settling time as we saw in the case of flap bending. Figure 4.31 confirms
that the best resistor for the tuned mode is not necessarily the optimum choice for vibra-
tion reduction. With the 35000 Ω resistor, the impulse response has a settling time of 1.27
seconds.
72
0 1 2 3 4 5 6 7−0.03
−0.02
−0.01
0
0.01
0.02
0.03Tip Displacement for Impulse Load
Time [s]
Dis
plac
emen
t [m
]
R = 35000 Ω
Figure 4.31: Lag Bending Impulse Response with RL-Shunt and 35000 Ω Resistor
This is the case for both flap and lag bending when the shunt circuit is tuned to the
second mode frequency. We have found that energy dissipation is attainable through either
R-Shunt or RL-Shunt circuits for both flap and lag bending motion. As expected, the size of
the resistor directly affects the damping potential of the system. The next chapter presents
a summary and comparison of all the systems we have investigated.
73
CHAPTER 5
DISCUSSION
5.1 SUMMARY OF RESULTS
Chapter 4 explored the energy dissipation capabilities of two different shunt circuits
applied to a rotating cantilever beam undergoing flap or lag bending. Both single-mode
and multi-mode approximations were included which resulted in the investigation thirteen
separate electromechanical systems. Table 5.1 reports the settling times for all thirteen
systems of interest. The type of bending, number of modes, shunt design, and resistor size
are the parameters that define each system.
74
Bending Modes Shunt Type Resistor Settling Time
Flap 1 R-Shunt 30000 Ω 1.84
Flap 1 RL-Shunt 30000 Ω 0.64
Flap 3 R-Shunt 30000 Ω 0.95
Flap 3 RL-Shunt (1st Mode Tuning) 25000 Ω 13.0
Flap 3 RL-Shunt (1st Mode Tuning) 120000 Ω 3.43
Flap 3 RL-Shunt (2nd Mode Tuning) 10000 Ω 1.15
Flap 3 RL-Shunt (2nd Mode Tuning) 30000 Ω 0.77
Lag 1 R-Shunt 50000 Ω 1.24
Lag 1 RL-Shunt 50000 Ω 0.57
Lag 3 R-Shunt 50000 Ω 1.31
Lag 3 RL-Shunt (1st Mode Tuning) 35000 Ω N/A
Lag 3 RL-Shunt (2nd Mode Tuning) 5000 Ω 5.36
Lag 3 RL-Shunt (2nd Mode Tuning) 35000 Ω 1.27
Table 5.1: Summary of Shunt Circuit Resistances and Settling Times for Impulse Responsesof Flap and Lag Bending
5.2 DISCUSSION OF SHUNT SYSTEMS
We will first compare the single-mode R-Shunt and single-mode RL-Shunt for both flap
and lag bending. The same size resistor was used in all three of these systems on purpose.
First, we notice that for both single-mode flap and lag, there is an significant decrease in
settling time when an RL-Shunt is used instead of an R-Shunt. This is not surprising as the
tuned RL-Shunt circuit maximizes the current flow when the beam vibrates at the target
frequency. Because we only assumed one mode, the maximum current will always be flowing
through the circuit and, therefore, across the resistor. This increases the amount of energy
75
dissipated as heat by the resistor, so we see a reduction of settling time of 65% for flap and
54% for lag.
Next, the difference between single-mode and multi-mode R-Shunts for both flap and lag
will be highlighted. Flap bending shows a large reduction in settling time when using three
modes rather than one. The frequency responses in Figure 4.18 show that, although the
30000 Ω resistor is not the best for the second and third modes, it still provides damping
at both modes. The differences between the frequency responses of the optimal resistor and
30000 Ω resistor are quite small for both the second and third modes. Therefore, we see
additional energy dissipation when we expand to three modes.
The difference between single-mode and multi-mode R-Shunts for lag bending, however,
is the opposite of flap bending. When the impulse response is expanded to include three
modes, we see an increase in settling time. Again, we consult the frequency responses of the
three-mode system in Figure 4.20. When resistors for the second and third modes, we notice
a larger separation between the optimal curve and the curve corresponding to the chosen
50000 Ω resistor. This separation results in the increase in settling time.
When comparing the single-mode RL-Shunt to the multi-mode RL-Shunts, we notice an
increase in settling times for both flap and lag bending, regardless of tuning. It has been
emphasized that if a structure is vibrating at a wide range of frequencies, an RL-Shunt circuit
will dissipate less energy than it would if the structure is vibrating at an isolated frequency.
Therefore, it is no surprise that the single-mode RL-Shunts have the lowest settling times of
all the systems. The single-mode approximation, however, is an unrealistic one. It is useful
when learning the fundamental electromechanical behavior of a beam/piezo with a shunt
circuit, but is too simple and does inaccurately describe the system’s behavior. Such is the
76
case for both R-Shunts and RL-Shunts.
The multi-mode RL-Shunts that are tuned to the first mode for flap and lag bending
highlight the inaccuracy of the single-mode approximation. With the optimal resistors for
the single-mode RL-Shunts for both flap and lag, the settling times are found to be 0.64
seconds and 0.57 seconds, respectively. Then the simulations are altered to include three
modes, the RL-Shunts are tuned to the first mode frequencies, and the optimal resistors for
the first mode are selected. At this point, the two models (single and multi mode) are both
tuned to the first mode with the optimal resistor included in the circuit. The only difference,
the number of modes, turns out to have quite an effect on the systems. For flap bending,
the settling time increases about 2000%. The lag bending multi-mode RL-Shunt never even
settles. These contradictory results clearly show the purpose of including multiple modes in
the model.
We found that the flap bending RL-Shunt tuned to the first mode has a settling time
of 13.0 seconds, a rather large value. It was pointed out in Section 4.7 that the optimal
resistor for the first mode frequency response may not necessarily be best for the second
and third modes. After finding a high settling time for the 25000 Ω resistor, we noticed
that the vibrations were quickly dampened at the beginning but slowly decreased after less
than a second. This suggested that the vibrations of another mode were not being reduced.
We looked at the frequency response at the second mode and found that the ideal resistor
near that frequency was 120000 Ω. This value corresponded to a frequency response with
less damping than the 25000 Ω resistor for the first mode frequency. The impulse response
generated for the 120000 Ω resistor showed much more energy dissipation, as the settling
time dropped to 3.43 seconds. The frequency response near the third mode did not change
77
much with varying resistance. Therefore, we only considered the second mode when choosing
an alternative resistance. These results reinforced the notion that the best resistor for an
RL-Shunt is not necessarily the one that shows the most damping at the targeted frequency.
As for lag bending, the RL-Shunt tuned to the first mode was much less effective than
it was for flap bending. The change in resistor affected the vibration reduction at the
beginning of the impulse response, as it did for flap bending. However, after the first mode
was damped out, the vibration amplitude stayed constant, as in Figure 4.28. There was no
settling time for the system because it was, effectively, an undamped oscillating beam. The
frequency response in Figure 4.27(c) shows minimal damping or change in damping with
varying resistance. Therefore, no effort was made to find an optimal resistance because this
RL-Shunt design eventually stops dissipating energy for all of the resistors.
Having discovered that the frequency response of the second mode has a direct effect on
the energy dissipation of both flap and lag RL-Shunts, we decided to tune the RL-Shunt to
the second mode frequency, rather than that of the first mode. We found that the impulse
responses had much smaller settling times when tuned to the second mode. For flap bending,
the optimal resistor in the vicinity of the second mode was 10000 Ω and gave a settling time
of 1.15 seconds. However, from our previous results, we knew that the ideal resistor for the
second mode was not ideal for the first or third. A 30000 Ω resistor was found to provide
more energy dissipation at the first mode frequency. The impulse response in Figure 4.26
shows that the new resistor gives a settling time of 0.77 seconds.
The same process was followed for the lag bending RL-Shunts tuned to the second mode.
Initially, a 5000 Ω resistor was used for the impulse response and the settling time was
5.36 seconds. We adjusted this value, as we did for the flap bending, after looking at the
78
first mode frequency response. The new 35000 Ω resistor lowered the settling time to 1.27
seconds.
It is worth noting that we were able to drastically reduce the settling time of the same
exact beam/piezo system solely by adjusting the electrical components in the shunt circuit.
We reduced the settling time by 12.23 seconds and 4.09 seconds for flap and lag bending,
respectively. It is also important to recognize that the inductances required to tune the
RL-Shunts to the second mode are much smaller than those for the first mode. The 766
H inductor for the first lag mode is unrealistically large compared to the second mode
inductance of 21.3 H. It would be impractical to install a 766 H inductor on a complex
structure like a helicopter rotor.
The results found in Chapter 4 show that energy dissipation of a rotating cantilever
beam is possible through piezoelectric shunt circuits. Variations in electrical components are
strongly related to the damping capabilities of the shunts. For both flap and lag bending, the
second mode tuned RL-Shunts had smaller settling times than the R-Shunts. This is because
the second mode is dominant during the impulse response. The various resistance values used
in the simulations were chosen somewhat arbitrarily. Although we chose an “optimal” resistor
for each system by looking at the frequency response, the true “best case” value for achieving
the most energy dissipation is most likely near our optimal resistance but slightly higher
or lower. The range of whole number resistances was chosen to simplify the comparison of
different systems and to emphasize the effects on the frequency responses. Energy dissipation
shown in the impulse responses in Chapter 4 confirms that passive vibration damping of a
rotating cantilever beam has been achieved through piezoelectric shunt circuits.
79
CHAPTER 6
CONCLUSION
6.1 DAMPING POTENTIAL
Vibration reduction of a rotating cantilever beam by means of a piezoelectric element
connected to a shunt circuit was investigated. First, the mathematical model for a rotating
cantilever beam with a piezoelectric element was developed using the Rayleigh-Ritz method
of assumed modes. The equations of motion were derived for both flap and lag bending. Next,
the equations pertaining to the electrical circuit of both a resistive shunt and a resistive-
inductive shunt were incorporated into the beam/piezo equations of motion. The state
space representation of the electromechanical system was derived to allow the generation of
frequency and impulse responses.
In order to verify our model, the frequency response was found for beam and piezoelectric
parameters from a previous study in the literature. This was a non-rotating beam with two
piezoelements connected in series to a single RL-Shunt. The vibration reduction for various
resistors was found to closely match the results in the journal paper.
The frequencies of a rotating beam were then found using the parameters of the Hart-II
Blade. Both single-mode and multi-mode approximations provided first mode flap and lag
frequencies similar to those found in the Hart-II data, which further verified our model.
The model was simulated for a single-mode with both an R-Shunt and an RL-Shunt. Each
80
shunt was assessed for damping potential of a number of resistors for both flap and lag
bending. The simulations were then repeated for a model assuming multiple modes. The
variations in resistance led to changes in vibration reduction which allowed for the selection
of an “optimal” resistor. The impulse responses were generated and the settling time was
reported as a figure of merit to assess energy dissipation in the electromechanical system.
Energy dissipation was apparent in each system after finding an appropriate resistor.
The piezoelectric shunt circuits were found to be an effective method for passive vibration
reduction in rotating cantilever beams.
6.2 APPLICATION TO HELICOPTER ROTORS
Passive damping provided by piezoelectric shunt circuits is ideal for complex dynamic
structures such as helicopter rotors. A piezoelectric element would add little mass or stiff-
ness to the rotor and the shunt circuit is a simple component that would require no external
power source, unlike active damping methods. The concerns for using the piezoelectric shunt
method on helicopter rotors arise from the geometric properties of the blade. More specif-
ically, the damping potential for lag displacements is significantly less for a true helicopter
rotor than the uniform beam used for this thesis. Unlike our rectangular uniform beam,
the cross-section of a helicopter blade is an airfoil. This means that the leading edge and
trailing edge have much less surface area to which a piezoelectric element could be mounted.
Additionally, the electromechanical coupling would be significantly less as a result of the
reduced area. In the equations of motion, electromechanical coupling directly relates the
displacement to the voltage generation in the shunt circuit. Therefore, the lag vibration
81
damping potential would be noticeably less than that of the flap vibration for a helicopter
rotor.
The separation of flap and lag bending of the cantilever beam led to uncoupled single DOF
equations of motion for both displacement directions. This helped us develop a fundamental
understanding of the shunt circuits and the responses of the rotating system with respect
to changes in electrical components. However, complex beam models for the dynamics of
helicopter rotors include structural coupling for flap, lag, and torsion. In a manner analagous
to our extension from single-mode to multi-mode simulations, the inclusion of coupled flap,
lag, and torsion accounts for important behavior in the helicopter rotor that would otherwise
be omitted. An electromechanical model for a rotor without structural coupling would
inaccurately predict the damping potential of an piezoelectric shunt circuit.
Regardless of the beam model, however, it is obvious that there will be strains on the
external surfaces of a vibrating helicopter rotor. If a piezoelectric shunt circuit is placed on
the rotor where strains are present, the potential damping capabilities of the shunt can be
investigated as we did in this study.
6.3 FUTURE WORK
Potential future work on this topic includes a number of concepts not considered in this
thesis. First, further investigation into the optimum resistance for an R-Shunt and RL-Shunt
is desirable. Although the various resistors chosen in this study can provide an idea of the
range in which this optimum lies, determining how to find the true “best case” is important
if the shunts are to be used to maximize the damping potential.
82
Developing and incorporating new shunt designs could allow for more energy dissipation.
Because the RL-Shunt we considered could only be tuned to one target mode, a shunt that
adapts to multiple vibrational frequencies could result in improved damping potential.
The size and location of the piezoelectric element have an immediate effect on the elec-
tromechanical coupling factor and, therefore, are parameters that could be optimized. Each
mode of vibration has an area of maximum bending moment at which a tuned RL-Shunt
could be placed. Also, more than one piezoelectric element could be placed along the span
of the beam.
Lastly, expanding the model to include coupling terms between the flap and lag displace-
ments would better approximate the true behavior of a rotating cantilever beam. Beyond
that, a model that accounts for flap-lag-torsion structural coupling would allow us to deter-
mine with great confidence if the piezoelectric shunt circuit is a viable method for reducing
vibrations.
83
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