Download Electrical Damping of a Piezoelectric Plate

Electrical Damping of a Piezoelectric Plate
John M. Thompson, PhD
Consulting Engineer
Abstract
Piezoelectric materials are substances that have strong coupling between structural displacements and
electrical loads. The application of an electrical load will cause the structure to deform in a manner similar
to a mechanical load. If the electrical load is applied rapidly, the structure will respond rapidly and will
begin vibrating. If these vibrations are undesirable, they may be eliminated by damping, including
electrical damping from a properly designed electrical circuit. This paper present the analysis of a
piezoelectric beam subjected to a stepped electrical voltage. The ANSYS piezoelectric circuit elements are
used to evaluate the electrical circuit that includes the piezoelectric beam.
Introduction
Piezoelectric materials are materials the exhibit a strong coupling between the mechanical degrees of
freedom and the electrical degrees of freedom. When a voltage potential is applied across a piezoelectric
material, the material will deform in a manner consistent with its electrical properties. And like mechanical
loadings, if the voltage is applied slowly, the deformations will occur slowly and the structure will attain its
final position quasi-statically. However, if the voltage is applied suddenly, the structure will respond in a
dynamic manner and will begin to vibrate about its equilibrium position. The time required for the
transient vibrations to dissipate will depend upon the damping characters of the material and upon any
external dampers that may be present in the structure.
Several options are available to reduce or eliminate the unwanted vibrations through damping. One option
would be to add a material with a high damping coefficient to the structure. The damping material could be
sandwiched between layers of the piezoelectric material or could be bonded to the surface. In either case,
the increased damping would cause the vibrations to be eliminated in a shorter time that the natural
piezoelectric structure. A second option would be to use an electrical circuit to modify the applied voltage,
thereby effectively damping the response of the piezoelectric structure. It is this second approach that is
presented in this paper.
Analysis
To investigate the effect of a damping circuit on a piezoelectric structure, a finite element model of that
structure was created The model consists of a piezoelectric bimorph cantilever beam using PLANE13 and
an electrical circuit using components using CIRC94. The piezoelectric bimorph model is taken from
Section 6.12 of the Coupled-Field Guide and the piezoelectric material properties from that example are
used here. The length of the beam was reduced from 100 mm to 50 mm to increase its natural frequency;
the thickness of the beam and the material properties remained the same. The beam is constructed as a
bimorph with the elements along the top polarized in the negative Y direction while the elements along the
bottom of the beam are polarized in the positive Y direction. Figure 1 shows the finite element model of
the piezoelectric bimorph. Figure 2 shows the details of the model including the element coordinate
systems, the boundary conditions at the fixed end of the beam and the coupling along the top and bottom of
the beam. The voltage applied across the beam is 100 volts.
Figure 1. Finite element model of a piezoelectric bimorph
Figure 2. Details at the fixed end of the beam (Figure2.jpg)
Static Analysis
The first analysis with the piezoelectric bimorph beam was performed as a static analysis to verify the
model and to determine the deflection of the beam due to the applied 100 volt load. The results of the
analysis are shown in Figure 3. The deflection of the tip of the beam is .822E-5 mm and this is the
expected equilibrium deflection of the beam once the transient behavior of the beam has damped out.
Figure 3. Static deflection of a piezoelectric bimorph
Modal Analysis
The second analysis with the piezoelectric bimorph beam was performed as a modal analysis to determine
the natural frequencies of the beam. The natural frequencies are need to estimate the required time constant
for the electrical circuit to be added to the simulation and to estimate the amount of natural damping that
might be present in the beam. The bottom electrode of the beam was grounded to 0 volts; the top electrode
was allowed to be free.
The results of the analysis showed that the fundamental frequency is 33.5 Hz and corresponds to flexural
bending of the beam, as shown in Figure 3. Frequencies for higher modes were 209.7, 586.1 and 1145.
Using the 33.5 Hz natural frequency and an assume natural damping of 1%, the stiffness damping factor for
the transient analyses to follow would be
Beta = .01 / Pi / 33.5 = 9.5E-5
This value of natural damping will be used for all of transient analyses that follow.
Transient Analysis without Electrical Circuit
The third analysis with the piezoelectric bimorph beam was performed as a transient analysis with natural
damping only. The potential of 100 volts was applied to the electrode at the top of the beam while the
bottom electrode was grounded and the beam was allowed to deform. The voltage potential was applied to
the ANSYS model in two load steps of 1e-6 seconds to insure that the initial conditions of zero velocity and
zero acceleration would be correct. After the two initial load steps were used to applied the potential,
subsequent load steps was used to determine the transient response of the beam to 0.5 seconds. The results
of the analysis is shown in Figure 4.
Figure 4. Transient analysis of piezoelectric bimorph w/o electrical damping
The results of the analysis show that the beam deforms and oscillates about an equilibrium position of
approximate .8E-5 mm as predicted by the static analysis. The analysis also shows that the beam oscillates
5 cycles in 0.15 seconds which gives a natural period of 0.03 seconds and a natural frequency of 1/0.03 =
33.3 Hz, which is consistent with the modal analysis of the beam. The results also show that at 0.5
seconds, the beam vibration amplitude is in excess of .4e-5 mm (half of the static deflection of the beam)
and will continue to oscillate for some time.
Transient Analysis with an RC Circuit
If the oscillations in Figure 4 are considered excessive, then additional damping will be required. One
method of applying that damping would be to limit the application of the voltage potential with a resistorcapacitor (RC) circuit as shown in Figure 5. With this type of circuit, the full voltage potential will be
mitigated initially and the full will be applied when the capacitor is charged. The time required to charge
the capacitor is known as the circuit time constant and is dependent upon the values of the resistor and the
capacitor. The time constant for an RC circuit is given as
tau = RC
We can use the standard rule of thumb that 5 time constants are required to fully charge the RC circuit.
Then if we pick a time for the circuit to be fully charged, we can calculate the values for the resistor and the
capacitor that will give us the performance that we want. Suppose we choose 0.03 seconds for the time
constant of the circuit. Then the time for the circuit to be fully changed would be 0.15 seconds.
Figure 5. Circuit components for an RC circuit
As a first estimate, we can choose electrical components such that the circuit would be fully change at time
= 0.03 seconds so that the loading time is the same as the natural frequency. Then the time constant for the
system would be 0.03/5 = 0.006 seconds. All that remains is to choose the values for the resistor and the
capacitor. If we choose a standard capacitor of 0.3 microfarads, then the resistor value would be
C = 0.3 microfarads
R = 0.006 / .3e-6 = 20,000 ohms.
These will be the nominal values. To study the effect of the electrical circuits, additional values that are
0.1, 0.5, 1, 2 and 5 times the nominal values will also be input and the output deflections of the beam will
be compared.
The results of this analysis where the capacitor is held constant at .3 microfarads and the resistor is
permitted to vary are shown in Figure 6. The results show that small values of resistance have little
damping effect and imply that a longer time constant may improve the response. The results also raise a
concern that the voltage applied to the top electrode is not as expected and that the response of the
piezoelectric beam is influencing that voltage. To verify that the applied voltage is ramping monotonically
and not oscillating with the beam, the voltage response at the top electrode was plotted in Figure 7. This
plot of the voltage is as expected and implies that the RC circuit is correctly mitigating the step applied
voltage. The results of the transient analysis with a 0.006 second time constant also imply that a longer
time constant will be required for this beam.
Figure 6. Beam response with a nominal time constant of 0.03
Figure 7. Voltage applied to the top electrode of beam
To study the effect of the longer time constant, a value of 0.03 seconds was chosen such that the charging
time for the beam will be 0.15 seconds. The nominal value for the capacitor was chosen to be 0.3
microfarads and the resistor was varied from the nominal value of 100,000 ohms. The results are shown in
Figure 8. Results for a resistor of 100,000 ohms and a varying the capacitance with a nominal capacitor of
.3e-6 are shown in Figure 9.
Figure 8. Beam response for RC circuit, variable resistance, c = 3e-7
Figure 9. Beam response for RC circuit, variable capacitance, R = 100000
Transient Analysis with an RL Circuit
In a similar manner, electrical damping can be achieved using an RL circuit. For an RL circuit, the time
constant is given by
tau = L/R.
Again, the process is to choose either the resistor or the inductor and then calculate the other component
based upon the assumed time constant. The electrical circuit is shown in Figure 10.
The results for the RL circuit are shown in Figure 11 and Figure 12. In both cases, one of the electrical
components is held constant at its nominal value and the other is allowed to vary with values of 0.1, 0.5, 1,
2 and 5 times the nominal value. The nominal value for the inductor is 1.5 Henrys and the nominal value
for the resistor is 50 ohms.
Figure 10. Electric components for an RL circuit
Figure 11. Beam response for RL circuit, variable resistance, H = 1.5
Figure 12. Beam response for RL circuit, variable inductance, R = 10
Transient Analysis with an RLC circuit
And, the two effects can be combined into a single electrical circuit similar to that shown in Figure 13. For
this example, the values for the electrical components was
R = 10,000 ohm
C = 10 microfarads
I = 2.5 Henry
The displacement of the end of the beam is shown in Figure 14. For this case, the values of the electrical
components was not varied and only the nominal values were used.
Figure 13. Electric components for the RLC circuit
Figure 14. Beam response for an RLC circuit
Conclusion
The results of the analysis show that the vibrations of a piezoelectric beam subjected to a step voltage can
be reduced significantly if the applied voltage is mitigated by other electrical components in a circuit.
The results also show that to eliminate vibration in a piezoelectric beam subjected to a step voltage requires
circuit time constants that are longer than the natural frequency of the beam. The reason for this is believed
to be the strong interaction coupling between the displacement and voltage degrees of freedom for
piezoelectric crystals. It can be demonstrated that the output voltage of the electrical circuit has the desired
value based upon the stepped input voltage and the values for the electrical components. But the
piezoelectric beam is not a passive device like an electrical motor and the response of the piezoelectric
beam in the electrical circuit will have an effect.
Appendix A – Input listing
/batch,list
/filnam,beam3daa
/title, Transient Analysis of a Piezoelectric Bimorph Beam
!!!! RC circuit, vary the resistance
!!!! nominal time constant = .15
!!!! nominal capacitance = .3e-6
cap=.3e-6
*dim,rc,array,5
rc(1)= 10000,50000,100000,200000,1000000
*dim,rest,array,600,5
*do,ii,1,5
parsav,all,beam3daa,par
/clear
parres,change,beam3daa,par
/com,
/PREP7
!
! Define problem parameters
!
!
- Geometry
!
L=50e-3
! Length, m
H=0.5e-3
! One-layer thickness, m
!
!
- Loading
!
V=100
! Electrode voltage, Volt
Uy=10.e-3
! Tip displacement, m
!
!
- Material properties for PVDF
!
E1=2.0e9
! Young's modulus, N/m^2
NU12=0.29
! Poisson's ratio
G12=0.775e9
! Shear modulus, N/m^2
d31=2.2e-11
! Piezoelectric strain coefficients,
C/N
d32=0.3e-11
d33=-3.0e-11
ept33=12
! Relative permittivity at constant
stress
!
dmping=.01
pi=acos(-1)
freq=1/.03
! estimated natural freq
bett=dmping/pi/freq
!
! Finite element model of the piezoelectric bimorph beam
!
local,11
! Coord. system for lower layer: polar
axis +Y
local,12,,,,,180
! Coord. system for upper layer: polar
axis -Y
csys,11
!!!!rect,0,L,-H,0
!!!!rect,0,L, 0,H
!!!!aglue,all
!!!!esize,H
!
!!!!et,1,PLANE223,1001,,0
stress
! Activate coord.
! Create area
! Create area
! Glue layers
! Specify the
!!!!tb,ANEL,1,,,1
!!!!tbda,1,1/E1,-NU12/E1,-NU12/E1
!!!!tbda,7,1/E1,-NU12/E1
!!!!tbda,12,1/E1
!!!!tbda,16,1/G12
system 11
for lower layer
for upper layer
element length
! 2-D piezoelectric element, plane
! Elastic compliance matrix
!!!!tb,PIEZ,1,,,1
!!!!tbda,2,d31
!!!!tbda,5,d33
!!!!tbda,8,d32
! Piezoelectric strain matrix
!!!!tb,DPER,1,,,1
!!!!tbdata,1,ept33,ept33
! Permittivity at constant stress
!!!!tblist,all
matrices
! List input and converted material
! -----------------------------------------------------------------------!
Alternative element type and material input
!
et,1,PLANE13,7,,2
! 2-D piezoelectric element, plane
stress
et,11,circu94,0
! resistor
et,12,circu94,2
! capacitor
et,13,circu94,4
! voltage source
et,14,circu94,1
! inductor
r,11,RC(ii)
r,12,cap
r,13,v
r,14,20
r,15,.1
!!!100000
!
mp,EX,1,E1
! Elastic properties
mp,NUXY,1,NU12
mp,GXY,1,G12
!
tb,PIEZ,1
! Piezoelectric stress matrix
tbda,2,0.2876e-1
tbda,5,-0.5186e-1
tbda,8,-0.7014e-3
!
mp,PERX,1,11.75
! Permittivity at constant strain
! -----------------------------------------------------------------------mp,dens,1,7500
!!!!type,1 $ esys,11
!!!!amesh,1
layer
!!!!type,1 $ esys,12
!!!!amesh,3
layer
!
n,1,,-H
n,51,L,-H
fill
ngen,3,100,1,51,1,,H
! Generate mesh within the lower
! Generate mesh within the upper
type,1
esys,11
e,1,2,102,101
esys,12
e,102,101,201,202
egen,50,1,-2
nsel,s,loc,x,L
*get,ntip,node,0,num,min
! Get
!
!!!!nelec = 10
!
surface
!!!!*dim,ntop,array,nelec
!!!!l1 = 0
!
!!!!l2 = L/nelec
!!!!*do,i,1,nelec
!
!!!!
nsel,s,loc,y,H
!!!!
nsel,r,loc,x,l1,l2
!!!!
cp,i,volt,all
!!!!
*get,ntop(i),node,0,num,min
electrode
!!!!
l1 = l2 + H/10
!!!!
l2 = l2 + L/nelec
!!!!*enddo
nsel,s,loc,y,-H
cp,1,volt,all
*get,nbot,node,0,num,min
!!!!d,nbot,volt,0
master node at beam tip
Number of electrodes on top
Initialize electrode locations
Define electrodes on top surface
! Get master node on top
! Update electrode location
! Define bottom electrode
! Ground bottom electrode
nsel,s,loc,y,H
cp,2,volt,all
*get,ntop,node,0,num,min
n,901,-H,-H
n,902,-H,+H
n,903,-H/2,-H*2
n,904,-h*2,H
type,13
real,13
!!!!e,901,nbot,903
type,11
real,11
e,904,902
! voltage source
! resistor
type,12
real,12
e,902,901
type,11
real,15
e,902,201
type,11
real,15
e,901,1
nsel,s,loc,x,0
d,all,ux,0,,,,uy
nsel,all
fini
/SOLU
antype,trans
tintp,,0.25,0.5,0.5
betad,bett
! capacitor
! resistor as jumper
! Clamp left end of bimorph
! Actuator simulation
!!!!d,ntop,volt,V
d,901,volt,0
d,904,volt,v
alls
!!!!*do,i,1,nelec
!!!!
d,ntop(i),volt,V
electrodes
!!!!*enddo
outres,all,all
time,1e-6
autots,off
nsub,1,1,1
solve
time,2e-6
solve
time,0.5
nsub,500,500,500
solve
fini
/POST26
nsol,2,node(L,0,0),uy
plvar,2
vget,rest(1,ii),2
! Save path in a file
fini
*enddo
save,beam3daa,db
/post26
nsol,2,node(L,0,0),uy
! Apply voltages to top
plvar,2
vput,rest(1,1),5
vput,rest(1,2),6
vput,rest(1,3),7
vput,rest(1,4),8
vput,rest(1,5),9
varnam,5, R=%RC(1)%
varnam,6, R=%RC(2)%
varnam,7, R=%RC(3)%
varnam,8, R=%RC(4)%
varnam,9, R=%RC(5)%
xvar,1
/title, Figure 8 - Beam response for RC circuit, variable resistance, C
= %cap%
plvar,5,6,7,8,9