How Membrane Loads Influence the Modal Damping of Flexural Structures
George A. LesieutreDepartment of Aerospace Engineering
Center for Acoustics and Vibration
Sandia National Laboratories
Albuquerque, NM
September 21, 2011
Overview
• Membrane loads
• Stiffness– Vibration modes
• Frequencies, mode shapes
• Damping– Data from the literature– Model development– Results
What Are Membrane Loads?
• In a 1-D or 2-D structural member that carries lateral loads: – Forces tangent to the midsurface– “Initial stress” or “pre-stress”
• Example members– Strings, Membranes– Beams, Plates, Shells
Importance in Structural Dynamics?
• Can provide lateral restoring force
• Can affect normal vibration modes– Frequencies – Mode shapes
Sample Applications
• Rotor blades
• Airplane fuselages
• Actuators; Acoustic transducers
• MEMS resonators
Beam Example: Spinning Blade
T (x) = ρA(ξ) ξΩ2 dξ
x
L
∫
T(x)
G.S. Bir, “Structural Dynamics Verification of RCAS,” NREL/TP-500-35328, February
2005.
Variation of spinning uniform blade modal freqs with rotor speed
First coupled mode, nonuniform blade (Ω=0)
First mode, nonuniform blade (Ω=50)
“CF Load”
Shell Example: Airplane Fuselage
• Fuselage vibration modes – Effects of pressurization
Baker, E.H., Hermann, G., “Vibrations of Orthotropic Cylindrical Sandwich Shells under Initial Stress,” AIAA J, v 4, n 6, 1966, 1063-1070.
w(x,ϑ ,t) =Wmnsinmπx
L⎛⎝⎜
⎞⎠⎟
cos nϑ( ) cos ωt( )
u(x,ϑ ,t) =Umn cosmπx
L⎛⎝⎜
⎞⎠⎟
cos nϑ( ) cos ωt( )
v(x,ϑ ,t) =Vmnsinmπx
L⎛⎝⎜
⎞⎠⎟
sin nϑ( ) cos ωt( )
ψ x (x,ϑ ,t) = Amn cosmπ x
L⎛⎝⎜
⎞⎠⎟
cos nϑ( ) cos ωt( )
ψ ϑ (x,ϑ , t) = Bmn sinmπ x
L⎛⎝⎜
⎞⎠⎟
sin nϑ( ) cos ωt( )
transverse shear
• Develop tunable piezo flexural transducers with high effective BWS product
• Use for transmission / reception of:– Frequency modulated (FM) pulses
• Short pulses for range resolution• High sweep rate for velocity resolution
• In-plane loads affect resonance
frequencies-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.5
0
0.5
ClampedSimply supported
DiskDisk
(f(frr-f-
f r0r0)/
f)/
f r0r0
N/NcrN/Ncr
++VVDD
++
++--
--
--VVCC
VVCC
Exploitation: Tunable Transducer
Calculations: Fuselage Damping
• Damping of thin sandwich shell
Boeing HSCT
Lesieutre, G.A., Wodtke, H.W., Zapfe,
J.A., Damped Composite Honeycomb
Sandwich Panels for High-Speed Aircraft
Interior Noise Reduction, Final Report
to Boeing Commercial Airplane Co., June 30,
1995.
Experiments: Rotor Blade Damping
Experiments in vacuum
Smith, C.B., Wereley, N.M., “Transient analysis for damping identification in rotating composite beams with integral damping layers,” Smart Materials and Structures, v 5, 1996, 540-550.
CLD treatment
Experiments: Fan Blade Damping
Experiments in vacuum
Kosmatka, J.B., and O. Mehmed, “Experimental Spin Testing of Integrally-Damped Composite Blades,” AIAA-98-1847.
Experiments: Tunable Piezoelectric Energy Harvester
Leland, E.S. and Wright, P.K., “Resonance Tuning Of Piezoelectric Vibration Energy Scavenging Generators Using Compressive Axial Preload,” Smart Materials and Structures, 2006, pp. 1413-1420.
Lesieutre, G.A., and C.L. Davis, “Can a Coupling Coefficient of a Piezoelectric Actuator be Higher Than Those of Its Active Material?,” JIMSS, Vol. 8, 1997, pp. 859-867.
With increasing compressive preload– Resonance frequency decreases– Modal damping increases– Coupling coefficient increases
• Micron-sized resonators offer advantages over larger counterparts– RF signal processing– Mass, pressure sensor– Size, power, sensitivity
• Spectral purity– High Q desirable: minimize (Meff/Q)
• Single-crystal Si, Si3N4, SiC
– RT Q ~ 10,000 to 100,000
MEMS Resonators
Verbridge, S.S., Shapiro, D.F, Craighead, H.G., Parpia, J.M., “Macroscopic Tuning Of Nanomechanics: Substrate Bending For Reversible Control Of Frequency And Quality Factor Of Nanostring Resonators,” Nano Letters, Vol. 7, No. 6, 2007, pp. 1728-1735.
Experiments: MEMS Resonators with Tension
• High Q attributed to tensile stress– RT Q > 106 measured (in vacuum)
• The precise mechanism by which tension increases Q even in the presence of increased material damping and boundary losses “remains unknown.”
Verbridge, S.S., Shapiro, D.F, Craighead, H.G., Parpia, J.M., “Macroscopic Tuning Of Nanomechanics: Substrate Bending For Reversible Control Of Frequency And Quality Factor Of Nanostring Resonators,” Nano Letters, Vol. 7, No. 6, 2007, pp. 1728-1735.
Damping Models
• Effects of membrane loads on modal damping of flexural structures
• Viscous damping– Strain-based– Motion-based
• Complex modulus (MSE)
&&x + 2ζω &x+ω 2 x=X(t)
Effect of Membrane Loads on Natural Frequencies: SS Beam
• Admits eigenfunctions:
• (Undamped) natural frequencies:
T
ρA&&w − T ′′w + EI ′′′′w = 0
Wm (x) =sinmπx
L⎛⎝⎜
⎞⎠⎟
ωm
2 =EI
mπ
L⎛⎝⎜
⎞⎠⎟
4
+ Tmπ
L⎛⎝⎜
⎞⎠⎟
2
ρA=
mπ
L⎛⎝⎜
⎞⎠⎟
4EI
ρA
⎛
⎝⎜⎞
⎠⎟1+
T
EI mπL( )
2
⎛
⎝⎜
⎞
⎠⎟
tension
ωm
= m2 π 2 EI
ρ AL4
⎛
⎝⎜⎞
⎠⎟
ω0
1 244 34 4
ωm 0
1 24 4 34 4
1 +T
EImπ
L⎛⎝⎜
⎞⎠⎟
2
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
1 2
= ωm 0
1 +T Pcr
m2
⎛⎝⎜
⎞⎠⎟
1 2
Beams (continued)
• Instability (buckling) possible– A result of compression
Pcr =−Tcr =π 2EI
L2
PP
critical load
-Pcr
T
ω1natural frequency
Effect of Membrane Loads on Damping: SS Beam (Viscous)
• Modal damping
T
ρA&&w + cs ′′′′&w − T ′′w + EI ′′′′w = 0
strain-based viscous damping
T Pcr
ζEI m =cEI
mπ
L⎛⎝⎜
⎞⎠⎟
2
2 ρ A EI( )1 2
ζ EI m 0
1 24 34
1
1+T
m2Pcr
⎛
⎝⎜⎞
⎠⎟
1 2 =ζ EI m0
1 +T Pcr
m2
⎛⎝⎜
⎞⎠⎟
1 2
−ω 2ρ A + EI(1 + i η EI
lossfactor
{ )mπ
L⎛⎝⎜
⎞⎠⎟
4
− Tmπ
L⎛⎝⎜
⎞⎠⎟
2⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
Am* (ω ) = Fm
Effect of Membrane Loads on Damping: Complex Modulus
• Modal damping
Modal damping evaluatedat resonance, decreases with T-1
T
T Pcr
complex modulus
ζm =η EI
2
1
1+T Pcr
m2
⎛⎝⎜
⎞⎠⎟
ωm
ω=
η EI
2
1
1+T Pcr
m2
⎛⎝⎜
⎞⎠⎟
1 2
ωm0
ω
Modal Loss Factor & Frequency vs. Tension & Mode Number
Modal tension stiffness factor
Normalized modal freq
Relative modal loss factor
αm =ω
m
2
ωm 0
2= 1 +
T Pcr
m2
⎛⎝⎜
⎞⎠⎟
ωm
ω0
= α m m2
ηm
η EI
=1
α m
Good qualitative agreement of theory with experiment
Experiments: MEMS Resonators with Tension
Verbridge, S.S., Shapiro, D.F, Craighead, H.G., Parpia, J.M., “Macroscopic Tuning Of Nanomechanics: Substrate Bending For Reversible Control Of Frequency And Quality Factor Of Nanostring Resonators,” Nano Letters, Vol. 7, No. 6, 2007, pp. 1728-1735.
Insight: Modal Strain Energy
• Damping of a mode of a structure is a weighted sum of the damping of its parts– Weighting factors are the fraction of strain
(potential) energy stored in each part
ηm =Vpart
VTOTAL
⎛
⎝⎜⎞
⎠⎟m
η partparts∑ with VTOTAL( )
m= Vpart( )
mparts∑
ηm =VEI
VTOTAL
⎛
⎝⎜⎞
⎠⎟m
η EI +VT
VTOTAL
⎛
⎝⎜⎞
⎠⎟m
ηT
mat’l loss factorloss factor assoc. w/ tension = 0
=VEI
VEI + VT
⎛
⎝⎜⎞
⎠⎟m
η EI =1
1 +T Pcr
m2
⎛⎝⎜
⎞⎠⎟
η EI
Summary and Conclusions
• Membrane loads affect the apparent lateral stiffness of flexural structures– Modify normal vibration modes (freqs, shapes)
• Tensile (compressive) membrane loads decrease (increase) modal damping– Strongest effect on the lowest vibration modes– Complex modulus: modal damping decreases in
direct proportion to the increase in tension– Viscous damping: slightly different, but similar– Consistent with available experimental data
• Alternate damping approaches?
More Than Frequency Shifts!
Acknowledgments
• In addition to the cited references, students and post-docs:– Hans-Walter Wodtke– Jeff Zapfe– Chris Davis– Chad Hébert– Julien Bernard– Jeff Kauffman– Mike Thiel
• Sponsors– Boeing (HSCT), ONR, NASA/Army (RCOE)
Effect of Membrane Loads on Damping: SS Beam (Viscous)
• Modal damping
T
ρA&&w + cv &w − T ′′w + EI ′′′′w = 0
motion-based viscous damping
ζV m =cV
2 ρ A EI( )1
2 mπ
L⎛⎝⎜
⎞⎠⎟
2
ζV m 0
1 24 4 4 34 4 4
1
1 +T
m2Pcr
⎛
⎝⎜⎞
⎠⎟
1 2 =ζ V m0
1 +T Pcr
m2
⎛⎝⎜
⎞⎠⎟
1 2
Effect of Membrane Loads on Damping: SS Beam (Damped)
• Modal damping
T
ρA&&w + c0 &′′w − T ′′w + EI ′′′′w = 0
ζm =c0
2 ρA EI( )1
2 1+T
EI mπL( )
2
⎛
⎝⎜
⎞
⎠⎟
12
Insensitive to frequency (if T=0)
Strings (1-D)
• Lateral stiffness due entirely to tension• Tension from sol’n of tangent problem
d T (x)
dx+ px(x) =0
Tangent Equilibrium
ρA(x)∂2w(x, t)
∂t 2−
∂
∂xT (x)
∂w(x, t)
∂x⎛⎝⎜
⎞⎠⎟
= pz (x, t)Lateral Eqn of Motion
Independent of lateral disps (linear)
Neglect dynamics
Membranes (2-D)
• Lateral stiffness due to membrane stress• Stress from sol’n of tangent problem
∂ N xx
∂x+
∂ N xy
∂y+ px (x, y) = 0
∂ N xy
∂x+
∂ N yy
∂y+ py (x, y) = 0
Tangent Equilibrium
ρh(x, y)∂2w
∂t 2−
∂
∂xN xx
∂w
∂x⎛⎝⎜
⎞⎠⎟
−∂
∂xN xy
∂w
∂y
⎛⎝⎜
⎞⎠⎟
−∂
∂yN xy
∂w
∂x⎛⎝⎜
⎞⎠⎟
−∂
∂yN yy
∂w
∂y
⎛⎝⎜
⎞⎠⎟
= pz (x, y, t)
Lateral Eqn of Motion
ρh∂2w
∂t 2− N ∇2 (w) = pz (x, y, t)
uniform tension
Beams (1-D)
• Lateral stiffness a combination of bending and tension– Tension can be negative (compression)
• Tension from sol’n of tangent problem– Like that of string
ρA(x)∂2w
∂t 2−
∂
∂xT (x)
∂w
∂x⎛⎝⎜
⎞⎠⎟
+∂2
∂x2EI(x)
∂2w
∂x2
⎛
⎝⎜⎞
⎠⎟= pz (x, t)
Lateral Eqn of Motion (planar)
Plates (2-D)
• Lateral stiffness a combination of bending and membrane stress– Stress(es) can be negative; instability possible
• Stress from sol’n of tangent problem– Like that of membrane
ρh∂2w
∂t 2+ D∇4w − N xx
∂2w
∂x2+ 2N xy
∂2w
∂x∂y+ N yy
∂2w
∂y2
⎛
⎝⎜⎞
⎠⎟= pz (x, y, t)
Lateral Eqn of Motion
Shells (2-D, curved)
• Curvature couples membrane & bending
Shallow cylindrical shell
a∂ Nxx
∂x+
∂ Nxθ
∂θ=0
a∂ Nxθ
∂x+
∂ Nθθ
∂θ=0
Tangent Equilibrium
ρh∂2w
∂t 2+ D∇4w +
1
aNθθ − N xx
∂2w
∂x2+
2
aN xθ
∂2w
∂x∂θ+
1
a2Nθθ
∂2w
∂θ 2
⎛
⎝⎜⎞
⎠⎟= pz (x,θ , t)
Lateral Eqn of Motion
ff (Hz) (Hz)
t t (s)(s)
WDWD
– initial carrying freq: f0 = 50 Hz
– carrying freq:– instantaneous freq:
Time-Frequency Analysis of FM Pulses
• Linear FM (LFM) pulse:– amplitude: A = 1 Vpk
– duration: = 1 s– frequency sweep: f = 50 Hz
tf
ftfm
+= 0)(
tf
ftfi
+= 2)( 0
tt (s) (s)
ff (Hz) (Hz)
ff (Hz) (Hz)
tt (s) (s)
22 ππ|V
(f)|
|V(f
)|22 (
V (V
22 /Hz)
/Hz)
vv (t)
(V
)(t
) (
V)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1
0
1
40 60 80 100 120 140 1600
1
2
3x 10-3
40 60 80 100 120 140 1600
0.5
1
Fixed (Q = 47)
Tunable vs. Fixed-Freq Transducer
750 800 850 900 950 100010501100115012000
0.02
0.04
0.06
0.08
0.1
0.12
Frequency (Hz)
Tim
e (
s)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14-5
-4
-3
-2
-1
0
1
2
3
4
5x 10
-3
Time (s)
u(t
) (m
/s)
Tunable (Q = 47)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14-6
-4
-2
0
2
4
6x 10
-3
Time (s)u
(t)
(m/s
)
Higher, uniform output over broad range
Tuning:
Track instantaneous frequency
Bernard, J. and G.A. Lesieutre, “Design and Realization of Frequency Agile Piezoceramic Transducers,” AIAA Adaptive Structures Forum, Atlanta, GA, April 3-6, 2000.
Prototype Transducers and Experiments
Electrical impedance measurement under bias voltages
• Coupling coefficient: Coupling coefficient: 0.20.2• Frequency shift: Frequency shift: 30% for 1 kV/mm30% for 1 kV/mm
0 100 200 300 400 500 600 700 800 900 10000.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
Vb (V)
Rel
ativ
e fr
eque
ncy
shif
t
#6
#5
#4
#2
#7
44 mm63.5 mm
12.7 mm