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

Airplane Fuselage (continued)

• Natural freqs of thin sandwich shell

• 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

More Than Frequency Shifts?

Do membrane loads affect modal damping?

If so, how?

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