Large Amplitude Vibrations and Modal Sensing of Intelligent Thin Piezolaminated Structures

Transcription

1 Large Amplitude Vibrations and Modal Sensing of Intelligent Thin Piezolaminated Structures S. Lentzen and R. Schmidt Insitut of General Mechanics, RWTH Aachen University

2 Contents MRT FOSD theory of shells Numerical procedure Modal array sensors Numerical examples Summary

3 Nonlinear FOSD theory of shells for small strains and moderate rotations Kinematic hypothesis: 0 t t 3 t V = V+Θ V % % % Assumptions: small strains: ε ij αβ = O ( ) αβ αβ αβ ε = ε +Θ ε + Θ ε 0 3 α 3 = α 3+Θ α 3 ε ε ε ε 33 = 0 2 ( ϑ ) with small rotations about the normal: moderate rotation of the normal: Green-Lagrange strain tensor: 2 ϑ ωαβ ω = O 2 ( ϑ ) α 3 = O( ϑ )

4 Strain-displacement relations Tangential strains: ε αβ = θαβ + ϕα ϕβ λ λ λ λ ε = ν + ν bα ϕλβ bβ ϕλα + ϕα bβν + ϕβ bαν bb λ κ b λ b λ ε αβ = α βν λν κ αν λ β βν λ α αβ α β β α λ λ Transverse shear strains: ε ϕ ν ν ϕ λ α3 = α+ α+ λα Linearised tangential strains: Linearised rotations: ϕ = ν b ν αβ λ and ε α 3 = 2 uuuuuu ν ν λ u α θ = ν + ν b ν 2 αβ α β β α αβ 3 αβ αβ 3 and , α b λ α = + α λ ϕ ν ν

5 Constitutive relations { D} = [ e]{ ε} + [ δ]{ E} T { S} = [ c]{ ε} [ e] { E} Direct piezoelectric effect Converse piezoelectric effect with [ e] = [ d][ c] and [ e] T = [ c][ d] T and ε σ 22 σ ε 22 D E S = = D = D E = E 2 2 { } τ { ε} 2ε { } { } τ 2ε 23 D E3 3 τ 2ε3

6 State of Equilibrium δw i = δw e ( T T { } { } { } { }) δ δε δ 0 Wi = S E D dv 0 V ( ) σ δw = ρ F A R δv dv + σ δv da D n δϕda i i i i i i i e i Q V A A σ Q Further Assumptions Electric field is only existent in transverse direction E = % { 0 0 E } 0 3 Electric potential is homogeneous between an electrode pair

7 Finite Element Implementation { F} + { F} i m i e [ M ]{ q&& } + [ D]{ q& } + { F} = { F} i e { Q} = { Q } i e () (2) { Q} + { Q} i m i e Central Differences {} q f ( q, q t t t, F i, t, M, D Δ ) = t+δ t { } ( ) ϕ = +Δ +Δ t t f q t t from (2) Implicit Methods (Newmark, ) Δ δwi = δq δϕ K {{ } { } }[ ] { } T T Δq T { Δϕ}

8 Finite Element Implementation, cont d [ K ] T Kqq Kq ϕ = T Kq ϕ K ϕϕ tangential stiffness matrix after static condensation of {φ}: [ ] * T K = K K K K T qq qϕ ϕϕ qϕ K K K T qϕ ϕϕ qϕ can be fully occupied e.g.: for an electrode pair covering the complete structure Solution: decoupled iterations

9 Modal Array Sensor Principle modal sensor signal α α 2 α n linear combiner φ φ 2 φ n sensors n i= α g = δ i ik jk structure [ G]{ α } = { e} with [ G] = gki j j

10 Cantilevered Beam with PZT Sensors 2 00 x 5 x mm E = 2GPa ν = mm 3 ρ = 2840 kg m 0 d 3 = m V δ 33 = F m E = 70GPa ν = ρ = 2700 kg m

11 load [0 3 N/m 2 ] Cantilevered Beam with PZT Sensors under Pressure Load linear nonlinear tip displacement [cm] sensor voltage [V] linear nonlinear

12 Gain Factor Distributions mode mode 3 α/α [-] max

13 FFT Analysis of the Modal Signals mode Hz 0 0 linear nonlinear Hz mode 3 linear nonlinear frequency [Hz]

14 Cantilevered Cylindrical Shell actuator E = 63GPa sensor R=300mm 50mm ν = ρ = 7600kg m 0 d 3 =.79 0 m V 8 δ 33 =.65 0 F m E = 20GPa ν = ρ = 7800kg m ξ = 0.002

15 Cantilevered Cylindrical Shell step line load N/m for ms hoop/radial tip displacement [mm] present Balamurugan et al. time [s]

16 Eigenfrequencies of the Clamped Cylindrical Shell mode mode 2 mode 3 mode 4 mode 5 mode 6 [Hz] [Hz] [Hz] [Hz] [Hz] [Hz] Balamurugan et al. ( sensor) Linear ( sensor) linear (5 sensors) nonlinear 800 N (5 sensors)

17 Sensor Voltage due to an Additional Hoop Line Force of N/m sensor voltages [ΔV] time [s]

18 FFT Analysis of the Modal Signals due to an Additional Hoop Line Force of N/m

19 Summary A geometrically nonlinear Finite Element has been presented to analyse piezolaminated shells Some issues concerning the FE implementation have been discussed Several issues concerning modal sensing of geometrically nonlinear vibrations with modal array sensors are discussed by means of some numerical examples

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