1 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. PASSIVE CONTROL OF VIBRATION AND WAVE PROPAGATION WAVE PROPAGATION IN SANDWICH PLATES WITH PERIODIC AUXETIC CORE Luca Mazzarella Mechanical Engineering Dept. Catholic University of America Washington, DC Massimo Ruzzene Mechanical Engineering Dept. Catholic University of America Washington, DC Panagiotis Tsopelas Civil Engineering Dept. Catholic University of America Washington, DC 20064

2 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD.  Introduction: wave propagation in 1D and 2D periodic structures;  Sandwich plates with periodic honeycomb core: Finite Element Modeling of unit cell Finite Element Modeling of unit cell Bloch reduction Bloch reduction Response to harmonic loading Response to harmonic loading  Performance of periodic sandwich plates: Configuration of the unit cell Configuration of the unit cell Phase constant surfaces Phase constant surfaces Contour plots Contour plots Harmonic response Harmonic response  Sandwich plate-rows with periodic honeycomb core: Theoretical Modeling Theoretical Modeling Transfer Matrix Transfer Matrix Propagation constants Propagation constants Dynamic stiffness matrix Dynamic stiffness matrix  Performance of periodic sandwich plate-rows: Configuration of the unit cell Configuration of the unit cell Propagation Patterns Propagation Patterns Structural response Structural response  Conclusions Outline

3 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Motivation Analysis is performed through the theory of 2-D PERIODIC STRUCTURES which are characterized by:  Frequency bands where elastic waves do not propagate  Directions where propagation of elastic waves does not occur Analysis of WAVE DYNAMICS in sandwich plates with core of two honeycomb materials alternating PERIODICALLY along the structure STOP/PASS BANDS “FORBIDDEN” ZONES x y Core B Core A

4 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Motivation GOAL: Evaluate characteristics of wave propagation for sandwich plates with periodic core configuration:  Determine stop/pass band pattern;  Determine directional characteristic and “forbidden zones” of response;  Evaluate influence of the cell and core geometry; 2D periodic structures behave as DIRECTIONAL MECHANICAL FILTERS

5 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Completely passive treatmentCompletely passive treatment Performance of traditional light-weight sandwich elements enhanced by directional filtering capabilitiesPerformance of traditional light-weight sandwich elements enhanced by directional filtering capabilities Improvement of the attenuation capabilities of periodic sandwich panels obtained through a proper selection of the core and cell configurationImprovement of the attenuation capabilities of periodic sandwich panels obtained through a proper selection of the core and cell configuration Stiffening geometric effect and change in mass density depending on core material geometryStiffening geometric effect and change in mass density depending on core material geometry External dimensions and weight not significantly affectedExternal dimensions and weight not significantly affected Sandwich plates with periodic auxetic core: Basic concepts

6 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. U L k-1 Cell k-1 Cell k F L k-1 F R F L k F R k Interface k U R k-1 U L k U R k Transfer matrix formulation: Y k : Y k : state vector T k : T k : transfer matrix i.e. i  : i th eigenvalue of Transfer Matrix T i  : i th eigenvalue of Transfer Matrix T log( i ) = PROPAGATION CONSTANT Introduction: plate-rows Periodic structure: Periodic structure: assembly of identical elementary components, or cells, connected to one another in a regular pattern. The plate-rows here considered are modeled as quasi-one-dimensional multi-coupled periodic systems  i  pass band (wave propagation)  i  stop band (wave attenuation)

7 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. 2D-periodic structure: 2D-periodic structure: cells connected to cover a plane nxnx nyny Unit cell x y LxLx LyLy Wave motion in the 2-D structure (Bloch’s Theorem): Motion of unit cell Propagation Constants Introduction: 2D plates

8 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Propagation Constants are complex numbers: (k=x,y) Phase Constant Attenuation Constant Condition for wave propagation: & Imposing the propagation constants allows obtaining the corresponding frequency of wave propagation: Phase Constant Surfaces Introduction: 2D plates

9 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Face sheets Honeycomb core A Honeycomb core B  l h  =h/l  =t/l (core A) (core B) l  h t t Geometric layout of regular (A) and auxetic (B) honeycomb structures Negative Poisson’s ratio behavior: Auxetic honeycombs with  =-60°,  =2 are characterized by a shear modulus which outcast up to five times the shear modulus of a regular honeycomb of the same material  Re-entrant geometry (AUXETIC SOLID) Honeycomb core

10 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Strain energy: Face sheets (extension + bending): Core (shear deformation): Core (shear deformation): Theoretical Modeling of the sandwich plate

11 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Theoretical Modeling of the sandwich plate Kinetic energy: Face sheets (translation + rotation): Core (translation + rotation): Core (translation + rotation):

12 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Equations of Motion: Theoretical Modeling of the sandwich plate

13 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Wave propagation in sandwich plate-rows Plate-Rows Outline of concepts described and methods applied SFEM is formulated from Transfer Matrix approachSFEM is formulated from Transfer Matrix approach Transfer Matrix obtained from distributed parameter model of sandwich plateTransfer Matrix obtained from distributed parameter model of sandwich plate Transfer Matrix is recast to obtain the Dynamic Stiffness Matrix of plate elementTransfer Matrix is recast to obtain the Dynamic Stiffness Matrix of plate element ASSEMBLED DYNAMIC STIFFNESS MATRIX STIFFNESS MATRIX TRANSFER MATRIX PASS / STOP BANDS TRANSFER MATRIX PASS / STOP BANDS RESPONSE OF STRUCTURE Core B Core A y x Simply supported edges

14 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. state space formulation state space formulation: Generalized Displacements Generalized Forces Equations of motion Transfer Matrix (k th part of the cell) Transfer Matrix formulation Only the dynamics along the x-axis has to be investigated The behavior along the y-axis is described by the harmonic exp(jk y y); k y =m  /L y (with m integer) is the wave number along the y-axis The analysis is performed independently for each harmonic m for the deformations along the y-axis: according to the CLT (Classical Laminated plate Theory).

15 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Face sheets Honeycomb core A Auxetic core B L xA L xB LyLy Cell configuration and Geometry:  Aspect ratio L x / L y =1/2  Length ratio L xA / L xB =1, 2, 1/2  Internal core geometry: Type A:  =1,  =30  Type B:  =2,  =-60  and  =-30   L y =1 m materialthickness Face sheet (1) Al 5 mm core (2) Nomex 9 cm Face sheet (3) Al 5 mm Unit cell LyLy L xB L xA Core “A” Core “B” Configuration of periodic plate-row: y x Simply supported edges  20 cells along S-S edges Forcing function F=F 0 sin(m  y/Ly)e j  t Output Performance of periodic sandwich plate-rows

16 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Performance of periodic sandwich plate-rows k y =  /L y L xA /L xB =1;  B =-60º L xA /L xB =1/2;  B =-60º Single cell eigenvalues 20 cells plate-row FRF

17 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. k y =2  /L y Performance of periodic sandwich plate-rows L xA /L xB =2;  B =-60º L xA /L xB =1;  B =-60º Single cell eigenvalues 20 cells plate-row FRF

18 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. k y =3  /L y Performance of periodic sandwich plate-rows L xA /L xB =1;  B =-60º L xA /L xB =1/2;  B =-60º 20 cells plate-row FRF Single cell eigenvalues

19 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Performance of periodic sandwich plate-rows Forcing harmonic [Hz] Forcing harmonic Forcing harmonic [Hz] Forcing harmonic [Hz] Propagation patterns k y =m  /L y ; m=1,…,10 Homogeneous core A L xA /L xB =1;  B =- 30º L xA /L xB =2;  B =- 60º L xA /L xB =1/2;  B =-30º [Hz]

20 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Cell analysis: {q R },{F R } {q T },{F T } {q L },{F L } {q I },{F I } {q LB },{F LB } {q RB },{F RB } {q RT },{F RT } {q LT },{F LT } {q B },{F B } Left Right Top Bottom Internal {q ij } generalized displacements at interface i,j {F ij } generalized forces at interface i,j Cell`s equation of motion: [K], [M]: cell`s stiffness and mass matrices where: 2D analysis: Bloch reduction

21 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Bloch`s Theorem: Relation between interface displacements (compatibility conditions) Relation between interface forces (equilibrium conditions) 2D analysis: Bloch reduction Reduced Mass and Stiffness Matrices: Cell`s Equation of Motion is reduced at: Frequency  of wave motion for the assigned set of propagation constants  x,  y with:

22 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Solution of Dispersion Relation: x/x/ y/y/  ( rad/s ) Phase Constant Surfaces are symmetric with respect to both  x,  y Phase Constant Surfaces are symmetric with respect to both  x,  y Analysis can be limited to the first quadrant of the  x,  y plane, within Analysis can be limited to the first quadrant of the  x,  y plane, within the [0,  ] range for  x,  y. First three phase constant surfaces for a sandwich plate with uniform core (A) represented over the first propagation zone ([0,  ] range for  x,  y ) 2D Wave propagation Phase Constant Surfaces:  x,  y )

23 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Face sheets Honeycomb core A Auxetic core B L yA L yB LxLx Cell configuration and Geometry:  Aspect ratio L x / L y_ =1  Length ratio L yA / L yB_ =1 and 2/3  Internal core geometry: Type A:  =1,  =30  Type B:  =2,  =-60  Performance of periodic sandwich plates materialthickness [mm] Face sheet (1) Al 1 core (2) Nomex 2 Face sheet (3) Al 1 Unit cell LxLx L yB L yA Core “A” Core “B” Configuration of periodic plate: Harmonic forcing

24 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. x/x/ y/y/  ( rad/s ) x/x/ y/y/ First phase constant surface First phase constant surface Contour plot Contour plot Phase Constant Surfaces The energy flow vector P at a given frequency  lies along the normal to the corresponding iso-frequency contour line in the k x, k y space, where k i =  i /L i, (i=x,y). The perpendicular to a given iso-frequency line for an assigned pair  x,  y corresponds to the direction of wave propagation (*)  x,  y corresponds to the direction of wave propagation (*) Homogeneous core, L x / L y_ =1 (*) Langley R.S., “The response of two dimensional periodic structures to point harmonic forcing” JSV (1996) 197(4),

25 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Contour plots x/x/ y/y/ y/y/ x/x/ Homogeneous core, L x / L y_ =1 Homogeneous core: No directional behavior expected No directional behavior expected Periodic core Periodic core (length ratio =1): Directional behavior expected Directional behavior expected above a “transition frequency” above a “transition frequency” Periodic core, L x / L y_ =1, L yA /L yB =1 Influence of the periodic core:

26 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Plate harmonic response Number of cells: N x = 20, N y = 20; 40x40 finite element grid Homogeneous Plate deformed configuration for excitation at  =7.5 rad/s L yA /L yB =1 L yA /L yB =2/3

27 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Number of cells: N x = 20, N y = 20; 40x40 finite element grid Plate deformed configuration for excitation at  =9.5 rad/s Plate harmonic response Homogeneous L yA /L yB =1L yA /L yB =2/3

28 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD. Plate deformed configuration for excitation at  =13 rad/s Plate harmonic response Homogeneous L yA /L yB =1L yA /L yB =2/3 Number of cells: N x = 20, N y = 20; 40x40 finite element grid

29 FEMCI Workshop 2002 May 23, NASA Goddard Space Flight Center, Greenbelt, MD.  Wave propagation in periodic sandwich plates and plate-rows is analyzed;  Auxetic and regular honeycomb cellular solids are utilized as core materials to generate impedance mismatch zones;  Analysis is performed through the combined application of the theory of periodic structures, the FE method, and the Transfer matrix and Spectral FE methods  The capability of the periodic core to generate stop bands for the propagation of waves along the plate-rows, and directional patterns for the propagation of waves along the plate plane has been assessed;  Analysis allows evaluating pass/stop bands propagation patterns, and the phase constant surfaces for the estimation of directional characteristics for wave propagation  The filtering capabilities are influenced by the geometry of the periodic cell;  Harmonic response shows directionality at specified frequencies, and confirms the propagation patterns  Completely passive treatment;  External dimensions and weight not significantly affected;  Improvement of the attenuation capabilities of periodic sandwich panels obtained through a proper selection of the core and cell configuration; Conclusions