# Presentation: 359647

## PASSIVE CONTROL OF VIBRATION AND

** ****WAVE PROPAGATION **
**IN SANDWICH PLATES **
**WITH PERIODIC AUXETIC CORE**

**Luca Mazzarella**
- Mechanical Engineering Dept.
- Catholic University of America
- Washington, DC 20064

**Massimo Ruzzene**
- Mechanical Engineering Dept.
- Catholic University of America
- Washington, DC 20064

**Panagiotis Tsopelas**
- Civil Engineering Dept.
- Catholic University of America
- Washington, DC 20064

## Outline

-  Sandwich plates with periodic honeycomb core:
-  Finite Element Modeling of unit cell
-  Bloch reduction
-  Response to harmonic loading
-  Performance of periodic sandwich plates:
-  Configuration of the unit cell
-  Phase constant surfaces
-  Contour plots
-  Harmonic response
-  Sandwich plate-rows with periodic honeycomb core:
-  Theoretical Modeling
-  Transfer Matrix
-  Propagation constants
-  Dynamic stiffness matrix
-  Performance of periodic sandwich plate-rows:
-  Configuration of the unit cell
-  Propagation Patterns
-  Structural response
-  Conclusions

**Outline**

## 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

## 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

## Completely passive treatment
Performance 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 configuration 
Stiffening geometric effect and change in mass density depending on core material geometry
External dimensions and weight not significantly affected

- Performance 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 configuration
- Stiffening geometric effect and change in mass density depending on core material geometry
- External dimensions and weight not significantly affected

- Sandwich plates with periodic auxetic core:

**Basic concepts**

## Introduction:  plate-rows

**Transfer matrix formulation:**

- Yk  :  state vector
- Tk  :  transfer matrix

- i.e.

- i : ith eigenvalue of Transfer Matrix T
- log(i ) = PROPAGATION CONSTANT

**Introduction:  plate-rows**

**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)

## Introduction: 2D plates

- Wave motion in the 2-D structure (Bloch’s Theorem):

**Introduction: 2D plates**

## Introduction: 2D plates

- (*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**

## Honeycomb core 

******

***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 **

## Theoretical Modeling of the sandwich plate

- _ __Face sheets_   (*extension *+ *bending*):

- _ __Core_  (s*hear deformation):*

**Theoretical Modeling of the sandwich plate**

## Theoretical Modeling of the sandwich plate

**Kinetic energy:**

- _ __Face sheets_   (*translation + rotation*):

- _ __Core_  (*translation + rotation):*

## Theoretical Modeling of the sandwich plate

**Theoretical Modeling of the sandwich plate**

## Wave propagation in sandwich plate-rows

- Plate-Rows

- Outline of concepts described and methods applied
- SFEM is formulated from Transfer Matrix approach
- Transfer Matrix obtained from distributed parameter model of sandwich plate
- Transfer Matrix is recast  to obtain the Dynamic Stiffness Matrix of plate element

**ASSEMBLED DYNAMIC**
** ****STIFFNESS MATRIX**

**   ****TRANSFER MATRIX                                 PASS / STOP  BANDS**

**RESPONSE OF STRUCTURE**

## Transfer Matrix formulation

**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).

## Performance of periodic sandwich plate-rows

- Face sheets

- Honeycomb core A

- Auxetic core B

*        **L**xA** ** *

*L**xB*

*L**y*

- 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

- 		material	thickness
- Face sheet (1)             Al             5 mm
- core (2)                    Nomex        9 cm
- Face sheet (3)             Al             5 mm

**Unit cell**

*L**y*

*     **L**xB*

*        **L**xA**  *

- Core “A”

- Core “B”

- Configuration of periodic plate-row:

- y

- x

- Simply supported edges

- Simply supported edges

-   20 cells along S-S edges

- Forcing function
- F=F0*sin*(*m****y/Ly*)*e**j****t*

-  Output

**Performance of periodic sandwich plate-rows**

## Performance of periodic sandwich plate-rows

## Performance of periodic sandwich plate-rows

**Performance of periodic sandwich plate-rows**

- LxA/LxB=2; **B=-60o

## Performance of periodic sandwich plate-rows

**Performance of periodic sandwich plate-rows**

## Performance of periodic sandwich plate-rows

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- Forcing harmonic

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- Propagation patterns**  **
- ky=*m*/Ly ; *m*=1,...,10

- Homogeneous core A

- LxA/LxB=1; **B=-30o

- LxA/LxB=2; **B=-60o

- LxA/LxB=1/2; **B=-30o

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## 2D analysis: Bloch reduction

- {qij} generalized displacements at interface i,j
- {Fij} generalized forces at interface i,j

- Cell`s equation of motion:

- where:

**2D analysis: Bloch reduction**

## 2D analysis: Bloch reduction

-  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:

## 2D Wave propagation

-  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
- 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 )

## Performance of periodic sandwich plates

- 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**

- 		material	thickness [mm]
- Face sheet (1)             Al                    1
- core (2)                    Nomex               2
- Face sheet (3)             Al                    1

**Unit cell**

*L**x*

*     **L**yB*

*        **L**yA**  *

- Core “A”

- Core “B”

- Configuration of periodic plate:

- Harmonic forcing

## Phase Constant Surfaces

-  First phase constant surface
-  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(*)

- 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), 447-469.

## Contour plots

***x*/

***y*/

***y*/

***x*/

- Homogeneous core, *L**x*/ *L**y_**=1*

- Homogeneous core:
-  No directional behavior expected

- Periodic core (length ratio =1):
-  Directional behavior expected
-    above a “transition frequency”

- Periodic core, *L**x*/ *L**y_**=1* , *L**yA**/L**yB*=*1*

**Influence of the periodic core:**

## Plate harmonic response 

**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

## Plate harmonic response 

**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*=1

*L**yA**/L**yB*=2/3

## Plate harmonic response 

**Plate harmonic response **

- Homogeneous

*L**yA**/L**yB*=1

*L**yA**/L**yB*=2/3

**Number of cells: N****x****= 20, N****y****= 20; 40x40 finite element grid**

## Conclusions

-  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**