Calculation of vibration damping of composite structures

From the GMC analysis together with the energy dissipation concept, we can have the damping properties of unidirectional composites by means of implementing simple loading, such as tension and pure shear on the RVE. However, when the composite structures are adopted for engineering applications, the vibration in general takes place in bending and torsional modes and the damping properties associated with these modes can not be estimated directly from the GMC approach. Here we adopted the two step simulation procedure to predict the damping behaviors of the composite structures with vibration motion. First, the basic material properties of the unidirectional composites, such as , , , et al, were evaluated using the GMC micromechanical model. In the second step, the material properties were considered as global material properties of equivalent element and utilized as inputs in the composites structures for the structural dynamic analysis. In other words, in the structural level, only the material properties of the composites prevail in the analysis. In this study, the structural dynamic analysis was carried out using the FEM approach in which the elements contain the damping and material properties of equivalent element. The detail analytical procedure was illustrated in Fig. 4.1.

E1 ψ₁₁ E₂ ψ₂₂

The modal damping capacity of composite structures can also be derived from

the strain energy dissipation concept. For a linear elastic material, the strain energy stored in a volume element is expressed as

{ } { }

where is the stiffness matrix of composites. The corresponding dissipated strain energy of a volume element can be written in terms of the specific damping capacity in the material principal directions as

[ ]

^C

where indicated the matrix form of damping properties of equivalent elements as shown in Tables 4.2-4.4.

[ ]

^ψ

⎥⎥

Therefore, with the energy dissipation and the strain energy of the equivalent element, the specific damping capacity of a volume element associated with a deformation can be written as follows

{ } [ ][ ] { }

In the finite element analysis, the strain field of a volume element can be expressed in terms of the nodal displacement in conjunction with the shape function as.

{ }

^{ε =}

[ ]

^B

{ }

^d (4.2.6)

a vibration motion, can be regarded as the mode shape of structure representing the relative nodal displacement of the element associated with its natural frequency.

Thus, once the model shapes of the composite structures were determined, the specific damping capacity of the composites structure can be determined by summating the specific damping capacity of each element calculated from Eq. (4.2.7). In the following, the mode shape of the composite structure will be evaluated from structural dynamics analysis together with finite element approach [13].

{ }

^d

From the principle of virtual work, the governing equation for composite structure with dynamic loading were derived by making the virtual work done by externally applied loads equal to the sum of virtual energy caused by inertial, dissipative, and internal forces for any virtual displacement. For a single element of volume V with surface area of S, this relation is written explicitly as

{ } { } { } { }

{ } { } { } { } { } { }

[

^{δu ρ u δu c u δε σ}

]

^dV

dS T δu dV

F δu

V

T T

T

V S

T T

∫

∫ ∫

+ +

=

+

&

&&

(4.2.8)

in which

{

and indicate the body forces and surface tractions, ρ and c denote the mass density and a damping parameter,

}

F

{ }

^T

{ }

^{δu and}

{ }

δε exhibit virtual displacements and their corresponding strains and

{ }

σ is the assumed stress existing in the body prior to virtual strains applied. For a undamped structure with free vibration or with clamp boundary conditions, all of virtual energy caused by the applied loading terms as well as the damping parameter, c were assumed to be zero, and thus the Eq. given in (4.2.8) was deduced as

{ } { }

^{δu ρ udV (e)}

{ } { }

^{δε σdV 0}

V T (e)

V

T +

∫

=

∫

^&& (4.2.9)

Here the superscript (e) designates that the integration is within a volume element. In the finite element method, the displacement field,

{ }

u and strain field

in the element can be represented by the nodal displacement as well as the shape function as

{ }

^ε

{ }

u =

[ ]

N

{ } { }

d u& =

[ ]

N

{ }

d&

{ }

u&& =

[ ]

N

{ }

d&&

{ }

ε =

[ ]

B

{ }

d (4.2.10)

where

[ ]

^N is the shape functions,

{ }

d is nodal displacement which is function of time,

[ ]

^B is the differentiation of shape function

[ ]

^{N , and}

{ }

dt

u& =du ,

{ }

²₂

dt u u&& =d

indicate the velocity and acceleration respectively. Substituting the Eq. (4.2.10) into Eq. (4.2.9) yields

{ }

δd ρ

[ ] [ ]

N NdV

{ }

d

[ ]

B

{ }

σ dV 0

(e)

V

(e)

V T T

T =

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

∫

^&& +

∫

(4.2.11)

By assuming the material is linear elastic,

{ }

σ in Eq. (4.2.11) can be substituted by . In addition, with the assistance of displacement and strain relation, i.e.,

, Eq. (4.2.11) is written as

{ }

^{σ =}

[ ]

^C

{ }

^ε

{ }

^{ε =}

[ ]

^B

{ }

^d

[ ]

m^(e)

{ }

d&& +

[ ]

k ^(e)

{ }

d =0 (4.2.12)

where the element mass matrix

[ ]

^{m (e)} and element stiffness matrix

[ ]

^k(e) is defined

as

[ ]

^=(e)

∫ [ ] [ ]

V T

(e) ρ N NdV

m (4.2.13)

[ ]

k

[ ] [ ][ ]

B C BdV

(e)

V T

(e)⁼

∫

(4.2.14)

Substituting Eq. (4.2.14) into Eq. (4.2.7) yields

{ } [ ] { } { }

d

[ ]

k

{ }

d 2

1

d k 2 d 1 ψ

(e) T

(e) ψ T

(e)= (4.2.15)

where

[ ]

k ^(e)_ψ representing the “energy dissipation stiffness matrix” is written as

[ ]

k

[ ] [ ][ ][ ]

B C ψ BdV

(e)

V T (e)

ψ ⁼

∫

(4.2.16)

For the global response, the structure mass matrix

[ ]

M and the structure stiffness matrix can be derived through the superposition of the element mass matrix and stiffness matrix

[ ]

, respectively by properly assigning each element matrix in the structure matrix depending on the structure node numbering. As a result, the equation of motion for the structure can be written as

[ ]

^K

[ ]

m^(e) k^(e)

[ ]

M

{ }

d&& +

[ ]

K

{ }

d =0 (4.2.17)

Moreover, the natural frequency and mode shape of the composites structures associated with each vibration mode can be evaluated by solving the eigenvalue problem of Eq. (4.2.17). In this study, the eigenvalue and eigenvector of Eq.

(4.2.17) corresponding to the natural frequency and modal shape of the structure, respectively were calculated by Matlab commercial code with “eig” command. It is worthy to mention that in the calculation of the mode shape of the composite structures, the effect of material damping was neglected and only the mass matrix and stiffness matrix were accounted for. From the definition of specific damping capacity, the modal damping capacity of the structure associated to each modal shape can be expressed in terms of the global stiffness matrix

{ }

^Φ

[ ]

K , the global energy dissipation stiffness matrix

[ ]

Kψ and the corresponding modal eigenvector

{ }

^Φ as

{ } [ ] { } { }

_i ^T

[ ] { }

_i

i ψ T i

i Φ K Φ

2 1

Φ K 2 Φ

1

ψ = (4.2.18)

where the index indicates the i

i

^th modal shape. It is noted that the global energy dissipation stiffness matrix is obtained from the superposition of the energy dissipation stiffness matrix given in Eq. (4.2.16). The code for calculating the modal shapes and damping capacity of composite structures is listed in Appendix C.

[ ]

K ψ

4.3 Discussions of the damping capacity of fiber composites with three different

在文檔中 探討不同纖維排列下複合材料之機械行為 (頁 55-61)