Modal Lumped Parameter Model of an ACLD System

The constitutive relationships considered for the beam are:

,  the free strain is	(38.1)

Where, the subscripts ‘b’, ‘v ’ and ‘c’ represent base beam, viscoelastic layer and constraining layer respectively. ‘E’ is the modulus of elasticity and ‘G’ the shear modulus. The strain terms are defined in the following equation.
Recall that the constraining layer is active and piezoelectric in nature. It is assumed to expand or contract
in the axial direction based on the sign of the applied voltage V. Following Fig 38.1, the corresponding strains at different layers may be defined as:

	(38.2)

Where, as shown in Fig. 38.1, u_b denotes the axial displacement of the host beam at the neutral axis, u_bv
and u_cv denote the axial displacement of the bottom and top of the constrained viscoelastic layer respectively and denotes the axial displacement at the mid-plane of the active constraining layer. Considering equilibrium of forces in the viscoelastic layer for an infinitesimal patch of length , at a distance from the center of the patch (located at a distance x_c from the origin); one can write

(38.3)

Where, F_c is the axial force in the constraining layer and b is the width of the beam. Using equations (38.1 and 2) one can get the transformed equation in terms of axial deflection of the top of the viscoelastic layer as:

	(38.4)

The coupling factor β signifies the interaction between piezoelectric layer and the host beam. Since, the shear modulus of the viscoelastic layer G is normally an order of magnitude lower than the elastic modulus of the active layer E_c, higher coupling occurs when the viscoelastic and the constraining layers are very thin.