Based on the Church-Hoff model, the nonlinear oscillations of a single encapsulated microbubble with a finite thickness shell are theoretically studied. The effects of viscoelasticity on radial oscillations and the fundamental and harmonic components are researched. The peaks of radial oscillations and magnitudes of power spectra of the fundamental and harmonic components all increase gradually with the shear modulus of shell varying from 0 to 10 MPa by an interval of 0. 1 MPa at the same shear viscosity, while they decrease as the shear viscosity increases from 0 to 1 Pa · s by an interval of 0. 01 Pa · s at the same shear modulus. The fluctuation ranges of subharmonic and ultraharmonic signals are much larger than both the fundamental and second harmonic components. It means that the effect of viscoelasticity on the subharmonic and ultraharmonic signals is greater than that on the fundamental and second harmonic components. So adjusting the viscoelasticity of the shell is a potential method to obtain a perfect microbubble contrast agent used for the subharmonic and ultraharmonic imaging. Four points with significant fundamental and harmonic components are chosen as an example： a shear viscosity of 0. 39 Pa · s with shear modulus of 3.9, 6. 6, and 8.6 MPa, respectively; a shear modulus of 6.6 MPa with a shear viscosity of 0.42 Pa · s.
A finite element model is presented for free-damped beam-stiffened plates. The nodes of the plate elements are treated as master-nodes, and the corresponding nodes of the beam elements are considered as slave-nodes. The stiffness and mass matrices of the elements are developed. Based on the analysis of the dynamic properties of the structures, modal loss factors are predicted by the modal strain energy method. Finally, an example is given to compare the results obtained from the proposed method with the results of the ANSYS software. The results show that the method in this paper is computationally efficient, simple and feasible with high precision and engineering practicability.
