Modal expansion analysis of nonlinear circumferential guided wave propagation in a circular tube

Abstract

Within the second-order perturbation approximation, the nonlinear effect of primary circumferential guided wave propagation in a circular tube is investigated using modal expansion analysis for waveguide excitation. The nonlinearity of the wave equation governing the wave propagation ensures the second-harmonic generation accompanying primary circumferential guided wave propagation. This nonlinearity may be treated as a second-order perturbation of the linear elastic response. The fields of the second harmonics of primary circumferential guided wave propagation are considered as superpositions of the fields of a series of double frequency circumferential guided wave (DFCGW) modes. Based on the momentum theorem and mathematical formulae of nonlinear stress tensor and its divergence under the cylindrical coordinate system, the mathematical expressions of the corresponding double frequency traction stress tensors and bulk driving forces are deduced for a certain primary circumferential guided wave mode. Subsequently, the equation governing the DFCGW mode expansion coefficient is established. Finally, the mathematical expression of second-harmonic field of the primary circumferential guided wave mode in a tube is derived. The results of the theoretical analyses and numerical calculations indicate that the degree of cumulative growth of the DFCGW mode with circumferential angle is obviously influenced by that of phase velocity matching between the primary and double frequency wave modes. It is found that the amplitude of the DFCGW mode can grow with circumferential angle when its phase velocity matches with that of the primary circumferential guided wave, and that the amplitude of the DFCGW mode will show a beat effect with circumferential angle when its phase velocity is not equal to that of the primary wave mode. The DFCGW mode, whose phase velocity matches with that of the primary wave mode, plays a dominant role in the field of second harmonic generated by the primary wave mode propagation, and the contribution of the other DFCGW modes to the said second-harmonic field is negligible after the primary wave mode has propagated some circumferential angle.

Influence of Change in Inner Layer Thickness of Composite Circular Tube on Second-Harmonic Generation by Primary Circumferential Ultrasonic Guided Wave Propagation

“…[18,19] However, it is difficult for the traditional linear CUGW to monitor minor change in material/structure of a circular tube. Based on the theoretical and experimental investigations of nonlinear effect of CUGW propagation in a single layer circular tube, [20,21] it has been experimentally validated that the minor change (early damage) in a tube material can be quantitatively assessed using the acoustic nonlinearity parameter for CUGW propagation. [22] In addition to minor change in the tube material itself, it can be expected that a minor change in geometrical parameter (e.g., inner layer thickness) may also obviously influence the efficiency of SHG of primary CUGW propagation.…”

“…1, within the second-order perturbation, the bulk driving force of double the fundamental frequency 𝐹 (𝑙) ] inside the composite circular tube, as well as the traction stress tensor of double the fundamental frequency 𝑃 (2𝜔) s = 𝑃 [𝑈 (𝑙) ] on the interfaces/surfaces of the composite circular tube, can be generated due to the convective nonlinearity and the inherent elastic nonlinearity of solid. [8,[20][21][22][23][24]28] According to the modal expansion approach, 𝐹 (2𝜔) b and 𝑃 (2𝜔) s are, respectively, assumed to be the bulk and surface sources for generation of a series of doublefrequency CUGW modes that constitute the secondharmonic field (denoted by 𝑈 (2𝜔,𝑙) (𝑟, 𝜃)) of the 𝑙th CUGW mode, namely, [8,[20][21][22][23][24]28]…”

“…Considering the initial condition for generation of the double-frequency CUGW, i.e., 𝐴 𝑚 (𝜃) = 0 when 𝜃 = 0, the expansion coefficient of the 𝑚th double-frequency CUGW mode can be formally given by [8,[20][21][22][23][24]28] 𝐴 𝑚 (𝜃) = 𝐴 𝑚 sin(∆𝑛𝜃) ∆𝑛 exp(𝑗∆𝑛𝜃) × exp[𝑗𝑛 (2𝜔,𝑚) 𝜃],…”

Experimental Observation of Cumulative Second-Harmonic Generation of Circumferential Guided Wave Propagation in a Circular Tube

“…[9−16] Moreover, the nonlinear effect of axial guided waves that propagate axially along the tubes has also been examined. [17,18] However, as an elementary mode of guided waves propagating along the circumference of the tubes (referred to as circumferential guided wave mode), [19] its nonlinear effect has not been investigated until recently by Gao et al [20] It is theoretically found that, under some conditions, the second harmonics of the primary (fundamental) circumferential guided wave propagation may have strong nonlinearity, i.e., the second harmonics may grow in amplitude with circumferential angles. Experimental observation of the strong nonlinear effect of the primary circumferential guided wave propagation is of substantial significance for possible practical applications.…”

“…Within the second-order perturbation approximation, the nonlinear effect of circumferential guided wave propagation in a circular tube has been investigated to date. [20] The fields of the second harmonics of primary circumferential guided wave propagation are considered as superpositions of the fields of a series of double frequency circumferential guided wave (DFCGW) modes. It is theoretically found that the amplitude of the DFCGW mode can grow with circumferential angle when its linear phase velocity matches with that of the primary circumferential guided wave.…”