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Rapid thermokinetics associated with laser-based additive manufacturing produces strong bulk crystallographic texture in the printed component. The present study identifies such a bulk texture effect on elastic anisotropy in laser powder bed fused Ti6Al4V by employing an effective bulk modulus elastography technique coupled with ultrasound shear wave velocity measurement at a frequency of 20 MHz inside the material. The combined technique identified significant attenuation of shear velocity from 3322 ± 20.12 to 3240 ± 21.01 m/s at 45$$^\circ$$ ∘ and 90$$^\circ$$ ∘ orientations of shear wave plane with respect to the build plane of printed block of Ti6Al4V. Correspondingly, the reduction in shear modulus from 48.46 ± 0.82 to 46.40 ± 0.88 GPa was obtained at these orientations. Such attenuation is rationalized based on the orientations of $$\alpha ^\prime$$ α ′ crystallographic variants within prior columnar $$\beta$$ β grains in additively manufactured Ti6Al4V.

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Parametrically excited systems are generally represented by a set of linear/nonlinear ordinary differential equations with time varying coefficients. In most cases, the linear systems have been modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. Although Floquét theory is applicable only to periodic systems, it is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to two typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are extremely close to the exact boundaries of the original quasi-periodic equations. The exact boundaries are detected by computing the maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. The coefficients of the parametric excitation terms are not necessarily small in all cases. ‘Instability loops’ or ‘Instability pockets’ that appear in the stability diagram of Meissner’s equation are also observed in one case presented here. The proposed approximate approach would allow one to construct Lyapunov-Perron (L-P) transformation matrices that reduce quasi-periodic systems to systems whose linear parts are time-invariant. The L-P transformation would pave the way for controller design and bifurcation analysis of quasi-periodic systems.