The harmonic balance (HB) method is widely used in the literature for analyzing the periodic solutions of nonlinear mechanical systems. The objective of this paper is to exploit the method for bifurcation analysis, i.e., for the detection and tracking of bifurcations of nonlinear systems. To this end, an algorithm that combines the computation of the Floquet exponents with bordering techniques is developed. A new procedure for the tracking of Neimark-Sacker bifurcations that exploits the properties of eigenvalue derivatives is also proposed. The HB method is demonstrated using numerical experiments of a spacecraft structure that possesses a nonlinear vibration isolation device.

Objectives:To characterize the clinical and genetic features of spinal bulbar muscular atrophy (SBMA), a rare neurodegenerative disorder caused by the expansion of a CAG repeat in the first exon of the androgen receptor gene, in the United Kingdom.Methods:We created a national register for SBMA in the United Kingdom and recruited 61 patients between 2005 and 2013. In our cross-sectional study, we assessed, by direct questioning, impairment of activities of daily living (ADL) milestones, functional rating, and subjective disease impact, and performed correlations with both CAG repeat size and degree of somatic mosaicism. Ten patients were deceased, 46 patients participated in the study, and 5 declined.Results:Subjects had an average age at onset of 43.4 years, and weakness onset most frequently occurred in the lower limbs (87%). Impaired mobility was the most frequently reported problem by patients, followed by bulbar dysfunction. Age distribution of the impairment of ADL milestones showed remarkable overlap with a Japanese study. We have identified a significant correlation between the number of CAG repeats and both age at onset and ADL milestones. Somatic mosaicism also showed a correlation with CAG expansion size and age at onset.Conclusions:Clinical features in SBMA show a substantial overlap when comparing populations with different genetic backgrounds. This finding has major implications, because multicenter trials will be necessary to obtain sufficient power in future clinical trials. Clinical-genetic correlations are strong in SBMA and should inform any clinical research strategy in this condition.

The nonlinear tuned vibration absorber (NLTVA) is a recently-developed nonlinear absorber which generalizes Den Hartog's equal peak method to nonlinear systems. If the purposeful introduction of nonlinearity can enhance system performance, it can also give rise to adverse dynamical phenomena, including detached resonance curves and quasiperiodic regimes of motion. Through the combination of numerical continuation of periodic solutions, bifurcation detection and tracking, and global analysis, the present study identifies boundaries in the NLTVA parameter space delimiting safe, unsafe and unacceptable operations. The sensitivity of these boundaries to uncertainty in the NLTVA parameters is also investigated.

Because nowadays structural engineers are willing to use or at least understand nonlinearities instead of simply avoiding them, there is a need for numerical tools performing analysis of nonlinear large-scale structures. Among these techniques, the harmonic balance (HB) method is certainly one of the most commonly used to study finite element models with reasonably complex nonlinearities. However, in its classical formulation the HB method is limited to the approximation of periodic solutions. For this reason, the present paper proposes to extend the method to the detection and tracking of codimension-1 bifurcations in the system parameters space. As an application, the frequency response of a spacecraft is studied, together with two nonlinear phenomena, namely quasiperiodic oscillations and detached resonance curves. This example illustrates how bifurcation tracking using the HB method can be employed as a promising design tool for detecting and eliminating such undesired behaviors.