We study the bifurcation structure of zonal flows on a rotating sphere. The setting of our problem is similar to the Kolmogorov problem on a flat torus, where the vorticity forcing is given by a single eigenfunction of the Laplacian. First we prove the global stability of two-jet zonal flow for arbitrary Reynolds number and the rotation rate of the sphere. Then we study the bifurcation structure of steady solutions arising from three-jet zonal flow. In the non-rotating case, we find that two steady travelling-wave solutions bifurcate from a three-jet zonal flow via Hopf bifurcation. As the Reynolds number increases, steady-travelling solutions arise via pitchfork bifurcation from the steady-travelling solutions. On the other hand, in the rotating case, we find saddlenode bifurcations and closed-loop branches. We carry out time integration to study the properties of unsteady solutions at high Reynolds numbers. In the non-rotating case, the unsteady solution is chaotic and it wanders around the steady-travelling solutions bifurcating from three-jet zonal flow. We show that a linear combination of the steady and steady-travelling solutions gives a good approximation of the zonal-mean zonal flow of the unsteady solution, suggesting that the chaotic solution at high Reynolds numbers exists mostly within a relatively low-dimensional space spanned by the steady and steady-travelling solutions, which become unstable at low Reynolds numbers.
Polyvinylidene Flouride (PVDF) is a film-type polymer that has been used as sensors and actuators in various applications due to its mechanical toughness, flexibility, and low density. A PVDF sensor typically covers an area of the host structure over which mechanical stress/strain is averaged and converted to electrical energy. This study investigates the fundamental “stress-averaging” mechanism for dynamic strain sensing in the in-plane mode. A numerical simulation was conducted to simulate the “stress-averaging” mechanism of a PVDF sensor attached on a cantilever beam subjected to an impact loading, taking into account the contribution of piezoelectricity, the cantilever beam’s modal properties, and electronic signal conditioning. Impact tests and FEM analysis were also carried out to verify the numerical simulation results. The results of impact tests indicate the excellent capability of the attached PVDF sensor in capturing the fundamental natural frequencies of the cantilever beam. There is a good agreement between the PVDF sensor’s output voltage predicted by the numerical simulation and that obtained in the impact tests. Parametric studies were conducted to investigate the effects of sensor size and sensor position and it is shown that a larger sensor tends to generate higher output voltage than a smaller one at the same location. However, the effect of sensor location seems to be more significant for larger sensors due to the cancelling problem. Overall, PVDF sensors exhibit excellent sensing capability for in-plane dynamic strain induced by impact loading.
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