An Inverse Force Analysis of a Planar Two-Spring System

A closed-form inverse force analysis was performed on a planar two-spring system. The two springs were grounded to pivots at one end and attached to a common pivot at the other. A known force was applied to the common pivot of the system, and it was required to determine all of the assembly configurations. By variable elimination, a sixth degree polynomial in the resultant length of one spring was derived, and from this, six real solutions of the point of application of force were obtained. Following this, the applied force was incremented along a line and the six paths of the moving pivot were tracked starting from the zero-load configurations. An analysis of these results showed stability phenomena indicating the workspace of this system contained regions of negative spring stiffness and points of catastrophe.

An Inverse Force Analysis of a Planar Two-Spring System

A closed-form inverse force analysis was performed on a planar two-spring system. The two springs were grounded to pivots at one end and attached to a common pivot at the other. A known force was applied to the common pivot of the system, and it was required to determine all of the assembly configurations. By variable elimination, a sixth degree polynomial in the resultant length of one spring was derived, and from this, six real solutions of the point of application of force were obtained. Following this, the applied force was incremented along a line and the six paths of the moving pivot were tracked starting from the zero-load configurations. An analysis of these results showed stability phenomena indicating the workspace of this system contained regions of negative spring stiffness and points of catastrophe.

A characterization of the uniform distribution on the circle in the analysis of directional data

It was conjectured some time ago by K. V. Mardia that if, for random samples of some fixed size N ≧ 2 from a given non-degenerate circular population, the sample mean direction and the sample resultant length are independently distributed, then the population must be uniformly distributed round the circle. In this paper it is shown that, apart from one minor exception, Mardia's conjecture is true in the case N = 2. No regularity conditions are necessary for the proof. The same problem has been studied, subject to regularity conditions, by Kent, Mardia and Rao (1976) for all sample sizes N ≧ 2 except N = 4.

A characterization of the uniform distribution on the circle in the analysis of directional data

It was conjectured some time ago by K. V. Mardia that if, for random samples of some fixed size N ≧ 2 from a given non-degenerate circular population, the sample mean direction and the sample resultant length are independently distributed, then the population must be uniformly distributed round the circle. In this paper it is shown that, apart from one minor exception, Mardia's conjecture is true in the case N = 2. No regularity conditions are necessary for the proof. The same problem has been studied, subject to regularity conditions, by Kent, Mardia and Rao (1976) for all sample sizes N ≧ 2 except N = 4.