A Mass-Spring-Damper Model of a Bouncing Ball (Conference proceeding)

The mechanical properties of a vertically dropped ball, represented by an equivalent mass-spring-damper model, are related to the coefficient of restitution and the time of contact of the ball during one bounce with the impacting surface. In addition, it is shown that the coefficient of restitution and contact time of a single bounce are related to the total number of bounces and the total time elapsing between dropping the ball and the ball coming to rest. For a ball with significant bounce, approximate expressions for model parameters, i.e., stiffness and damping or equivalently natural frequency and damping ratio, are developed. Experimentally based results for a bouncing pingpong ball are presented

A new method of dropping the time ball

Mr. Robert Henry, in reading a paper on " A new method of dropping the time ball," referred briefly to the existing means for letting Hobart know when it is supposed to be 1 o'clock. In his opinion, provided the arrangements were all in good working order, the present method should be effective. But to ensure absolute accuracy it was desirable that the process should be automatic as far as possible, and with this object in view he submitted a scheme for dropping the ball, and also firing the gun electrically. He described the details, and acknowledged indebtedness to Captain Parker, R.N., for the idea as to the manner for releasing the time ball

A Mass-Spring-Damper Model of a Bouncing Ball

The mechanical properties of a vertically dropped ball, represented by an equivalent mass-spring-damper model, are shown to be related to impact parameters. In particular, the paper develops relationships connecting the mass, stiffness and damping of a linear ball model to the coefficient of restitution and the contact time of the ball with the surface during one bounce. The paper also shows that the ball model parameters are functions of quantities readily determined in an experiment: (i) the height from which the ball is dropped from rest, (ii) the number of bounces, and (iii) the time elapsing between dropping the ball and the ball coming to rest. For a ball with significant bounce, approximate expressions are derived for the model parameters as well as for the natural frequency and damping ratio. Results from numerical and experimental studies of a bouncing ping-pong ball are presented

The Computational Complexity of Duality

We show that for any given norm ball or proper cone, weak membership in its dual ball or dual cone is polynomial-time reducible to weak membership in the given ball or cone. A consequence is that the weak membership or membership problem for a ball or cone is NP-hard if and only if the corresponding problem for the dual ball or cone is NP-hard. In a similar vein, we show that computation of the dual norm of a given norm is polynomial-time reducible to computation of the given norm. This extends to convex functions satisfying a polynomial growth condition: for such a given function, computation of its Fenchel dual/conjugate is polynomial-time reducible to computation of the given function. Hence the computation of a norm or a convex function of polynomial-growth is NP-hard if and only if the computation of its dual norm or Fenchel dual is NP-hard. We discuss implications of these results on the weak membership problem for a symmetric convex body and its polar dual, the polynomial approximability of Mahler volume, and the weak membership problem for the epigraph of a convex function with polynomial growth and that of its Fenchel dual.Comment: 14 page

Uniform infinite planar triangulation and related time-reversed critical branching process

We establish a connection between the uniform infinite planar triangulation and some critical time-reversed branching process. This allows to find a scaling limit for the principal boundary component of a ball of radius R for large R (i.e. for a boundary component separating the ball from infinity). We show also that outside of R-ball a contour exists that has length linear in R.Comment: 27 pages, 5 figures, LaTe