Lateral deflection of archery arrows

A whole mechanical and mathematical model of an arrow–bow motion system, which accounts for arrow deflection in the lateral plane, has been created. The arrow is modelled as a shaft within the framework of the technical theory of vibration of beams with particles at the ends. The model takes into consideration the mechanical properties of a string, bow limbs and a grip as an oscillator of concentrated elastic and inertial elements connected with the feathered end of the arrow. Experimental methods for measuring arrow stiffness in sport archery practice are discussed with reference to the dynamic stability of the internal ballistics. Theoretical investigation of natural modes and frequencies of bow and arrow vibration has been conducted as a study of a boundary problem for the differential equation of the fourth order. The solution of the equation has been found in a form of polynomial series using a method of successive approximations. Data for the first four natural frequencies and modes have been obtained and practical conclusions have been drawn. An archer’s paradox and the spine phenomenon are explained using the results of mathematical modelling. As a result of modelling and computer simulations, an engineering method for matching bow and arrow parameters has been proposed. Comparative results for the wrong and right combination of these parameter values for the modern sport bow and two arrows are presented.