Daniel Bernoulli (1742)

Abstract

Daniel Bernoulli had written Euler in 1735 of his work on the (transversally) vibrating rod (see p. 70). Around 1740, he finally wrote a paper on the subject.¹ In this work, he obtains the pendulum condition as â d⁴y/ dz⁴ = y, and the boundary conditions appropriate for the rod clamped at one end; he solves it approximately by both the series method of undetermined coefficients and in the form²

$$y\, = \,A\,\cosh rz\, + \,B\,\sinh \,rz\, + \,C\cos \,rz\, + \,D\,\sin rz,\,\hat a\, = \,1/{r^4}$$

(13.1)

, for the fundamental solution; in the latter form he finds approximate values of r for the higher modes; he effectively obtains the flexural rigidity e (where â = ae/p) in terms of the deflection of the rod under a certain force at the free end; thus he predicts the fundamental frequency of a needle, clamped at one end, which he confirms experimentally. After completing this work, Bernoulli took up the problem of the vibrating rod that is free at both ends, apparently inspired to understand the vibrations of carillon rods. He wrote a second paper³ that treats this case in detail, calculating both nodal points and frequencies for the first five modes. He reports on experiments in which the different modes are induced by gently holding a rod at predicted nodal points and heard to vibrate at predicted frequencies. In doing these experiments, Bernoulli hears superimposed modes and he gives a theoretical argument that superposition can occur.