An experimental investigation of forced vibrations

Abstract

The solution of the differential equation y″ + 2 Ry′ + n 2 y = E cos pt is written in a new form which clearly exhibits many important facts thus far overlooked by theoretical and experimental investigators. Writing s = n − p, and Δ n = n − √ n 2 − R 2, it is found: ( a) When s ≠ Δ n, there are “beats,” and the first “beat” maximum is greater than any later maximum while the first “beat” minimum is less than any later “beat” minimum. The “beat” frequency is (s − Δn) 2π . ( b) When n 2 − p 2 = R 2, there are no “beats,” and the resultant amplitude grows monotonically from zero to the amplitude of the forced vibration, ( c) At resonance, when n = p, we still have maxima which occur with a frequency Δn 2π in a damped system. ( d) The absence of “beats” is neither a sufficient nor a necessary condition for resonance in a damped system. In the experimental investigation the upper extremity of a simple pendulum was moved in simple harmonic motion and photographic records obtained of the motion of the pendulum bob. Different degrees of damping were used, ranging from very small to critical. The experimental results are in excellent agreement with theory.