The simple harmonic motions are given by
The resultant displacement is
Equating the real and imaginary parts, we get
The resultant motion of Eqn. (2.11) is simple harmonic with amplitude and phase angle given by
When N is large and ϕ is small, we may write
and the phase difference between the first component vibration x1 and Nth component vibration xN is nearly equal to 2θ.
The resultant amplitude may be obtained by the vector polygon method (Fig. 2.1). The polygon OABCD is drawn with each side of length a and making an angle φ with the neighbouring side. The resultant has the amplitude OD with the phase angle = ∠ DOA with respect to the first vibration.
Special Cases
(i) We consider the special case when there is superposition of a large number of vibrations xi of very small amplitude a but continuously increasing phase. The polygon will then become an arc of a circle and the chord joining the first and the last points of the arc will represent the amplitude of the resultant vibration (Fig. 2.2). When the last component vibration is at A, the first and the last component vibration are in opposite phase and the amplitude of the resultant vibration = OA = diameter of the circle. When the last component vibration is at B, the first and the last component vibrations are in phase, the polygon becomes a complete circle and the amplitude of the resultant vibration is zero.
(ii) When the successive amplitudes of a large number of component vibrations decrease slowly and the phase angles increase continuously the polygen becomes a spiral converging asymptotically to the centre of the first semicircle.
