The mean‐value theory predicts the mean line through the logarithmically recorded frequency‐response curve of a complex vibrator, the height of the resonance peaks, and the height of the minima that occur between every two resonance peaks. It predicts the mean and the extremes in the frequency‐ response curve from the first resonance of the vibrator to very high frequencies. The computations are based on the mode masses and on the density of the resonances. They are simple and they give considerable insight into the dynamics of complex vibrators. Part I of this paper presents the basics of the mean‐value theory. Part II deals with the variation of the vibration velocity with the distance from the driver and with the frequency. Part III is concerned with shells without and with stiffeners and with rib‐stiffened plates. Because it is always the component of the characteristic velocity that is transmitted by discontinuities (as though the parts of the vibrator behind them were infinitely large) that determines the geometric mean values of the response, the effect of ribs and other appendages on the vibration can be predicted without much computation. As an example of the application of the theory, the vibration isolation that can be obtained by spring mounting a motor on a ring welded into a shell is computed in detail and the predictions for the ring excited shell are compared with the measurements.