T1-25 3.  The displacements (max. deflections) of the mounts can be calculated from Equation (45) for any given single disturbing force or torque. If several force/torques act simultaneously, vector addition of forces in different directions is required, and Equation (45) cannot be used. 4.  The case of a horizontal disturbing force has not been considered in this presentation. 5.  Other things being equal, the best arrangement for the mounts is to arrange them so that their resultant force passes through the center of gravity of the equipment and that its line of action is a principal axis. If there is a resultant torque about the center of gravity, its direction should be about a principal axis through the center of gravity. However, if this arrangement is impractical, it need not be adhered to. 9.0    COMPLEX DRIVING FORCES When the disturbing forces are neither sinusoidal nor suddenly applied, the vibration analysis becomes more compli- cated. While it is more difficult to give general guidelines or methods of analysis, one can consider every force-time variation as composed of components of different frequencies, each being a multiple of the basic (usually driving) frequency. Math- ematically, this is known as expanding an arbitrary function into a Fourier series. Once these frequency components (har- monics) are determined, each one being sinusoidal at a different frequency, any component can be analyzed like a sinusoi- dal force. This can provide at least some understanding of the vibration phenomenon. Often the lowest-frequency (funda- mental) component predominates and is the most important component to analyze. It is possible, however, that the design of the vibration isolation system will appear unfeasible on the basis of an analysis of only the fundamental component, whereas the exact analysis would show that a vibration isolation mounting can be useful; i.e., sometimes an analysis of components of several frequencies may be required [1]. This, however, may be quite difficult. In such cases, resolving an arbitrary force- time variation into several harmonics can provide some insight. The following represents data in the Fourier series (decomposition into several harmonics) of some representative force- time variations in Figure 38, which are neither sinusoidal nor sudden. Each force is assumed to be a periodic function of the time; l = t/T, where t is pulse width, T is the process period; 3 = fundamental frequency. The Fourier expansions for these forcing functions are given in Table 2. Figure 38    Typical Periodic Nonsinusoidal Vibratory Processes t y h 2h Square wave h t t 2t t T y h 2h Saw tooth h t y 2h Repeated step t