Figure 1 depicts the schematic of a two degree-of-freedom double mounting system (a) and compares that to the schematic of single mounting system (b). As in any isolation application, the goal is to isolate the base from the vibration of the machine caused by the perturbation force F, i.e., lowering the force transmitted to the base Ft , while avoiding excessive vibration of the mass (bouncing) due to shock excitation at the base (x_base) common in vehicular (including luxury watercrafts) applications.
Figure 1(a) Double
mounting system
Figure 1(b) Single
mounting system
To have the same static deflection as that of single mount, the mounts used in a double mounting application are twice as stiff at the ones that would have been used in an equivalent single mounting application; note that the 2 similar mounts in series provide the same stiffness as that of a single mount twice as soft.
Considering that the dynamics associated with the inertia forces (including the ones associated with the auxiliary mass) are negligible at low frequencies, isolation effectiveness of double mounting at low frequencies is very much the same as that of single mounting. At higher frequencies the auxiliary mass M2 affects the dynamics of the double mount system, enhancing its isolation capabilities
As expected and clear from Figures 1(a) and 1(b) the addition of the auxiliary mass and the 2nd mount, doubles the degrees of freedom (and thus the number of resonances) of double mounting systems compared to those of single mounting ones. Figure 2 shows the magnitude of the frequency response functions mapping the machine vibration force (F) and shock excitation (x_base) to the transmitted force (Ft) and machine displacement (x), in a double mount system (with two similar mounts) and compares them to those of a single mount system; the magnitude of Ft/F which is the same as x/x_base (also called ‘transmissibility’) is depicted in Figure 2(a) and magnitude of x/F (also called ‘receptance’) is depicted in Figure 2(b). The doubling of a resonant frequencies in double mount systems is apparent in Figure 2; note that there are twice as many peaks in the double mounting frequency response function traces (red and blue traces) compared to that of a single mounting system (black trace). For proper isolation, the 2nd peak must be placed at a frequency lower than the lowest excitation frequency (for example, half the shaft rpm frequency in case of Diesel engine), necessitating the use of very large auxiliary mass.
As stated above, the main advantage of double mounting is its high effectiveness in lowering the transmission of vibration at high frequencies; see Figure 2(a). In fact at high enough frequencies, the vibration isolation effectiveness of double mounting systems is twice as good as that of single mounted systems. Considering that structure-borne noise occurs normally at higher frequencies than those of perceptible vibration,, double mounting with a reasonably sized auxiliary mass (with about 30-40% of the weight of the isolated machine itself) is an attractive solution for abating the transmission of such noise.
The transmission of low-frequency vibration and low-frequency structure-borne noise in double mounted isolation systems are generally no better than those of equivalent single mounted isolation system. Moreover, in a double mounting system the motion of the isolated machine in response to perturbation forces (the receptance) depicted by Figure 2(b) is no smaller (better) than that of single mounting. This is true at all frequencies, especially at the 2nd resonant frequency, where the motion of double mounted machine is excessively higher (worse) than that of single mounted machine, due to resonance. The side effects of the enhancement in high-frequency effectiveness is the creation of a 2nd resonance in transmitted vibration as well as the machine motion evident from the ‘transmissibilily’ and ‘receptance’ plots of Figures 2(a) and 2(b).
Another drawback of extending high-frequency vibration isolation effectiveness farther into low frequencies by increasing the size of the auxiliary mass ( M2 in Figure 1(a) ) is the weight penalty which could be an important issue in vehicular applications.
Figure 2 Transmissibility and receptance FRFs of a double mounting system (with two auxiliary masses) and a single mounting system
Figure 3 depicts the ratio of the two natural frequencies vs. the size of the 2nd (auxiliary) mass m2 (as a fraction of the main (machine) mass, m1) for a two degree of freedom double mounted isolation system. Evident from Figure 3, lower 2nd natural frequencies require excessively large auxiliary masses. In other words, extending the benefits of double mounting to lower frequencies requires unacceptably large auxiliary masses. For example, reducing the natural frequency ratio by a factor of 2, from 6 to 3, requires the auxiliary mass to increase by a factor of 10 from 10% of the main mass to 100% of the main mass.
Figure 3 Natural frequency ratio
