As the name implies, double mounting (also called two-stage mounting) is an isolation practice in which two mounts separated by an auxiliary mass are used at each mounting location.
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.
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.
Figure 1(a) Double mounting system
Figure 1(b) Single mounting system
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.
The main advantage of double mounting is its high effectiveness in lowering the transmission of vibration at high frequencies; see Figure 2(a). But, 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 excessive weight penalty which is an important issue in vehicular applications.
Figure 2 Transmissibility and receptance FRFs mapping the machine vibration force, F, and shock excitation at the base, x_base, to the force transmitted to the base, Ft, and displacement of the machine, x, in 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
As stated above the schematics and simulations presented above, are for simplified one and two degree of freedom systems representing single and double mounted isolation arrangements, respectively. In actuality, and assuming all the masses (the machine mass and the auxiliary mass) are rigid and the isolators (mounts) are massless, a single mounted machine has 6 degrees of freedom (x,y,z linear motions and roll, pitch, yaw rotary motions) and thus 6 resonances and a double mounted machine has 12 degrees of freedom and thus 12 resonances. Two of such modes are depicted in Figure 4. To achieve the smallest transmissibility, the mounted system (single or double) should be designed so that its highest resonant frequency is placed well below the lowest of the machine forcing frequencies. In addition, in marine applications the lowest resonant frequency of the mounted system should be placed well above the rolling and pitching frequencies of the motion of the vessel.
Figure 4 A sample (2) of the 12 vibration modes of a double mounted system
Considering that neither the auxiliary mass in a double mounted system is infinitely rigid nor the mounts are massless, the mounting system will exhibit a number of undesirable high frequency modes due to the lack of rigidity in the auxiliary mass. These are in addition to the 12 rigid body modes of such mounting system, stated above.
Two of such modes are depicted in Figure 5. Note that in these modes, all of the vibration motion is due to the bending deformation of the auxiliary mass, not the motion of the isolated machine.
Figure 5 A sample (2) of the 12+ vibration modes of a double mounted system associated with the auxiliary mass
The isolation effectiveness of double mounting isolation could get reduced by these flexible body modes at medium to high frequencies. The frequency response functions (FRFs) of Figure 6 show the transmissibilty (a) and receptance (b) measured at a mounting foot of a double mounted (blue traces) and single mounted (red traces) machine.
As in the simplified two degree-of-freedom approximation of double mounting system of Figure 1(b) and its corresponding transmissiblity of Figure 2(a), the multi degree-of-freedom double mounting has high effectiveness in lowering the transmission of vibration at high frequencies except at the resonant frequencies corresponding to the vibration modes of the auxiliary mass. At these frequencies , e.g. 25 and 125 Hz in Figure 6(a), a double mounted system transmits more vibration (has higher transmissibility) than its single mounting system equivalent. This vibration transmission problem gets exacerbated when any of these frequencies matches any of the perturbing frequencies of the isolated machine.
Also, as in simplified two degree-of-freedom approximation the receptance of a double mounting system is the same as that of single mounting at most frequencies and larger (worse) than single mounting system at resonant frequencies corresponding to the vibration modes of the auxiliary mass; see Figure 6(b).
Figure 6 Transmissibility (a) and receptance (b) FRFs of a double mounted system (blue traces) and single mounted system (red traces)
To minimize the undesirable effects of the dynamics of the auxiliary mass on the isolation effectiveness of a double mounted system, utmost attention should be paid to the design of the auxiliary mass so that its resonant frequencies are well beyond the operational frequencies.
As discussed above, the added mounting stage more than doubles the degrees of freedom of an isolation system. Despite its deceiving simple look, proper implementation of double mounting and hence benefiting from its high frequency isolation advantage requires thorough understanding of a) the spatial dynamics associated with the double mounting system, and b) the flexible body vibration of the auxiliary mass. As shown in Figure 6(a), although the trend in high frequency transmissibility attenuation of double mounting is still twice as high as that of single mounting, at certain discrete frequencies it might be the opposite. Extreme care should be taken to avoid matching these discrete frequencies to any of the higher order harmonic of the vibration of the machine being isolated. Addressing all the concerns associated with double mounting, a few of which discussed above, would make properly (not haphazardly) designing and implementing double mounting, costly.
Also, somewhat straightforward to realize double mounting while a machine is being installed, it is a major challenge to retrofit an existing single mount isolation system to a double mount one.
Raising an existing isolated machine to place a large mass underneath it, could be challenging in some retrofit applications. Besides, the height of a double mount isolation mechanism (two rubber mounts and a mass in between) might be difficult to accommodate in some retrofit applications.