Parametric Study Damping Variation Stiffness Variation Adding Mass Double (Two Stage) Mounting

The impact of varying the parameters of a mounted system (damping, stiffness, and mass), one parameter at a time, on the isolation effectiveness in a one degree of freedom system is analyzed. The spring-mass-dashpot system of Figure 1 is frequently used as a one degree of freedom approximation of an isolation system. The goal is to isolate the base from the vibration of the mass caused by the excitation 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. The spring dashpot pair, known as Voight model, is commonly used to approximate the dynamics of commonly used viscoelastic isolators such as rubber mounts.

Figure 1 A simple mounting system

Figures 2(a), 3(a), and 5(a) depict the magnitudes of the frequency response functions (FRF) mapping the vibration excitation force (F) and shock excitation (x_base) to the transmitted force Ft and mass displacement x, commonly known as transmissibility, for varying a) damping, b) stiffnesses, and c) mass (by adding a dead weight to the mass), respectively. Figures 2(b), 3(b), and 5(b) depict the magnitudes of the frequency response functions (FRF) mapping the vibration excitation force (F) to the mass displacement x, for the same parameter variations.

Damping Variation Clear from the transmissibility plot of Figure 2(a), lower damping does not affect neither the transmission of force from the vibrating mass (e.g. diesel generator) to the base nor the transmission of shock inputs at the base to the mass at low frequencies, increase them around the resonant frequency, and lower these transmissions at high frequencies. This is why in most isolation applications the mount is selected to be highly underdamped with the resonant frequency of the isolated system less than 70% of the lowest vibration and shock excitation frequencies of the system. Of course, when either or both excitations (vibration and shock) are broadband (like discontinuous shock excitation common in marine applications), the above stated guideline will not be effective; broadband excitation will set off the resonance of the mounted system causing the mass to bounce. Figure 2(b) indicates that variation in damping only influences the transmission of vibration force to the motion of the mass at resonance; the lower the damping the more severe the resonance. The reason practitioners do not embrace the obvious solution to the resonance problem, i.e., adding damping, is that damping deteriorates the high-frequency vibration isolation effectiveness of the mount resulting in transmission of noise. Figure 2 Transmissibility and displacement Frequency response functions for different damping ratios.

Stiffness Variation Figure 3(a) indicates that the softer mounts (mounts with lower stiffness) decrease both the transmission of force from the vibrating mass to the base and the transmission of shock inputs at the base to the mass over a reasonably large range in frequencies (which normally coincide with the frequency range of interest in most mounting applications), including resonant frequency. Figure 3(b) indicates that soft mounting adversely affects the transmission of vibration force to the motion of the mass at lower frequencies, including at resonance; the lower the stiffness the larger the low-frequency motion of the mass caused by vibration forces. This adverse effect of soft-mounting would be tolerable if a) the excitation is harmonic (not impulsive such as shock inputs) and b) the resonant frequency of the isolated system is below the lowest excitation frequencies of the system. Figure 3 Transmissibility and displacement Frequency response functions for different stiffnesses. Frequently, e.g. in marine vibration isolation applications, the isolated object is excited by both periodic vibration, e.g., the force of firing in a diesel engine, and impulsive shock, e.g., the base excitation in a marine diesel generator or any other vehicular engine applications. The base excitation has normally a broadband spectrum, so there is always some energy at the resonant frequency(ies) of the isolated system. If the mounts are highly underdamped, the vibration amplitude at this (these) frequency(ies) becomes excessive. This problem can be addressed by adding damping to the mount. But in vibration isolators, as stated earlier, broadband damping which results in high frequency vibration (and noise) transmission is avoided.

Adding Mass

Increasing the mass of the isolated object by adding an auxiliary mass to it, in order to vary the attributes of the isolation system, is a common high-frequency vibration isolation practice. One way of realizing this scheme is to install the vibrating machine on a massive concrete block and isolate the block from the foundation by rubber or neoprene isolators. A schematic of a simple mounted system with an auxiliary mass, m, added to the isolated body (machine/equipment), M, is shown in Figure 4.

Figure 4 A simple mounting system with added mass (m)

Figure 5(a) indicates that the addition of extra mass to the mounted object, enhances the high-frequency shock and vibration isolation attributes. It does not change the low-frequency dynamics of the system, except at resonant frequency. It also lowers the resonant frequency of the system while making the system more oscillatory (underdamped); note that damping ratio is inversely proportional to the square root of the mass. Again, as in soft-mounting the adverse effect on resonane would be tolerable if a) the excitation is harmonic (not impulsive such as shock inputs) and b) the resonant frequency of the isolated system is below the lowest excitation frequencies of the system. Clear from Figure 5, the high-frequency isolation enhancement of added mass will be realized when a sizeable amount of it, e.g., at least half the mass of the machine being isolated, is used. To put this in perspective, for isolating a 170 KVA diesel generator weighting about 2 tons, one needs to attach a slab of steel weighting at least 1 ton to the machine to make the effort worthwhile. The inconvenience of implementing this solution, especially in retrofit applications, not to mention the potential problems associated with excessive static displacement of the mount due to the added mass is clear. In addition this solution has a considerable weight penalty which is highly objectionable in vehicular/marine applications. Figure 5 Transmissibility and displacement Frequency response functions for different added masses (m).

Double (Two Stage) Mounting A somewhat common practice for enhancing the high frequency vibration isolation, is to place another mount between the added mass m, see Figure 4, and the isolated object M. This arrangement, called double mounting is discussed in detail here.

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