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Constant-natural-frequency isolators outperform conventional mounts and cost less than active-damping systems.
In a recent article in Machine Design, "Shaking up vibration models," the discussion centered on a revised isolation -system model that represents a vibrationsensitive object such as a cutting-tool holder or CMM measuring head, mounted on the machine bed or frame. External vibrations create harmful effects on the object by exciting vibratory relative motion between the tool or measuring head and the workpiece or part.
The discussion showed that satisfying the following equation:
leads to adequate vibration isolation in sensitive systems. Here, f^sub n^ = the system natural frequency, δ = damping (log decrement) of the mount, Δ = allowable vibration amplitude in the work zone, a^sub 0^ = floor displacement, and µ.= transmissibility from the bed into the object work zone at frequency f.
Dynamic coupling
The revised model describes objects supported by one vibration isolator with both floor and bed vibrations acting on the same axis. In contrast, most real-life objects use several mounts or vibration isolators. Such setups can experience so-called "dynamic coupling," with vertical floor vibrations inducing vertical and horizontal vibrations in the bed, and vice versa. Floors usually vibrate vertically and horizontally, with horizontal vibration amplitudes -30% lower than vertical amplitudes.
Most commercial vibration isolators have much lower horizontal stiffness than vertical stiffness. Thus, the isolators significantly attenuate horizontal floor vibrations and do not transform them into horizontal vibrations of the bed. This is beneficial because precision objects are usually much more sensitive to horizontal than vertical vibrations. Therefore, the main source of undesirable horizontal bed vibrations is horizontal vibrations transformed from vertical floor vibrations - due to dynamic coupling in vibration isolation systems.
However, satisfying the following conditions eliminates dynamic coupling between vertical (Z-axis) floor vibrations and horizontal X and V- axis) vibrations of an object supported by n isolators:
Here i = the isolators number (1 < i < n); k = stiffness of the i-th isolator in the vertical (Z) direction; a^sub xi^, and a^sub y^, are X and Y coordinates of the i-th isolator if the coordinate systems origin is at the objects center of gravity (eg).
A natural way to comply with this equation is to determine the distribution of the object weight...