In the last issue of MACHINE DESIGN, we examined methods to predict how vibration-isolation systems would perform. (“Shaking up vibration models,” Nov. 6, 2008, p. 63 to 68.) This discussion centered on a revised isolation-system model that represents a vibration-sensitive 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 work piece or part.
Who, What, Where
Advanced Antivibration Components Div., Designatronics Inc., New Hyde Park, N.Y.
Edited by Kenneth J. Korane, [email protected]
Additional technical data is available at vibrationmounts.com/Catalog.html#Request
For more in-depth discussion, see E.I. Rivin, "Passive Vibration Isolation," 2003, ASME Press.
The discussion showed that satisfying the following equation:
leads to adequate vibration isolation in sensitive systems. Here, fn = the system natural frequency, § = damping (log decrement) of the mount, Δo = allowable vibration amplitude in the work zone, ao = floor displacement, and f = transmissibility from the bed into the object work zone at frequency f.
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 selhorizontal 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 Y-axis) vibrations of an object supported by n isolators:
Σikziaxi = 0, Σikziayi= 0 (2)
Here i = the isolator’s number (1<i< n); kzi = stiffness of the i-th isolator in the vertical (Z) direction; axi, and ayiare X and Y coordinates of the i-th isolator if the coordinate system’s origin is at the object’s center of gravity (cg).
A natural way to comply with this equation is to determine the distribution of the object weight W between mounting points, and then use isolators with stiffness constants kzi proportional to the fraction Wi of the total weight acting on the i-th mounting point. So if Wireplaces each constant stiffness kzi, then the summation equation becomes an identity because distances axi and ayi are measured from the cg.
Thus, if vertical stiffness of the i-th isolator/mount is proportional to Wi, or
kzi = AWi, and A = a constant, (3)
then equation (2) is satisfied because it reflects the moment equilibrium of the object in vertical planes X-Z and Y-Z.
Most commercial, passive vibration isolators have constant stiffness (CS) kz. Correlation (3) is critical to satisfy equation (2), but it is impossible to achieve with conventional CS isolators. Here are some reasons:
1. The desired vertical natural frequency determines total vertical stiffness kz of all isolators, or
Individual isolators should have stiffness proportional to the weight loads acting on them. To determine weight distribution, we must know cg. Unless dealing with simple objects, you can determine cg using detailed design drawings, CAD software, or experimentally. However, experiments are not feasible for heavy machinery. And machine manufacturers seldom provide cg information. Further complicating matters, for production machinery with heavy moving parts (tables, stages, and gantries) or with widely differing workpiece masses, cg can shift significantly while a machine is running. Usually, users determine cg by “expert judgment” with at least ±10 to 15% error.
2. With cg known or assumed, weight distribution can be calculated only for a statically determinate case when the object mounts on three (n = 3) isolators. Most real vibration-sensitive objects sit on more than three supports. Finding weight distribution in such statically indeterminate cases requires additional information or assumptions.
Weight distribution can change if the object’s mounting surfaces and the floor are not flat, if mounting surfaces vary in stiffness, or if the object is not leveled properly. Differences in weight distribution between mounting points before and after leveling may exceed ±35%.
Other causes for indeterminacy of the weight distribution are the above-mentioned movements of heavy parts and variations in workpiece weights. Conservatively, actual weight distribution between the mounting points can deviate from estimates as much as ±50%.
3. After establishing weight distribution, engineers must select isolators that satisfy equations (2) and (3). Among different-size commercial CS mounts, ratios of static stiffness coefficients kz vary between ~ 1.2 to 2.0, or a ±1.26 stiffness variation. The accepted tolerance on rubber hardness is ±5 durometer units, equivalent to ±17% variation in stiffness. Often, different mounts in the same product line use different rubber blends with different dynamic-to-static stiffness ratios Kdyn. Kdyn usually ranges between 1.5 and 4.0, resulting in additional ±1.5× stiffness variation.
Considering all these uncertainties (cg position, weight distribution, rubber hardness, static and dynamic stiffnesses), the stiffness of isolators selected according to the calculated/assumed weight distribution conservatively varies at least ±1.15 × 1.5 × 1.26 × 1.17 × 1.5 ±4×. Thus, condition (2) is not satisfied even in the crude first approximation, leading to strong coupling between vertical and horizontal/rocking vibratory modes in vibration-isolation systems using CS isolators. As a result, engineers must use much softer isolators than suggested by equation (1). Because soft isolators are not desirable, designers often turn to expensive and rather unreliable active isolators.
A better answer may be so called “constant natural frequency” (CNF) isolators, which reliably satisfy equation (2) without experiments or cumbersome calculations. CNF isolator stiffness is not constant but proportional to the weight load. Thus, it automatically satisfies the decoupling conditions, making CNF isolators passive smart elements. An ideal CNF isolator has stiffness in all directions (not just the Z direction) proportional to the load along its main axis, as expressed by (3). Commercial CNF isolators do not deviate from this proportionality more than ±10 to 15%, so they easily satisfy the decoupling condition (2) for Z–direction vibrations.
Testing has verified these results. One case involved a precision surface grinder 20 times more sensitive to horizontal than to vertical floor vibrations. It was tested on CS and on CNF isolators with the floor vibrating at a constant 5-μm amplitude ( Z direction ) and frequency range of 9 to 35 Hz.
The accompanying graphic shows relative vibration amplitudes between the grinding wheel and machined surface with the machine installed on five CS isolators with natural frequency fvz= 15 Hz (isolator manufacturer’s recommendation) and on five CNF isolators, fvz = 20 Hz. Relative vibrations are ~30% lower (0.25 versus 0.35 μm) when the machine mounts on 2× stiffer CNF isolators.
No explanation for this effect could be found, other than the dynamic decoupling CNF isolators provide. In light of equation (1), isolators made from high-damping rubber can allow for a further increase of fz. Thus, isolators combining CNF characteristics and high damping can be as effective as low fz (low effective stiffness) conventional CS isolators and may, in many cases, replace active isolators.
Inside CNF isolators
Many of these isolators use volumetric incompressibility of rubberlike materials which deform in compression only if they can bulge on free (unloaded) surfaces. The exible element is a monolithic rubber block molded in a relatively simple cavity, but its core consists of two quasi-independent rubber rings. The rings are bonded to top and bottom metal covers. A calibrated clearance gap separates the inner surface of the lid and outer surface of the rubber ring. Another calibrated gap is between the inner surface of the rst ring and outer surface of the second ring.
At low loads, all four sides of the rings are free to bulge, resulting in relatively low sti ness. As weight increases, the bulge on the outer surface of the rst ring touches the lid, and the bulge on the ring’s inner surface contacts the bulge on the outer surface of the second ring. Thus, the gaps gradually ll with rubber, constraining expansion and increasing sti ness.
A nut transmits loads to the leveling bolt. And a rubber disc attaches the nut to the top cover to permit vertical displacements during leveling.
The Advanced Antivibration Components Div. of Designatronics Inc. uses this design for three Uniload models of CNF isolators with natural frequency fz ≈ 10 Hz. 100-mm OD isolators are made from two blends of oil-resistant rubber. One, of black rubber, has Durometer H = 60 and a load range of 550 to 1,900 lb. The other, brown-rubber model, has Durometer H = 40 and load range of 300 to 1,000 lb. A heavy-duty, 146-mm OD isolator made of black rubber handles loads from 1,600 to 8,000 lb.
Engineers can design CNF isolators for other natural frequencies, both lower and higher than 10 Hz, using principles described in U.S. Patent 5,934,653. Additional technical data is available at vibrationmounts.com/Catalog.htm#Request.