Machine Design

# Shaking up vibration models

Here’s a better way to look at vibration-isolation systems.

Who, What, Where

Authored by Eugene I. Rivin
Components Div.,
Designatronics Inc.,
New Hyde Park, N.Y.

Edited by Kenneth J. Korane,
[email protected]

Key points
• Traditional vibration-isolation models are too simple to generate accurate results.
• More realistic models and associated equations lead to more-accurate vibration isolation.

Resources
Designatronics Inc., designatronics.com

Additional technical data is available at vibrationmounts.com/Catalog.htm#Request

For a more-detailed analysis, see E.I. Rivin, “Passive Vibration Isolation,” 2003, ASME Press.

The idea is simple: Protect sensitive objects from vibrating floors and foundations. So is the basic vibration-isolation system model, found in most every vibration textbook, and shown in the accompanying graphic. An object, modeled as a solid block, sits on a low-stiffness vibration isolator that, in turn, mounts on the vibrating support surface.

This system with mass m and stiffness kv has natural frequency:

where ωn = the angular natural frequency.

Based on this model, the “Absolute transmissibility” graph shows how vibrations transmitted from the floor to the object depend on supporting-surface vibration frequency (expressed as ω/ωn) and isolator damping (expressed as a fraction ζ of the critical damping ζc of the system). Thus, the object vibrates less than the supporting surface if vibration frequency ω ≥ 1.41ωn. Lower-stiffness vibration isolators decrease the natural frequency ωn and transmit less vibration to the object.

The graph also shows that increasing isolator damping ζ reduces an object’s vibration amplitude at resonance (which is not an important regime for isolating precision objects) and at ω < 1.41ωn, but lessens isolation at frequencies ω > ωn.

This theory is attractive because it is straightforward, and engineers use it extensively for practical applications. Two basic conclusions from the above discussion are:

1. Softer isolators transmit less vibration from supporting surfaces to objects, so any level of transmission attenuation is possible. For example, a system can have 98% attenuation if ω > 7ωn, but this requires vibration isolators with low damping (such as metal springs or a lightly filled rubber).

2. Unfortunately, low-stiffness isolators result in fn = ωn/2π ≤ 2 to 3 Hz. They are not practical because objects become sensitive to even a light touch or other minor excitations, such as machinetool tables reversing directions or stages repositioning photolithography tools.

Isolation options
The basic system’s natural frequency can be accurately calculated using:

where Δ = the static deflection of the spring in cm. Thus, for fn = 2 to 3Hz, Δ = 6.25 to 2.8 cm – quite significant static deflections.

Generally, engineers want vibration isolators with the greatest allowable stiffness for a given application. But choosing passive isolators per the basic model results in unacceptably low stiffness in cases that need significant vibration attenuation. So designers turn to servocontrolled or “active” vibration isolators.

Active isolators are expensive and not as reliable as passive isolators. Another approach, also costly, attaches the object to a massive “inertia block” that, in turn, mounts on vibration isolators. Because mass m increases, proportionally stiffer isolators produce the same low ωn. Besides being expensive to construct, inertia blocks take a lot of valuable floor space and require considerable time and effort to move.

Realistic modeling
The problem is that the basic vibration-isolation model is too simplistic and does not properly represent typical vibration-sensitive objects. Important issues to consider are real-life characteristics of vibration isolation systems and dynamic coupling between vertical and horizontal vibratory motions. Examining these two issues lets us reformulate the design model and improve performance of vibration-isolation systems.

The basic system implicitly assumes the object is a solid block. But machine tools, photolithography equipment, and metrology devices like coordinate-measuring machines (CMMs) are dynamic devices. And external vibrations generate internal vibrations in the working zone: for instance, between the cutting tool and workpiece, or the CMM’s measuring stylus and part being measured.

We can depict such objects as shown in the “Revised model.” Here MU represents the upper member, such as the production machine’s toolholder or the CMM’s measuring head. MB represents the mass and km the structural stiffness of the bed or frame. Usually, MB >> MU. External vibrations create an objectionable effect on the object, noted as vibratory relative motion xrbetween the tool or measuring head and the workpiece or part.

The “Relative transmissibility” graph shows how relative motion xr depends on external vibration frequency and on the object’s structural parameters (structural natural frequency ωm and damping ζ, expressed as a fraction of the critical damping).

The object is rather insensitive to low-frequency vibration — masses MU and MB move pretty much together. Sensitivity further decreases as structural-resonance frequency ωm = 2πfm increases. The greatest sensitivity is at fm; at higher frequencies relative motion stabilizes and does not significantly depend on frequency. The revised model can be constructed for vertical, horizontal transverse, and horizontal-longitudinal directions. Well-designed precision equipment usually has high fm and less vibration sensitivity at frequencies f < fm.

Real-world considerations
Extensive studies of floor vibrations and the structural dynamics of typical machines show that maximum vibration amplitudes and frequency ranges are quite consistent for similar types of facilities.

For instance, the worst case for manufacturing plant floors is, conservatively, a spectrum with constant displacement amplitudes of 3 μm for vertical and 2 μm for horizontal vibrations in the 3 to 20-Hz range. Vertical vibration amplitudes ≤ 0.15 μm and horizontal ≤ 0.05 μm in the 3 to 12-Hz frequency range conservatively represent the worst vibration environment in microelectronics fabricating facilities (“fabs”). In many cases, there is no need to measure actual plant-floor vibrations — these numbers and ranges are generally appropriate.

At frequencies below these ranges, floor vibrations produce negligible relative motion in the working zone. At frequencies above the higher limits, floor-vibration intensity declines fast.

The most important lowest (“fundamental”) structural natural frequencies range from fm ≈ 10 Hz for ultraprecision photolithography tools to fm ≈ 20 to 50 Hz for reasonably designed precision machine tools, microscopes, and similar equipment.

Such systems require isolation at these frequency ranges and maximum floor displacement amplitudes. Also, most precision devices are much more sensitive to horizontal than vertical vibrations. For machine tools the difference is 3 to 10; for ultraprecision equipment the difference is from ~1.5 to 2.0 for optical microscopes to about 10 for microlithography steppers.

Because commercial vibration isolators have one-third to one-tenth the stiffness in the horizontal direction, compared to the vertical, transmission of horizontal floor vibrations to the bed is negligible.

On the other hand, vertical floor vibrations are usually more intense and isolators have relatively high vertical stiffness, so vertical floor vibrations transmitted to the object’s bed are important. Due to dynamic coupling between vertical and horizontal vibrations in the vibration-isolation system, vertical vibrations from the floor induce dangerous horizontal vibrations in the bed.

New model
Based on all this, we can develop a criterion for isolating vibration-sensitive objects. The four-part graphic “Vibration characteristics of sensitive objects” illustrates how floor vibration propagates into the work zone of sensitive objects.

The “Displacement” graph conservatively approximates peak floor-vibration amplitudes at manufacturing plants or fabs, with maximum peak amplitudes not exceeding ao.

The second graph, “Floor to bed,” presents three cases:

1. A relatively stiff mount (a metal jack or a rigid rubber mount) supports the object and resonant frequency is relatively high.

2. The mount is much softer, resulting in lower-frequency resonance but with the same resonance amplitude (having the same damping) as in case 1.

3. The mount had the same stiffness as in case 2, but with much higher damping and, thus, significantly lower resonance amplitude.

The third graph, “Bed to work area,” plots absolute vibratory motion transmitted from the bed to the work zone (as shown in “Relative transmissibility.”)

In other words, floor vibration filtered by the absolute transmissibility function becomes vibrations in the bed. And bed vibrations, filtered by the relative transmissibility function, become the relative vibrations in the working zone.

Relative vibration amplitude (determined by the object’s specifications) cannot exceed the specified amplitude Δo in the “Relative displacement” graph. Case 1 (rigid mount) exceeds Δo at and near two resonance peaks, one at the isolation natural frequency fn and another at the structural frequency fs. The second resonance is usually at low floor-vibration frequencies and, therefore, less dangerous.

The “Relative displacement” plot shows there are two ways to isolate an object with amplitude in the work zone aw < Δo. One reduces mount stiffness and, thus, fn , seen in “Floor to bed.” The second increases mount damping, as in “Bed to work area.” In-depth analysis shows that satisfying the following equation leads to adequate vibration isolation:

where δ = damping (log decrement) of the mount, and μf = transmissibility from the bed into the object work zone at frequency f.

Thus, the resonance of the object and vibration isolator is the vibration-isolation system’s working regime. And second, increasing damping in vibration isolators improves isolation and lets designers use stiffer isolators and/ or higher natural frequency isolation systems to produce the required isolation.

In the next issue of Machine Design, we’ll explore some practical applications of these findings.

External vibrations can cause unacceptable motion between a cutting tool and workpiece. Good antivibration systems minimize the harmful effects.

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