*by Ron Newman*

Vibration monitoring and analysis are recognized as key components of a well-founded reliability centered maintenance (RCM) program. Machinery health and long-term reliability are interpolated from trends in vibration amplitude; however, often these levels increase to dangerously high values with little or no warning! Resonance is frequently identified as the root cause. Understanding the physics of natural frequency, and the tools and methods used to recognize and confirm a resonant condition are essential skills for the vibration analyst.

All structures exhibit natural frequencies similar to the ringing of a tuning fork. In a practical sense, machines and machine components can be viewed as a summation of tuning forks, or simple single degree of freedom, mass, spring, or damping systems. Viewing the dynamic behavior of just one of these mass, spring, damping units illustrates how the elements interact to determine natural frequency. Effectively, when subjected to force(s), each element – mass, stiffness and damping – is working to restore the natural “at rest” state and is more or less successful depending on frequency. Below the natural frequency, the resulting response (vibration) is largely controlled by mass, whereas above the natural frequency, the response is governed by the stiffness. Unfortunately, precisely at the natural frequency, the mass and stiffness effects are 180 degrees out of phase with each other, so in theory, the vibration would go to infinity were it not for damping. The basic relationship for the mass/spring/damping model is defined in the Figure 1 equation.

ω = angular frequency in radians/sec (radian is the angle subtended by an arc length equal to the radius)

π = Pi (~ 3.14)

f = frequency in cycles/sec (Hz)

k = stiffness

m = mass

**Figure 1: Generally, natural frequency (ω**_{n}) is the square root of stiffness (k) divided by mass (m)

_{n}) is the square root of stiffness (k) divided by mass (m)

It can be seen from Figure 2 that by increasing the number in the denominator (mass), the resulting natural frequency would be reduced.

###### Figure 2: Adding mass (m) lowers the natural frequency

Conversely, increasing the number in the numerator (stiffness), the natural frequency would increase, as shown in Figure 3.

###### Figure 3: Adding stiffness (k) raises the natural frequency

Damping serves to minimize or reduce the amplitude at the natural frequency and is often regarded as a measure of the frictional energy that is defined by the molecular characteristics of the structure. Again, in a practical sense, the tuning fork is a lightly damped structure versus, for example, a heavily damped tabletop. See Figure 4.

###### Figure 4: Relative response magnitude (left) and phase (right) with damping factors (?) of 0 (no damping) to 1.0 (critically damped)

A condition of resonance occurs when a machine’s natural frequency coincides with a forcing frequency within the operation of the equipment. The forcing frequency results from the dynamic forces generated by the rotating elements and can be either discrete, i.e., 1x, 2x, 3x … nthx due to unbalance, misalignment, looseness, blade or vane pass, etc., or broadband, i.e., widely distributed noise due to severe looseness, flow turbulence, cavitation, sliding/rubbing contact, frictional energy, etc.

**Figure 5: Forcing frequencies may be discrete (left) or broadband (right)**

Unlike our single degree of freedom model, machines will have several degrees of freedom, each with their own dynamic characteristics, so it should not be too hard to imagine that one or more of these natural frequencies may be excited, resulting in a serious resonant condition.

###### Figure 6: Resonance will occur when machine speed coincides with a natural frequency peak (green curve)

The problem is exacerbated by the modern VFD motor. Because it operates over a wide speed range, the likelihood that a natural frequency will be excited increases substantially. Vibration can increase by a factor of 10 to 30 at this critical speed. Left unresolved, component or complete catastrophic failure is inevitable.

Note: Large turbo-machinery will often reveal a rotor critical speedreferring to the elastic deformation of the shaft at a natural frequency. General plant machinery,such as fans, pumps, blowers, rolls, etc., will exhibit a critical speed due to a structural resonance -- that is a resonance of the bearing pedestal, fan shroud, pump volute or piping, base, drive belts, or any of a number of structural support members.

A common method that is useful in identifying a possible resonant condition is comparing the vibration levels at a specific measurement point in each of the axis – horizontal, vertical, axial. When a large ratio (≥ 10:1) is found, for example between the horizontal and vertical, resonance should be suspected.

**Figure 7: Suspect resonance when amplitude ratios reach ≥ 10:1**

Dramatic changes in vibration amplitude arising from moderate speed variations, as well as broken welds or cracks, are sure signs of fatigue due to resonance. The vibration analyst has a number of diagnostic tools to confirm a resonant condition.

- Bump test
- Two-channel impact test
- Run-up/coast-down measurement
- Order tracking

## BUMP TEST

**Figure 8: Bump test**

The single channel bump test provides the user with information regarding the dominant natural frequency. It is simple to perform and requires no additional hardware beyond a data collector with sufficient functionality. NOTE: The amplitude of the peak is only relative to the impact, i.e., it is not a measure of the “gain” or “amplification” at the resonant frequency and no phase data is acquired.

## TWO-CHANNEL IMPACT TEST

**Figure 9: Two-channel impact test**

The two-channel impact test measures both the input to the structure (force) and the output response of the structure (acceleration). The ratio of output to input gives the frequency response function. Here, unlike with the single channel bump test, the user acquires phase data AND the amplification factor.

## RUN-UP/COAST DOWN MEASUREMENT

**Figure 10: Run-up/coast down measurement**

The run-up/coast down is another method used to highlight a resonant condition at a specific operating speed. The graph plots amplitude vs. frequency vs. RPM.

## ORDER TRACKING

**Figure 11: Order tracking**

Order analysis plots amplitude and phase vs. RPM. The data can be represented as a BODE plot (left) or NYQUIST plot (right) with amplitude and phase plotted as vectors in polar coordinates.

Resolving the problem may mean a simple modification or an expensive reengineering of the equipment. Initial, low-cost options include minimizing the forcing frequencies. As an example, often the primary forcing frequency is due to a multiple of running speeds (1x, 2x, etc.), but balancing the rotor to a better quality grade and/or employing a laser alignment will yield good results. It should be noted that field balancing and laser shaft alignment are also key components of a precision maintenance strategy. Another economical solution is to operate the machine at a different speed.

When structural modifications are required, additional measurement data is essential to ensure changes will be effective from both an engineering and cost point of view. Modal analysis and operating deflection shape (ODS) seem to be competing methodologies to the casual observer. In truth, modal analysis is principally useful to the design and development engineer needing to know the structural properties of the device. It characterizes the structure in terms of natural frequency, damping and shape independent of the excitation.

An operating deflection shape tells the user what is really happening in-situ. ODS is the weighted summation of the modal responses, the weighting depends on the excitation or forcing frequencies, i.e., it is a combination of the forcing functions and the structural modes.

**Figure 12: Operating deflection shape (left) is the weighted summation of the modal responses (right)**

Commercially available software will aid the operator in the creation of a simple computer model of the structure’s geometry, identifying the measurement points and collecting the data. The model is animated so the user can visually see where significant nodes lie, thus optimizing the modifications required. In the sense that these modifications are attempting to shift the natural frequency higher or lower, there are preferred methods to accomplish the task. ODS provides the user with information that will be valuable in identifying points on the structure, where, as an example, stiffening (raising the natural frequency) will be most effective. Other less commonly used options to resolve a resonance problem include additional damping and/or a dynamic absorber. Since damping is an integral property of the structure, unless the aim is basic noise reduction of a resonant panel, shroud, coupling guard, cowling, etc., damping is less practical for large machinery. Dynamic absorbers have been used successfully to minimize vibration on large vertical boiler feedwater pumps. The design is critical to their success.

In the end, the objective is to ensure a smoothly running plant. Understanding the issue of resonance is one more step along the way to a world-class maintenance organization.

**Figure 13: Vibration severity according to ISO 10816-3**

## REFERENCES

- Pruftechnik -
*An Engineers Guide to Shaft Alignment/Vibration Analysis/Balancing* - Pruftechnik -
*Vibration Handbook* - Bruel & Kjaer - Technical review(s) – 1987-1, 1994-2, 1973-3, 1973-4
- Bruel & Kjaer – Application note(s) - How to Determine the Modal Parameters of Simple Structures (Svend Gade, Henrik Erlufsen, Hans Konstantin-Hansen) - Mobility Measurements (Kevin Gatzwiller, Henrik Erlufsen)
- Bruel & Kjaer - Primers - Structural Testing Parts 1 & 2
- Mobius - CAT II / CAT III course notes
- International Standards Organization (ISO) - 10816-3

## About the Author

*Ron Newman is in Technical Sales & Support at Pruftechnik, Canada. He has over 35 years in the field of machine vibration & acoustics. Ron is an ISO Level III BSc in Mechanical Engineering Technology. www.pruftechnik.ca*