# Resonance and Magnification in Piping System Vibration

By Vincent A. Carucci

From a practical standpoint, the most important problem with forced vibration in a piping system is resonance. Resonance occurs when the vibration forcing frequency is at or very close to an acoustical or mechanical natural frequency of the system. Since most structures and piping systems have very little damping, the vibration amplitude becomes very high if resonance occurs.

If resonance occurs, it will usually be at the fundamental (i.e., first) or one of the other lower order natural frequencies. Two types of resonance must be considered: mechanical and acoustical.

## Mechanical Resonance

Real systems have some damping which eventually reduces the vibration amplitude. Damping also helps determine the peak system response to an exciting force at a given frequency. This peak amplitude can be approximated by using the magnification factor (MF). The MF is the ratio of the dynamic deflection to the static deflection that would occur in the system if the forces were statically applied. The MF is a function of the forcing frequency, mechanical natural frequency, and the viscous damping coefficient (see Figure 1).

When the damping is low, the MF will be high at the natural frequency. The MF will typically be in the range of 10 to 25 at resonance.

## Acoustical Resonance

Acoustical resonance in a piping system occurs when reflected pressure pulses from a piping discontinuity (e.g., closed valve, tee, elbow) travel back to and arrive at their source in time to join in phase with the next pulse. The resultant larger pressure pulse travels down the pipe, is reflected back again, increases in size again, and so on. The final amplitude of this reinforcement process is limited by the dynamic friction forces in the piping system.

In acoustic resonance, the system responds with large amplitude pressure surges when it is excited by relatively small amplitude pressure fluctuations that occur near an acoustical natural frequency.

This is analogous to the large amplitude deflections that are caused by a mechanical resonance. As with mechanical resonance, the fundamental acoustic natural frequency is typically the easiest to excite.

The MF in the case of acoustic resonance represents the amplitude of pressure response at some point in the acoustic system to the amplitude of pressure excitation. The qualification “at some point” is important since there are locations in an acoustic system (i.e., nodes) where the pulsations are relatively small even at resonance. However, there is high pressure magnification at other points. Since acoustic damping from viscous drag forces and heat conduction is small, the MF for an acoustic resonance can be very high, ranging from 10 to 40.

## Pipe Failure Considerations

Eliminating or isolating vibration sources is the ideal solution to a vibration problem. However, it is often impossible to do this in a cost-effective manner. Therefore, most process plant piping systems vibrate to some extent. Acceptable vibration limits must be established to determine if a particular vibrating pipe is a potential problem that must be resolved, or whether it can be left alone.

## Fatigue Stress Considerations

Steady-state piping vibrations are usually evaluated with respect to their effect on the fatigue life of the piping. The allowable stress values must be determined from fatigue curves due to the large number of cycles encountered in steady-state vibration. The number of cycles to failure is a function of the mean stress, alternating stress, material, environment, and the geometry of the piping component.

## Transient Vibration Considerations

Transient vibrations are generally evaluated considering excessive surge pressures, pipe deflections, or reaction loads. Fatigue is a less important concern in this case because of the low number of dynamic transient events expected. However, fatigue must be considered if the number of cycles becomes significant.

The primary concern with transient vibrations is the possibility of unacceptable overpressures within the piping and connected equipment due to the dynamic surge pressures that are typical of water-hammer events. Peak dynamic surge pressures are additive to the static operating pressure. The total pressure can then exceed the specified design pressure for the system. Significant overpressure due to hydraulic surge may cause catastrophic failure of the piping system, as well.

Other types of damage that can occur include the failure of pipe supports, restraints, or small diameter branch connections, as well as the overloading of attached equipment. Therefore, designing a piping system for these latter effects is based primarily on controlling pipe movements, and assuring that the support system and equipment can absorb the transient reactions.