1. What Is Resonance?

Most structures and machines undergo natural oscillations, and therefore periodic external forces acting on them can cause resonance. Resonance is often referred to as oscillations at the natural frequency or at the critical frequency. Resonance is the phenomenon of a sharp increase in the amplitude of forced oscillations, which occurs when the frequency of external excitation approaches the resonant frequencies determined by the properties of the system. The increase in oscillation amplitude is only a consequence of resonance — the cause is the coincidence of the external (excitation) frequency with the internal (natural) frequency of the vibrating system (rotor-bearing).

Resonance is the phenomenon whereby at a certain frequency of the excitation force, the vibrating system becomes particularly responsive to the action of that force. System parameters such as low stiffness and/or weak damping, acting on the rotor machine at the resonant frequency, can lead to the occurrence of resonance. Resonance does not necessarily lead to machine breakdown or component failure, except when defects in the machine cause vibration, or when a nearby installed machine "induces" vibration at the same frequency as the natural frequencies.

This is comparable to the oscillations of a bell or a drum. In the case of a bell (Fig. 1), all its energy is in the potential form when it is stationary and at the highest points of its trajectory, and as it passes through the lowest point at maximum velocity, the energy converts to kinetic. Potential energy is proportional to the mass of the bell and the height of the lift relative to the lowest point; kinetic energy is proportional to the mass and the square of the velocity at the measurement point. That is, if you strike the bell, it will resonate at a specific frequency (or frequencies). If it is at rest, it will not oscillate at the resonant frequency.

Resonance is a property of the machine whether it is running or not. It should be noted that the dynamic stiffness of the shaft when the machine is rotating can differ significantly from the static stiffness when the machine is stopped, while the resonance changes only insignificantly.

There is an established rule, based on practical experience, which states that resonant frequencies measured during machine shutdown (coastdown) are approximately 20 percent lower than the forced vibration frequencies. Resonant frequencies of individual machine assemblies and parts — such as the shaft, rotor, casing, and foundation — are oscillations at their natural frequencies.

After machine installation, the resonant frequencies may change their values due to changes in system parameters (mass, stiffness, and damping), which after connecting all the mechanisms of the machine into a single unit may increase or decrease. Additionally, dynamic stiffness, as noted above, can shift the resonant frequencies when machines operate at nominal rotation speed. Most machines are designed so that the rotor does not have the same natural frequency as the shaft. A machine consisting of one or two mechanisms should not be operated at a resonant frequency. However, with wear and changes in clearances, the natural frequency very often shifts toward the operating rotation speed, causing resonance.

The sudden appearance of oscillations at a defect frequency — such as a loosened fit or other fault — can cause the machine to vibrate at its resonant frequency. In this case, machine vibration will increase from an acceptable level to an unacceptable one if the oscillations are caused by the resonance of machine assemblies or elements.

2. Resonance During Startup and Shutdown (Fig. 2)

Resonance can be observed during machine startup, when it passes through the resonant frequency (Fig. 2). As the rotation speed increases, the amplitude will grow to its maximum value at the resonant frequency (n_res) and decrease after passing through it. When the rotor passes through resonance, the vibration phase changes by 180 degrees. At resonance, system oscillations are shifted in phase by 90 degrees relative to the oscillations of the excitation force.

The 180-degree phase shift is often observed only on rotors that have a single correction plane (Fig. 3, left). More complex "shaft/rotor-bearing" systems (Fig. 3, right) have a phase shift that lies in the range of 160° to 180°. Whenever a vibration analysis specialist observes a high oscillation amplitude, they should assume that its rise to an unacceptable level may be related to system resonance.

3. Rotor Configurations (Fig. 3)

The vibration behavior of a rotor depends critically on its geometry and how it is supported. A simple rotor with a single correction plane (an overhung disk) shows a clean 180° phase shift through resonance. A more complex system — such as two connected rotors through a cardan shaft — exhibits multiple coupled modes and the phase shift may deviate from the ideal 180°.

4. Mass, Stiffness, and Damping (Figs. 4–7)

Mass, stiffness, and damping — these are the three parameters of the vibrating system that affect the frequency and increase the amplitude of oscillations at resonance.

Mass characterizes the properties of the body and is a measure of its inertia (the greater the mass, the less acceleration it acquires under the action of a periodic force), which causes its oscillations.

Stiffness is a property of the system that opposes the inertial forces arising as a result of mass forces.

Damping is a property of the system that reduces the energy of oscillations by converting it to thermal energy due to friction in the mechanical system.

where f_n — natural frequency, k — stiffness, m — mass, ζ — damping ratio, Q — quality factor (amplification at resonance), A_res — resonance amplitude, F₀ — excitation force amplitude.

To reduce resonance, the system parameters are selected so that its resonant frequencies are positioned as far as possible from possible external excitation frequencies. In practice, so-called dynamic vibration absorbers, or dampers, are used for this purpose.

The interactive simulator below (replacing static Figs. 4–7 from the original article) shows the Amplitude-Frequency Characteristic (AFC) of a simple vibrating system consisting of mass, spring, and damper. Adjust the parameters to observe these effects in real time:

An analogy can be drawn with a guitar string. The tighter you pull the string on the guitar (more stiffness), the higher the tone (resonant frequency) rises — until the string breaks. If you use the thickest string (greater mass), the tone it produces will be lower.

5. Measuring Resonance (Fig. 8)

One of the most common methods for measuring the resonant frequency of a structure is impact excitation using an instrumented hammer.

The impact on the structure, in the form of an input strike, excites small disturbing forces over a certain frequency range. The oscillations created by the impact represent a transient, short-duration energy transfer process. The spectrum of the impact force is continuous, with maximum amplitude at 0 Hz and subsequent decrease with increasing frequency.

The impact duration and the spectrum shape during impact excitation are determined by the mass and stiffness of both the impact hammer and the machine structure. When using a relatively small hammer on a hard structure, the stiffness of the hammer tip determines the spectrum. The hammer tip acts as a mechanical filter. By selecting the stiffness of the hammer tip, one can choose the frequency range of investigation.

When using this measurement technique, it is very important to strike different points of the structure, since not all resonant frequencies can always be measured by striking and measuring at one and the same point. When determining machine resonance, both points — the impact point and the measurement point — must be verified (tested).

If the hammer has a soft tip, the main quantity of output energy will excite oscillations at low frequencies. A hammer with a hard tip delivers little energy at any specific frequency, except that its output energy will excite oscillations at high frequencies. The response to the hammer strike can be measured with a single-channel analyzer, provided the machine is stopped and disconnected.

6. Amplitude–Phase Frequency Characteristic — APFC (Fig. 9)

Machine resonance can be determined using a single-channel analyzer as an increase in oscillation amplitude at the resonant frequency and by the 180-degree phase change when passing through resonance — if amplitude and phase of oscillations are measured at the rotation frequency during machine startup (run-up) or shutdown (coastdown). The characteristic constructed on the basis of these measurements is called the Amplitude-Phase Frequency Characteristic (APFC).

Analysis of the APFC (Fig. 9) allows the vibration analysis specialist to identify the resonant frequencies of the rotor.

If the phase does not change when passing through a suspected resonance, then the amplitude increase may be related to random excitation and is not a rotor resonance. In such cases, in addition to vibration measurements during run-up/coastdown, it is recommended to perform an "impact test".

When using a multi-channel vibration analyzer, the resonance of a structure can be determined with great accuracy by measuring input and output signals from the system at the same time, while controlling the vibration phase and coherence collected during the same time period. Coherence is a dual-channel function used to evaluate the degree of linearity between input and output signals of the system. This means that resonant frequencies can be identified significantly faster.

7. Some Considerations About Machine Resonance

Attention should be paid to the analysis of different types of machines and their operating modes, which may complicate resonance testing:

Due to differences in structural stiffness in the horizontal and vertical directions, the resonant frequency will differ depending on the direction. Therefore, resonances may be most strongly manifested in a particular direction.

As previously discussed, resonant frequencies differ when the machine is running vs. when it is stopped (switched off). Vertical equipment, as a rule, causes a great deal of concern, since during operation of such equipment there is always resonance that occurs during operation of a cantilever-mounted electric motor.

Some machines have a large mass, and therefore cannot be excited with a hammer — alternative excitation methods are required to determine the actual resonant frequencies. Sometimes, on very large machines, a vibrator is used that is tuned to a specific frequency range, because the vibrator has the ability to deliver large amounts of energy at each individual frequency when oscillating.

And one final consideration — before conducting resonance testing, it is very useful to first measure the background vibration level (the response to random excitation from the surrounding environment). This will help prevent an error in determining the diagnosis (system resonance) based on the maximum oscillation amplitude at a certain frequency above the background level.

8. Summary

In this article we discussed the influence of resonant frequencies on machine vibration. All structures and machines have resonant frequencies, but resonance does not affect the machine if there are no frequencies that excite it. If the machine's vibration is excited by its own natural frequency, then there are three options for detuning the system from resonance:

Option 1. Shift the frequency of the disturbing force away from the resonant frequency.

Option 2. Shift the resonant frequency away from the frequency of the disturbing force.

Option 3. Increase the damping of the system to reduce the resonance amplification factor.

Options 2 and 3 usually require some structural modifications that cannot be performed unless modal analysis and/or finite element study have been carried out on the structure.