1.　What is vibration ?

　When considering the concept of vibration, one might envision objects oscillating or exhibiting a rattling or shakiness upon contact. However, from an engineering perspective, let us explore the definition of vibration in the context of machine condition monitoring and diagnostics. To do so, we will refer to the ISO 2041 standard ⁽¹⁾, which provides the terminology utilized in mechanical vibration and condition monitoring.

　ISO 2041 defines “vibration” as “mechanical oscillations about an equilibrium point”, and further defines “oscillation” as “variation, usually with time, of the magnitude of a quantity with respect to a specified reference when the magnitude is alternately greater and smaller than the specified reference”. Accordingly, the “mechanical vibration” under discussion can be defined as “a variation, usually with time, of the magnitude of a quantity of motion or position of a mechanical system with respect to a specified reference when the magnitude is alternately greater and smaller than the specified reference”.

　For example, if the rotor of a rotating machine swings as it rotates, and from a certain direction it appears to move closer and further away repeatedly, this is called “shaft vibration”. Furthermore, if the shaft vibration is transmitted to the casing via the bearings, and the position of a certain part of the casing repeatedly fluctuates over time, this is called “casing vibration”, and both of these can be referred to as “machine vibration”.

　So, what is the repetition speed of “the magnitude is alternately greater and smaller than the specified reference” in mechanical vibration, in other words, the vibration frequency (number of repetitions per second: Hz)? In the field of machine condition monitoring and diagnosis, the measurement target typically ranges from a few Hz to 10 kHz.

　In reality, vibration phenomena can exhibit complex behavior, which can be considered a synthesis of “simple harmonic vibration” at multiple frequencies. Therefore, it is essential to understand “simple harmonic vibration” as a fundamental concept. According to ISO 2041, “simple harmonic vibration (sinusoidal vibration)” is defined as “periodic vibration where the values of the vibration parameters can be described as sinusoidal functions of the independent time variable”. In this definition, the single harmonic vibration can be expressed by equation (1).

\begin{equation}
y=y_{0}\sin \left( \omega t+\varphi _{0}\right) \tag{$1$}
\end{equation}

　Where

\begin{alignat}{2}
& y & \; & \text{is the simple harmonic vibration;} \\
& y_{0} & \; & \text{is the amplitude;} \\
& ω & \; & \text{is the angular frequency;} \\
& t & \; & \text{is the independent variable time;} \\
& \varphi _{0} & \; & \text{is the initial phase angle of the vibration.} \\
\end{alignat}

　When the vibration frequency is \(f\), the angular frequency ω is as in equation (2).

\begin{equation}
\omega =2\pi f \tag{$2$}
\end{equation}

　Figure 1-1 shows a simple system consisting only of an ideal spring and a weight with no damping as an example of single harmonic vibration. At first, the weight is pulled downward by a displacement \(D\) from the equilibrium point (the zero position), i.e., to the minus \(D\) position. When the weight is released from this position, it repeatedly moves up and down between plus \(D\) and minus \(D\) with the equilibrium point at its center, and the time variation of this motion shows a sinusoidal waveform as shown in the figure. This motion shows a single harmonic vibration, where \(f\) is the oscillation frequency and \(T\) is the period.

　If we denote the amplitude of the vibration displacement as \(D\) and the displacement of the simple harmonic vibration as \(d\), we can replace \(y\) in equation (1) by d and \(y_{0}\) by \(D\) to express the equation as in equation (3).

\begin{equation}
　d=D sin⁡( ωt+\varphi_{0} )=D sin⁡( 2πft+\varphi_{0}) \tag{$3$}
\end{equation}

　As shown in Figure 1-1, the amplitude \(D\) is the peak value of the vibration displacement. When measuring the displacement of mechanical vibration, it is common to express it in terms of the peak-to-peak value of \(2D\), which is the difference between the positive peak value and the negative peak value.

2. Vibration monitoring parameters

　There are three types of vibration monitoring parameters that are used: Displacement, Velocity, and Acceleration. In this column, the amplitude of displacement is represented as \(D\), velocity as \(V\), and acceleration as \(A\). The relationship between these parameters is explained.

　In any given motion, differentiating the displacement will yield the velocity, and differentiating the velocity will yield the acceleration. This principle applies to simple harmonic vibration as well. Differentiating equation (3) for the displacement \(d\) of simple harmonic vibration yields the velocity \(v\) in equation (4), and further differentiating the velocity \(v\) yields the acceleration \(a\) in equation (5). As demonstrated in Figures 1-2, each vibration monitoring parameter, including displacement, velocity, and acceleration, is interconnected with differentiation and integration.

\begin{equation}
v =\omega D\cos \left( \omega t+\varphi _{0}\right)=V\cos \left( \omega t+\varphi _{0}\right) \tag{$4$}
\end{equation}
\begin{equation}
a=-\omega ^{2}D\sin \left( \omega t+\varphi _{0}\right)=-A\sin \left( \omega t+\varphi _{0}\right) \tag{$5$}
\end{equation}

　Therefore, by replacing the part of the sine function that repeats between ±1 with the maximum value of 1 in equations (3) to (5), as shown in equations (6) to (8), the relationship between “displacement \(D\)”, “velocity \(V\)” and “acceleration \(A\)” can be expressed in a relationship equation that focuses on amplitude and frequency alone.

\begin{equation}
D=V/\omega =V/2\pi f=A/\left( 2\pi f\right) ^{2} \tag{$6$}
\end{equation}
\begin{equation}
V=\omega D=2\pi fD=A/2\pi f \tag{$7$}
\end{equation}
\begin{equation}
A=\omega V=2\pi fV=\left( 2\pi f\right) ^{2}D \tag{$8$}
\end{equation}

　Figure 1-3 also shows the amplitudes of displacement and acceleration for the case of an object vibrating with a constant vibration velocity of 10 mm/s peak amplitude, even if the vibration frequency changes, by applying equations (6) and (8). Note that the vibration energy is proportional to the square of the vibration velocity, but this relationship is independent of frequency, so Figures 1-3 can be said to represent a vibrating object whose vibration energy is constant with respect to frequency.

　As illustrated in Figures 1-3, the amplitude of vibration displacement is inversely proportional to frequency and increases with decreasing frequency. Conversely, the amplitude of acceleration is directly proportional to frequency and increases with frequency. This indicates that acceleration is generally suitable for measuring high-frequency vibrations, such as those associated with rolling bearing and gear malfunctions, while displacement is better suited for low-frequency vibrations, such as shaft vibrations.

　Table 1-1 shows the vibration monitoring parameters and the approximate applicable frequency range.

Table 1-1:　Vibration monitoring parameters, applicable frequencies, and target machines

Monitoring parameter	Applicable frequency range	Target machines Applicable abnormal phenomena	Examples of applicable transducers
Displacement	1 Hz to 200 Hz	Large rotating machines supported by journal bearings Imbalance, Misalignment, Looseness	Eddy Current Type Displacement Transducer FK-202F
Velocity	10 Hz to 1 kHz	General purpose machines supported by rolling element bearings Imbalance, Misalignment, Looseness, Gear abnormality	Piezoelectric Type Velocity Transducer CV-86
Acceleration	1 kHz or higher	General purpose machines supported by rolling element bearings Bearing abnormality, Gear abnormality	Piezoelectric Type Accelerometer CA-302
NOTE The applicable frequency range is provided as a general guideline. There may be instances where it is applied beyond the range shown in the table. In addition, the applicable target machines and abnormal phenomena are presented as a reference to the main ones, and there may be instances where they are applied differently. The response frequencies of the transducers shown as examples cover a wider range than the applicable frequency range of the monitoring parameters shown in the table. Therefore, the applications of these transducers are not limited to the frequency range shown in the table.

3. Importance of vibration measurement in condition monitoring and diagnosis

Condition monitoring and diagnosis of rotating machinery includes primary effect parameters such as inputs and outputs to the machines (e.g., power, flow, pressure, speed, torque, etc.) and secondary effect parameters such as vibration, noise, temperature, and lubricant condition that change as the machine operates. For instance, the ISO 17359 standard⁽²⁾ provides a comprehensive list of parameters applicable to condition monitoring for various machinery types, including electric motors and steam turbines. As shown in Table 1-2, which is an excerpt from the list of parameters in Table A.1 of ISO 17359, vibration is one of the monitoring parameters that can be widely applied to the various types of machines listed in it.

Tables B.2 to B.10 of ISO 17359 detail the measurement parameters suitable for the identified faults for each machine type shown in Table 1-2. These show that vibration is very effective as a measurement parameter for machine condition monitoring, as vibration is an abnormal symptom or indicator of many faults. Tables 1-3 to 1-6 present examples of measurement parameters and techniques for faults in electric motors, steam turbines, pumps, and compressors. These tables demonstrate that vibration is an abnormal symptom or indicator of numerous fault occurrences.

As listed, vibration monitoring and analysis and diagnosis techniques are the most widely used condition monitoring and diagnosis technology for a wide range of abnormal phenomena in various rotating machinery.

In the next and subsequent columns, we will outline the condition monitoring system for large, high-speed rotating machinery supported by journal bearings and the condition monitoring system for general-purpose rotating machinery supported by rolling element bearings.

Table 1-2:　Examples of condition monitoring parameters by machine type

Table 1-3:　Example of electric motor faults matched to measurement parameters and techniques

Table 1-4:　Example of steam turbine faults matched to measurement parameters and techniquese

Table 1-5:　Example of pump faults matched to measurement parameters and techniques

Table 1-6:　Example of compressor faults matched to measurement parameters and techniques

References

(1)	ISO 2041:2018, “Mechanical vibration, shock and condition monitoring – Vocabulary”, International Organization for Standardization, 2018.
(2)	ISO 17359:2018, “Condition monitoring and diagnostics of machines – General guidelines”, International Organization for Standardization, 2018.