VIBRATION MEASUREMENT BASICS
based on materials from DLI (edited by Smirnov V.A.)

What is vibration?

Vibration - These are mechanical vibrations of the body.
The simplest type vibrations is the oscillation or repeated movement of an object about an equilibrium position. This type of vibration is called general vibration, because the body moves as a whole and all its parts have the same speed in magnitude and direction. The equilibrium position is the position in which the body is at rest or the position that it will occupy if the sum of the forces acting on it is zero.
The oscillatory motion of a rigid body can be completely described as a combination of six simplest types of motion: translational in three mutually perpendicular directions (x, y, z in Cartesian coordinates) and rotational relative to three mutually perpendicular axes (Ox, Oy, Oz). Any complex movement of the body can be decomposed into these six components. Therefore, such bodies are said to have six degrees of freedom.
For example, a ship can move in the direction of the stern-bow axis (straight ahead), rise and fall up and down, move in the direction of the starboard-port axis, and also rotate about a vertical axis and experience roll and pitch.
Let's imagine an object whose movements are limited to one direction, for example, the pendulum of a wall clock. Such a system is called a system with one degree of freedom, because The position of the pendulum at any time can be determined by one parameter - the angle at the fixing point. Another example of a single degree of freedom system is an elevator, which can only move up and down along a shaft.
Body vibration is always caused by some forces excitement. These forces can be applied to an object from the outside or arise within it. Next we will see that the vibration of a particular object is completely determined by the strength of excitation, its direction and frequency. It is for this reason that vibration analysis can reveal the excitation forces during machine operation. These forces depend on the state of the machine, and knowledge of their characteristics and laws of interaction makes it possible to diagnose defects in the latter.

The simplest harmonic oscillation

The simplest oscillatory movements existing in nature are elastic rectilinear oscillations of a body on a spring (Fig. 1).

Rice. 1. An example of the simplest oscillation.

Such a mechanical system has one degree of freedom. If you move the body some distance from the equilibrium position and release it, the spring will return it to the equilibrium point. However, the body will acquire a certain kinetic energy, overshoot the equilibrium point and deform the spring in the opposite direction. After this, the speed of the body will begin to decrease until it stops at the other extreme position, from where the compressed or extended spring will again begin to return the body back to the equilibrium position. This process will be repeated again and again, with a continuous flow of energy from the body (kinetic energy) to the spring (potential energy) and back.
Figure 1 also shows a graph of body movement versus time. If there were no friction in the system, then these oscillations would continue continuously and indefinitely with constant amplitude and frequency. In real mechanical systems such ideal harmonic motions do not occur. Any real system has friction, which leads to a gradual attenuation of the amplitude and converts the vibration energy into heat. The simplest harmonic movement is described by the following parameters:
T - period of oscillation.
F - oscillation frequency, = 1/T.
Period - this is the time interval that is required to complete one cycle of oscillation, that is, it is the time between two successive moments of crossing the zero point in one direction. Depending on the speed of the oscillations, the period is measured in seconds or milliseconds.
Oscillation frequency - the reciprocal of the period, determines the number of oscillation cycles per period, it is measured in hertz (1Hz = 1/second). When rotating machines are considered, the fundamental frequency corresponds to the rotational speed, which is measured in rpm (1/min) and is defined as:

= F x 60,

Where F- frequency in Hz,
because There are 60 seconds in a minute.

Oscillation equations

If we plot the position (displacement) of an object experiencing simple harmonic oscillations along the vertical axis of the graph, and time along the horizontal scale (see Fig. 1), then the result will be a sinusoid described by the equation:
d=D sin(t),
Where d-instantaneous displacement;
D-maximum displacement;
= 2F - angular (cyclic) frequency, =3.14.

This is the same sine curve that is well known to everyone from trigonometry. It can be considered the simplest and main temporary implementation of vibration. In mathematics, the sine function describes the dependence of the ratio of the leg to the hypotenuse on the value of the opposite angle. With this approach, a sine curve is simply a graph of a sine wave versus the magnitude of the angle. In vibration theory, a sine wave is also a function of time, but one cycle of vibration is sometimes also considered as a 360-degree phase change. We will talk about this in more detail when considering the concept of phase.
The speed of movement mentioned above determines the speed of change in body position. The rate (or speed) of change of a certain quantity relative to time, as is known from mathematics, is determined by the derivative with respect to time:

=dd/dt=Dcos(t),
where n is the instantaneous speed.
From this formula it is clear that the speed during a harmonic oscillation also behaves according to a sinusoidal law, however, due to the differentiation and transformation of sine into cosine, the speed is shifted in phase by 90 (that is, by a quarter of a cycle) relative to the displacement.
Acceleration is the rate of change of velocity:

a=d /dt= - 2 Dsin(t),
where a is the instantaneous acceleration.
It should be noted that the acceleration is out of phase by another 90 degrees, as indicated by the negative sine (that is, 180 degrees relative to the displacement).

From the above equations it can be seen that speed is proportional to displacement times frequency, and acceleration is proportional to displacement times square of frequency.
This means that large displacements at high frequencies must be accompanied by very large velocities and extremely large accelerations. Imagine, for example, a vibrating object that experiences a displacement of 1 mm at a frequency of 100 Hz. The maximum speed of such an oscillation will be equal to the displacement multiplied by the frequency:
=1 x 100 =100 mm With
Acceleration equals displacement times the square of the frequency, or
a = 1 x (100) 2 = 10000 mm s 2 = 10 m s 2
The acceleration of gravity g is equal to 9.81 m/s2. Therefore, in units of g, the acceleration obtained above is approximately equal to
10/9.811 g
Now let's see what happens if we increase the frequency to 1000 Hz
=1 x 1000 = 1000 mm s =1 m/s,
a = 1 x (1000) 2 = 1000000 mm/s 2 = 1000 m/s 2 = 100 g
Thus, we see that high frequencies cannot be accompanied by large displacements, since the enormous accelerations arising in this case will cause destruction of the system.

Dynamics of mechanical systems

A small compact body, for example a piece of marble, can be represented as a simple material point. If you apply an external force to it, it will begin to move, which is determined by Newton's laws. In a simplified form, Newton's laws state that a body at rest will remain at rest unless acted upon by an external force. If an external force is applied to a material point, then it will begin to move with acceleration proportional to this force.
Most mechanical systems are more complex than a simple point of matter, and they will not necessarily move under the influence of a force as a single unit. Rotary machines are not absolutely solid and their individual components have different rigidities. As we will see later, their reaction to external influence depends on the nature of the influence itself and on the dynamic characteristics of the mechanical structure, and this reaction is very difficult to predict. Problems of modeling and predicting the response of structures to known external influences are solved using using finite element method (FEM) and modal analysis. Here we will not dwell on them in detail, since they are quite complex, however, to understand the essence of vibration analysis of machines, it is useful to consider how forces and structures interact with each other.

Vibration amplitude measurements

The following concepts are used to describe and measure mechanical vibrations:
Maximum Amplitude (Peak) - this is the maximum deviation from the zero point, or from the equilibrium position.
Sweep (Peak-Peak) is the difference between the positive and negative peaks. For a sine wave, the peak-to-peak amplitude is exactly twice the peak amplitude, since temporary implementation in this case it is symmetrical. However, as we will soon see, this is not true in general.

RMS amplitude ( SKZ) equal to the square root of the mean square of the vibration amplitude. For a sine wave, the RMS value is 1.41 times less than the peak value, however, this ratio is valid only for this case.
SKZ is an important characteristic of vibration amplitude. To calculate it, it is necessary to square the instantaneous values ​​of the oscillation amplitude and average the resulting values ​​over time. To obtain the correct value, the averaging interval must be at least one oscillation period. After this, the square root is taken and the RMS is obtained.

SKZ should be used in all calculations relating to power and vibration energy. For example, AC 117V (we are talking about the North American standard). 117 V is the rms voltage value that is used to calculate the power (W) consumed by appliances connected to the network. Let us recall once again that for a sinusoidal signal (and only for it) the root mean square amplitude is equal to 0.707 x peak.

Concept of phase

Phase is a measure of the relative time shift of two sinusoidal oscillations. Although phase is a time difference in nature, it is almost always measured in angular units (degrees or radians), which are cycle fractions fluctuations and, therefore, do not depend on the exact value of its period.

1/4 cycle delay = 90 degree phase shift

Concept of PHASE

The phase difference between two oscillations is often called phase shift . A 360-degree phase shift represents a time delay of one cycle, or one period, which essentially means that the oscillations are completely synchronized. A phase difference of 90 degrees corresponds to a shift of oscillations by 1/4 cycle relative to each other, etc. The phase shift can be positive or negative, that is, one temporary implementation can lag behind another or, conversely, advance it.
Phase can also be measured relative to a specific point in time. An example of this is the phase of the unbalanced component of the rotor (heavy place), taken relative to the position of some of its fixed points. To measure this value it is necessary to form rectangular impulse corresponding to a specific reference point on the shaft. This pulse can be generated by a tachometer or any other magnetic or optical sensor sensitive to geometric or light inhomogeneities on the rotor, and is sometimes called a tacho pulse. By measuring the delay (advance) between the cyclic sequence of tacho pulses and the vibration caused by the imbalance, we thereby determine their phase angle.

Phase angle can be measured relative to a reference point both in the direction of rotation and in the direction opposite to rotation, i.e. either as a phase delay or as a phase advance. Various equipment manufacturers use both approaches.

Vibration units

So far we have considered vibration displacement as amplitude measure vibrations. The vibration displacement is equal to the distance from the reference point, or from the equilibrium position. In addition to vibrations in coordinates (displacement), a vibrating object also experiences fluctuations in speed and acceleration. Velocity is the rate at which a position changes and is usually measured in m/s. Acceleration is the rate of change of velocity and is usually measured in m/s 2 or in units of g (acceleration due to gravity).
As we have already seen, the displacement graph of a body experiencing harmonic oscillations is a sinusoid. We also showed that the vibration velocity in this case obeys a sinusoidal law. When the displacement is maximum, the speed is zero, since in this position there is a change in the direction of movement of the body. It follows that temporary implementation speed will be shifted in phase by 90 degrees to the left relative to the temporal implementation of the displacement. In other words, the velocity is 90 degrees ahead of the displacement in phase.
Remembering that acceleration is the rate of change of speed, it is easy, by analogy with the previous one, to understand that the acceleration of an object experiencing a harmonic vibration is also sinusoidal and equal to zero when the speed is maximum. Conversely, when the speed is zero, the acceleration is maximum (the speed is changing most rapidly at this moment). Thus, acceleration is 90 degrees ahead of speed in phase. These relationships are shown in the figure.

There is another vibration parameter, namely, the rate of change of acceleration, called sharpness (jerk) .
Sharpness - this is the sudden cessation of deceleration at the moment of stopping, which you feel when you brake in a car without releasing the brake pedal. For example, elevator manufacturers are interested in measuring this value, because elevator passengers are sensitive to changes in acceleration.

A quick reference to amplitude units

In the figure shown, the same vibration signal is presented in the form of vibration displacement, vibration velocity and vibration acceleration.

Note that the displacement plot is very difficult to analyze at high frequencies, but the high frequencies are clearly visible in the acceleration plot. The speed curve is the most uniform in frequency among the three. This is typical for most rotary machines, but in some situations the displacement or acceleration curves are most uniform. It is best to choose units of measurement for which the frequency curve appears flattest: thereby providing maximum visual information to the observer. Vibration velocity is most often used for machine diagnostics.

Complex vibration

Vibration is movement caused by an oscillatory force. In a linear mechanical system, the frequency of vibration coincides with the frequency of the exciting force. If several exciting forces with different frequencies act simultaneously in a system, then the resulting vibration will be the sum of vibrations at each frequency. Under these conditions the resulting temporary implementation there will be no more hesitation sinusoidal and can be very difficult.
In this figure, high- and low-frequency vibrations are superimposed on each other and form a complex time realization. In simple cases like this, it is fairly easy to determine the frequencies and amplitudes of individual components by analyzing the waveform (time realization) of the signal, but most vibration signals are much more complex and much more difficult to interpret. For a typical rotary machine, it is often very difficult to extract the necessary information about its internal state and operation by studying only temporary vibration occurrences, although in some cases the analysis of the latter is quite a powerful tool, as we will discuss further in the section on machine vibration monitoring.

Energy and power

To excite vibration, energy must be expended. In the case of machine vibration, this energy is generated by the engine of the machine itself. Such an energy source can be an alternating current network, an internal combustion engine, a steam turbine, etc. In physics, energy is defined as the ability to do work, and mechanical work is the product of a force and the distance over which this force acts. The unit of measurement of energy and work in the International System (SI) is Joule. One Joule is equivalent to a force of one Newton acting over a distance of one meter.
The portion of a machine's energy due to vibration is usually not very large compared to the total energy required to operate the machine.
Power is work performed per unit of time, or energy expended per unit of time. In the SI system, power is measured in Watts, or Joules per second. One horsepower is equivalent to 746 watts. Vibration power is proportional to the square of the vibration amplitude (similarly, electrical power is proportional to the square of voltage or current).
In accordance with the law of conservation of energy, energy cannot arise from nothing or disappear into nowhere: it passes from one form to another. The vibration energy of a mechanical system gradually dissipates (that is, turns) into heat.

When analyzing the vibration of a more or less complex mechanism, it is useful to consider the sources of vibrational energy and the paths through which this energy is transmitted within the machine. Energy always moves from a source of vibration to an absorber, where it is converted into heat. Sometimes this path can be very short, but in other situations the energy can travel long distances before being absorbed.
The most important absorber of machine energy is friction. A distinction is made between sliding friction and viscous friction. Sliding friction occurs due to the relative movement of various parts of the machine relative to each other. Viscous friction is created, for example, by a film of oil lubricant in a plain bearing. If the friction inside the machine is low, then its vibration is usually high, because Due to the lack of absorption, vibration energy accumulates. For example, machines with rolling bearings, sometimes called antifriction bearings, tend to vibrate more than machines with plain bearings, in which the lubricant acts as a significant energy absorber. The absorption of vibration energy due to friction also explains the use of rivets instead of welded joints in aviation: riveted joints experience small movements relative to each other, due to which vibration energy is absorbed. This prevents vibration from developing to destructive levels. Such structures are called heavily damped. Damping is essentially a measure of vibration energy absorption.

Natural frequencies

Any mechanical structure can be represented as a system of springs, masses and dampers. Dampers absorb energy, but masses and springs do not. As we saw in the previous section, the mass and the spring form a system that resonates at its characteristic natural frequency. If you impart energy to such a system (for example, push a mass or pull back a spring), then it will begin to oscillate at its own frequency, and the vibration amplitude will depend on the power of the energy source and on the absorption of this energy, i.e. damping inherent in the system itself. The natural frequency of an ideal mass-spring system without damping is given by:

where Fn - Natural frequency;
k is the elasticity coefficient (stiffness) of the spring;
m - mass.

It follows that with increasing spring stiffness, the natural frequency also increases, and with increasing mass, the natural frequency decreases. If the system has damping, and this is true for all real physical systems, then the natural frequency will be slightly lower than the value calculated using the above formula and will depend on the amount of damping.

The set of spring-mass-damper systems (that is, the simplest oscillators) that can model the behavior of a mechanical structure are called degrees of freedom. The vibration energy of the machine is distributed between these degrees of freedom depending on their natural frequencies and damping, as well as depending on the frequency of the energy source. Therefore, vibration energy is never distributed evenly throughout the machine. For example, in a machine with an electric motor, the main source of vibration is the residual imbalance of the engine rotor. This results in noticeable levels of vibration at the motor bearings. However, if one of the machine’s natural frequencies is close to the rotor’s rotation frequency, then its vibrations can be high even at a fairly large distance from the engine. This fact must be taken into account when assessing machine vibration: the point with the maximum vibration level is not necessarily located near the excitation source. Vibratory energy often travels long distances, such as through pipes, and can cause real havoc when encountering a distant structure whose natural frequency is close to that of the source.
The phenomenon of coincidence of the frequency of the exciting force with the natural frequency is called resonance. At resonance, the system oscillates at its own frequency and has a large oscillation range. At resonance, the oscillations of the system are shifted in phase by 90 degrees relative to the oscillations of the exciting force.
In the pre-resonant zone (the frequency of the exciting force is less than the natural frequency), there is no phase shift between the oscillations of the system and the exciting force. The system moves with the frequency of the exciting force.
In the zone after resonance, the oscillations of the system and the exciting force are in antiphase (shifted relative to each other by 180 degrees). There are no resonant amplitude enhancements. As the excitation frequency increases, the vibration amplitude decreases, but the phase difference of 180 degrees remains for all frequencies above the resonant one.

Linear and nonlinear systems

To understand the mechanism of vibration transmission inside a machine, it is important to understand the concept of linearity and what is meant by linear or nonlinear systems. So far we have used the term linear only in relation to the amplitude and frequency scales. However, this term is also used to describe the behavior of any systems that have an input and an output. By system we mean here any device or structure that can perceive excitation in some form (input) and give an appropriate response to it (output). Examples include tape recorders and amplifiers that convert electrical signals, or mechanical structures, where at the input we have an exciting force, and at the output we have vibration displacement, speed and acceleration.

Definition of Linearity

A system is called linear if it satisfies the following two criteria:
If input x causes output X in the system, then input 2x will produce output 2X. In other words, the output of a linear system is proportional to its input. This is illustrated in the following figures:

If input x produces output X, and input y produces output Y, then input x+y will produce output X+Y. In other words, a linear system processes two simultaneous input signals independently of each other, and they do not interact with each other within it. It follows, in particular, that a linear system does not produce a signal at the output with frequencies that were not present in the input signals. This is illustrated in the following figure:

Note that these criteria do not require that the output be analog or similar in nature to the input. For example, the input may be electric current, and the output may be temperature. In the case of mechanical structures, in particular machines, we will consider the vibration force as the input, and the measured vibration itself as the output.

Nonlinear systems

No real system is completely linear. There is a wide variety of nonlinearities that are present to varying degrees in any mechanical system, although many of them behave almost linearly, especially at weak inputs. A system that is not completely linear has frequencies at its output that were not at the input. An example of this is stereo amplifiers or tape recorders, which generate harmonics input signal due to the so-called nonlinear (harmonic) distortion, deteriorating playback quality. Harmonic distortion is almost always greater at higher signal levels. For example, a small radio sounds quite clear at low volume levels, but begins to crackle when the volume is turned up. This phenomenon is illustrated below:

Many systems have a nearly linear response to a weak input signal, but become nonlinear at higher levels excitement. Sometimes there is a certain threshold of the input signal, a slight excess of which leads to strong nonlinearity. An example would be signal clipping in an amplifier when the input level exceeds the permissible voltage or current swing of the amplifier's power supply.

Another type of nonlinearity is cross-modulation, where two or more input signals interact with each other and produce new frequency components, or modulation sidebands, that were not present in either one. The side bands in vibration spectra are associated with modulation.

Nonlinearities of rotary machines

As we already mentioned, the vibration of a machine is actually a response to forces caused by its moving parts. We measure vibration at different points of the machine and find the force values. By measuring the frequency of vibration, we assume that the forces causing it have the same frequencies, and its amplitude is proportional to the magnitude of these forces. That is, we assume that the machine is a linear system. In most cases, this assumption is reasonable.

However, as the machine wears, its clearances increase, cracks and looseness appear, etc., its response will deviate increasingly from the linear law, and as a result, the nature of the measured vibration may become completely different from the nature of the exciting forces.

For example, an unbalanced rotor acts on a bearing with a sinusoidal force at a frequency of 1X, and there are no other frequencies in this excitation. If the mechanical structure of the machine is nonlinear, then the exciting sinusoidal force will be distorted, and in the resulting vibration spectrum, in addition to the 1X frequency, its harmonics will appear. The number of harmonics in the spectrum and their amplitude are a measure of the nonlinearity of the machine. For example, as a sliding bearing wears out, the number of harmonics in its vibration spectrum increases and their amplitude increases.
Flexible connections with misalignment are non-linear. This is why their vibration characteristics contain a strong second harmonic of the reverse frequency (i.e. 2X). Wear of a misaligned clutch is often accompanied by a strong third harmonic of the rotation frequency (RF). When forces of different frequencies interact within a machine in a non-linear manner, modulation occurs and new frequencies appear in the vibration spectrum. These new frequencies, or side stripes. present in the spectra of defective gears, rolling bearings, etc. If the gear is eccentric or has some kind of irregular shape, then the turning frequency will modulate the frequency of tooth mesh, resulting in sidebands in the vibration spectrum. Modulation is always a nonlinear process in which new frequencies appear that were absent in the exciting force.

Resonance

Resonance is a state of the system in which the frequency excitement close to natural frequency design, that is, the frequency of oscillations that this system will perform when left to itself after being removed from a state of equilibrium. Typically, mechanical structures have many natural frequencies. In the event of resonance, the vibration level can become very high and lead to rapid destruction of the structure.
Resonance appears in the spectrum in the form of a peak, the position of which remains constant as the speed of the machine changes. This peak can be very narrow or, conversely, wide, depending on the effective damping structures at a given frequency.
To determine whether a machine has resonances, you can perform one of the following tests:

 Test blow (bump test) - The car is hit with something heavy, for example a mallet, while vibration data is recorded. If the machine has resonances, then its damping vibration will highlight its own frequencies.
Acceleration or Coasting - the machine is turned on (or turned off) and vibration data and tachometer readings are simultaneously taken. When the machine speed approaches the natural frequency of the structure, temporary implementation vibrations will appear strong highs.
Speed ​​Variation Test - the speed of the machine is changed in a wide range (if possible), taking vibration data and tachometer readings. The data obtained is then interpreted in the same way as in the previous test. The figure shows an idealized mechanical resonance response curve. The behavior of a resonating system under the influence of an external force is very interesting and slightly contrary to everyday intuition. It strictly depends on the excitation frequency. If this frequency is lower than its natural frequency (that is, located to the left of the peak), then the entire system will behave like a spring, in which the displacement is proportional to the force. In the simplest oscillator, consisting of a spring and a mass, it is the spring that will determine the response to excitation by such a force. In this frequency region, the behavior of the structure will coincide with everyday intuition, responding to a large force with a large displacement, and the displacement will be in phase with the force.

In the region to the right of the natural frequency the situation is different. Here mass plays a decisive role, and the entire system reacts to force, roughly speaking, in the same way as a material point would. This means that the acceleration will be proportional to the applied force, but the amplitude of the displacement will be relatively constant as the frequency changes.
It follows that the vibration displacement will be in antiphase with the external force (since it is in antiphase with vibration acceleration): when you put pressure on the structure, it will move towards you and vice versa!
If the frequency of the external force exactly matches the resonance, then the system will behave completely differently. In this case, the reactions of the mass and the spring will cancel out, and the force will only be seen as damping, or friction, of the system. If the system is weakly damped, then the external influence will be like pushing air. When you try to push him, he gives in to you easily and weightlessly. Therefore, at the resonant frequency you will not be able to apply much force to the system, and if you try to do this, the vibration amplitude will reach very large values. It is damping that controls the movement of the resonant system at its natural frequency.
At natural frequency the phase shift ( phase angle) between the excitation source and the response of the structure is always 90 degrees.
For machines with long rotors, such as turbines, natural frequencies are called critical speeds. It is necessary to ensure that in the operating mode of such machines their speeds do not coincide with critical ones.

Test hit

Test hit is a good way to find natural frequencies machines or structures. Impact testing is a simplified form of mobility measurement that does not use a torque hammer and therefore does not determine the amount of force applied. The resulting curve will not be correct in any precise sense. However, the peaks of this curve will correspond to the true values ​​of the natural frequencies, which is usually sufficient to assess the vibration of the machine.

Performing a Impact Test using an FFT analyzer is extremely easy. If the analyzer has a built-in negative delay function, then its trigger is set to a value of about 10% of the length of the time record. The car is then hit with a heavy tool with a fairly soft surface near the location of the accelerometer. For impact, you can use a standard measuring hammer or a piece of wood. The mass of the hammer should be approximately 10% of the mass of the machine or structure being tested. If possible, the FFT analyzer's time window should be exponential to ensure that the signal level is zero at the end of the time record.
On the left is a typical impact response curve. If the analyzer does not have a trigger delay function, a slightly different technique can be used. In this case, the Hann window is selected and 8 or 10 averagings are specified. Then the measurement process is started, while simultaneously hitting the hammer randomly until the analyzer finishes measuring. The density of impacts must be evenly distributed over time so that the frequency of their repetition does not appear in the spectrum. If a three-axis accelerometer is used, natural frequencies in all three axes will be recorded.

In this case, to excite all modes of vibration, make sure that the shocks are applied at 45 degrees to all axes of accelerometer sensitivity.

Frequency analysis

To get around analysis limitations in the time domain, frequency or spectral analysis of a vibration signal is usually used in practice. If the temporary implementation has a schedule in time domain, then the spectrum is a graph in frequency domain. Spectral analysis is equivalent to converting a signal from the time domain to the frequency domain. Frequency and time are related to each other by the following relationship:

Time=1/Frequency
Frequency=1/Time

The bus schedule clearly reveals the equivalence of information representations in the time and frequency domains. You can list the exact departure times of the buses (time domain), or you can say that they leave every 20 minutes (frequency domain). The same information looks much more compact in the frequency domain. A very long time schedule is compressed into two lines in frequency form. This is very significant: events that occupy a large time interval are compressed in the frequency domain to individual bands.

Why is frequency analysis needed?

Note that in the above figure, the frequency components of the signal are separated from each other and are clearly expressed in the spectrum, and their levels are easy to identify. This information would be very difficult to extract from the temporary implementation.

The following figure shows that events that overlap each other in the time domain are separated into separate components in the frequency domain.

The temporary implementation of vibration carries a large amount of information that is invisible to the naked eye. Some of this information may be in very weak components, the magnitude of which may be less than the thickness of the graph line. However, such weak components can be important in identifying developing faults in a machine, such as bearing defects. The very essence of condition-based diagnostics and maintenance is the early detection of incipient faults, therefore, it is necessary to pay attention to extremely low levels of vibration signal.

In the spectrum shown, the very weak component represents a small developing fault in the bearing, and it would go undetected if we were analyzing the signal in the time domain, that is, focusing on the overall level of vibration. Since RMS is simply the general level of fluctuation over a wide frequency range, therefore, a small disturbance at the bearing frequency may go unnoticed in the change in the RMS level, although this disturbance is very important for diagnosis.

How is frequency analysis performed?

Before we begin the procedure for performing spectrum analysis, let's take a look at the different types of signals we have to work with.

 From theoretical and practical points of view, signals can be divided into several groups. Different types of signals have different types of spectra, and to avoid errors when performing frequency analysis, it is important to know the characteristics of these spectra.

Stationary signal

First of all, all signals are divided into stationary And non-stationary . Stationary signal has time-constant statistical parameters. If you look at a stationary signal for a few moments and then return to it again after some time, it will look essentially the same, that is, its overall level, amplitude distribution, and standard deviation will be almost the same. Rotary machines generally produce stationary vibration signals.
Stationary signals are further divided into deterministic and random. Random (non-stationary) signals unpredictable in their frequency composition and amplitude levels, but their statistical characteristics are still almost constant. Examples of random signals are rain falling on a roof, jet noise, turbulence in a gas or liquid flow, and cavitation.

Deterministic signal

Deterministic signals are a special class of stationary signals . They maintain a relatively constant frequency and amplitude composition over a long period of time. Deterministic signals are generated by rotary machines, musical instruments and electronic oscillators. They are in turn divided into periodic And quasiperiodic . The temporal implementation of a periodic signal is continuously repeated at equal intervals of time. The frequency of repetition of a quasiperiodic time shape varies over time, but to the eye the signal appears periodic. Rotary machines sometimes produce quasi-periodic signals, especially in belt driven equipment.
Deterministic signals - This seems to be the most important type for analyzing machine vibrations, and their spectra are similar to those shown here:
Periodic signals always have a spectrum with discrete frequency components called harmonics or harmonic sequences. The term harmonic itself comes from music, where harmonics are integer multiples of the fundamental (reference) frequency.

Non-stationary signal

Non-stationary signals are divided into continuous and transient. Examples of a non-stationary continuous signal are the vibration produced by a jackhammer or artillery cannonade. By definition, a transient signal is a signal that begins and ends at the zero level and lasts a finite time. It can be very short or quite long. Examples of transient signals are the blow of a hammer, the noise of a flying airplane, or the vibration of a car during acceleration and coasting.

Examples of time implementations and their spectra

Below are examples of time realizations and spectra that illustrate the most important concepts in frequency analysis. Although these examples are in a sense idealized, since they were obtained using an electronic signal generator followed by processing with an FFT analyzer. However, they determine some characteristic features inherent in the vibration spectra of machines.

A sinusoidal wave contains only one frequency component, and its spectrum is a single point. Theoretically, a true sinusoidal oscillation exists unchanged for an infinite time. In mathematics, a transformation that takes an element from the time domain to an element in the frequency domain is called the Fourier transform. This transformation compresses all the information contained in a sine wave of infinite duration down to a single point. In the spectrum above, the only peak has a finite rather than zero width, which is due to the error of the numerical calculation algorithm used, called the FFT (see below).
In a machine with a rotor imbalance, a sinusoidal exciting force occurs with a frequency of 1X, that is, once per revolution. If the response of such a machine were completely linear, then the resulting vibration would also be sinusoidal and similar to the time realization above. In many poorly balanced machines, the temporary implementation of vibrations really resembles a sine wave, and in the vibration spectrum there is a large peak at the frequency 1X, that is, at the reverse frequency.

The following figure shows the harmonic spectrum of a periodic oscillation such as a clipped sinusoid.
This spectrum consists of components separated by a constant interval equal to 1/(oscillation period). The lowest of these components (the first after zero) is called the fundamental one, and all the rest are called its harmonics. Such an oscillation was obtained using a signal generator, and, as can be seen from examining the time signal, it is asymmetrical with respect to the zero axis (equilibrium position). This means that the signal has a constant component, which turns into the first line on the left in the spectrum. This example illustrates the ability of spectral analysis to reproduce frequencies down to zero (zero frequency corresponds to a constant signal or, in other words, the absence of oscillations).
Typically, when vibration analysis of machines, it is not advisable to perform spectral analysis at such low frequencies for a number of reasons. Most vibration sensors do not provide correct measurements down to 0 Hz, and only special accelerometers, such as those used in inertial navigation systems, allow this. For machine vibrations, the smallest frequency of interest is usually 0.3X. In some machines this may be below 1 Hz. Special techniques are required to measure and interpret signals below the 1 Hz range.
When analyzing the vibration characteristics of machines, it is not so rare to see temporary implementations cut off like the one shown above. This usually means that there is some kind of looseness in the car, and something is restricting the movement of the loose element in one of the directions.
The signal shown below is similar to the previous one, but it has a cut on both the positive and negative sides.

As a result, the time graph of the oscillation (time implementation) turns out to be symmetrical. Signals of this type can occur in machines in which the movement of weakened elements is limited in both directions. In this case, the spectrum of the periodic signal will also contain harmonic components, but these will only be odd harmonics. All even harmonic components are missing. Any periodic symmetrical oscillation will have a similar spectrum. The spectrum of a square waveform would also look similar to this.

Sometimes a similar spectrum is found in a very loose machine in which the movement of the vibrating parts is limited on each side. An example of this is an unbalanced machine with loose mounting bolts.
The short pulse spectrum obtained using a signal generator is very wide.

Please note that its spectrum is not discrete, but continuous. In other words, the signal energy is distributed over the entire frequency range, rather than concentrated on several individual frequencies. This is typical for non-deterministic signals such as random noise. and transient processes. Note that, starting from a certain frequency, the level is zero. This frequency is inversely proportional to the pulse duration, so the shorter the pulse, the wider its frequency content. If an infinitely short impulse existed in nature (mathematically speaking, - delta function ), then its spectrum would occupy the entire frequency range from 0 to +.
When examining a continuous spectrum, it is usually impossible to tell whether it belongs to a random signal or transitional. This limitation is inherent in Fourier frequency analysis, so when faced with a continuous spectrum it is useful to study its temporal implementation. When applied to the analysis of machine vibration, this makes it possible to distinguish between impacts that have pulsed temporary implementations and random noise caused, for example, cavitation.
A single pulse such as this is rarely found in rotary machines, however in the impact test this type of excitation is used specifically to excite the machine. Although its vibration response will not be such a classically smooth curve as shown above, it will nevertheless be continuous over a wide frequency range and have peaks at the natural frequencies of the structure. This means that impact is a very good type of excitation for identifying natural frequencies, since its energy is distributed continuously over a wide frequency range.
If a pulse having the above spectrum is repeated with a constant frequency, then
the resulting spectrum, which is shown here, will no longer be continuous, but consisting of harmonics of the pulse repetition frequency, and its envelope will coincide with the shape of the spectrum of a single pulse.

Such signals are produced by bearings with defects (potholes, scratches, etc.) on one of the rings. These pulses can be very narrow, and they always cause the appearance of a large series of harmonics.

Modulation

Modulation is called nonlinear a phenomenon in which several signals interact with each other in such a way that the result is a signal with new frequencies that were not present in the original ones.
Modulation is the bane of audio engineers because it causes modulation distortion that plagues music lovers. There are many forms of modulation, including frequency and amplitude modulation. Let's look at its main types separately. Frequency modulation (FM), shown here, is the variation of the frequency of one signal under the influence of another, which usually has a lower frequency.

The modulated frequency is called the carrier frequency. In the presented spectrum, the component with maximum amplitude is the carrier, and other components that are similar to harmonics are called sidebands. The latter are located symmetrically on both sides of the carrier with a step equal to the value of the modulating frequency. Frequency modulation is often found in the vibration spectra of machines, especially in gears, where the frequency of tooth engagement is modulated by the rotation frequency of the wheel. It also occurs in some acoustic speakers, although at a very low level.

Amplitude modulation

The frequency of the temporal implementation of an amplitude-modulated signal appears to be constant, but its amplitude oscillates with a constant period

This signal was obtained by rapidly varying the output gain of an electronic signal generator during the recording process. A periodic change in the amplitude of a signal with a certain period is called amplitude modulation. The spectrum in this case has a maximum peak at the carrier frequency and one component on each side. These additional components are the side stripes. Please note that unlike frequency modulation, which results in a large number of sidebands, amplitude modulation is accompanied by only two sidebands, which are located symmetrically relative to the carrier at a distance equal to the value of the modulating frequency (in our example, the modulating frequency is the frequency with which gain knob when recording a signal). In this example, the modulating frequency is significantly lower than the modulating or carrier frequency, but in practice they often turn out to be close to each other (for example, on multi-rotor machines with similar rotor speeds). Also, in real life, both the modulating and modulating signals have more complex shapes than the sine waves shown here.

The relationship between amplitude modulation and sidebands can be visualized in vector form. Let's imagine a time signal in the form of a rotating vector, the magnitude of which is equal to the amplitude of the signal, and the angle in polar coordinates is equal to the phase. The vector representation of a sine wave is simply a vector of constant length rotating around its origin at a speed equal to the frequency of the oscillation. Each cycle of temporary implementation corresponds to one revolution of the vector, i.e. one cycle is 360 degrees.

Amplitude modulation of a sinusoidal oscillation in vector representation looks like the sum of three vectors: the carrier of the modulated signal and two sidebands. The sideband vectors rotate, one a little faster, and the other a little slower than the carrier.

Adding these sidebands to the carrier causes changes in the amplitude of the sum. In this case, the carrier vector appears stationary, as if we were in a coordinate system rotating with the carrier frequency. Note that when the sideband vectors rotate, a constant phase relationship is maintained between them, so the sum vector rotates at a constant frequency (with the carrier frequency).

To represent frequency modulation in this way, it is enough to introduce a slight change in the phase relationships of the side vectors. If the lateral vector of a lower frequency is rotated 180 degrees, then frequency modulation will occur. In this case, the resulting vector swings back and forth around its origin. This means an increase and decrease in its frequency, that is, frequency modulation. It should also be noted that the resulting vector varies in amplitude. That is, along with frequency modulation there is also amplitude modulation. To obtain a vector representation of pure frequency modulation, it is necessary to introduce a set of side vectors that have precisely defined phase relationships with each other. Equipment vibration almost always contains both amplitude and frequency modulation. In such cases, some of the sidebands may be added out of phase, resulting in the upper and lower sidebands having different levels, that is, not being symmetrical with respect to the carrier.

Beats

The timing implementation shown is similar to amplitude modulation, but in reality it is just the sum of two sine waves with slightly different frequencies, called a beat.

Because these signals are slightly different in frequency, their phase difference varies from zero to 360 degrees, which means that their total amplitude will either be amplified (signals in phase) or weakened (signals out of phase). The beat spectrum contains components with the frequency and amplitude of each signal, and there are no side bands at all. In this example, the amplitudes of the two source signals are different, so they do not completely cancel out at the zero point between the maxima. Beating is a linear process: it is not accompanied by the appearance of new frequency components .
Electric motors often generate vibrational and acoustic signals that resemble beating, in which the frequency of the false beat is equal to twice the slip frequency. In reality, this is amplitude modulation of a vibration signal with double the slip frequency. This phenomenon in electric motors is sometimes also called beating, probably because it makes the mechanism sound like an out-of-tune musical instrument, “beating.”

This example of beats is similar to the previous one, but the levels of the adding signals are equal, so they completely cancel each other at the zero points. Such complete mutual destruction is very rare in real vibration signals of rotary equipment.
We saw above that beats and amplitude modulation have similar temporal implementations. This is indeed true, but with a slight correction - in the case of beats, a phase shift occurs at the point of complete mutual destruction of the signals.

Logarithmic frequency scale

So far we have considered only one type of frequency analysis, in which the frequency scale was linear. This approach is applicable when the frequency resolution is constant over the entire frequency range, which is typical for the so-called narrowband analysis, or analysis in frequency bands with a constant absolute width. This is exactly the kind of analysis that is performed, for example, by FFT analyzers.
There are situations where frequency analysis is needed, but the narrowband approach does not present the data in the most useful form. For example, when the adverse effects of acoustic noise on the human body are studied... Human hearing reacts not so much to the frequencies themselves, but to their relationships. The frequency of a sound is determined by the pitch of the tone perceived by the listener, with a change in frequency of two times being perceived as a change in pitch of one octave, regardless of what the exact values ​​of the frequencies are. For example, a change in the frequency of a sound from 100 Hz to 200 Hz corresponds to an increase in pitch by one octave, but an increase from 1000 to 2000 Hz also corresponds to a shift of one octave. This effect is so accurately reproduced over a wide frequency range that it is convenient to define an octave as a frequency band in which the upper frequency is twice as high as the lower, although in everyday life the octave is only a subjective measure of change in sound.

To summarize, we can say that the ear perceives a change in frequency proportional to its logarithm, and not to the frequency itself. Therefore, it is reasonable to choose a logarithmic scale for the frequency axis of acoustic spectra, which is done almost everywhere. For example, the frequency characteristics of acoustic equipment are always given by manufacturers in the form of graphs with a logarithmic frequency axis. When performing frequency analysis of sound, it is also common to use a logarithmic frequency scale.

The octave is such an important frequency range for human hearing that so-called octave band analysis has become established as a standard type of acoustic measurement. The figure shows a typical octave spectrum using center frequency values ​​according to international ISO standards. The width of each octave band is approximately 70% of its center frequency. In other words, the width of the analyzed bands increases in proportion to their central frequencies. The level in dB is usually plotted along the vertical axis of the octave spectrum.

It could be argued that the frequency resolution of octave analysis is too low for machine vibration studies. However, it is possible to define narrower bands with constant relative width. The most common example of this is the one-third octave spectrum, where the bandwidth is approximately 27% of the center frequencies. Three one-third octave bands fit into one octave, so the resolution in such a spectrum is three times better than octave analysis. When normalizing vibration and noise of machines One-third octave spectra are often used.
An important advantage of analysis in frequency bands with a constant relative width is the ability to represent a very wide frequency range in a single graph with a fairly narrow resolution at low frequencies. Of course, this suffers in resolution at high frequencies, but this does not cause problems in some applications, such as troubleshooting machines.
For machine diagnostics, narrowband spectra (with constant absolute bandwidth) are very useful for detecting high-frequency harmonics and sidebands, but such high resolution is often not required for detecting many simple machine faults. It turns out that the vibration velocity spectra of most machines roll off at high frequencies, and therefore constant relative bandwidth spectra tend to be more uniform over a wide frequency range. This means that such spectra make better use of the dynamic range of the instruments. One-third octave spectra are narrow enough at low frequencies to reveal the first few harmonics of the rotation frequency, and can be effectively used for fault detection through trending.
It should be recognized, however, that the use of constant relative bandwidth spectra for vibration diagnostic purposes is not very widely accepted in industry, with the possible exception of a few noteworthy examples, such as the submarine fleet.

Linear and logarithmic amplitude scales

It may seem that it is best to examine vibration spectra on a linear amplitude scale, which gives a true representation of the measured vibration amplitude. When using a linear amplitude scale, it is very easy to identify and evaluate the highest component in the spectrum, but smaller components may be completely missed or, at best, great difficulty will arise in estimating their magnitude. The human eye can detect components in the spectrum that are approximately 50 times lower than the maximum, but anything less than this will be missed.
A linear scale may be used if all significant components are approximately the same height. However, in the case of machine vibration, incipient faults in parts such as bearings generate signals with very small amplitude. If we want to reliably track the development of these spectral components, then it is best to plot the logarithm of the amplitude on the graph, and not the amplitude itself. With this approach, we can easily plot and visually interpret signals that differ in amplitude by 5000, i.e. have a dynamic range at least 100 times greater than linear scale allows.

Different types of amplitude representation for the same vibration characteristic (linear and logarithmic amplitude scales) are presented in the figure.
Note that on a linear amplitude scale spectrum, the large peaks are very clear to read, but the low level peaks are difficult to see. When analyzing machine vibration, however, it is often the small components in the spectrum that are of interest (for example, when diagnosing rolling bearings). Do not forget that when monitoring vibration, we are interested in the increase in the levels of specific spectral components, indicating the development of an emerging fault. A motor ball bearing may develop a small defect on one of the rings or on the ball, and the vibration level at the corresponding frequency will initially be very small. But this does not mean that it can be neglected, because the advantage of condition-based maintenance is that it allows you to detect a malfunction at the initial stage of development. It is necessary to monitor the level of this small defect to predict when it will become a significant problem requiring intervention.
Obviously, if the level of the vibration component corresponding to some defect doubles, then great changes have occurred with this defect. The power and energy of a vibration signal is proportional to the square of the amplitude, so doubling it means that four times more energy is dissipated into vibration. If we try to track the spectral peak with an amplitude of about 0.0086 mm/s, we will have a very difficult time because it will be too small compared to the much higher components.

The 2nd of the given spectra shows not the vibration amplitude itself, but its logarithm. Since this spectrum uses a logarithmic amplitude scale, multiplying the signal by any constant simply shifts the spectrum upward without changing its shape or the relationships between the components.
As you know, the logarithm of the product is equal to the sum of the logarithms of the factors. This means that if a signal's gain is changed, it does not affect the shape of its spectrum on a logarithmic scale. This fact greatly simplifies the visual interpretation of spectra measured at different gains - the curves simply shift up or down on the graph. When using a linear scale, the shape of the spectrum changes sharply as the gain of the device changes. Note that although the vertical axis in the above graph uses a logarithmic scale, the amplitude units remain linear (mm/s, in/s), which means more zeros after the decimal point.
And in this case, we received a huge advantage for visual assessment of the spectrum, since the entire set of peaks and their relationships now became visible. In other words, if we now compare the logarithmic vibration spectra of a machine whose bearings are experiencing wear, we will see an increase in the levels of only the bearing tones, while the levels of other components will remain unchanged. The shape of the spectrum will immediately change, which can be detected with the naked eye.

The following figure shows the spectrum, where decibels are plotted along the vertical axis. This is a special type of logarithmic scale that is very important for vibration analysis.

Decibel

A convenient form of logarithmic representation is the decibel, or dB. Essentially, it is a relative unit of measurement that uses the ratio of amplitude to some reference level. Decibel (dB) is determined by the following formula:

Lv= 20 lg (U/Uo),

Where L= Signal level in dB;
U is the vibration level in conventional units of acceleration, velocity or displacement;
Uo is the reference level corresponding to 0 dB.

The concept of decibel was first introduced into practice by Bell Telephone Labs back in the 20s. It was originally used to measure relative power loss and signal-to-noise ratio in telephone networks. Soon the decibel began to be used as a measure of sound pressure level. We will denote the level of vibration velocity in dB as VdB (from the word Velocity speed), and define it as follows:

Lv= 20 lg (V/Vo),
or
Lv= 20 lg (V/(5x10 -8 m/s 2))

A reference level of 10 -9 m/s 2 is sufficient for all machine vibration measurements in decibels to be positive. The specified standardized reference level is consistent with the International SI System, but is not recognized as a standard in the United States or other countries. For example, the US Navy and many American industries use 10 -8 m/s as a reference value. This leads to the fact that the American readings for the same vibration velocity will be 20 dB lower than in SI. (The Russian standard uses a reference vibration velocity level of 5x10 -8 m/s, so Russian readings Lv another 14 dB lower than American ones).
Thus, the decibel is a logarithmic relative unit of vibration amplitude, which allows for easy comparative measurements. Any 6 dB increase in level corresponds to a doubling of amplitude, regardless of the original value. Likewise, any 20 dB change in level means a tenfold increase in amplitude. That is, with a constant amplitude ratio, their decibel levels will differ by a constant number, regardless of their absolute values. This property is very convenient when tracking vibration development (trends): an increase of 6 dB always indicates a doubling of its value.

DB and amplitude ratios

The table below shows the relationship between dB level changes and the corresponding amplitude ratios.
We strongly recommend using decibels as the vibration amplitude unit, as it provides much more information than linear units. In addition, the logarithmic scale in dB is much clearer than the logarithmic scale with linear units.

Level change in dB	Amplitude ratio	Level change in dB	Amplitude ratio
			1000
			3100
	10 La in AdB, taken in accordance with the Russian standard, will be 20 dB higher than the American one). It turns out that at 3.16 Hz vibration velocity levels in Vd B and vibration acceleration in AdB coincide (in the American system this occurs at a frequency of 159.2 Hz). The formulas below determine the relationships between the levels of vibration acceleration, speed and displacement in AdB, VdB and DdB respectively: L V = L A - 20 log(f) + 10, L V = L D + 20 log(f) - 60, L D = L A - 20 log(f 2) + 70, NOTE Acceleration and Velocity in linear units can be obtained from the corresponding levels using the formulas: NOTE Note that for temporal implementations in the time domain, linear amplitude units are always used: the instantaneous value of the signal can be negative and therefore cannot be logarithmed. 90 92 94 96 98 100 102 104 106 108 110 112 114 116 118	1,6 2 2,5 3,2 4 5 6,3 7,9 10 13 16 20 25 32 40	120 122 124 126 128 130 132 134 136 138 140 142 144 146 148	50 63 79 100 130 160 200 250 320 400 500 630 790 1000 1300

Source text provided by Octava+

How is vibration measured?

To quantitatively describe the vibration of rotating equipment and for diagnostic purposes, vibration acceleration, vibration velocity and vibration displacement are used.

Vibration acceleration

Vibration acceleration is the vibration value directly related to the force that caused the vibration. Vibration acceleration characterizes the dynamic force interaction of elements inside the unit that caused this vibration. Usually displayed as amplitude (Peak) - the maximum absolute value of the acceleration in the signal. The use of vibration acceleration is theoretically ideal, since the piezoelectric sensor (accelerometer) measures acceleration and does not need to be specially converted. The disadvantage is that there are no practical developments for it on standards and threshold levels, there is no generally accepted physical and spectral interpretation of the features of the manifestation of vibration acceleration. It is successfully used in diagnosing defects of an impact nature - in rolling bearings and gearboxes.

Vibration acceleration is measured in:

• meters per second squared [m/sec 2 ]
• G, where 1G = 9.81 m/s 2
• decibels, the level should be 0 dB. If not specified, then the value 10 -6 m/s 2 is taken (ISO 1683:2015 standard and GOST R ISO 13373-2-2009)

How to convert vibration acceleration to dB?

For standard level 0 dB = 10 -6 m/s 2:

AdB = 20 * log10(A) + 120

AdB – vibration acceleration in decibels

A – vibration acceleration in m/s 2

120 dB – level 1 m/s 2

Vibration velocity

Vibration velocity is the speed of movement of the controlled point of the equipment during its precession along the measurement axis.

In practice, it is usually not the maximum value of vibration velocity that is measured, but its root mean square value, RMS. The physical essence of the RMS vibration velocity parameter is the equality of the energy impact on the machine supports of a real vibration signal and a fictitious constant one, numerically equal in value to the RMS. The use of the RMS value is also due to the fact that previously vibration measurements were carried out with pointer instruments, and according to their operating principle they are all integrating, and show exactly the root-mean-square value of the alternating signal.

Of the two widely used in practice representations of vibration signals (vibration velocity and vibration displacement), the use of vibration velocity is preferable, since this is a parameter that immediately takes into account both the movement of the controlled point and the energy impact on the supports from the forces that caused the vibration. The information content of vibration displacement can be compared with the information content of vibration velocity only if, in addition to the amplitude of vibrations, the frequencies of both the entire vibration and its individual components are taken into account. In practice, doing this is very problematic.

To measure RMS vibration velocity are used. More complex devices (vibration analyzers) also always have a vibration meter mode.

Vibration velocity is measured in:

• millimeters per second [mm/sec]
• inches per second: 1 in/s = 25.4 mm/sec
• decibels, the level should be 0 dB. If not specified, then, according to GOST 25275-82, the value 5 * 10 -5 mm/sec is taken (According to the international standard ISO 1683:2015 and GOST R ISO 13373-2-2009, 0 dB is taken as 10 -6 mm/sec)

How to convert vibration velocity to dB?

For standard level 0 dB = 5 * 10 -5 mm/sec:

VdB = 20 * log10(V) + 86

VdB – vibration velocity in decibels

lg10 – decimal logarithm (logarithm to base 10)

V – vibration velocity in mm/s

86 dB – level 1 mm/s

Below are the vibration velocity values ​​in dB for. It can be seen that the difference between adjacent values ​​is 4 dB. This corresponds to a difference of 1.58 times.

mm/s	dB
45	119
28	115
18	111
11,2	107
7,1	103
4,5	99
2,8	95
1,8	91
1,12	87
0,71	83

Vibration movement

Vibration displacement (vibration displacement, displacement) shows the maximum limits of movement of the controlled point during the vibration process. Typically displayed in peak-to-peak (double amplitude, Peak-to-peak, Peak to peak). Vibration displacement is the distance between the extreme points of movement of an element of rotating equipment along the measurement axis.

Vibration- movement of a point (or body) around its original position, repeating exactly at certain intervals (periodically). The simplest form of periodic oscillation is harmonic vibrations, the graph of which versus time is a sinusoid (see Fig. 1). The time between two subsequent, exactly similar positions of an oscillating point (or body) is called period of oscillation (T).

Frequency fluctuations is related to the period through the relation:

As for the magnitude of the fluctuation, it can be described, according to GOST 10816-1-99, by three main parameters: vibration displacement ( s ) , vibration speed ( v ) And vibration acceleration ( a ) . These parameters have certain mathematical relationships to each other when considering harmonic (simple) vibrations. If the vibration of a point (or body) has a purely longitudinal form of vibration along one axis (X), then instantaneous displacement (vibration displacement) from the starting position can be described by the mathematical equation:

where is the angular frequency;

– maximum point offset(or body) from the original position;

t- time.

The change in time offset is speed (vibration velocity) movement of a point (or body). Therefore, vibrations can also be described in terms of speed (v)

Thus, vibration displacement can be converted into velocity through differentiation.

Differentiation is accompanied by multiplication of amplitude by frequency, therefore vibration velocity amplitude at a certain frequency is proportional to the displacement (s) multiplied by frequency (f). With a fixed offset, the speed will double as the frequency increases, and if the frequency is increased by 10 times, the speed will increase by 10 times.

The change in the speed of movement of a point (or body) in time is acceleration (vibration acceleration) movements:

That is, in order to obtain acceleration from speed, another differentiation is necessary, which means another multiplication by frequency. Therefore, the acceleration at a fixed displacement will be proportional to the square of the frequency.

According to Newton's second law, force equals mass times acceleration. Therefore, for a given displacement, the force will be proportional to the square of the frequency. That's why on in practice they do not encounter oscillations where large accelerations are accompanied by large displacements, there are simply no such very large forces that would be extremely destructive.

As can be seen from the above equations, the shape and period of oscillation remains the same regardless of whether displacement, velocity or acceleration are considered.

It should be noted that instantaneous values s, v, a differ in phase. So the velocity is ahead of the displacement by phase angle 90 0 (in equation) and acceleration advances the speed by a phase angle of 90 0 (in the equation). As a characterizing quantity we used peak oscillation amplitude value, that is. Applying Peak Value vibration amplitudes effective when considering harmonic (simple) vibrations.

The values ​​of vibration displacement, vibration velocity and vibration acceleration in standard units of measurement are related by the following equations:

When considering oscillations (Fig. 2), other amplitude values ​​are used.

Average arithmetic absolute value vibration amplitude characterizes the overall vibration intensity and is determined by the formula:

The average value of the oscillation amplitude is used when analyzing oscillations over a very long period of time (days, several days), mainly in stationary equipment monitoring systems. Therefore, this value is not of particular practical interest.

Another value of the oscillation amplitude is root mean square value (SKZ). RMS is an important characteristic of vibration amplitude. To calculate it you need to square instantaneous values ​​of oscillation amplitude, and average the resulting values ​​over time. To obtain the correct value, the averaging interval must be at least one oscillation period. After this, the square root is taken and the RMS is obtained.

For purely harmonic vibrations(vibration contains only one vibration frequency) the relationship between peak, average and mean square amplitude values are determined by the following formulas:

In a more general form, these relationships can be described as follows:

Odds F f And Fc are called shape factor and crest factor, respectively. These coefficients give an idea of ​​the waveform of the vibration being studied.

For purely harmonic vibrations these coefficients are equal:

The oscillations encountered in practice are not purely harmonic oscillations, although many of them may be periodic. Figure 3 shows an example of a typical oscillation encountered in practice.

By determining the peak, average and root mean square values ​​of this vibration, as well as its shape and amplitude coefficients, you can obtain a lot of useful information and, as a result, talk about the non-harmonic nature of the vibration. However, it is almost impossible, based on this information, to predict possible defects caused by vibration in the structural elements of a machine or mechanism. Therefore you need to use others

Vibration parameters in various units of measurement can be recalculated not only using the above formulas, but also using vibration conversion calculators, which are offered by both foreign and domestic companies. In Fig. 4 you see one of these calculators. To get acquainted with its work, you can download it to your disk and run it.

R.S. If any of you, dear reader, did not quite understand this article because you are not good with mathematics, then I recommend that you first study this issue using the book : . In this book, all the material is presented in ordinary language, without a single formula.

For a temporary signal

Converting vibration values ​​from one representation to another and back is quite simple if you have a time signal.

To convert vibration velocity into vibration acceleration and vibration displacement into vibration velocity, it needs to be differentiated.

To convert vibration acceleration into vibration velocity and vibration velocity into vibration displacement, the signal must be integrated.

In devices this is done by hardware integrators. In a computer program this is done using mathematical methods.

For example, the simplest formulas:

A i =(V i -V i-1)/dt

V i =(A i-1 +4*A i +A i+1)*dt/6 (Simpson’s method)

dt - step between signal samples

A i - i-th sample of vibration acceleration signal

V i - i-th sample of the vibration velocity signal

We must not forget that when integrating we do not know the constant component of the signal. That is, we cannot obtain a constant displacement (gap) from vibration velocity.

For integral parameters

If the value is “read” from the scale of a pointer device or from the digital indicator of the device, then greater restrictions are imposed on mutual conversions. Conversions can be performed only for those vibration signals that contain vibrations of only one frequency f. In this case, the following expressions are valid:

V = A /(2*3.14*f)*1000 /1.4142	V=112.5*A/f
V = S *(2*3.14*f)/1000 /2/1.4142	V = 0.00222 * S * f
A = V *(2*3.14*f)/1000 *1.4142	A = 0.00888 * V * f
A = S *(2*3.14*f)/1000 *(2*3.14*f)/1000 /2	A = 0.00002 * S * f 2
S = V /(2*3.14*f)*1000 *1.4142*2	S=450*V/f
S = A /(2*3.14*f)*1000 /(2*3.14*f)*1000 *2	S = 50712 * A / f 2

coefficient 2: translation Peak<->Scope

These seemingly simple formulas must be used with caution, since in practice there are almost never purely sinusoidal signals of the same frequency. A real vibration always contains several frequencies.

For spectrum

To convert the vibration velocity spectrum into a vibration acceleration spectrum, you need to multiply each harmonic amplitude (each count) of the spectrum by (2*Pi*f) and rotate the phase by an angle of -90°. The vibration displacement is also converted into vibration velocity.

A i = V i *(2*3.14*f i) /1000

V i = S i *(2*3.14*f i) /1000

Re i = Im i *(2*3.14*f i) /1000

Im i = -Re i *(2*3.14*f i) /1000

For reverse translation (vibration acceleration->vibration velocity, vibration velocity->vibration displacement), you need to divide each harmonic amplitude by (2*Pi*f) and rotate the phase by an angle of +90°.

V i = A i /(2*3.14*f i) *1000

S i = V i /(2*3.14*f i) *1000

For the complex spectrum the following formulas are used:

Re i = -Im i /(2*3.14*f i) *1000

Im i = Re i /(2*3.14*f i) *1000

Additionally, a factor of 1000 must be taken into account due to the µm transition<->mm/s<->m/s 2 and Peak conversion factors<->SKZ<->Scope.

The graphs show the amplitude spectra of vibration acceleration, vibration velocity and vibration displacement of one signal.

Not enough information?

Write me your question, I will answer you and add useful information to the article.

The most informative method of obtaining data on the technical condition of mechanical equipment currently is to analyze the parameters of the vibration signal. To solve various levels of practical and research problems the following are used:

• noise analysis of mechanisms;
• measurement of the general vibration level;
• measurement of vibration parameters;
• analysis of the vibration signal spectrum and analysis of time realizations.

Let us first consider the nature of the occurrence of mechanical vibrations using the example of a single-mass system (). The parameters of this system are:

• weight ( m);
• hardness ( c);
• damping coefficient ( h).

System oscillations are possible under the influence of force ( F), variable relative to the direction of oscillation. Force F may be constant, but the parameters of the contacting surfaces can cause its periodic change. For example, the force of gravity interacting with the worn surface of a bearing as the shaft rotates serves as a source of vibration. The frequency response of the vibrations will indicate the nature of the damage.

The parameters of the oscillatory process are determined by the following equation, in which k– frequency of natural oscillations of the system, ε – parameter that determines the damping properties of the system:

Damage to the mechanical system can lead to changes in:

• rigidity (for example, wear of parts, loosening of threaded connections);
• damping coefficient (in case of cracks);
• influencing forces (when the roughness of contacting surfaces changes).

Vibration processes can be divided into stationary (determined in time) and non-stationary (not defined in time). Stationary processes can be periodic, harmonic or polyharmonic and non-periodic - almost periodic, transitional, and also random. Periodic oscillations– oscillations in which each value of a fluctuating quantity is repeated at equal time intervals. The simplest periodic signal is a harmonic oscillation.

Harmonic vibrations– oscillations in which the values ​​of the fluctuating quantity change over time according to the law of sine or cosine ():

S(t) = A × sin(ω × t + φ),

Where A– amplitude of oscillations; φ – initial phase of oscillations; ω – angular velocity.

Figure 2.7 – Harmonic oscillatory process

For harmonic vibrations: A, φ , ω = const. For almost harmonic vibrations: A, φ , ω – slowly changing functions of time, some of them may be constant, some increasing or decreasing. For example, amplitude, angular velocity when starting or stopping a mechanism.

Polyharmonic oscillations can be represented as the sum of two or more harmonic oscillations (harmonics), the frequencies of the harmonics are multiples of the fundamental frequency ().

Figure 2.8 – Polyharmonic oscillatory process

Random processes unpredictable in their parameters (frequency, amplitude), but retain their statistical characteristics (average value, dispersion) throughout the entire observation process. For example: cavitation in the flow part of the pump, the noise of a running engine.

Non-stationary processes are divided into continuous and short-term. These are processes whose probabilistic characteristics are functions of time. For example: impact processes, the appearance of damage, cracks during operation.

Vibration is classified:

• by nature:
  • mechanical;
  • aerohydrodynamic;
  • electromagnetic;
  • electrodynamic);
• by structural unit:
  • rotary;
  • scapular;
  • bearing;
  • serrated

Parameters of periodic oscillations

1. Vibration frequency:

   f = 1/T(Hz),

   Where T– period (time of a full cycle of oscillations), s; ω = 2 × π × f– angular velocity. Allows you to identify the source of vibration and damage.

2. Vibration movement S(µm) – displacement component describing vibration. Vibration displacement as a diagnostic parameter is of interest in cases where it is necessary to know the relative displacement of object elements or deformation.
3. Vibration velocity V(mm/s) – derivative of vibration displacement with respect to time. Vibration velocity is used to determine the technical condition of machines when measuring the overall vibration level. This parameter is associated with the energy of mechanical vibrations aimed at destroying parts.
4. Vibration acceleration A(m/s 2) – derivative of vibration velocity with respect to time. Vibration acceleration is used to determine the degree of damage and impact force in rolling bearings and gears.

Interrelation of vibrational quantities in harmonic processes:

S = V × 10 3 / (2 × π × f) = a × 10 6 / (2 × π × f) 2;
V = 2 × π × f × S × 10 -3 = a × 10 3 / (2 × π × f);
a = (2 × π × f) 2 × S × 10 -6 = 2 × π × f × V × 10 -3.

Basic characteristics of oscillatory and vibration processes

Oscillation range– the difference between the largest and smallest values ​​of the oscillating quantity in the considered time interval (double amplitude).

Peak value– is defined as the greatest deviation of the oscillatory value from the average position.

Arithmetic mean of instantaneous vibration values characterizes the overall vibration intensity.

Mean square value– the square root of the arithmetic mean or the integral mean value of the square of the fluctuating quantity in the time period under consideration.

Crest factor (peak factor)– the ratio of the peak value to the root mean square value of the measured parameter.

Measurements vibration movements(peak or amplitude, range of oscillations) is carried out in the low-frequency range 2-400 Hz. Approximate values ​​of vibration displacement are indicated in.

Table 2.2 – Vibration displacement values ​​and technical condition

Rotation speed, min -1	Vibration displacement amplitude, µm
	Great	Fine	satisfactorily	needs correction	dangerous
300	0-27	27-70	70-140	140-260	> 260
500	0-25	25-60	60-125	125-240	> 240
600	0-22	22-56	56-118	118-240	> 230
1000	0-18	18-45	45-100	100-200	> 200
1500	0-15	15-40	40-85	85-170	> 170

Measuring overall vibration levels

When determining the values ​​of the general vibration level, the root-mean-square value of the vibration velocity is measured in the frequency range of 10-1000 Hz. This complies with the requirements of the ISO 10816 standard. Measurements are regulated in three mutually perpendicular directions: vertical, horizontal and axial. During normal operation, the horizontal component has a maximum value, and the axial component has a minimum value. The vibration velocity for most mechanisms should not exceed 4.5 mm/s.

Vibration velocity values ​​that define the boundaries of states:

• up to 4.5 mm/s – satisfactory;
• 4.5-10.0 mm/s – bad;
• over 10.0 mm/s – emergency.

The values ​​are given for operation under load.