ISO 20816 1 2016 © ISO 2016 Mechanical vibration — Measurement and evaluation of machine vibration — Part 1 General guidelines Vibrations mécaniques — Mesurage et évaluation des vibrations de machin[.]
General
Overview
This clause outlines the recommended measurements, procedures, and operating conditions for assessing machine vibration, allowing for evaluation in line with the general criteria and principles established in Clause 6.
Vibration measurements
Measuring vibration on non-rotating components or assessing relative shaft vibration is a standard practice in machinery maintenance The choice of measurement type for the protection system typically relies on the expertise of the machine manufacturer.
Frequency range
The measurement of vibration shall be broad band, so that the frequency spectrum of the machine is adequately covered.
The frequency range required for evaluating machine integrity varies based on the machine type; for instance, assessing rolling element bearings necessitates higher frequencies compared to machines that utilize only fluid-film bearings.
Guidelines for instrumentation frequency ranges for specific machine classes are given in the appropriate parts of ISO 20816.
NOTE 1 In the past, broad-band measurements in the range 10 Hz to 1 000 Hz were the intended metric for full-load acceptance testing This might not meet the requirements of a condition monitoring scheme and might need to be modified for the purposes of vibration monitoring and diagnostics.
NOTE 2 Vibration condition monitoring and diagnostics of machines are described in ISO 13373.
On certain equipment, e.g gearboxes and rolling element bearings, it can be appropriate to use a different frequency range for acceptance purposes.
Types of measurements
Vibration measurements on non-rotating parts
Vibration measurements on stationary components are typically performed using a seismic transducer, which detects the absolute velocity or acceleration of the structural elements to which it is attached, such as the bearing housing.
Relative shaft vibration measurements
Relative shaft vibration measurements utilize a non-contacting transducer to detect the vibratory displacement between the shaft and its supporting structural member, such as the bearing housing.
Absolute shaft vibration measurements
Absolute shaft vibration measurements can be performed using two primary methods: first, by employing a shaft-riding probe equipped with a seismic transducer, such as a velocity type or accelerometer, to directly measure absolute shaft vibration; second, by utilizing a non-contacting transducer to assess relative shaft vibration in conjunction with a seismic transducer that measures support vibration Both transducers should be positioned closely to ensure they experience the same absolute motion in the measurement direction, with their conditioned outputs combined to yield an accurate measurement of absolute shaft motion.
To ensure accurate results, it is crucial to use the same datum time reference for outputs from both seismic and non-contacting transducers.
Measurement parameters
Measurement quantities
This document outlines key measurement quantities for vibration analysis: vibration displacement is quantified in micrometres, vibration velocity is expressed in millimetres per second, and vibration acceleration is measured in metres per square second.
The use, application and limitations of these quantities are discussed in Clause 6.
The relationship between broad-band acceleration, velocity, and displacement is complex, lacking a straightforward correlation Similarly, there is no simple connection between peak, peak-to-peak, root-mean-square, and average values of vibration A.1 provides a brief discussion on these complexities and establishes precise relationships among these quantities when the harmonic content of the vibration waveform is understood.
To prevent misunderstandings and guarantee accurate interpretation, it is essential to clearly specify the measurement quantity along with its unit, such as peak-to-peak displacement in åm (1 åm = 10^{-6} m) and root mean square (r.m.s.) velocity in mm/s.
NOTE Vibration is a vector quantity and therefore, when comparing two different values, it can be necessary to consider the phase angle between them (see Annex D).
For measuring the vibration of non-rotating components, the preferred metric is root mean square (r.m.s.) velocity, whereas for shaft vibration, peak-to-peak displacement is the favored measurement.
Since this document applies to both relative and absolute shaft vibration measurements, displacement is further defined as follows:
— relative displacement which is the vibratory displacement of the shaft with reference to support structure, such as a bearing housing or machine casing;
— absolute displacement which is the vibratory displacement of the shaft with reference to an inertial reference system.
It should be clearly indicated whether displacement values are relative or absolute.
Relative and absolute displacements are further defined by several different displacement quantities, each of which is now in widespread use These include the following:
S max maximum vibratory displacement in the plane of measurement, measured from time- integrated mean position, see Formula (A.10);
S (p–p) peak-to-peak vibratory displacement in the direction of measurement defined as
S (p–p)max maximum peak-to-peak vibratory displacement in the plane of measurement.
Shaft vibration can be measured using various displacement quantities, which must be clearly identified to ensure accurate interpretation according to the criteria outlined in Clause 6 The relationships among these quantities are illustrated in Figure A.3.
Vibration magnitude
The result of measurements made with an instrument which complies with the requirements of Clause 5 is called the vibration magnitude at a specific measuring position and direction.
When assessing the broad-band vibration of rotating machinery, it is standard to use the root mean square (r.m.s.) value of vibration velocity, as it correlates with vibration energy However, some may opt for alternative measures such as displacement, acceleration, or peak values instead of r.m.s values This necessitates the use of different criteria that may not be directly related to those based on r.m.s values.
For shaft vibration measurement, it is common practice to consider peak-to-peak values.
Vibration severity
Measurements are typically taken at multiple positions and in one to three directions, resulting in a range of vibration magnitude values The highest broad-band magnitude value recorded under specified machine support and operating conditions is referred to as vibration severity.
For many machine types, a single vibration severity value effectively represents their vibratory condition However, this method may not suffice for certain machines, necessitating an independent assessment of vibration severity at multiple measurement positions across various locations.
Measuring positions
Positions for measurements on non-rotating parts
Measurements on non-rotating components should be conducted on the bearings, bearing support housing, or other structural elements that significantly react to the dynamic forces from the rotating parts at the bearing sites, as these locations are crucial for characterizing the machine's overall vibration Typical measurement locations are illustrated in Figures 1 through 5.
To assess the vibrational behavior at each measurement location, it is essential to conduct measurements in three orthogonal directions The comprehensive set of measurements illustrated in Figures 1 to 5 is typically necessary only for acceptance testing.
Operational monitoring typically involves measurements in the radial direction, which includes horizontal transverse and vertical orientations Additionally, axial vibration measurements may be taken, specifically for thrust bearings, as they are crucial at locations where direct axial dynamic forces are transmitted.
Detailed recommendations for specific machine types are provided in the additional parts of ISO 20816.
Figure 1 — Measuring points for pedestal bearings
Figure 2 — Measuring points for housing-type bearings
Figure 3 — Measuring points for small electrical machines
Figure 4 — Measuring points for reciprocating engines close to the bearing locations
Figure 5 — Measuring points for vertical machine sets
Positions for measurements on rotating shafts
For effective measurements on rotating shafts, it is essential to position transducers to evaluate lateral movement at critical points It is advisable to place two transducers at or near each machine bearing for both relative and absolute measurements These transducers should be radially mounted in the same transverse plane, ideally within ±5° of a radial line Additionally, it is recommended to position both transducers 90° ± 5° apart on the same bearing half, ensuring consistency in placement across all bearings.
A single transducer can effectively replace the conventional pair of orthogonal transducers at each measurement plane, provided it delivers sufficient data regarding shaft vibration.
To accurately assess total non-vibration runout, it is essential to conduct specific measurements that account for shaft surface metallurgical non-homogeneities, local residual magnetism, and mechanical runout Additionally, it is important to recognize that anisotropic rotors, like two-pole generators, may produce misleading runout signals due to the influence of gravity.
Figure 6 — Measuring points for measurements on rotating shafts
Figure 7 — Mounting of non-contacting probes for the measurement of shaft relative vibration
4.4.2.2 Positions for relative shaft vibration measurements
Non-contacting relative vibration transducers are typically installed in tapped holes within the bearing housing or on rigid brackets near the bearing housing or shell When mounted inside the bearing, it is crucial to position the transducers to avoid disrupting the lubrication pressure wedge Alternative axial mounting arrangements can be implemented, but they will require different vibration assessment criteria Additionally, for transducers mounted on brackets, it is essential that the bracket is devoid of natural frequencies that could negatively impact the transducer's ability to accurately measure relative shaft vibration.
The shaft surface at the transducer location must be smooth and devoid of geometric discontinuities, metallurgical non-homogeneities, and local residual magnetism to prevent false signals While an electroplated or metallized surface may be permissible, calibration differences should be considered It is essential that the total combined electrical and mechanical runout, as measured by the transducer, does not exceed 25% of the allowable vibration displacement or 6 μm, whichever is greater For existing machines not originally designed for shaft vibration measurements, alternative runout criteria may be necessary.
4.4.2.3 Positions for absolute shaft vibration measurements using combined seismic and non-contacting relative vibration transducers
Using a combination of seismic and non-contacting relative vibration transducers allows for the measurement of absolute shaft vibration This is achieved by integrating the signal from the seismic transducer to convert its acceleration or velocity output into displacement, and then summing the displacement outputs from both transducers.
NOTE 1 In order to avoid incorrect results, it is important to ensure that the same datum time reference is used for the outputs from both the seismic and non-contacting transducers.
The mounting and other requirements for the non-contacting transducer are as specified in 4.4.2.2
To ensure accurate measurement of absolute shaft vibration, the seismic transducer must be securely mounted to the machine structure, such as the bearing housing, in proximity to the non-contacting transducer Both transducers should experience the same absolute vibration of the support structure in the measurement direction, and their sensitive axes must be aligned parallel to each other This alignment allows for the effective summation of their conditioned signals.
NOTE 2 Information on mounting accelerometers for seismic measurements is given in ISO 5348.
NOTE 3 In the past, absolute shaft vibration measurements have also been performed using a shaft-riding mechanism with a seismic transducer.
Figure 8 — Mounting of non-contacting and seismic probes for the measurement of shaft absolute vibration
Machine support structure for acceptance testing
General
The acceptance criteria should be agreed between the customer and manufacturer.
In-situ tests
When acceptance is carried out in situ, the support structure shall be that supplied for the machine
In this case, it is important to ensure that all the major components of the machine and structure are installed when the test is carried out.
Valid comparisons of vibration for machines of the same type but on different foundations or sub- foundations can only be made if the foundations concerned have similar dynamic characteristics.
In a test facility
Various types of machines undergo acceptance testing on a test bed, which may have different support structure characteristics compared to the installation site The support structure plays a crucial role in influencing the measured vibrations It is essential to ensure that the natural frequencies of the entire test setup do not align with the machine's rotational frequencies or any of its significant harmonics.
The test setup is considered adequate if the vibration levels recorded at the machine feet or base frame near the bearing support do not exceed 50% of the vibration levels measured at the corresponding bearing in both horizontal and vertical directions Furthermore, the arrangement must not significantly alter any of the major resonance frequencies.
In cases where a substantial support resonance is detected during acceptance testing and cannot be mitigated, it may be necessary to conduct vibration acceptance testing on the fully installed machine in its operational environment.
Acceptance tests for certain machines, such as small electrical machinery, can be conducted when they are mounted on a resilient support system, as outlined in IEC 60034-14 It is essential that all rigid-body mode frequencies of the machine's support system are below half of the lowest significant excitation frequency of the machine This can be achieved by using a resilient support baseplate or by suspending the machine on a soft spring.
Machine operating conditions
Shaft vibration measurements must be conducted once the agreed normal operating conditions—such as speed, load, temperature, and pressure—are met Any additional vibration measurements taken under different conditions will not be considered for evaluation as per Clause 6.
Evaluation of vibration from other sources
When vibration levels surpass the recommended limits, it is essential to measure environmental vibrations with the machine turned off to determine if they significantly contribute to the observed vibrations If environmental vibrations exceed one-third of the recommended limits, efforts should be made to mitigate their magnitude.
The instrumentation must be specifically designed to function effectively in its intended environment, considering factors such as temperature and humidity For measuring vibration severity, refer to ISO 2954, while ISO 10817-1 provides specifications for instruments that measure shaft vibration.
Proper installation of vibration transducers is crucial to maintain accurate vibration response characteristics of the machine According to ISO 5348, guidelines for mounting accelerometers are also relevant for velocity transducers.
Modern instrumentation offers various methods for delivering measurement values, with acceptance criteria defined by r.m.s velocity in mm/s for non-rotating components and peak-to-peak displacement in micrometres for rotating parts These criteria can be adjusted to align with the units commonly used on-site, following the same assumptions integrated into the distributed control systems.
An effective measurement system should include on-line calibration capabilities for the readout instrumentation and provide isolated outputs for additional analysis as needed.
General
Overview
This clause outlines the fundamental criteria and principles for assessing machine vibration The evaluation standards pertain to both operational monitoring and acceptance testing, focusing solely on the vibration generated by the machine rather than any external vibrations.
The evaluation criteria for machine vibration are influenced by various factors and can differ greatly between machine types, and even among different rotors within the same system It is crucial to establish appropriate criteria specific to each machine to avoid misapplication of standards across different types.
EXAMPLE Evaluation criteria for a high-speed compressor operating in a petrochemical plant are likely to be different from those for large turbo-generators.
This clause outlines the evaluation criteria for measuring both non-rotating parts and rotating shafts, without specifying vibration values When both measurement procedures are applicable, the more restrictive one will typically take precedence Detailed criteria for various classes and types of machinery can be found in the relevant sections of ISO 20816.
The criteria outlined in this clause specifically pertain to vibrations caused by the excitation of rotating elements They do not apply to the assessment of electromagnetic vibrations that occur at twice the line frequency, which are transmitted from the generator stator core to the bearings.
Types of measurement on rotating shafts
6.1.2.1 There are two principal factors by which shaft vibration is judged: a) absolute vibration of the shaft; b) vibration of the shaft relative to the structural elements.
6.1.2.2 If the evaluation criterion is the change in shaft vibration, then a) when the vibration of the structure, on which the shaft-relative transducer is mounted, is small (i.e less than 20 % of the relative shaft vibration), either the relative shaft vibration or absolute shaft vibration may be used as a measure of shaft vibration, and b) when the vibration of the structure, on which the shaft-relative transducer is mounted, is 20 % or more of the relative shaft vibration, the absolute shaft vibration shall be measured and, if found to be larger than the relative shaft vibration, it shall be used as the measure of shaft vibration.
6.1.2.3 If the evaluation criterion is the dynamic load on the bearing, the relative shaft vibration shall be used as the measure of shaft vibration.
6.1.2.4 If the evaluation criterion is stator/rotor clearance, then a) when the vibration of the structure, on which the shaft-relative transducer is mounted, is small (i.e less than 20 % of the relative shaft vibration), the relative shaft vibration shall be used as a measure of clearance absorption, and b) when the vibration of the structure, on which the shaft-relative transducer is mounted, is 20 % or measure of clearance absorption unless the vibration of the structure, on which the shaft-relative transducer is mounted, is not representative of the total stator vibration In this latter case, special measurements are required.
6.1.2.5 The shaft vibration associated with a particular evaluation range depends on the size and mass of the vibrating body, the characteristics of the mounting system and the output and use of the machine
When specifying shaft vibration ranges for a specific class of machinery, it is essential to consider the various purposes and circumstances involved Additionally, relevant product specifications should be referenced where applicable.
Factors affecting evaluation criteria
When specifying evaluation criteria for vibration measurements, several key factors must be considered These include the measurement's purpose, such as maintaining running clearances versus avoiding excessive dynamic loads on bearings Additionally, the type of measurement—whether it involves non-rotating parts or shaft vibration—plays a crucial role Other important considerations are the quantities measured, measurement positions, shaft rotational speed, bearing type, clearance, and diameter Furthermore, the machine's function, output, and size, along with the stiffness of bearings, pedestals, and foundations, as well as rotor mass and stiffness, are essential in determining effective vibration measurement criteria.
The diverse factors involved in machine evaluation prevent the establishment of universal criteria applicable to all machines Instead, various criteria based on operational experience are essential for different machines, serving primarily as guidelines However, there are instances where machines may function safely outside these general recommendations or fail to operate even when vibration values fall within acceptable limits.
Types of evaluation criteria
General
Vibration severity in different machine classes is evaluated using two key criteria: the first focuses on the magnitude of broad-band vibrations observed, while the second assesses variations in magnitude, regardless of whether these changes indicate increases or decreases.
Criterion I: Vibration magnitude at rated speed under steady
For non-rotating components, it is essential to establish limits for absolute vibration magnitude to ensure dynamic loads on bearings remain acceptable and that vibration transmission to the support structure and foundation is minimized The maximum vibration magnitude recorded at each bearing or pedestal is evaluated against four established zones based on international standards This recorded maximum is referred to as vibration severity.
The vibration of a machine is influenced by its size, the properties of the vibrating body, the mounting system, and its intended purpose Therefore, it is essential to consider these factors when determining vibration measurement ranges for various machine types For most machines, regardless of bearing type, measuring the broad-band root mean square (r.m.s.) vibration velocity on structural components, like bearing housings, effectively reflects the operational conditions of the rotating shaft elements, ensuring their reliable performance.
Vibration velocity is often adequate for assessing vibration severity across various machine speeds, but relying solely on a single velocity value can result in excessive vibration displacements, especially in low-speed machines where the once-per-revolution component prevails In high-speed machines or those with high-frequency vibrations, constant-velocity criteria may lead to unacceptable accelerations Therefore, evaluation criteria based on velocity are typically represented in a manner that defines upper and lower frequency limits, indicating that allowable vibration velocity varies with frequency below a certain threshold and above another, while a constant-velocity criterion is applicable within the intermediate frequency range.
NOTE The r.m.s velocity values for non-rotating parts listed in Table C.1 refer to this constant-velocity region.
The precise nature of the evaluation criteria and the values of f l , f u , f x and f y for specific machine types are given in the additional parts of ISO 20816.
Many machines exhibit broad-band vibration predominantly at a single frequency, typically the shaft rotational frequency The permissible vibration levels can be determined from Figure 9, which illustrates the vibration velocity associated with that specific frequency.
For less-common machines exhibiting significant vibratory energy beyond the breakpoints \( f_x \) and \( f_y \), several measurement approaches can be utilized Firstly, broad-band displacement can be measured for energy below \( f_x \), while broad-band acceleration is applicable for energy above \( f_y \), ensuring that the allowable vibration metrics align with the velocity indicated in Figure 9 Secondly, a frequency analyser can determine the velocity, displacement, or acceleration across the frequency spectrum, with the equivalent broad-band velocity calculated using Formula (A.2) and appropriate weighting factors for components outside the \( f_x \) and \( f_y \) range It is crucial to evaluate this value against the constant velocity between \( f_x \) and \( f_y \), as direct comparisons with Figure 9 may be misleading unless the broad-band vibration is dominated by a single frequency Lastly, a composite broad-band measurement can be performed using an instrument with weighting networks that reflect the shape of Figure 9, which should also be assessed relative to the constant velocity between \( f_x \) and \( f_y \).
The evaluation criteria for specific machine types are outlined in the supplementary sections of ISO 20816, with Section C.2 offering further guidance For certain machines, it may be essential to establish additional criteria beyond those illustrated in Figure 9, as noted in Section 6.5.3.
Figure 9 — General form of vibration velocity evaluation criteria
For rotating shafts, it is essential to establish limits on vibration magnitude to ensure dynamic loads on bearings remain acceptable, maintain adequate radial clearance, and minimize vibration transmission to the support structure and foundation The maximum vibration magnitude at each bearing is evaluated against four established zones based on international standards.
Figure 10 illustrates the relationship between allowable peak-to-peak shaft vibration and operating speed range It is widely recognized that as the operating speed of a machine increases, the permissible vibration limits tend to decrease However, the specific values and their rate of change with speed differ across various machine types.
Y peak-to-peak shaft vibration
It is crucial to recognize that vibration values at zone boundaries and the associated speed ranges differ among various machine types Selecting the appropriate criteria is essential to prevent inaccurate extrapolation.
Figure 10 — Generalized example of evaluation criteria for shaft vibration
The evaluation zones outlined in ISO 20816 enable a qualitative assessment of machine vibrations during steady-state conditions at normal operating speeds, offering guidance on potential actions It's important to note that the categorization and number of zones may vary depending on the specific type of machine.
Zone A: The vibration of newly commissioned machines normally falls within this zone.
NOTE The effort required to achieve vibration within zone A can be disproportionate and unnecessary.
Zone B: Machines with vibration within this zone are normally considered acceptable for unrestricted long-term operation.
Machines operating in Zone C exhibit vibrations that render them unsuitable for prolonged continuous use While they can function for a limited time under these conditions, it is essential to address the issue promptly when the opportunity for maintenance arises.
Zone D: Vibration values within this zone are normally considered to be of sufficient severity to cause damage to the machine.
Numerical values for zone boundaries outlined in ISO 20816 serve as guidelines to prevent gross deficiencies and unrealistic requirements Specific machine features may necessitate the use of adjusted zone boundary values, either higher or lower In these instances, it is essential to provide a rationale and confirm that operating with elevated vibration levels will not jeopardize the machine's safety.
NOTE For the vibration evaluation for non-rotating parts of those machines for which no specific International Standard exists, see Annex C.
Acceptance criteria must be mutually agreed upon by the machine manufacturer and the customer, with prior negotiations recommended While evaluation zones help establish these criteria, the numerical values for the zone boundaries are not meant to function as definitive acceptance specifications.
Contractual acceptance tests shall be carried out under clearly defined duration and operating parameters, e.g load, temperature, pressure.
After major component replacement, maintenance or service activities, acceptance criteria shall take into account the scope of activity and the vibration behaviour prior to servicing.
Criterion II: Change in vibration magnitude
This criterion evaluates changes in vibration magnitude compared to a previously established reference value Even if zone C of Criterion I has not been reached, a significant increase or decrease in vibration may necessitate action These changes can be either instantaneous or gradual, signaling potential damage or warning of impending failure Criterion II focuses on changes in vibration magnitude under steady-state operating conditions.
When applying Criterion II, it is essential to ensure that vibration measurements are collected from the same transducer location and orientation, and under similar operating conditions Any significant deviations from normal vibration levels must be thoroughly investigated to prevent potential hazards.
ISO 20816 outlines the criteria for evaluating vibration changes for monitoring purposes It's important to recognize that certain changes may go undetected unless discrete frequency components are specifically monitored.
Operational limits
General
For long-term operation, it is common practice for some machine types to establish operational vibration limits These limits take the form of ALARMS and TRIPS.
Alarms serve as crucial indicators that a specific vibration threshold has been exceeded or a significant change has taken place, prompting the need for potential remedial action Typically, when an alarm is triggered, operations may continue temporarily while investigations are conducted to determine the cause of the vibration change and to outline necessary corrective measures.
TRIPS are critical for determining the vibration threshold that, if surpassed, may lead to machine damage When the TRIP value is exceeded, it is essential to take prompt action to mitigate the vibration or to shut down the machine to prevent further issues.
Different operational limits, reflecting differences in dynamic load, bearing/seal clearances and support stiffness may be specified for different measurement positions and directions.
Where appropriate, guidelines for specifying ALARM and TRIP criteria for specific machine types are given in other parts of ISO 20816.
Setting of ALARMS
ALARM values can significantly fluctuate for different machines, either increasing or decreasing Typically, these values are established in relation to a baseline determined from prior experience specific to the measurement position or direction of each machine.
In the absence of a defined baseline, the initial ALARM setting for a new machine should be determined based on experience with similar machines or in relation to accepted values Over time, as a steady-state baseline is established, the ALARM setting must be adjusted to reflect this new information.
When the steady-state baseline of a machine changes, such as after an overhaul, it is essential to adjust the ALARM settings accordingly This allows for the establishment of distinct operational ALARM settings for various bearings on the machine, which take into account the variations in dynamic load and bearing support stiffness.
Setting of TRIPS
The TRIP values are crucial for assessing the mechanical integrity of machines, reflecting specific design features that allow them to endure abnormal dynamic forces Typically, these values remain consistent across machines of similar design and are not directly associated with the steady-state baseline values used for setting alarms.
Additional factors
Vibration frequencies and vectors
This document focuses on the evaluation of broad-band vibration, omitting frequency components and phase, which is generally sufficient for acceptance testing and operational monitoring However, for specific machine types, incorporating vector information in vibration assessments may be beneficial.
Vector change information is essential for identifying and characterizing alterations in a machine's dynamic state Broad-band vibration measurements may overlook these changes, as illustrated in Annex D.
The specification of criteria for vector changes is beyond the scope of this document.
Vibration sensitivity of the machine
Vibration levels on a specific machine can be highly sensitive to variations in steady-state operational conditions While this sensitivity may not always be critical, there are instances where satisfactory vibration measurements under certain conditions can become unacceptable if those conditions shift.
In situations where the vibration sensitivity of a machine is uncertain, it is essential for the customer and supplier to come to a mutual agreement regarding the need for testing or theoretical evaluation, as well as the scope of such assessments.
Techniques for rolling element bearings
Alternative methods for assessing the condition of rolling element bearings, aside from broad-band vibration measurements, are being developed and are detailed in Annex B However, defining evaluation criteria for these additional methods is not covered in this document.
A.1 Vibration of non-rotating parts
The use of root mean square (r.m.s.) velocity to assess the vibratory response of various machine types has proven effective over the years For simple alternating waveforms composed of distinct harmonic components with known amplitude and phase, and minimal random vibration or shock, Fourier analysis allows for the precise correlation of fundamental quantities such as displacement, velocity, acceleration, peak, peak-to-peak, r.m.s., and average through established mathematical relationships This annex summarizes several of these key relationships.
From measured vibration velocity versus time records, the r.m.s value of the velocity, v rms , can be calculated by Formula (A.1): v T v t t
(A.1) where v(t) is the time-dependent vibration velocity;
T is the sampling time which is longer than the period of any of the major frequency compo- nents of which v(t) is composed.
Acceleration, velocity, and displacement magnitudes can be determined for various frequencies through spectral analyses When the peak-to-peak displacement values in micrometres, r.m.s velocity values in millimetres per second, and r.m.s acceleration values in metres per square second are known, along with their corresponding frequencies in hertz, the associated r.m.s velocity characterizing the motion can be calculated using the formula: \$$v_{rms} = \sqrt{(s_1 f_1)^2 + (s_2 f_2)^2 + \ldots + (s_n f_n)^2}\$$
When vibration consists of two significant frequency components resulting in beats with root mean square (r.m.s.) values of \( v_{\text{min}} \) and \( v_{\text{max}} \), the approximate r.m.s vibration velocity, \( v_{\text{rms}} \), can be calculated using the formula: \[v_{\text{rms}} = \frac{1}{2}(v_{\text{max}} + v_{\text{min}})\]
The vibration velocity of a single frequency component allows for the evaluation of peak-to-peak displacement, denoted as \( s_i \) in micrometers, using Formula (A.4).
� ν ν� p (A.4) where v i is the r.m.s value of the vibration velocity, in millimetres per second, of the component with frequency, f i , in hertz.
Y r.m.s velocity, mm/s a peak-to-peak displacement, àm b r.m.s acceleration, m/s 2
Figure A.1 — Relationship between acceleration, velocity and displacement for single- frequency harmonic components
The mean values of shaft displacement in two orthogonal directions, x and y, relative to a reference position, are determined by time-based integrals, as illustrated in Formulas (A.5) and (A.6).
In the equation 2 d (A.6), the variables x(t) and y(t) represent the time-dependent alternating displacement values in relation to a reference position The duration of integration, denoted as t2 - t1, is significantly larger than the period of the lowest frequency vibration component.
Absolute vibration measurements involve a fixed reference position in space, while relative vibration measurements indicate the shaft's mean position relative to stationary components at the measurement site Variations in these mean values can arise from several factors, including movements in bearings or foundations and changes in oil film characteristics, which typically occur gradually compared to the vibration components that contribute to the alternating mean values.
In general, the time-integrated mean position in any direction differs from the position calculated as half the sum of the maximum and minimum displacement values However, when shaft vibration is sinusoidal and at a single frequency, the shaft center traces an elliptical path In this case, the time-integrated mean position aligns with the position determined by averaging the maximum and minimum displacement values.
2 fixed reference axes x and y x(t), y(t) time-dependent alternating values of displacement relative to the reference position x y , time-integrated mean values of shaft displacement
0 time-integrated mean position of orbit
K instantaneous position of shaft centre
Figure A.2 — Kinetic orbit of shaft
0 time-integrated mean position of orbit x y , time-integrated mean values of shaft displacement
K instantaneous position of shaft centre
P position of shaft for maximum displacement from time-integrated mean position
S 1 instantaneous value of shaft displacement
S max maximum value of shaft displacement from time-integrated mean position 0
S A1 , S B1 instantaneous values of shaft displacement in directions of transducers A and B, respectively
S (p–p)max maximum value of peak-to-peak displacement
S B(p–p) peak-to-peak values of shaft displacement in directions of transducers A and B
NOTE In this example sketch, S A(p–p) = S (p–p) since S A(p–p) > S B(p–p)
Figure A.3 — Definition of shaft displacement quantities
A.2.2 Peak-to-peak displacement of shaft vibration
In shaft vibration measurements, the key focus is on the alternating values that define the orbit's shape For instance, in the kinetic shaft orbit illustrated in Figure A.3, two transducers, A and B, positioned 90° apart, are utilized to accurately measure the shaft's vibration.
At a specific moment, the shaft center aligns with point K on the orbit, resulting in an instantaneous shaft displacement of S₁ from the mean position In the directions of transducers A and B, the instantaneous displacements from the mean position are Sₐ₁ and Sᵦ₁, respectively The relationship between these displacements is defined by Formula (A.7).
The values of S 1 , S A1 and S B1 vary with time as the shaft centre moves around the orbit; the corresponding waveforms measured by each transducer are shown in Figure A.3.
NOTE If the orbit is elliptical, then these waveforms are pure sine waves of the same frequency.
The peak-to-peak displacement value for transducer A, denoted as \$S_A(p–p)\$, is determined by the difference between its maximum and minimum displacements, with a similar definition applying to transducer B, \$S_B(p–p)\$ It is important to note that \$S_A(p–p)\$ and \$S_B(p–p)\$ are generally unequal and can vary from measurements taken in other radial directions Therefore, the peak-to-peak displacement value is influenced by the measurement direction.
Measurement quantities are independent of the absolute mean position, making it unnecessary to utilize systems that measure both mean and alternating values.
Peak-to-peak displacement is the quantity which has been used most frequently for monitoring vibration of rotating machines.
Measuring peak-to-peak displacement in two orthogonal directions is straightforward, but determining the maximum peak-to-peak displacement and its angular position, as illustrated in Figure A.3, poses challenges Practically, alternative measurement methods are utilized to approximate this maximum displacement value For more accurate assessments, a detailed analysis of the shaft orbit using an oscilloscope is essential The three prevalent methods for achieving satisfactory approximations are outlined in sections A.2.2.2 to A.2.2.4.
A.2.2.2 Method A: Resultant value of peak-to-peak shaft displacement values measured in two orthogonal directions
The value of S (p–p)max can be approximated from Formula (A.8):
Using Formula (A.8) as an approximation for vibrations primarily at rotational frequency tends to overestimate the value of \$S(p–p)_{max}\$ by about 40% This maximum error is observed in circular orbits and decreases as the orbit flattens, reaching zero error in the case of a straight line orbit.
A.2.2.3 Method B: Taking the maximum value of peak-to-peak shaft displacement values measured in two orthogonal directions
The value of S (p–p)max can be approximated from Formula (A.9):
S (p–p)max = S A(p–p) or S B(p–p) (A.9) whichever is greater.
The use of Formula (A.9) as an approximation when the vibration is predominantly at rotational frequency generally under-estimates the value of S (p–p)max , with a maximum error of approximately 30 %.
The maximum error occurs for a flat orbit and progressively reduces as the orbit becomes circular, with a zero error when the orbit is circular.