BS ISO 2041:2009

Mechanical vibration, shock and condition monitoring. Vocabulary. | Status: Superseded, Withdrawn

Scope This International Standard defines terms and expressions unique to the areas of mechanical vibration, shock and condition monitoring.

1 General

1.1 displacement relative displacement time varying quantity that specifies the change in position of a point on a body with respect to a reference frame Note 1 to entry: The reference frame is usually a set of axes at a mean position or a position of rest. In general, a rotation displacement vector, a translation displacement vector, or both can represent the displacement. Note 2 to entry: A displacement is designated as relative displacement if it is measured with respect to a reference frame other than the primary reference frame designated in a given case. Note 3 to entry: Displacement can be:

• — oscillatory, in which case simple harmonic components can be defined by the displacementamplitude (and frequency), or
• — random, in which case the root-mean-square (rms) displacement (and band-width and probability density distribution) can be used to define the probability that the displacement will have values within any given range.

Displacements of short time duration are defined as transient displacements. Non-oscillatory displacements are defined as sustained displacements, if of long duration, or as displacement pulses, if of short duration. 1.2 velocity relative velocity rate of change of displacement Note 1 to entry: In general, velocity is time-dependent. Note 2 to entry: The reference frame is usually a set of axes at a mean position or a position of rest. In general, a rotation velocity vector, a translation velocity vector, or both can represent the velocity. Note 3 to entry: A velocity is designated as relative velocity if it is measured with respect to a reference frame other than the primary reference frame designated in a given case. The relative velocity between two points is the vector difference between the velocities of the two points. Note 4 to entry: Velocity can be:

• — oscillatory, in which case simple harmonic components can be defined by the velocityamplitude (and frequency), or
• — random, in which case the root-mean-square (rms) velocity (and band-width and probability density distribution) can be used to define the probability that the velocity will have values within any given range.

Velocities of short time duration are defined as transient velocities. Non-oscillatory velocities are defined as sustained velocities, if of long duration. 1.3 acceleration relative acceleration rate of change of velocity Note 1 to entry: In general, acceleration is time-dependent. Note 2 to entry: The reference frame is usually a set of axes at a mean position or a position of rest. In general, a rotation acceleration vector, a translation acceleration vector, or both and the Coriolis acceleration can represent the acceleration. Note 3 to entry: An acceleration is designated as relative acceleration if it is measured with respect to a reference frame other than the inertial reference frame designated in a given case. The relative acceleration between two points is the vector difference between the accelerations of the two points. Note 4 to entry: In the case of time-dependent accelerations, various self-explanatory modifiers, such as peak, average, and rms (root-mean-square), are often used. The time intervals over which the average or root-mean-square values are taken should be indicated or implied. Note 5 to entry: Acceleration can be:

• — oscillatory, in which case simple harmonic components can be defined by the accelerationamplitude (and frequency), or
• — random, in which case the rms acceleration (and band-width and probability density distribution) can be used to define the probability that the acceleration will have values within any given range.

Accelerations of short time duration are defined as transient accelerations. Non-oscillatory accelerations are defined as sustained accelerations, if of long duration, or as acceleration pulses, if of short duration. 1.4 standard acceleration due to gravity g n unit, 9,806 65 metres per second-squared (9,806 65 m/s2) Note 1 to entry: Value adopted in the International Service of Weights and Measures and confirmed in 1913 by the 5th CGPM as the standard for acceleration due to gravity. Note 2 to entry: This “standard value” (gn = 9,806 65 m/s2 = 980,665 cm/s2 ≈ 386,089 in/s2 ≈ 32,174 0 ft/s2) should be used for reduction to standard gravity of measurements made in any location on Earth. Note 3 to entry: Frequently, the magnitude of acceleration is expressed in units of gn. Note 4 to entry: The actual acceleration produced by the force of gravity at or below the surface of the Earth varies with the latitude and elevation of the point of observation. This variable is often expressed using the symbol g. Caution should be exercised if this is done so as not to create an ambiguity with this use and the standard symbol for the unit of the gram. 1.5 force dynamic influence that changes a body from a state of rest to one of motion or changes its rate of motion Note 1 to entry: A force could also change a body’s size or shape if the body resists motion. Note 2 to entry: The newton is the unit of force. One newton is the force required to give a mass of one kilogram an acceleration of one metre per second squared. 1.6 restoring force reaction force caused by the elastic property of a structure when it is being deformed 1.7 jerk rate of change of acceleration 1.8 inertial reference system inertial reference frame coordinate system or frame which is fixed in space or moves at constant velocity without rotational motion and thus, not accelerating 1.9 inertial force reaction force exerted by a mass when it is being accelerated 1.10 oscillation variation, usually with time, of the magnitude of a quantity with respect to a specified reference when the magnitude is alternately greater and smaller than the specified reference cf. Note 1 to entry: Variations with time such as shock processes or creeping motions are also considered to be oscillations in a more general sense of the word. 1.11 environment aggregate, at a given moment, of all external conditions and influences to which a system is subjected cf. induced environment, natural environment 1.12 induced environment conditions external to a system generated as a result of the operation of the system 1.13 natural environment conditions generated by the forces of nature and the effects of which are experienced by a system when it is at rest as well as when it is in operation 1.14 preconditioning climatic and/or mechanical and/or electrical treatment procedure which may be specified for a particular system so that it attains a defined state 1.15 conditioning climatic and/or mechanical and/or electrical conditions to which a system is subjected in order to determine the effect of such conditions upon it 1.16 excitation stimulus external force (or other input) applied to a system that causes the system to respond in some way 1.17 response (of a system) output quantity of a system 1.18 transmissibility non-dimensional complex ratio of the response of a system in forced vibration to the excitation Note 1 to entry: The ratio may be one of forces, displacements, velocities or accelerations. Note 2 to entry: This is sometimes known as a transmissibility function. 1.19 overshoot when the maximum transient response exceeds the desired response Note 1 to entry: If the output of a system is changed from a steady value A to a steady value B by varying the input, such that value B is greater than A, then the response is said to overshoot when the maximum transient response exceeds value B. Note 2 to entry: The difference between the maximum transient response and the value B is the value of the overshoot. This is usually expressed as a percentage. 1.20 undershoot when the minimum transient response falls below the desired response Note 1 to entry: If the output of a system is changed from a steady value A to a steady value B by varying the input, such that value B is less than A, then the response is said to undershoot when the minimum transient response is less than value B. Note 2 to entry: The difference between the minimum transient response and the value B is the value of the undershoot. This is usually expressed as a percentage. 1.21 system set of interrelated elements considered in a defined context as a whole and separated from their environment 1.22 linear system system in which the magnitude of the response is proportional to the magnitude of the excitation Note 1 to entry: This definition implies that the principle of superposition can be applied to the relationship between output response and input excitation. 1.23 mechanical system system comprising elements of mass, stiffness and damping 1.24 foundation structure that supports a mechanical system Note 1 to entry: It can be fixed in a specified reference frame or it can undergo a motion. 1.25 seismic system system consisting of a mechanical system attached to a reference base by one or more flexible elements, with damping normally included Note 1 to entry: Seismic systems are usually idealized as single-degree-of-freedom systems with viscous damping. Note 2 to entry: The natural frequencies of the mass as supported by the flexible elements are relatively low for seismic systems associated with displacement or velocity transducers, and are relatively high for acceleration transducers, as compared with the range of frequencies to be measured. Note 3 to entry: When the natural frequency of the seismic system is low relative to the frequency range of interest, the mass of the seismic system may be considered to be at rest over this range of frequencies. 1.26 equivalent system system that can be substituted for another system for the purpose of analysis Note 1 to entry: Many types of equivalence are common in vibration and shock technology:

1. a) a torsional system equivalent to a translational system;
2. b) an electrical or acoustical system equivalent to a mechanical system, etc.;
3. c) equivalent stiffness;
4. d) equivalent damping.

1.27 degrees of freedom minimum number of generalized coordinates required to define completely the configuration of a mechanical system Note 1 to entry: This applies to mechanical systems, not to be confused with statistical degrees of freedom. Note 2 to entry: It is often referred to by the acronym DOF. 1.28 discrete system lumped parameter system mechanical system in which the mass, stiffness, and/or damping elements are discretely located 1.29 single-‌degree-‌of-‌freedom system SDOF system requiring only one coordinate to define completely its configuration at any instant 1.30 multi-‌degree-‌of-‌freedom system system for which two or more coordinates are required to define completely the configuration of the system at any instant 1.31 continuous system mechanical system in which the mass, stiffness, and/or damping properties are spatially distributed rather than discretely located Note 1 to entry: The configuration of a continuous system is specified by a function of a continuous spatial variable, or variables, in contrast to a discrete or lumped parameter system that requires only a finite number of coordinates to specify its configuration. 1.32 centre of gravity point through which the resultant of the weights of its component particles passes without resulting in moment for all orientations of the body with respect to a gravitational field Note 1 to entry: If the field is uniform, the centre of gravity coincides with the centre of mass (1.33). 1.33 centre of mass point of a body where the first moment of the overall mass with reference to a Cartesian coordinate system is equal to the first moments of mass of all points of the body Note 1 to entry: This is the point at which an object is in balance in a uniform gravitational field. 1.34 principal axes of inertia three mutually perpendicular axes intersecting each other at a given point about which the products of inertia of a solid body are zero Note 1 to entry: If the point is the centre of mass of the body, the axes and moments are called central principal axes and central principal moments of inertia. Note 2 to entry: In balancing, the term “principal inertia axis” is used to designate the one central principal axis (of the three such axes) most nearly coincident with the shaft axis of the rotor and is sometimes referred to as the balance axis or the mass axis. 1.35 moment of inertia sum (integral) of the product of the masses of the individual particles (elements of mass) of a body and the square of their perpendicular distances from the axis of rotation 1.36 product of inertia sum (integral) of the product of the masses of the individual particles (elements of mass) of a body and their distances from two mutually perpendicular planes 1.37 stiffness ratio of change of force (or torque) to the corresponding change in translational (or rotational) deformation of an elastic element cf. dynamic stiffness 1.38 compliance reciprocal of stiffness cf. dynamic compliance 1.39 neutral surface neutral surface of a beam in simple flexure surface in which there is no strain Note 1 to entry: It should be stated whether or not the neutral surface is a result of the flexure alone, or whether it is a result of the flexure and other superimposed loads. 1.40 neutral axis neutral axis of a beam in simple flexure line or plane in a beam where the longitudinal stress, tensile or compressive is zero 1.41 transfer function mathematical representation of the relationship between the input and output of a linear time-invariant system Note 1 to entry: A transfer function is usually a complex function defined as the ratio of the Laplace transforms of the output to the input of a linear time-invariant system. Note 2 to entry: It is usually given as a function of frequency, and is usually a complex function. See response (1.17), transmissibility (1.18) and transfer impedance (1.50). 1.42 complex excitation excitation expressed as a complex quantity with amplitude and phase angle Note 1 to entry: The concepts of complex excitations and responses were evolved historically in order to simplify calculations. The actual excitation and response are the real parts of the complex excitation and response. If the system is linear, the concept is valid because superposition holds in such a situation. Note 2 to entry: This term should not be confused with excitation by a complex vibration, or vibration of complex waveform. The use of the term “complex vibration” in this sense is deprecated. 1.43 complex response response of a system expressed as a complex quantity with amplitude and phase angle from a specified excitation Note 1 to entry: See the notes under complex excitation (1.42). 1.44 modal analysis vibration analysis method that characterizes a complex structural system by its modes of vibration, i.e. its natural frequencies, modal damping and mode shapes, and based on the principle of superposition 1.45 modal matrix linear transformation matrix which consists of the eigen vectors or modal vectors of a system Note 1 to entry: It renders the system both inertially and elastically uncoupled, i.e. the modal mass and modal stiffness matrices are transformed into diagonal matrices. 1.46 modal stiffness stiffness element associated with a specified mode of vibration 1.47 modal density number of modes per unit bandwidth Note 1 to entry: Modal density is a measure widely used in structural dynamics as a diagnostic tool in assessing vibration power flow in complex, structural systems. It can play a crucial role in determining changes in vibration power flow that may be a precursor to fatigue failure in some part of the structure, or a metric used in structural condition monitoring evaluations. In addition to these applications, it is a parameter required by the Statistical Energy Analysis method for evaluating the high-frequency response of complex structures and in selecting appropriate vibration-control methods and devices. 1.48 mechanical impedance complex ratio of force to velocity at a specified point and degree-of-freedom in a mechanical system Note 1 to entry: The force and velocity may be taken at the same or different points and degrees-of-freedom in the system undergoing simple harmonic motion. Note 2 to entry: In the case of torsional mechanical impedance, the terms “force” and “velocity” should be replaced by “torque” and “angular velocity”, respectively. Note 3 to entry: In general, the term “impedance” applies to linear systems only. Note 4 to entry: The concept is extended to non-linear systems where the term “incremental impedance” is used to describe a similar quantity. 1.49 direct mechanical impedance driving point mechanical impedance complex ratio of the force to velocity taken at the same point or degree-of-freedom in a mechanical system during simple harmonic motion Note 1 to entry: See the notes under mechanical impedance (1.48). 1.50 transfer (mechanical) impedance complex ratio of the force applied at point i, in a specified degree-of-freedom in a mechanical system, to the velocity at another point j in a specified direction or degree-of-freedom in the same system, during simple harmonic motion Note 1 to entry: See the notes under mechanical impedance (1.48). 1.51 free impedance ratio of the applied excitation complex force to the resulting complex velocity with all other connection points of the system free, i.e. having zero restraining forces Note 1 to entry: Historically, often no distinction has been made between blocked impedance and free impedance. Caution should, therefore, be exercised in interpreting published data. Note 2 to entry: Free impedance is the arithmetic reciprocal of a single element of the mobility matrix. While experimentally determined free impedances could be assembled into a matrix, this matrix would be quite different from the blocked impedance matrix resulting from mathematical modelling of the structure and, therefore, would not conform to the requirements for using mechanical impedance in an overall theoretical analysis of the system. 1.52 blocked impedance impedance at the input when all output degrees of freedom are connected to a load of infinite mechanical impedance Note 1 to entry: Blocked impedance is the frequency-response function formed by the ratio of the phasor of the blocking or driving-point force response at point i, to the phasor of the applied excitation velocity at point j, with all other measurement points on the structure “blocked”, i.e. constrained to have zero velocity. All forces and moments required to fully constrain all points of interest on the structure need to be measured in order to obtain a valid blocked impedance matrix. Note 2 to entry: Any changes in the number of measurement points or their location will change the blocked impedances at all measurement points. Note 3 to entry: The primary usefulness of blocked impedance is in the mathematical modelling of a structure using lumped mass, stiffness and damping elements or finite element techniques. When combining or comparing such mathematical models with experimental mobility data, it is necessary to convert the analytical blocked impedance matrix into a mobility matrix or vice versa. 1.53 frequency-‌response function frequency-dependent ratio of the motion-response Fourier transform to the Fourier transform of the excitation force of a linear system Note 1 to entry: Excitation can be harmonic, random or transient functions of time. The test results obtained with one type of excitation can thus be used for predicting the response of the system to any other type of excitation. Note 2 to entry: Motion may be expressed in terms of velocity, acceleration or displacement; the corresponding frequency-response function designations are mobility, accelerance and dynamic compliance or impedance, effective (i.e. apparent) mass and dynamic stiffness, respectively (see Table 1). 1.54 mobility mechanical mobility complex ratio of the velocity, taken at a point in a mechanical system, to the force, taken at the same or another point in the system Note 1 to entry: Mobility is the ratio of the complex velocity-response at point i to the complex excitation force at point j with all other measurement points on the structure allowed to respond freely without any constraints other than those constraints which represent the normal support of the structure in its intended application. Note 2 to entry: The term “point” designates both a location and a direction. Note 3 to entry: The velocity response can be either translational or rotational, and the excitation force can be either a rectilinear force or a moment. Note 4 to entry: If the velocity response measured is a translational one and if the excitation force applied is a rectilinear one, the units of the mobility term will be m/(N·s) in the SI system. Note 5 to entry: Mechanical mobility is the matrix inverse of mechanical impedance. 1.55 direct (mechanical) mobility driving-‌point (mechanical) mobility complex ratio of velocity and force taken at the same point in a mechanical system Note 1 to entry: Driving-point mobility is the frequency-response function formed by the ratio, in metres per Newton second, of the velocity-response complex amplitude at point j to the excitation force complex amplitude applied at the same point with all other measurement points on the structure allowed to respond freely without any constraints other than those constraints which represent the normal support of the structure in its intended application. 1.56 transfer (mechanical) mobility mechanical mobility where the velocity and the force are considered at different points of the system 1.57 dynamic compliance frequency-dependent ratio of the spectrum, or spectral density, of the displacement to the spectrum, or spectral density, of the force 1.58 dynamic stiffness complex ratio of the force, taken at a point in a mechanical system, to the displacement, taken at the same or another point in the system Note 1 to entry: The terms “dynamic elastic constant” and “dynamic spring constant” are sometimes used. Note 2 to entry: The dynamic stiffness may be dependent upon strain (amplitude and frequency), strain-rate, temperature or other conditions. Note 3 to entry: The dynamic stiffness, k*, of a linear translational single-degree-of-freedom system characterized by the equation m d 2 x d t 2 + c d x d t + kx = F where F = F0 eiωt is equal to k = (F0 + mω2x0 − iωcx0)/ x0 c is the linear (viscous) damping coefficient; e is the base of natural logarithms; F0 is the forceamplitude; i = − 1 ; k is the elastic (spring) constant; m is the mass; t is the time; x is the displacement; x0 is the displacementamplitude; ω is the angular frequency. Table 1 — Equivalent definitions to be used for various kinds of output/input ratios Motion expressedas displacement Motion expressedas velocity Motion expressedas acceleration a Dynamic compliance is also called receptance. b Mobility is sometimes called mechanical admittance. c Accelerance has unfortunately been called inertance in some publications. Inertance is not a standard term and is not acceptable because it is in conflict with the common definition of acoustic inertance and is also contrary to the implication carried by the word inertance. Term Dynamic compliancea Mobilityb Accelerancec Symbol xi/Fj Yij = vi/Fj ai/Fj Unit m/N m/(N·s) m/(N·s2) = kg−1 Boundary conditions Fj = 0; i ≠ j Fj = 0; i ≠ j Fj = 0; i ≠ j See Figure 3 1 2 Comment Boundary conditions are easy to achieve experimentally. Term Dynamic stiffness Blocked impedance Blocked effective mass Symbol Fi/xj Zij = Fi/vj Fi/aj Unit N/m N·s/m N·s2/m = kg Boundary conditions Xj = 0; i ≠ j vj = 0; i ≠ j Aj = 0; i ≠ j Comment Boundary conditions are very difficult or impossible to achieve experimentally. Term Free dynamic stiffness Free impedance Effective (apparent) mass (free effective mass) Symbol Fj/xi Fj/vi = 1/Yij Fj/ai Unit N/m N·s/m N·s2/m = kg Boundary conditions Fj = 0; i ≠ j Fj = 0; i ≠ j Fj = 0; i ≠ j Comment Boundary conditions are easy to achieve, but results shall be used with great caution in system modelling. Figure 1 — Mobility plot a Downwards sloping lines are used for mass. b Upwards sloping lines are used for stiffness. X frequency, in hertz (Hz) Y1 phase angle, in degrees Y2 mobility magnitude, in decibels (dB), [ref. 1 m/(N·s)] Figure 2 — Accelerance magnitude plot corresponding to the mobility graph plotted in Figure 1 a Upwards sloping lines represent stiffness. b Horizontal lines represent mass. X frequency, in hertz (Hz) Y accelerance, in decibels (dB), [ref. 1 m/(N·s2)] Figure 3 — Dynamic compliance magnitude plot corresponding to the mobility graph plotted in Figure 1 a Horizontal lines represent stiffness. b Downwards sloping lines represent mass. X frequency, in hertz (Hz) Y dynamic compliance, in decibels (dB), [ref. 1 m/N] 1.59 dynamic mass complex ratio of force to acceleration 1.60 accelerance frequency-dependent ratio of the spectrum, or spectral density, of the acceleration to the spectrum, or spectral density, of the force 1.61 spectrum description of a quantity as a function of frequency or wavelength 1.62 level (of a quantity) logarithm of the ratio of the quantity to a reference of the same kind Note 1 to entry: The base of the logarithm, the reference quantity and the kind of level shall be specified. Note 2 to entry: Examples of kinds of levels in common use are electric-power level, sound-pressure level, and voltage-squared level. Note 3 to entry: The definition is expressed symbolically as: L = log r q q 0 L is the level of the kind determined by the kind of quantity under consideration, measured in units of logr; r is the base of the logarithms and the reference ratio; q is the quantity under consideration; q0 is the reference quantity of the same kind. Note 4 to entry: A difference in the levels of two like quantities q1 and q2 is described by the same formula because, by the rules of logarithms, the reference quantity is automatically divided out as follows: log r q 1 q 0 − log r q 2 q 0 = log r q 1 q 2 Note 5 to entry: In vibration terminology, the term “level” is sometimes used to denote amplitude, average value, root-mean-square value, or ratios of these values. These uses are deprecated. 1.63 bel unit of level when the base of the logarithm is 10 Note 1 to entry: Use of the bel is restricted to levels of quantities proportional to power. See also the notes under level (1.62) and decibel (1.64). 1.64 decibel dB one tenth of a bel Note 1 to entry: The magnitude of a level in decibels is ten times the logarithm to the base 10 of the ratio of power-like quantities, i.e. L = 10 lg X 2 X 0 2 = 20 lg X X 0 Note 2 to entry: Examples of quantities that qualify as power-like quantities are sound-pressure squared, particle-velocity squared, sound intensity, sound-energy density and voltage squared. Thus, the bel is a unit of sound-pressure-squared level; however, it is common practice to shorten this to sound-pressure level because ordinarily no ambiguity results from so doing.