Vibration and Shock Handbook Glossary Every so often, a reference book appears that stands apart from all others, destined to become the definitive work in its field. The Vibration and Shock Handbook is just such a reference. From its ambitious scope to its impressive list of contributors, this handbook delivers all of the techniques, tools, instrumentation, and data needed to model, analyze, monitor, modify, and control vibration, shock, noise, and acoustics. Providing convenient, thorough, up-to-date, and authoritative coverage, the editor summarizes important and complex concepts and results into “snapshot” windows to make quick access to this critical information even easier. The Handbook’s nine sections encompass: fundamentals and analytical techniques; computer techniques, tools, and signal analysis; shock and vibration methodologies; instrumentation and testing; vibration suppression, damping, and control; monitoring and diagnosis; seismic vibration and related regulatory issues; system design, application, and control implementation; and acoustics and noise suppression. The book also features an extensive glossary and convenient cross-referencing, plus references at the end of each chapter. Brimming with illustrations, equations, examples, and case studies, the Vibration and Shock Handbook is the most extensive, practical, and comprehensive reference in the field. It is a must-have for anyone, beginner or expert, who is serious about investigating and controlling vibration and acoustics.
Glossary A B vibration Vertical vibration of an aircraft represented by the sum of the vibrations measured at the two sides of the cockpit A B vibration Roll vibration of an aircraft represented by the difference in the vibrations measured at the two sides of the cockpit A-weighted sound level Sound pressure level that has been weighted according to A-weighting networks, commonly used in measurements that relate to human response to noise (dBA) See Sound level Abnormality scaling The process of scaling the level of abnormality in each vibration feature between and to signify the degree of deviation from its normal value Absolute acceleration shock response spectrum This is an SRS for acceleration See Response spectrum (RS) and Shock response spectrum (SRS) Absolute error The absolute value of the difference between the exact solution and the computed solution, or between the true value and the measured value Absolute limit Conditions that could result in catastrophic failure Absorber See Vibration absorber Absorption coefficient Fraction of the incident energy absorbed by a sound surface AC Alternating current AC bridge A bridge circuit that uses an AC excitation It is more stable than a DC bridge G-1 © 2005 by Taylor & Francis Group, LLC G-2 Vibration and Shock Handbook AC motor A motor that uses an AC excitation Single-phase and multiphase, induction, and synchronous varieties are available Acceleration Rate of change of velocity with respect to time along a specified axis (m/sec2 or g) or about an axis (rad/sec2) Second derivative of displacement Gives both magnitude and direction (vector) Accelerogram A recording of the acceleration of the ground during an earthquake Accelerograph An instrument (accelerometer) that records the acceleration of the ground during an earthquake Accelerometer Sensor or transducer measuring acceleration; output (typically voltage) is proportional to acceleration A piezoelectric accelerometer contains one or more piezo-electric crystal elements, which produce electric charges or voltages when stressed in tension, compression, or shear Can be two or three axial The output can be in digital in some cases, with proper transducers Acceptability index The percentage of times an adjustment set will result in the vibration satisfying the specification Accuracy How well a measurement (reading) or a computed result represents the true (correct) value Accuracy of a value and accuracy of an instrument are related because the latter can determine the former Acoustic absorption coefficient A ratio (fraction) expressing the ability of a material to absorb acoustic power Acoustic emission (AE) Sound or ultrasound pulses generated during crack initiation or propagation in materials and coatings from being subjected to stress Acoustic enclosure A rigid box surrounding noise sources to provide noise isolation for machinery or equipment Acoustic holography A method using phase and amplitude information over a closed surface to reconstruct a sound field in the region outside the surface Acoustic impedance Acoustic pressure acting on a surface divided by the volume (or particle) velocity at the surface Assumes a plane wave is falling perpendicular to the surface May assume acoustic material of infinite depth on which the wave is applied Acoustic power The rate of acoustic energy flow emitted or radiated by a specified sound or noise source (W) Acoustic resonance Condition of peak (maximum) response of a fluid-borne system to a periodic driving pressure oscillation Typically, the frequency at which the system tends to oscillate naturally © 2005 by Taylor & Francis Group, LLC Glossary G-3 Across wind response Response perpendicular to the direction of wind Action plan When measured values of noise or vibration violate the local or national regulatory limits, “action plans” shall be developed for correcting of the situation Active bridge A bridge circuit (with four bridges; e.g., Wheatstone bridge) that uses an operational amplifier (an active device, which uses an external power source) Active control See Active vibration control Active damping Controlled damping where the damping force is generated by an actuator (with an external power source) in response to sensing (e.g., active suspensions and magnetorheology) Active device A device (hardware) that needs an external power source to operate Active filter Uses an operational amplifier (an active device), as opposed to a passive filter, which uses only passive devices Active gauge The strain gauge mounted on the deforming member and forms a part of the bridge circuit See Dummy gauge The bridge output is proportional to the change in the active gauge (automatically compensated for ambient changes by the dummy gauges) Active mount An engine mount that uses actuators (with external power) for providing proper suspension and isolation (force and motion) Active resonator absorber See Vibration absorber and Resonator absorber A resonator absorber that utilizes an active device to generate resonance (see Resonance) within the absorber subsection Active suspension A suspension system (of a vehicle) that uses actuators (with external power) with feedback control for providing proper suspension Active vibration control A vibration control method in which an external force (utilizing external power) is applied to control vibrations ADC An analog-to-digital converter, which samples an analog signal at some frequency (sampling rate) and converts the sampled data into a digital form Adiabatic process A process that takes place with no heat transfer with the environment © 2005 by Taylor & Francis Group, LLC G-4 Vibration and Shock Handbook Adjustable suspension element Typically includes variable rate damper and stiffness Aeroelastic model The model that simulates the dynamic properties of buildings to capture the motion-dependent loads Air-borne sound A sound that is transmitted from one structure to another Aliasing distortion The error that results due to sampling (discretization) of an analog signal Along-wind response Response in the direction of wind Amplitude Peak magnitude of variation (displacement, velocity, acceleration, etc.) of each oscillation of vibration The maximum value of a sinusoidal quantity A measure of floor vibration Amplitude-dependent damping Nonlinear form of damping where its value depends on the amplitude but not frequency of motion, e.g., fluid friction or aerodynamic damping or whose magnitude is proportional to the square of the velocity; nonhysteretic internal damping Amplitude ratio The ratio of forced vibration amplitude to the static amplitude Analog device A device with analog circuitry (hardware) but no digital components It deals with analog or timecontinuous signals but not digital signals Analog-to-digital conversion See ADC Anelasticity Property of real solids responsible for internal friction (prefix “an” means “other than”) Anisotropic damping where there are differences depending on directions relative to crystalline axes Angular frequency Frequency of a periodic quantity, measured in radians per unit time Cyclic frequency multiplied by 2p: The usual symbol is v: Angular velocity The rate of change of angular position (i.e., orientation) of an object (radians per second [rad/sec], revolutions per minute [rpm], etc.) Annular clearance The uniform gap between the rotor and stator of a turbine or compressor motor or some rotating machine Antialiasing filter A low-pass filter with its cut-off set near half the sampling rate of the filtered signal in order to minimize the aliasing error © 2005 by Taylor & Francis Group, LLC Glossary G-5 Appendages Additional secondary devices Additional nonstructural elements to the main structure to increase the building’s function to the occupants Asymptotic stability Stability representation where the response naturally decays to zero Asymptotic stability borderline Maximum depth-of-cut below which chatter will not occur for all spindle speeds Attenuate Decrease To make small Attenuation of muffler Decrease in sound power between two points in a muffler system (noise suppressor) Autocorrelation Correlation of a signal with itself, with a phase shift (reference time shift) expressed as a function of the shift Inverse Fourier transform of the power spectrum Autonomous system Self-contained system with internal control (no external control) A system whose parameters are timeinvariant (i.e., not vary with time) Autotuning Automatic tuning (see tuning) of controller parameters to achieve desired performance (e.g., in vibration control) Average tuning iteration number (ATIN) The average number of iterations taken for tuning an aircraft Average value A representative value of a data set The centriod of the data The approximate point at which the sum of moments of all the data values is zero A first-order statistic of random data Mean value Expected value Mathematical expectation Axi-symmetric wave A sound wave generated by a point dipole When propagation of sound wave is described by polar coordinates, its magnitude is independent of the angle about the given axis Axial Along the shaft centerline Along the axis Back e.m.f An electromotive force (e.m.f.) is a voltage generated in a conductor located in a magnetic field when there is a relative motion (in an orthogonal direction) The back e.m.f is the voltage generated in a moving coil that is connected to a voltage source (like motor rotor windings) and located in a magnetic field, where the generated e.m.f opposes the applied voltage Backlash Mechanical looseness in between meshed gear wheels due to unavoidable clearances between meshing teeth © 2005 by Taylor & Francis Group, LLC G-6 Vibration and Shock Handbook Baffled space The idealized boundary condition to model the sound field generated by a compact source embedded in a plane large in size compared to wavelength, e.g., a radiation field generated by a circular vibrating piston mounted flush with the surface Balance A condition where the rotating centerline between bearings coincides with that which defines the center of mass distribution See Static balancing and Dynamic balancing Balance weight adjustments Weights added to or removed from the rotor hub to tune vibrations through changes in blade mass Balance weights are measured in grams or ounces, with positive adjustments representing the addition of weight Balancing A procedure for adjusting the mass distribution of a rotor by adding or removing weight, with the goal of achieving less vibration amplitude at rotational speed See Static balancing and Dynamic balancing Band-pass filter A filter that allows through the signal components within a given range of frequencies, stopping (filtering) those above and below that band Bandstop filter A filter that suppresses a given range of frequencies, transmitting only those above and below that band Also called band-elimination filter Also called a notch filter when the band is very small Bandwidth Operating frequency range of a device The difference between the upper- and lower-cut-off frequencies at which the signal is attenuated by dB The difference between the limiting frequencies within which performance of a device falls within specified limits with respect to some characteristic A measure of speed of response of a device Barkhausen effect Jerkiness in the magnetization curve of a ferrous specimen Base isolation See Isolation An engineering technology that uncouples the motion of a structural system from the ground motion by inserting an isolation layer under the structure Basis Countable set of linearly independent functions or vectors that can represent a general function or a vector space Bathtub curve Curve (based on anecdotal and statistical data) describing the frequency of failures Applicable to an individual machine or population of machines of the same type Beam A structural (vibrating) component that is long and slender and has some flexibility (bending or flexure) Commonly modeled by Euler–Bernoulli equation or Timoshenko equation © 2005 by Taylor & Francis Group, LLC Glossary G-7 Bearing A support for the ends of a shaft, a rotating member, etc May be dry or lubricated Will introduce some damping, but it should be minimized for proper operation Linear bearing is one that provides a guideway for rectilinear motion (rather than rotary motion) Beats The alternating and significant rise and fall of vibration amplitude caused by two sources vibrating at nearly equal frequencies (or engines or rotating machines rotating at nearly equal speeds) Bernoulli –Euler beam A common analytical model (partial differential equation with respect to time and length coordinate) used to analyze slender beams Rotatory inertia and deformation due to shear are neglected in this model (holds for slender beams), unlike the Timoshenko beam model Bicoherence Second-order correlation measure between signals Biorthogonal wavelets Symmetric wavelets with compact support created by two sets of wavelets; one set for decomposition and one set for reconstruction Bisection method A nonlinear root-finding method which repeatedly bisects the interval to find a root For this method to be successful, the initial interval has to be such that the function is positive at one endpoint and negative at the other Blast mitigation Physical defensive measures taken to reduce the damaging effects of explosions from devices introduced by hostile parties Includes both exterior hardening (wall, etc.) and internal measures taken to improve the survival of the protected resources (e.g., using more stable chemicals in a protected process) Bode plot The pair of plots of amplitude versus frequency and phase versus frequency Usually, the amplitude is given in dB (or in log scale) as a nondimensional magnification factor of a transfer-function magnitude The frequency is given in log scale Boom A long, overhanging pole or beam like structure As in a crane or a robot, a boom can be used to position and manipulate objects at a remote location Boundary conditions (BCs) The conditions (force, moment, motion, deformation, etc.) at the boundary locations of an object (e.g., beam, plate, structure) used for solving the governing equations of the object These are conditions expressed with respect to spatial coordinates (space) rather than time Also require initial conditions (ICs) which are conditions at the initial time, for a complete solution of the system equations Boundary layer A thin stationary layer of liquid with a hypothetical boundary next to a solid surface where much of the velocity changes (gradients) occur © 2005 by Taylor & Francis Group, LLC G-8 Vibration and Shock Handbook Bracing Temporary supports of a structure using struts in vertical and lateral directions during seismic rehabilitation of the structure Bridge circuit An electric circuit with four arms having resistances or impedances, and input (source) and an output Where pure resistances are used, we have a Wheatstone bridge Hydraulic circuits also may be represented as bridge circuits, and analyzed using similar techniques Bridge balance The condition (the relation between bridge resistances or impedances) such that the output of the bridge circuit is zero (even when the source provides an excitation) Bridge constant The constant parameter relating a variation in a bridge arm (resistance or impedance) to the corresponding bridge output, starting with a balanced bridge Broadband (wideband) process A random process where most of the energy is in a wideband of frequency Broadband excitation An excitation with a significant range of frequency (almost a white noise spectrum) Buffer A device provided at an isolation bearing which will be hit by the moving plate of an isolation bearing to limit the bearing displacement during extreme earthquakes Bulk modulus Change in volume divided by initial volume gives the bulk strain The bulk stress (or pressure) divided by the bulk strain is the bulk modulus Bump Simple shock that is generally repeated many times when testing, e.g., half-sine, 10g; 16 msec, 3000 bumps (shock) per axis, bumps per second Bump test See Bump An impact test A test carried out on an object (equipment, structure, etc.) using a bump impact Butterworth filter A low-pass filter with many poles (i.e., high order) to provide good roll-off (i.e., sharp cut-off) Calibration Adjusting some parameters or conditions (using knobs, keys, computer interface, etc.) of a device so that its performance (output or reading) is accurate with known engineering units Cantilever A flexible beam fixed at one end (zero displacement and zero slope) and free at the other (zero bending moment and zero shear force) Carrier A sufficiently high-frequency signal that can carry another (transient) signal of lower frequency for maintaining the accuracy of the carried signal during signal transmission and analysis © 2005 by Taylor & Francis Group, LLC Glossary G-9 Carrier frequency Frequency of the carrier signal Carrier signal See Carrier Cartesian coordinates Orthogonal (perpendicular) coordinates in a rectangular coordinate frame with a common origin Cascade connection Connection in series of two or more devices Cascade plots Successive spectra plotted with respect to time and displayed in a three-dimensional manner Also known as Waterfall plots Catastrophic failure Sudden and complete failure of a machine Cannot be analyzed by conventional methods of nonlinear system analysis Causality Cause–effect behavior of a dynamic system The numerator order (or the input order) cannot be greater than the denominator order (or the output order) of the system transfer function This determines the physical realizability of the system Center of gravity The point representing the average position of weight in a body The point of intersection of orthogonal axes, relative to which the first moment of weight is zero Typically the same as the center of mass Denoted by COG or CG Center of mass The point representing the average position of matter (mass) in a body The point of intersection of orthogonal axes, relative to which the first moment of mass is zero Typically the same as the center of gravity, denoted by COG or CG Center of stiffness The point of intersection of orthogonal axes relative to which the first moment of stiffness is zero Centered difference Finite difference approximation that uses the Taylor series expansions on either side of the point in question Centrifugal acceleration Acceleration of a rotating mass (e.g., attached to a string or arm) directed towards the center rotation Equal to v2 r where v is the angular speed of rotation and r is the radius of rotation Centrifugal compressor A compressor (a machine in which air or gas is compressed) by the mechanical action of rotating vanes or impellers The flow is in a radial direction through the impeller, which imparts velocity (kinetic energy) to the fluid Passing through a diffuser section at exit, the kinetic energy is converted into pressure (potential energy) © 2005 by Taylor & Francis Group, LLC G-10 Vibration and Shock Handbook Centrifugal pump See Centrifugal compressor As the impeller of the pump (enclosed in a casing or volute with an inlet and a discharge connection), kinetic energy is imparted to the fluid by the rotating impeller and is converted into pressure at the outlet Centripetal acceleration See Centrifugal acceleration (synonymous) Centroid Same as the center of mass or center of gravity for practical purposes Cepstra Plural of cepstrum Equivalent to spectra, but in the “quefrency domain” rather than the frequency domain Cepstrum The inverse Fourier spectrum of the logarithm of the Fourier spectrum of a signal Change limit Warning levels that provide early warning well in advance of the absolute limit Channel A signal line in a piece of equipment or hardware Both input channels and output channels are important The capability and complexity increase as the number of channels increases Chaos Deterministic form involving the “butterfly effect” (work of E Lorenz) Characteristic equation See Characteristic polynomial Characteristic polynomial The polynomial, when set to zero (characteristic equation) and solved, gives the eigenvalues (or poles or characteristic roots) of a dynamic system The denominator polynomial of the transfer function of a dynamic system The roots are complex in general The imaginary parts give the natural frequencies The real parts give the reciprocal of time constants or damping level Charge amplifier An electronic amplifier used to convert the high-output impedance of a device (e.g., piezoelectric accelerometer) into low-output impedance so that it is compatible with common signal acquisition and readout devices (processors, controllers, displays, recorders, etc.) Chatter An instability in a machine tool, which occurs at high frequencies, causing poor machining quality, increasing tool wear, generating noise, and causing other problems to the machine Chatter frequency Dominant frequency of vibration during the occurrence of chatter (unstable vibration in a machine tool) © 2005 by Taylor & Francis Group, LLC G-60 Vibration and Shock Handbook Significant wave height A statistical description of random waves Average of the highest one third of all waves or four times the variance Consistently very close to the observed wave height Simple beam See Simply supported beam A beam restrained at its end only against vertical movement Simple harmonic motion See Simple harmonic vibration Simple harmonic vibration A motion such that the displacement, velocity and acceleration are pure sinusoidal functions of time Sometimes called “harmonic motion.” Simple oscillator A single-DoF mass –spring–damper system (or an LRC electric circuit), which can produce damped oscillations at one frequency (damped natural frequency) An undamped simple oscillator is a harmonic oscillator, which can produce a periodic oscillation described mathematically by either of the harmonic trigonometric functions (sine or cosine), and has only one frequency component (undamped natural frequency) Simple (or perfect) shock Shock whose signal can be represented exactly in simple mathematical terms Standards specify generally one of the three following: half-sine (approached by a versed sine waveform); terminal peak sawtooth; and rectangular shock (approached by a trapezoid waveform) Simply supported beam A beam pinned (or simply supported) at its ends so that the displacement and bending moment at these ends will be zero Slopes and shear forces will not be zero Sine beat A signal formed by adding two sine signals whose frequencies nearly equal, so as to generate beats Such a signal is used for excitation in some types of vibration testing Sine-dwell A vibration test where a shake applies a sinusoidal excitation of a specific frequency to the test object over a specific duration, changes the frequency, and repeats the dwell in specific frequency steps Sine sweep A vibration test procedure where a harmonic excitation is applied to a test object (using an exciter or shaker) and the frequency of excitation is slowly and continuously varied at a specified rate Single-category-based classifier (SCBC) The pattern classifier that performs abnormality scaling by comparing the value of its feature with its no-fault value Single-degree-of-freedom (single-DoF) system A system with only one coordinate (strictly, one incremental coordinate to include nonholonomic systems) to define completely the configuration of the system at any instant Single plane balancing See Static balancing, which is synonymous © 2005 by Taylor & Francis Group, LLC Glossary G-61 SISO Single-input–single-output system A system with just one input and one output Skeleton Wavelet transform values along a ridge Sliding bearing A type of base isolator that protects a structure from ground excitations by using a sliding frictional interface at the base SNR Signal-to-noise ratio, determined by both electronic and mechanical noise (but not thermal noise due to the fluctuation –dissipation theorem) Soft foot A condition that occurs when one of a machine’s hold-down bolts are not tight enough to resist the dynamic forces exerted by the machine Soft story A story of a building that is more flexible than the other stories so that the stories above this story are isolated from the ground shaking Softened compression stress– strain relationship Viewing the shear action on a membrane element as a two-dimensional problem, the compressive strength of concrete in one direction is reduced by cracking due to tension in the perpendicular direction Softening parameter When a membrane element is subjected to shear, the compressive strength of concrete is reduced to approximately 40 to 60% Soil –structure interaction A dynamic interaction between a structure and its supporting soil, in which dynamic properties of one will affect the response of the other Solenoid A linear (rectilinear) two-position (on– off) actuator A ferromagnetic rod surrounded by a coil of many turns An electric current through the coil will generate a flux that will move the rod, thereby turning on the solenoid A version (linear motor) is able to provide continuous movements Sone Unit of the loudness that is defined as the loudness of a 1000 jHz pure tone with a sound pressure level of 40 dB This relationship is expressed as S ẳ 2LL 240ị=10 ; S (sone) and L (phons) Sound absorbing material Porous materials such as glass fiber blankets used to attenuate sound propagating in ducts to improve sound transmission loss and to reduce standing waves in air spaces Sound bridge Structures such as studs or stud bolts connecting the plates added to a noise source in insulation lagging or the plates of a double wall They can considerably reduce insertion loss of the lagging or transmission loss of the double wall © 2005 by Taylor & Francis Group, LLC G-62 Vibration and Shock Handbook Sound insulation lagging Thin plates separated by airspace or sound absorbing materials added to noise sources such as pipes, ducts and machines to isolate their noise Sound intensity The time-average rate of energy flow per unit area, denoted by the vector I (W/m2) Sound level Sound pressure level that has been weighted according to particular weighting curves: A- B-, C-weighting networks and associated levels The frequency response and decibel conversions from a flat response for each of these weightings are available Sound power level The acoustic power radiated by a source with respect to the international reference of 10212 (watts), expressed in decibels (dB) Sound pressure The difference between the instantaneous pressure at a fixed point in a sound field and the pressure at the same point without the sound (Pa) Alternatively, this may be defined as the ratio (dB) of the instantaneous pressure at a fixed point in a sound field with respect to a reference of £ 1026 (Pa), for which a pure tone of sound is assumed Span The distance between two consecutive supports of a beam, truss, bridge, guideway, etc Spatial coupling Effect of multiple support motions occurring when the attached equipment is mounted at multiple locations of the structure Specific fan noise level Ratio (dB) of sound power generated from a fan due to its pressurized flow with respect to some standard power level Specifications Conditions that must be satisfied by operating or design parameters during operation or design of a product in order to satisfy the objectives of the product Spectral density See Power spectral density Fourier transform of the autocorrelation function of a random signal (random process) It shows how the variance of the system is distributed over frequency It may also be expressed as a “spectral energy density” since the unit of the variance often has the unit of energy Spectrogram A graph of time vs frequency, e.g., the Gabor distribution Spectrum A description of a quantity as a function of frequency or wavelength Fourier transform of a signal is called the Fourier spectrum (or simply a spectrum) © 2005 by Taylor & Francis Group, LLC Glossary G-63 Spherical wave When propagation of sound wave is described by radial (polar) coordinates, the sound field is represented by a “spherical wave,” which is generated by a point monopole Spike energy meter An accelerometer tuned to have its resonant frequency excited by the impacts in a deteriorating bearing, thus giving an early warning of an impending failure Spindle speed selection In machining: a technique where the spindle speed is chosen to avoid or suppress chatter Spindle speed variation In machining: a technique where the spindle speed is continuously adjusted to suppress chatter Spring The basic flexibility element characterized by its stiffness, force, or deflection Stores elastic potential energy In three-dimensional problems, a spring may assume three orthogonal stiffness parameters SRSS Square-root of sum of squares of a set of values SSE See Safe shutdown earthquake Stability Stability of a dynamic system: If the natural (unforced) response of a system decays to zero, then the system is asymptotically stable If for a finite excitation, the system response is bounded, we have bounded-input –bounded-output (BIBO) input Stability of a numerical method: Numerical errors not grow without bound Stability lobe diagram Chatter stability chart in the process parameter space Standardized shock response spectrum See Shock response spectrum (SRS) Standardized or reduced SRS plotted in dimensionless coordinates The abscissa is the product of natural frequency and shock duration The ordinate is the ratio spectrum value or shock pulse amplitude Standoff distance Distance away from a specific structure that determines the amount of damage that would occur if an explosion of a specific size was detonated For standoff threats, the distance away from a potential vantage point with direct line of sight to the asset State space formulation A generalized analytical model for a dynamic system expressed as a minimum set of first-order ordinary differential equations (state equations) The associate variables are the state variables that may include structural displacements and velocities, which will represent the dynamic state of the system Static balancing See Balancing Balancing achieved on a single plane of vibration perpendicular to the axis of rotation © 2005 by Taylor & Francis Group, LLC G-64 Vibration and Shock Handbook Static loading Constant loading of a member Stationary process A random process where the statistics can be considered independent of time Stationary process is used synonymously with weakly stationary process where the first-order statistics such as the mean and the variance are independent of time, and the second-order statistics such as the autocorrelation function depend on the difference in time Statistically determinate A structure is considered statistically determinate if all the support reactions and internal forces in the members can be determined solely by the equations of static equilibrium Stator The stationary structure surrounding the rotating member (rotor) of a rotating electrical or mechanical machine Steady-state error Offset The error in the response of a system under steady-state conditions Steady-state vibration Steady-state vibration exists in a system if the velocity of each particle is a continuous periodic quantity Stiffness Characteristic parameter of a spring element Ratio of change of force (or torque) to the corresponding change in translational (or rotational) deflection of an elastic element, i.e., force required to deflect an elastic element through a unit distance Typical units are Newtons/meter (N/m) Stiffness asymmetry The stiffness (or flexibility) is not symmetrical in two or more orthogonal directions Stiffness law Transmission loss characteristic of a single plate that the transmission loss depends on frequency and equivalent stiffness of the plate below the first-order natural frequency of the plate Stiffness matrix The matrix multiplied by the displacement vector in the Newtonian form of differential equations of motion of a mechanical (vibrating) system, expressed in the vector-matrix form Stochastic process An indexed collection of correlated (in general) random variables Stochastic resonance Counterintuitive increase of SNR when the noise level is raised Only possible in nonlinear systems Stokes law Viscous friction force on a sphere of radius R, given by 6mRv, where m is the fluid’ viscosity Strain Change in length divided by length A nondimensional measure of the physical deformation or deflection resulting from stress (force per unit area) © 2005 by Taylor & Francis Group, LLC Glossary G-65 Strain gauge A miniature resistive element (metal foil or semiconductor) whose resistance changes when stretched (deformed), which can be measured using a Wheatstone bridge circuit When properly mounted on (i.e., bonded to) a structural member, it can measure the local direct strain in the member Stress–strain curve of a steel bar in concrete A steel bar in concrete is stiffened by tensile stress of concrete It relates the average stress to the average strain of a large length of bar crossing several cracks Stretched exponentials Discovered with Leyden jars (capacitors), a multiexponential model used by some to explain features of anelasticity Stribeck damping A sophisticated friction model where the coefficient of friction drops rapidly and then increases gradually (somewhat linearly) as the relative velocity increases from zero Stroke The maximum rectilinear movement that is possible in a device Strouhal frequency Frequency at which the vortices are shed from a fixed cylinder subjected to a constant velocity flow It is proportional to the diameter of the structure and inversely proportional to the stream velocity Structural influence The coefficient that represents the proximity effect of the accelerometer generating a feature of a machine (gearbox) component Structural wall An in-plane wall to reduce the relative interstory distortions of a building caused by seismic-induced motions Structure-based connectionist network (SBCN) In machine monitoring for fault detection, this network takes advantage of the integration capability of neural networks, but avoids the need for supervised training In the SBCN, the network weights are defined according to the “structural” knowledge of the monitored machine (e.g., gearbox) and the type of fault represented by various vibration features The structural influences in the SBCN, which represent the proximity effect of faults on accelerometers, are determined according to the RMS value of the frequency response of a simplified lumped-mass model of the monitored machine (e.g., gearbox) Structure-borne noise Type of noise that is based on the generation and propagation of timewise varying motions and forces in solid bodies, as well as on the associated sound radiation Subharmonic Sinusoidal quantity of a frequency that is an integral submultiple of a fundamental frequency Subharmonic resonancy A phenomenon observed in sliding isolation where the isolated system possesses extra harmonic resonant frequencies in the low-frequency range in addition to the original ones as a result © 2005 by Taylor & Francis Group, LLC G-66 Vibration and Shock Handbook Substructured online hybrid test A test in which only inelastic parts of a structure are experimentally tested, while the remaining elastic parts are modeled in a computer Both parts are interconnected using online communications so that the complete structural responses are obtained Substructuring Modeling, analysis and design development method where subsystems are developed first and assembled together (interconnected) through appropriate connecting elements The analysis is simplified by using interconnection laws and making use of the results for the subsystems Subsystem structure A structure that is regarded as a subsystem in the analyzing process of statistical energy analysis (SEA) Sweep rate The rate at which the frequency is changed in a sine-sweep test Symmetric matrix The corresponding elements of a symmetric matrix about its main diagonal are identical Specifically, aij ¼ aji where a is an element, i is the row number, and j is the column number The transpose does not change the matrix in this case Synchronous averaging Averaging of the vibration signal synchronous with the running speed of the machinery being monitored Synchronous whirling See Whirling The angular velocity of whirling rotation of the rotor geometric center is equal to the rotational speed (spin) of the rotor shaft Tachometer A speed sensor See DC tachometer AC tachometers have a primary coil with an AC signal (carrier signal), a secondary coil whose induced signal is demodulated to determine the speed, and a rotor that is either a permanent magnet of a closed conduction (as in induction motor) Thermal bearings Flexible bearings used in long span bridges for accommodating the changes of deck length due to varying temperature to protect bridge piers from receiving large lateral forces Terminal peak sawtooth shock (TPS) Same as Final peak sawtooth shock (FPS) A simple shock for which the acceleration –time curve has the shape of a triangle where acceleration increases linearly up to a maximum value and then instantly decreases to zero Test factor Where the shock test specification is derived from measurements of the real environment, this is the coefficient applied to mean SRS of the measured shock pulses to take into account the variability of the resistance of the test item and the very small number of specimens tested (often only one specimen) Test response spectrum See TRS Thermoelastic damping Energy dissipation (damping) due to the anharmonic, interatomic potential independent of defects © 2005 by Taylor & Francis Group, LLC Glossary G-67 Threshold The minimum value a stimulus may have to create a desired effect Threshold of hearing This is defined for binaural listening as the lowest sound pressure in free field that one can just hear as the signal reduced Threshold speed The speed at which a stimulus is perceptible or can produce an effect Throttling Reducing the fluid flow in a system by a valve or regulator Time constant The inverse of the real part (disregarding the negative sign) of an eigenvalue Inversely proportional to the speed of response and the level of stability Time-dependent effects Normally referred to material properties that are influenced by motion history Time domain Domain of analysis of a dynamic (vibrating) system where the independent variable is time (t) In this domain, vibration data (and systems) are represented, displayed or analyzed as a function of time Time-domain analysis Analysis of dynamic (vibration) systems and response (motion) results, which are described with respect to time Time-domain simulation Computer simulation of the response of a dynamic system as a function of time Technique to analyze chatter where delay-differential equations defining the machining forces and structural vibrations are numerically simulated in the time domain Time-history analysis The analysis of a response of a dynamic system (e.g., structure) with respect to time Time waveform index A single number representative of a waveform, calculated using a raw vibration signal and used for trending and comparisons TNT equivalent The weight of trinitrotoluene (TNT) that has an equivalent energy output to that of a specified weight of another explosive compound Tool chatter Self-excited vibration between the cutting tool and the workpiece due to a cutting tooth machining over a portion of the workpiece that was machined by a previous tooth of the cutter This causes uneven chip thickness and consequently, excessive self-excited vibrations Tool condition monitoring (TCM) A method by which the condition or state of cutting tools can be monitored © 2005 by Taylor & Francis Group, LLC G-68 Vibration and Shock Handbook Tool condition monitoring system (TCMS) A combination of sensors and algorithms to achieve TCM Tool wear The loss of material on the edges of a cutting tool due to machining Tooth passing frequency Frequency at which the cutting tool teeth pass a given point during a spindle revolution Tooth passing period Time interval for two successive cutting tool teeth to pass a given point during a spindle revolution Tornado A violently rotating column of air in contact with, and extending between, a convective cloud and the surface of the Earth It is the most destructive of all storm-scale atmospheric phenomena They can occur anywhere in the world given the right conditions, especially after landfall of hurricanes Torque whirl A whirling instability that can occur in overhung, long, slender, rotors with high load torque values See Whirling Torsional response Response causing a twisting motion Torsional stiffness Torque applied to a member divided by the resulting angular displacement (twist) of the member Torsional vibrations Oscillatory angular motion causing twisting of a shaft (carrying a rotor) in a rotating machine Tracking analyzer Devise used to record and analyze data from machines with variable speed Transducer A device that senses a physical quantity is called a sensor A transducer is a device that converts the sensed signal (measurand) into a form (typically, an electrical output signal of sufficient value) proportional to the measured variable Often, the terms sensor and transducer are used interchangeably Transfer function The ratio of the output of a dynamic system to its input in the Laplace domain or frequency domain (see Frequency domain and FRF) assuming zero initial conditions Transient-response analysis Sometimes called time-history analysis, this determines the dynamic response of a dynamic system (e.g., structure, machine) under the action of time-varying excitation Excitation forces are explicitly defined in the time domain Computed response usually includes the time-varying displacements, velocities, accelerations, strains, and stresses Transient vibration The vibratory motion of a mechanical system as it changes from one state to another as a function of time This may be a decaying or growing signal but not a steady vibration It may consist of forced or free vibration or both © 2005 by Taylor & Francis Group, LLC Glossary G-69 Transmissibiliy Transmitted vibration (force or motion) divided by an applied vibration (force or motion) Inverse of Isolation Transmission loss The difference between incident and transmitted acoustic intensity levels It is used to rate the sound insulating properties of a wall or some other structure or device For a muffler, this is the ratio between the energy in incident sound wave at the muffler inlet and energy transmitted at the muffler outlet into perfectly absorbing termination Transmitted power An acoustic power that is transmitted from a subsystem to another subsystem Trapezoidal shock Simple shock for which the acceleration –time curve grows linearly up to a given value, remains constant during a certain time, and then decreases linearly to zero Trim tab adjustments Trim tabs are adjustable surfaces on the trailing edge of the rotor blades of a rotating machine (e.g., jet engine) They affect the aerodynamic pitch moment of their foils and consequently, their vibration characteristics Tab adjustments are measured in thousandths of an inch (thou), with positive and negative changes representing upward and downward tabbing, respectively TRS Test-response spectrum The response spectrum of the excitation signal that is applied to a test object during testing Truncation error The error created by deleting a signal after a finite time period to obtain a finite segment before digitally analyzing it This is equivalent to multiplying the signal by a rectangular window The error can be reduced (or ripples in the frequency spectrum of the truncated signal can be suppressed) by using other types of window functions with smoothing at the two ends Tuning The process of adjusting the controller parameters in order to achieve the required performance (e.g., vibration control) Unbalance (or imbalance) A condition when the center of mass of a rotating component is not coincident with the center of rotation Unbalance response The displacement vibratory response of a rotor to its mass imbalance as a function of rotor speed Uncertainty factor Where the shock test specification is derived from measurements of a real environment, this is the coefficient applied to the mean SRS of the measured shock pulses to take into account the variability of the shock environment and the variability of the resistance of the material © 2005 by Taylor & Francis Group, LLC G-70 Vibration and Shock Handbook Uncertainty principle The product of uncertainties of conjugate variables is greater than or equal to a constant, which in quantum mechanics is Planck’s constant Also known by the name Heisenberg Undamped natural frequency The frequency of free vibration resulting from only elastic and inertial forces of the system while neglecting damping Undamped system A system where there is no energy dissipation It has energy storage capability (e.g., inertia elements to store kinetic energy and springs to store elastic potential energy) If there is no external force applied on the system, then the system will experience free steady vibrations Motion of the system are established by an initial disturbance (i.e., initial conditions) Undetermined coefficient method Use a trial solution consisting of the sum of possible set of functions multiplied by unknown coefficients Substitute in the equation to be solved and equate like terms to determine the unknown coefficients Uplift force Force perpendicular to the flow Van der Pol oscilaltor Classic nonlinear electronic oscillator Variable frequency drives A controlling device to vary the frequency of AC supplied to an electric motor in order to vary proportionately the speed of rotation of the motor Variable structure control (VSC) A class of nonlinear switching controllers whose structure consists of different phases Typically associated with a discontinuity function (e.g., signum function) When the system response enters a different phase, a different controller (corresponding to the particular phase) is switched on VCC Voltage to current converter Velocity A vector quantity that specifies the magnitude and direction of the rate of change of displacement with time along a specified axis with respect to a reference frame (m/sec) First derivative of displacement with respect to time It may refer to angular motion as well as to uniaxial (rectilinear) motion Where the reference frame is not inertial, the velocity is often designated “relative velocity.” Velocity change Variation of the velocity during a shock, i.e., integral of the acceleration time-history signal over the shock duration Velocity of sound Velocity of sound wave propagation In air, it is proportional to the square root of ambient pressure and of the inverse of ambient density, or it is proportional to the square root of the absolute temperature For example, in stationary air, the velocity is 340.4 m/sec at 158C © 2005 by Taylor & Francis Group, LLC Glossary G-71 Velocity shock Shock motion characterized by a sudden velocity change of the support (base) of a system Versed-sine (or Haversine) shock Simple shock for which the acceleration –time curve has the form of one period of the curve representative of the function [1 cos( )], with this period starting from the zero value of the function It is thus a signal ranging between two minima of a sine wave VFC Voltage to frequency converter Vibration The oscillating, reciprocating, or other periodic motion of a rigid or elastic body or medium (such as a floor) when its position or state of equilibrium has been changed Vibration absorber A secondary system (usually a mass–spring –damper trio with zero or low damping) to protect the primary device from vibrating Typically, the secondary system vibrates at its natural frequency (resonance) generating an opposing force to suppress vibration in the primary system Vibration absorption Addition of a vibration absorber to protect the primary device from vibrating Vibration analyzer The device that could monitor and analyze the machine conditions using the machine’s vibration signals Vibration control Means of controlling vibration using external or internal forces and loads Vibration features Aspects of vibration that can illuminate the effect of individual faulty components of a machine (e.g., gearbox) on vibration Usually, vibration features are defined as functions of specific frequencies affected by the components Vibration isolation See Isolation Isolating the source of vibration from the concerned system protecting a device from vibration of its point of attachment Motion isolation and force isolation are of interest Vibration measurements For a bladed rotating machine (e.g., jet engine, turbine, compressor), obtained at the frequency of once per blade revolution (1 per rev) by two accelerometers, A and B, attached to the sides of the cockpit The vibration data is vectorially combined into two components: A ỵ B, representing the vertical (heave) vibration of the system (e.g., aircraft), and A B, representing its roll vibration Vibration meter Small, handheld, inexpensive, simple-to-use, self-contained device that gives an overall vibration level reading Vibration mode See Mode shape For a flexible plate, the vibration represents the particular spatial configuration of vibration amplitude (i.e., mode shape) with phase information of the moving elements, and the associated natural frequency The actual vibration of the plate is obtained by the superposition of the © 2005 by Taylor & Francis Group, LLC G-72 Vibration and Shock Handbook distinct vibration modes For continuous (i.e., distributed-parameter) systems (e.g., plate), there are infinite distinct modes, many of which at the high-frequency end can be neglected depending on the frequency range of operation Vibration modeling The process of converting an engineering vibration problem into a mathematical model, whereby the major vibration characteristics of the original problem can be accurately predicted Vibration mounts Devices that are placed at the base (or legs) of a machine or equipment to reduce its vibration problems and specifically, to reduce the vibration forces transmitted from the machine or equipment to the support structure and its environment A key purpose is vibration isolation Inertia blocks may be attached to the machine base and the mounts Vibration response The output of a vibrating system for an arbitrary input Vibration signature A unique frequency response or pattern that corresponds to a particular input or excitation signal Vibration standards National and international vibration standards and guidelines published by recognized professional organizations and regulatory agencies The objective of these publications is to establish and control quality, safety, durability and reliable performance of rotating machinery Viscoelasticity Model of anelasticity where an elastic solid also has fluid like (viscous) properties Viscous damping The dissipation of energy that occurs when a particle in a vibrating system is resisted by a force that has a magnitude proportional to the magnitude of the relative velocity of the particle and a direction opposite to the direction of the particle Volume coefficient of elasticity of air Generally, isothermal conditions prevail for frequencies below 100 Hz, whereas adiabatic conditions prevail above kHz In between these limits, there is a volume coefficient of elasticity (similar to bulk modulus) that varies between 1.0 £ 105 and 1.4 £ 105 N/m2 Volute One of the major components of a centrifugal pump forming the casing around the impeller, which channels the flow exiting the impeller into the outlet of the pump while converting the velocity energy generated by the impeller into pressure energy Vortex Any circular or rotary flow in the atmosphere that possesses vorticity Vulnerability The expected damage given a specified value of a hazard parameter © 2005 by Taylor & Francis Group, LLC Glossary G-73 Wake Region surrounded by the shear layers originating from separation points Wake excitation Excitation caused by the vortices in the wake Waterfall plots Successive spectra plotted with respect to time and displayed in a three-dimensional manner Also known as Cascade plots Wavelength In acoustics, the distance between successive two peaks in the repetitive spatial variation of the pressure or the particle local velocity Wavelet Functions having representations in the time and frequency domains of small support set Wavelet analysis Similar to Fourier analysis, wavelet analysis is the breaking up of a time waveform into shifted and scaled versions of a wavelet A wavelet is a waveform of limited duration with an average value of zero Wavelet transform Representation of a function as a series of wavelets Wear-in failures Failures that occur at the beginning of a machine’s useful life, usually characterized by a relatively high rate of failure Wearout failures Failures that occur as a machine reaches the end of its design life WECPNL Acronym of weighted equivalent continuous perceived noise level It is very close to LAeq in concept and it would not be difficult to change the assessment index from WECPNL to LAeq WECPNL is used for the assessment of aircraft noise Whipping A term used to describe relatively large motion associated with whirling of a rotor Typically, occurs when the whirling speed matches the critical speed of the rotor Whirling Condition when the rotor shaft bends so that the center of mass of the rotor rotates about the fixed axis through the bearings while spinning about the shaft geometric axis See Synchronous whirl White noise process A random process where the energy is equally distributed over the entire frequency domain It is a mathematical simplification rather than a real process Wideband Corresponding to a wide (large) frequency band © 2005 by Taylor & Francis Group, LLC G-74 Vibration and Shock Handbook Wideband process A random process where most of the energy is in a wideband of frequency Wiener–Khinchine relationship Relates the autocorrelation function to the spectral density of a random process Young’s modulus See Modulus of elasticity Zener model Classic description of damping that involves relaxation processes — named after the individual who also contributed to a well-known voltage controlling diode Zero drift See Drift The drift in a device under conditions of zero output Zero period acceleration See ZPA Zero shift Slow variation of the average of a signal due to the fact that high accelerations make the crystal of the accelerometer work in a temporarily nonlinear field This shift can affect the calculation of the shock response spectrum Zeros The roots of the numerator polynomial equation of a transfer function Together with poles, they represent the entire system (transfer function) and govern the system behavior (stability, phase, speed of response, percentage overshot, rise time, etc.) ZPA Zero period acceleration The acceleration shown on a RS or shock SRS at infinite frequency This can be used as a representative parameter of a signal (e.g., earthquake ground motion) for testing and designing a system © 2005 by Taylor & Francis Group, LLC ...G-2 Vibration and Shock Handbook AC motor A motor that uses an AC excitation Single-phase and multiphase, induction, and synchronous varieties are available... The device that generates and applies an excitation to the test object in dynamic (vibration) testing © 2005 by Taylor & Francis Group, LLC G-24 Vibration and Shock Handbook Expansion joint A... occur © 2005 by Taylor & Francis Group, LLC G-8 Vibration and Shock Handbook Bracing Temporary supports of a structure using struts in vertical and lateral directions during seismic rehabilitation