9

Vibrations and Non-linear Oscillations

English


9.1 ALIASING
Phenomenon, caused by a loss of high frequency content of an analogue signal, which can occur whenever this signal is sampled (at a rate) less than twice the maximum component frequency.
9.2 AMPLITUDE (OF HARMONIC VIBRATION) , COMPLEX
Complex parameter, modulus of which equals the amplitude and the argument equals phase of harmonic vibration.
9.3 ANTIRESONANCE
Minimum amplitude response to a simple harmonic excitation between two consecutive natural frequencies of a system.
9.4 ATTRACTOR
Set of points or a subspace in phase space toward which a time history of motion, for different initial conditions, approaches after transients die out.
9.5 ATTRACTOR, CHAOTIC [STRANGE]
Attractor characterized by the most complex form of bounded posttransient recurrent behaviour with random characteristics observed in a deterministic dissipative dynamical system.
9.6 ATTRACTOR, PERIODIC
Closed orbit in the phase space, of an asymptotically stable time continuous dynamical system, satisfying recurrence by returning precisely to its starting point after its period
9.7 ATTRACTOR, POINT
Stable equilibrium point in the phase space of a dynamical system.
9.8 BANDWIDTH [NOISE], EFFECTIVE
Bandwidth of an ideal band-pass filter that would pass the same amount of power from a white noise source, as the described filter.
9.9 BANDWIDTH, NOMINAL
Spacing between frequencies at which a band-pass filter attenuates the signal by 3 dB.
9.10 BASIN [DOMAIN] OF ATTRACTION
Set of initial conditions in phase space which leads to a particular long-time motion or attractor.
9.11 BIFURCATION
Qualitative change in the topology of the attractor-basin phase portrait (multiple points generated in Poincaré mapping), realisable under the quasistatic variation of a control parameter across its critical value.
9.12 BIFURCATION, GLOBAL
Bifurcation whose effects are not restricted to the neighbourhood of a point or cycle in phase space.
9.13 BIFURCATION, HOPF
Emergence of a limit cycle oscillation from an equilibrium state as some system parameter is varied, characterized by complex conjugate pair of linear eigenvalues whose real part passes through zero.
9.14 BIFURCATION, LOCAL
Bifurcation whose effects are restricted to the neighbourhood of a point or cycle in phase space.
9.15 BIFURCATION, SADDLE-NODE
Bifurcation occuring if a saddle and a node coalesce, characterized by a real linear eigenvalue passing through zero whereby a system is forced to jump dynamically to a distant attractor.
9.16 BOUNDARY, BASIN
Trajectories (in phase space) initialized on a basin boundary flow towards a saddle solution which attracts within the boundary but repels across it.
9.17 CEPSTRUM
Inverse Fourier transform of a logarithmic spectrum.
9.18 CHAOS
Loose generic term for a complex, seemingly irregular behaviour of deterministic dynamical systems characterised by a sensitive dependence on initial conditions and a broad-band noise power spectrum.
9.19 CHARACTERISTIC, AMPLITUDE-FREQUENCY
Dependence of the amplitude of a forced vibration upon the (angular) frequency of a harmonic excitation.
9.20 CHARACTERISTIC, AMPLITUDE-PHASE (-FREQUENCY) [ARGAND’S]
Dependence of the complex amplitude of a harmonic forced vibration upon the angular frequency of a harmonic excitation.
9.21 CHARACTERISTIC, PHASE-FREQUENCY
Dependence of the difference between phases of a harmonic forced vibration and a harmonic excitation upon its angular frequency.
9.22 COEFFICIENT [FACTOR], LOSS
Measure of damping capacity of a system during forced harmonic vibration, expressed by dimensionless ratio of damping energy dissipated per cycle to 2π times strain energy at the maximum displacement.
9.23 CURVE, BACKBONE [SKELETON]
Dependence between amplitude and frequency of free vibration of an undamped non-linear system.
9.24 CURVE, RESONANCE
Plot of a characteristic parameter of the response of an excited system in a certain vicinity of its resonance versus a chosen parameter of the system or excitation. Note: special kind of a resonance curve is e.g. an amplitude-frequency characteristic.
9.25 CYCLE, LIMIT
Closed phase trajectory of a non-linear, non-conservative autonomous system (see also non-linear system). Note: In the dynamical systems literature it also includes forced periodic motions (see also Hopf bifurcation, periodic attractor).
9.26 DAMPING, COULOMB
Dissipation of energy at Coulomb friction.
9.27 DAMPING, HYSTERETIC [RATE-INDEPENDENT LINEAR]
Damping due to hysteretic damping force during forced harmonic vibration of a system, causing loss of energy per cycle independent of excitation frequency.
9.28 DAMPING, NEGATIVE
Damping dependent on velocity, when input of energy into a vibration system takes place.
9.29 DAMPING, PROPORTIONAL
Damping of a multi-degree of freedom linear system when the damping matrix is a linear combination of the inertia and stiffness matrices.
9.30 DAMPING, STRUCTURAL
Damping due to external friction (mostly dry friction) in constraints of a system.
9.31 DENSITY, POWER SPECTRAL
Mean-square value of a time variable quantity per unit bandwidth.
9.32 DIVERGENCE
Instability, typified by exponentially increasing displacement from the original state of a system.
9.33 DOUBLING, PERIOD
Sequence of periodic vibrations, in which the period doubles as some control parameter of the problem is varied. These frequency-halving bifurcations occur at smaller and smaller intervals of the control parameter (cascade of bifurcations).
9.34 EIGENVALUES, LINEAR
Eigenvalues of the linearized or linear system used also to examine stability of the system in the neighbourhood of an equilibrium point.
9.35 ENVELOPE, LIMIT
Envelope to a set of amplitude-frequency characteristics for different values of a system parameter to which the envelope is invariant.
9.36 EQUATION, DUFFING’S
Archetype equation of a driven oscillator with a cubic {polynomial} non-linear restoring force.
9.37 EQUATION, HILL
Linear differential equation of the second order having periodic coefficients.
9.38 EQUATION, MATHIEU
Special case of Hill equation having harmonic coefficients.
9.39 EQUATION, VAN DER POL
Second-order differential equation with linear restoring force and non-linear damping, which exhibits a limit cycle behaviour (self-excited oscillations).
9.40 ERROR, BIAS [SYSTEMATIC]
Systematic deviation of a determined value from the true value due to imperfections of the measuring device, method, environment, operator.
9.41 EXCITATION, EXTRAPARAMETRIC
In a rotating system with variable moment of inertia, special kind and combination of external and parametric excitation, dependent nonlinearly upon unknown both position angle and angular velocity of the system.
9.42 EXCITATION, FORCE {MOMENT}
Excitation of a mechanical system by an excitation force {excitation moment}.
9.43 EXCITATION, KINEMATIC
Excitation of a mechanical system due to a prescribed motion of arbitrary points of the system.
9.44 EXCITATION, PARAMETRIC
Excitation of a mechanical system due to change of system parameters periodically in time, independently of its motion.
9.45 FILTER [WAVE FILTER]
Device or algorithm to transform signal.
9.46 FILTER, BAND-PASS
Filter with a single transmission band extending from non-zero lower cutoff [corner] frequencies to finite upper cutoff [corner] frequencies.
9.47 FILTER, CONSTANT BANDWIDTH
Band-pass filter whose bandwidth (in Hertz) is constant and independent of the arithmetic center frequency.
9.48 FILTER, DIGITAL
Filter which operates on digital data.
9.49 FILTER, HIGH-PASS
Filter with a transmission band starting at a non-zero lower cutoff [corner] frequency and extending to (theoretically) infinite frequency.
9.50 FILTER, LOW-PASS
Filter with a transmission band starting at a finite cutoff [corner] frequency and extending (down) to zero frequency.
9.51 FILTER, PROPORTIONAL [CONSTANT PERCENTAGE] BANDWIDTH
Band-pass filter whose bandwidth is proportional to the geometric center frequency.
9.52 FLUTTER
Self-excited aeroelastic oscillations generated as a consequence of Hopf bifurcation instability.
9.53 FORCE {MOMENT}, EXCITATION
Time variable force {moment} acting upon a mechanical system independently of its state.
9.54 FORCE, HYSTERETIC DAMPING
Internal force within an elasto-dissipative material with magnitude proportional to displacement but with the same direction as velocity of a material element.
9.55 FRACTAL
Geometric property of a set of points in a multidimensional space having the quality of self-similarity at different length scales.
9.56 FREQUENCY, AMPLITUDE RESONANT
Frequency of a forced vibration at which the maximum amplitude of a system response occurs.
9.57 FREQUENCY, ANGULAR [CIRCULAR]
Product of the frequency of a simple harmonic quantity and the factor 2π .
9.58 FREQUENCY, CENTER
Arithmetic {geometric} bandwidth center of a band-pass filter (i.e. the mid point on the respectively linear {logarithmic} scale).
9.59 FREQUENCY, DAMPED NATURAL
Frequency of the free vibration of a damped linear system.
9.60 FREQUENCY, PHASE RESONANT
Frequency of a forced vibration at which phase is 90 degrees.
9.61 INSENSIBILITY
Phenomenon {behaviour} described by a non-linear static characteristic with a dead band.
9.62 INTEGRATOR
Electrical circuit used for converting a (vibratory) acceleration signal to one which is proportional to velocity or displacement.
9.63 INTERMITTENCY
Type of chaotic motion in which long unpredictable time intervals of almost regular, periodic or steady motion are followed by bursts of random like motion.
9.64 ISOLATION, VIBRATION
Reduction in response of a mechanical system to an excitation, achieved by use of a resilient interface.
9.65 ISOLATOR, VIBRATION
Resilient interface designed to attenuate the transmission of vibration in a frequency range.
9.66 LINEARIZATION
Efficient method for solution of non-linear systems in the first approximation, based on replacing the non-linear mathematical equations by the linear ones.
9.67 MAP, SPECTRAL [WATERFALL PLOT], [CASCADE PLOT]
Three-dimensional plot of frequency spectra versus another variable (usually time or machine speed).
9.68 METER, VIBRATION [VIBROMETER]
Instrument for measuring and indicating the magnitude of vibration in terms of displacement, velocity and acceleration.
9.69 MODULUS, DYNAMIC
Ratio of stress to strain during harmonic forced vibration of a linear material.
9.70 MOTION, CHAOTIC
Type of motion that is very sensitive to changes in initial conditions, unpredictable in the range of a chaotic attractor.
9.71 MOTION, GLOBAL
Motion between and among equilibrium points that is not confined to a small region of phase space.
9.72 MOTION, LOCAL
Motion that does not wander far from an equilibrium point.
9.73 NOISE, WHITE
Broadband noise whose energy per unit bandwidth is constant.
9.74 NUMBER, IMPACT
Number of impacts of the system motion during one period of excitation force.
9.75 OCTAVE
Frequency interval between two frequencies with a ratio of two.
9.76 PARAMETER, CONTROL
Parameter which governs a dynamical system and remains constant (or slowly variable) during the motions of the system.
9.77 PHENOMENON, SYNCHRONIZATION
Phenomenon, at which frequency of a self-excited vibration changes, due to the action of external or parametric excitation, near resonance towards the external excitation frequency.
9.78 PLOT [DIAGRAM], BODE
Plot of (logarithmic) gain and phase versus frequency for a transfer function.
9.79 PLOT [DIAGRAM], CAMPBELL [CASCADE PLOT*], [WATER-FALL PLOT*]
Plot of  resonance frequencies versus excitation frequency for a system at polyharmonic excitation (used to check for coincidence of vibration sources).
9.80 POINT, EQUILIBRIUM [FIXED]
1. For time-continuous dynamical systems, point in phase space towards which a solution may approach as transients decay. 2. For time-discrete dynamical systems, a finite set of points where the system visits each point in a sequential manner as a difference equation is iterated.
9.81 POINT, HYPERBOLIC
Equilibrium point of a dynamical system characterized by a phase portrait which is structurally stable against perturbations, having its linear eigenvalues in the stable domains.
9.82 POINT, SADDLE
Unstable equilibrium point not directly observable in a physical system, as it repels trajectories in some phase directions but attracts in others, having real eigenvalues with at least one positive and one negative eigenvalue.
9.83 PORTRAIT, PHASE [STATE]
Set of phase trajectories corresponding to various initial conditions of a dynamical system.
9.84 RANGE, DYNAMIC
Ratio of the largest to the smallest signals that can be measured on a certain device.
9.85 REPELLOR
Unstable steady-state solution {equilibrium, cycle, etc.} that repels all adjacent motions.
9.86 RESONANCE, COMBINATION
Resonance at which excitation frequency is close to a linear combination of natural frequencies of the linearized system.
9.87 RESONANCE, EXTRAPARAMETRIC
Resonance sustained by an extraparametric excitation, manifesting in its pronounced form by two mutually intersecting branches of a resonance curve.
9.88 RESONANCE, INTERNAL
Resonance occurring when some natural frequencies of the linearized system are mutually in a ratio of small integer numbers.
9.89 RESONANCE, PARAMETRIC
Amplitude response exponentially increasing in time occurring in a dynamical system due to the time periodic change of any system parameters. Regions of instability for small periodic changes of the system parameters are related to the natural frequencies of the corresponding conservative system with zero intensity of parametric excitation (single, summation or difference parametric resonances).
9.90 SAMPLING, STROBOSCOPIC
Reading of values of physical quantities of an oscillating system in regular time intervals, at the period of the driving excitation.
9.91 SECTION [MAP], POINCARÉ
Sequence of points in phase space generated by penetration of a continuous evolution trajectory through a generalized surface or plane in the space.
9.92 SELF-EXCITATION
Excitation due to inflow of energy from an internal nonoscillating source generated by the motion of the system.
9.93 SEPARATRIX
Invariant boundary of dimension n(1 that separates regions in an n-dimensional phase space.
9.94 SEVERITY, VIBRATION
Criterion for predicting possible danger relative to specific values and/or parameters which characterize a vibration.
9.95 SEVERITY (OF A MACHINE), VIBRATION
Set of maximum rms values of vibration velocity, measured at significant points of a machine (such as bearings, mountings).
9.96 SHIFT [DIFFERENCE] (OF HARMONIC VIBRATIONS), PHASE
Phase difference of two harmonic vibrations with equal frequencies.
9.97 SIGNATURE (OF A MACHINE)
Signal frequency spectrum which is specific to a particular machine or component, system or subsystem, at specific instant under specific operating conditions.
9.98 SPACE {PLANE}, PHASE [STATE]
Abstract mathematical space {plane} the coordinates of which are the state variables.
9.99 SPECTRUM, AUTO (POWER)
Spectrum the magnitude of which represents power and phase is zero, defined as the Fourier transform of the input, times its complex conjugate.
9.100 SPECTRUM (OF A MACHINE), BASELINE
Vibration spectrum taken when a rotating machine is new or in good working conditions, used as a reference for later analysis.
9.101 SPECTRUM, SHOCK (RESPONSE)
Spectrum of the maximum responses of a series of specified systems (usually linear single degree-of-freedom) to an applied shock as a function of their natural frequencies.
9.102 STATE (OF A DYNAMICAL SYSTEM)
Quantitative description of behaviour of a dynamical system at any time, determined by the values of state variables.
9.103 STATE, RECURRENT
Particular state of a dynamical system when after sufficient time the system returns arbitrary close to this state.
9.104 STIFFNESS, COMPLEX DYNAMIC
Ratio of the complex amplitude of a harmonic excitation force to the complex response amplitude of a simple harmonic vibration.
9.105 THEORY, CATASTROPHE
Theory which studies dependence of the number and type of equilibrium points upon the parameters near their critical values.
9.106 TONES, COMBINATION
In acoustics and vibration, frequencies that appear as a linear combination of two fundamental frequencies.
9.107 TRAJECTORY, PHASE [STATE]
Path in a phase space {plane}.
9.108 TRANSFORM, FOURIER
Reversible integral transformation of a time function into a corresponding frequency function.
9.109 TRANSFORM, DISCRETE FOURIER
Version of the Fourier transform applicable to a finite number of discrete samples.
9.110 TRANSFORM, FAST FOURIER (FFT)
Algorithm for computation of the discrete Fourier transform in a fast and efficient way (from sampled time data to discrete frequency components).
9.111 VARIABLES (OF A DYNAMICAL SYSTEM), STATE
Minimum set of variables which completely describe state of a dynamical system. Note: The state variables can be uniquely determined from the mathematical model of a dynamical system provided that their initial conditions are given.
9.112 VECTOR, STATE
Vector whose components are state variables of a dynamical system.
9.113 VIBRATION, ALMOST-PERIODIC [QUASIPERIODIC]
Vibration close to a periodic one whose harmonic components have nearly commensurable frequencies.
9.114 VIBRATION, COMBINED
Periodic forced vibration with harmonics whose frequencies are rational multiples of frequency of a certain harmonic excitation.
9.115 VIBRATION, DAMPED
Vibration of a mechanical system with presence of damping.
9.116 VIBRATION, DECAYING
Free damped vibration of a mechanical system when peak-to-peak values of its response variable exponencially diminish in time.
9.117 VIBRATION, DIVERGENT
Vibration of a mechanical system with time-increasing peak-to-peak values of the response variable or its derivative.
9.118 VIBRATION, NON-LINEAR
Vibration of a non-linear system.
9.119 VIBRATION, PARAMETRIC
Vibration caused and maintained by a parametric excitation.
9.120 VIBRATION, PERIODIC
Vibration whose amplitude pattern repeats itself after its period.
9.121 VIBRATION, SELF-EXCITED [SELF-INDUCED]
Vibration caused and maintained by a self-excitation.
9.122 VIBRATION, SUBHARMONIC
Periodic forced vibration with harmonics the frequencies of which are fractions of the frequency of a certain harmonic excitation.
9.123 VIBRATION, SUPERHARMONIC [ULTRAHARMONIC]
Periodic forced vibration with harmonics the frequencies of which are integer multiples of frequency of a certain harmonic excitation.
9.124 VIBRATIONS, ANTI-PHASE
Synchronous vibrations having at any time phase differences of 180 degrees.
9.125 VIBRATIONS, IN PHASE
Synchronous vibrations the phases of which are the same at any instant.