Glossary

Amplitude

Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system.

Anharmonicity

Anharmonicity is the deviation of a system from being a harmonic oscillator. An oscillator that is not oscillating in simple harmonic motion is known as an anharmonic oscillator where the system can be approximated to a harmonic oscillator and the anharmonicity can be calculated using perturbation theory. If the anharmonicity is large then other numerical techniques have to be used.

Attenuation

In physics, attenuation (in some contexts also called extinction) is the gradual loss in intensity of any kind of flux through a medium.

Bandwidth

Bandwidth is the difference between the upper and lower frequencies in a contiguous set of frequencies. It is typically measured in hertz, and may sometimes refer to passband bandwidth, sometimes to baseband bandwidth, depending on context. Passband bandwidth is the difference between the upper and lower cutoff frequencies of, for example, an electronic filter, a communication channel, or a signal spectrum.

Black-Nichols Plot

See Nichols Plot.

Bode Plot

A Bode plot is a graph of the transfer function of a linear, time-invariant system versus frequency, plotted with a log-frequency axis, to show the system's frequency response. It is usually a combination of a Bode magnitude plot, expressing the magnitude of the frequency response gain, and a Bode phase plot, expressing the frequency response phase shift.

Control

Control theory is an interdisciplinary branch of engineering and mathematics, that deals with the behavior of dynamic systems. The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain reference over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system.

Control Variable

In Simcenter Amesim, the control variable is the input variable that defines excitation for the linearized system. It is used for example as the input of the transfer functions. After linearization, observer variables are included in the input vector u of the standard state-space representation form: dx/dt = Ax + Bu & y = Cx + Du.

Critically-Damped Oscillator

An oscillator system which is critically damped (? = 1) returns to equilibrium as quickly as possible without oscillating. This is often desired for the damping of systems such as doors.

Damping

In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system.

Damping Coefficient

The damping coefficient is the value in [N/(m/s)] that produces damping. For example, it can be the viscous damping coefficient (usually noted: c or R in [N/(m/s)]).

Damping Ratio

In engineering, the damping ratio is a dimensionless measure ([null] or [%]) describing how oscillations in a system decay after a disturbance. Many systems exhibit oscillatory behavior when they are disturbed from their position of static equilibrium. The damping ratio is a measure of describing how rapidly the oscillations decay from one bounce to the next.

Eigenvalue, Eigenvector, Eigenspace

In mathematics, eigenvalue, eigenvector, and eigenspace are related concepts in the field of linear algebra. Eigenvalues, eigenvectors and eigenspaces are properties of a matrix. They give important information about the matrix, and can be used in matrix factorization. If the action of a matrix on a (nonzero) vector changes its magnitude but not its direction, then the vector is called an eigenvector of that matrix. Each eigenvector is, in effect, multiplied by a scalar, called the eigenvalue corresponding to that eigenvector. The eigenspace corresponding to one eigenvalue of a given matrix is the set of all eigenvectors of the matrix with that eigenvalue.

Fourier Transform

In mathematics, the Fourier Transform (often abbreviated FT) is an operation that transforms one complex-valued function of a real variable into another. In such applications as signal processing, the domain of the original function is typically time and is accordingly called the time domain. The domain of the new function is typically called the frequency domain, and the new function itself is called the frequency domain representation of the original function. It describes which frequencies are present in the original function. The Fourier Transform decomposes a function into oscillatory functions. The term "Fourier Transform" refers both to the frequency domain representation of a function, and to the process or formula that "transforms" one function into the other.

Frequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.

Gain

A gain is defined as the mean ratio of the signal output of a system to the signal input of the same system. It may also be defined on a logarithmic scale, in terms of the decimal logarithm of the same ratio ("dB gain").

Laplace Transform

In physics and engineering, the Laplace transform is used for analysis of linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. In this analysis, the Laplace transform is often interpreted as a transformation from the time-domain, in which inputs and outputs are functions of time, to the frequency-domain, where the same inputs and outputs are functions of complex angular frequency, in radians per unit time. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.

Linear Algebra

Linear algebra is a branch of mathematics concerned with the study of vectors, with families of vectors called vector spaces or linear spaces, and with functions that input one vector and output another, according to certain rules. These functions are called linear transformations and are often represented by matrices. An elementary application of linear algebra is to the solution of a system of linear equations in several unknowns. Nonlinear mathematical models can often be approximated by linear ones.

Linearization

Process to linearize a non-linear system around an operating point. A linear system equivalent to the non-linear system around its operating point is obtained as output of the linearization process. The linearization can be a numerical linearization with small perturbations applied to the system around the operating point (such as in Simcenter Amesim) or a formal linearization with the derivation of the analytical non-linear equations (most of the time, to be done manually, which is quite fastidious).

Logarithmic Decrement

Logarithmic decrement, ?, is used to find the damping ratio of an under-damped system in the time domain. The logarithmic decrement is the natural log of the amplitudes of any two successive peaks.

LTI Systems

Linear Time-Invariant (LTI) systems such as transfer function models, zero/pole/gain models, or state-space models.

Modal Analysis

Modal Analysis is used to the natural mode shapes and frequencies of a system during free vibration.

Modal Shape

Modal shapes (or mode shapes) are the representation of the eigenvectors of a dynamic system for its different eigenvalues. It gives a direct view of the contributions of the observer variables on the shape of a mode of the system (usually characterized by its frequency in Hz).

Nichols Plot

A Nichols plot is a plot used in signal processing in which the logarithm of the magnitude is plotted against the phase of a frequency response on orthogonal axes. This plot combines the two types of Bode plot - magnitude and phase - on a single graph, with frequency as a parameter along the curve. This plot is also used in closed loop control systems which have negative feedback. In these systems the plot is useful for assessing the absolute and relative stability of the system.

Non-Linearity

Real systems have all kinds of non-linearities: dead-band, backlash, Coulomb friction, hysteresis, quantization, saturation, kinematic non-linearities, and many others.

Nyquist Plot

A Nyquist plot is used in automatic control and signal processing for assessing the stability of a system with feedback. It is represented by a graph in polar coordinates in which the gain and phase of a frequency response are plotted. The plot of these phasor quantities shows the phase as the angle and the magnitude as the distance from the origin. This plot combines the two types of Bode plot - magnitude and phase - on a single graph, with frequency as a parameter along the curve.

Observer Variable

In Simcenter Amesim, the control variable is the output variable that defines observation for the linearized system. It is used for example as the output of the modal shapes, or as the output of the transfer functions in combination with control variables (inputs). After linearization, observer variables are included in the output vector y of the standard state-space representation form: dx/dt = Ax + Bu & y = Cx + Du.

Oscillation

Oscillation is the repetitive variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes is used to be synonymous with "oscillation".

Over-Damped Oscillator

An oscillator system which is over-damped (? > 1) returns (exponentially decays) to equilibrium without oscillating. Larger values of the damping ratio ? return to equilibrium slower.

Period

The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency.

Phase

The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0.

Phase Difference

Two oscillators that have the same frequency and different phases have a phase difference, and the oscillators are said to be out of phase with each other. The amount by which such oscillators are out of step with each other can be expressed in degrees from 0° to 360°, or in radians from 0 to 2?. If the phase difference is 180 degrees (? radians), then the two oscillators are said to be in anti-phase. If two interacting waves meet at a point where they are in anti-phase, then destructive interference will occur.

Pole

In the mathematical field of complex analysis, a pole of a meromorphic function is a certain type of singularity that behaves like the singularity of (1 / zn) at z = 0. This means that, in particular, a pole of the function f(z) is a point a such that f(z) approaches infinity as z approaches a.

Pole-Zero Plot

In mathematics, signal processing and control theory, a pole-zero plot is a graphical representation of a rational transfer function in the complex plane which helps to convey certain properties of the system such as stability, region of convergence, minimum phase.

Q-Factor

In physics and engineering, the quality factor or Q-factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency. A higher Q-factor indicates a lower rate of energy loss relative to the stored energy of the oscillator; the oscillations die out more slowly. A pendulum suspended from a high-quality bearing, oscillating in air, has a high Q-factor, while a pendulum immersed in oil has a low one. Oscillators with high quality factors have low damping so that they ring longer. Sinusoidally driven resonators having higher Q factors resonate with greater amplitudes (at the resonant frequency) but have a smaller range of frequencies around that frequency for which they resonate.

Resonance

In physics, resonance is the tendency of a system (usually a linear system) to oscillate with larger amplitude at some frequencies than at others. These are known as the system's resonant frequencies (or resonance frequencies). At these frequencies, even small periodic driving forces can produce large amplitude oscillations. Resonances occur when a system is able to store and easily transfer energy between two or more different storage modes (such as kinetic energy and potential energy in the case of a pendulum). However, there are some losses from cycle to cycle, called damping. When damping is small, the resonant frequency is approximately equal to a natural frequency of the system, which is a frequency of unforced vibrations. The physical systems have multiple, distinct, resonant frequencies.

Root

In mathematics, a root (or a zero) of a real-, complex- or generally vector-valued function ƒ is a member x of the domain of ƒ such that ƒ(x) vanishes at x, that is, x such that f(x)=0. In other words, a "root" of a function ƒ is a value for x that produces a result of zero ("0").

Root Locus

The root locus plot is a real part / imaginary part plot which is used to follow the trajectories of the system eigenvalues when parameters are changed. It can be used in open-loop to easily identify the frequency and damping ratio with change of design parameters, or in closed-loop to visualize the stability of the system.

State-Space representation

State-space models rely on linear differential or difference equations to describe the system dynamics. Continuous-time models are of the form: dx/dt = Ax + Bu & y = Cx + Du, where x is the state vector and u and y are the input and output vectors. Such models may arise from the equations of physics, from state-space identification, or by state-space realization of the system transfer function.

State Variable

A state variable is one of the set of variables that describe the "state" of a dynamic system. Intuitively, the state of a system describes enough about the system to determine its future behavior. In Simcenter Amesim, after linearization, state variables are included in the state vector x of the standard state-space representation form: dx/dt = Ax + Bu & y = Cx + Du.

Transfer Function

A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a (linear time-invariant) system. The transfer function is commonly used in the analysis of single-input single-output filters, for instance. It is mainly used in signal processing, communication theory, and control theory. The term is often used exclusively to refer to linear, time-invariant systems (LTI). Most real systems have non-linear input/output characteristics, but many systems, when operated within nominal parameters have behavior that is close enough to linear that LTI system theory is an acceptable representation of the input/output behavior.

Under-Damped Oscillator

An oscillator system which is under-damped (? < 1) oscillates (with a slightly different frequency than the undamped case) with the amplitude gradually decreasing to zero.

Zero

In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0. An important property of the set of zeros of a holomorphic function of one variable (that is not identically zero) is that the zeros are isolated.