Download Appendix A Glossary

Appendix A
Glossary
Absolute motion - a motion in an inertial space related to a fixed frame of reference.
Acceleration (angular) - a time derivative of angular velocity.
Acceleration (linear) - the rate of change of velocity, the derivative of velocity vector v with respect to time t, the second time derivative of displacement vector u or
of the position x with respect to time.
Amplitude - the greatest deviation of the instantaneous value of a periodic quantity
from its mean, esp. maximum, value of a single harmonic quantity.
Aperiodic motion - a motion that is not periodic, especially in an overdamped vibratory motion. Uni-directional motion towards a position of an equilibrium.
Applied / Constraint forces - the applied forces are forces imposed to the system
by coupling elements (force actuators, springs, dampers) and forces which can be
described by physical laws (gravity force, etc.). The constraint forces are imposed
to the system by kinematic constraint elements (joints, bearings, actuators that prescribe motion of the system).
Belt friction - when a belt passes over a rough pulley, the tensions in the belt on the
two sides of the pulley, say T1 and T2 , will differ. When slip is about to occur, then
T1 T2 f , where T1 > T2, f is the coefficient of friction is the angle of wrap in
radians and is the base of natural logarithm.
Bending moment - see internal stress resultants.
Boundary conditions - spatial conditions imposed on the solution of partial differential equations in time and space. Together with initial conditions (prescribing
values of kinematic quantities at the beginning of the process) they allow determining the arbitrary constants that occur in the solution.
Boundary value problem - finding the solution of a differential equation where
conditions are given at the boundaries of a certain interval or region, e.g. displacements or surface traction forces of a body are prescribed at its supporting points or
at surface.
= e
e
476
APPENDIX A. GLOSSARY
477
Center of curvature - see radius of curvature.
Center of gravity (weight, mass) - see centroid of a discrete or continuous quantity.
Centroid of a continuous quantity - the centroid of a continuous quantity (line,
surface, volume, mass, weight) pointed to by a vector r can be found by a uniform
manner (using line element L, area element A, volume element V mass element
m or weight element G) within the specified region. For example for the centroid
of mass of a body we have r
r m= m, where r is a positional vector
(function of spatial coordinates) within the body .
Centroid of discrete quantities - similarly as for the centroid of a continuous quann
n
tity we could write r
i=1 ri qi = i=1 qi , where qi is the i-th quantity assumed
to be concentrated and fixed to a point with the positional vector ri .
Closed form, solution - if exact solution of a mathematical model can be found
using pure analytical methods, it is called a closed form solution.
Coefficient of kinetic friction - is the ratio of kinetic friction force to the normal
force. The kinetic friction force is the tangential force between two bodies after
motion begins. It is less than the static friction force.
Coefficient of static friction - is the ratio of the limiting friction force to the normal force. The limiting friction force is the tangential force between two bodies
that occurs when the motion is about to happen.
Complementary function - a solution to a differential equation with the zero righthand side.
Computational mechanics - could be explained as a way to solve the problems
of mechanics. It can be viewed as an intersection of laws of mechanics, rules of
numerical mathematics and programming methods. With fast development of computers both numerical and programming methods are being rapidly evolved. This
allows tackling complicated time dependent problems of mechanics with geometrical and material non-linearities.
Constrained particle - a particle, whose motion is limited, constrained, by prescribed conditions.
Constraint - a restriction on the motion of a system.
Constraint elements - mechanical links between bodies and/or the frame, which
kinematically restrict the motion. The forces caused by these elements are called
reaction forces or constraint forces. Typical examples are bearings joints and displacement actuators.
Continuous model, system - see Discrete / Continuous system.
Continuum - is a model that ignores the molecular or atomic structure of matter
and assumes that the properties of an infinitely small particle are the same, regardless of its size.
Continuum mechanics - is concerned with formulation of equations describing
the motion and mechanical and thermal behavior of matter and with the solution
d
d
=P
d
=R d
P
d
R
d
d
APPENDIX A. GLOSSARY
478
of these equations for prescribed initial and boundary conditions. Usually solid
and fluid continua are being distinguished and treated by somewhat different approaches.
Coriolis acceleration - a component of the absolute acceleration of a particle due to
its motion by the relative velocity v with respect to a frame rotating by the angular
velocity ω. It is defined by the cross product ω v.
Couple of forces - consists of two parallel forces equal in magnitude, but oppositely
directed.
Coupling elements - mechanical links between bodies and/or the environment
which impose a force or a torque. They do not restrict the motion kinematically
like constraint elements. The imposed forces are of applied type. Typical examples
are springs, dampers, force actuators. Inertia properties are sometimes neglected.
Cycle - whole sequence of the periodic quantity during one period.
Damped vibration, system see Undamped / Damped vibration, system.
Damping - a process in which energy is dissipated.
Deformable body - a body (mechanical system, structure) which deforms under the
action of an applied force. Within a scope of solid continuum it is a model characterized by a proper constitutive relation between force and displacement and stress
and strain.
Degrees of freedom, dof, (number of) - number of independent generalized coordinates required to uniquely define the configuration of a system at any instant of
time.
Descriptor form - representation of the governing equations of a constrained system as a coupled set of equations of motion plus constraint equations. The number
of dynamical equations exceeds the number of degrees of freedom. The constraint
equations may be expressed in terms of positions and orientations (position level), in
terms of velocities (velocity level) or in terms of accelerations (acceleration level).
Differential - algebraic equation system, DAE - a set of coupled differential and
algebraic equations.
Direction cosines - the components of a unit vector. The individual components
are cosines of the angles made by the vector with the coordinate axes.
Discrete / Continuous model - a large class of real mechanical systems can be
modelled by a finite number of degrees of freedom. A mass particle attached to
a massless spring is an example of discrete model. Other systems, such as those
that include deformable members with continuously distributed mass, have infinite
number of degrees of freedom. A cantilever beam is an example of a continuous
model.
Dispersion - is said to occur when the velocity of propagation of a harmonic wave
motion depends on its frequency.
Dispersive / Nondispersive medium, system - for a system admitting solution in
the form u A ! ei(kx !t) , where the amplitude A ! is a function of frequency
2
= ( )
( )
APPENDIX A. GLOSSARY
479
!, x and t are space and temporal variables respectively, the so-called dispersive
relation ! = ! (k ); k = 2=, with k; being the wave number and wave length
respectively, can be found. If the frequency ! in the dispersive relation is proportional to the wave number k then the system is called nondispersive. Otherwise it is
dispersive. Periodic waves propagate in nondispersive systems by the same velocity
regardless of their frequencies.
Displacement (angular) - a displacement of a rotating radius vector measured by
angular coordinates (usually in degrees or radians).
Displacement (linear) - the difference of two positional vectors of the same particle, say , related to two instants of time, say t
t and t. The displacement is
defined by uA
xA t+t xA txA . The term linear is sometimes used as a
contrast to angular.
Dynamics - is a branch of mechanics where due emphasis is paid to inertial effects
as opposed to statics where such effects are ignored.
Eigenvalue - see standard and generalized eigenvalue problem.
Eigenvector - see standard and generalized eigenvalue problem.
Energy - a scalar quantity that measures the capacity of a body (mechanical system,
structure) for doing work.
Energy (kinetic) - the energy of a body (mechanical system, structure) due solely
to its motion in space.
Energy (potential) - the energy stored in a body (mechanical system, structure) by
virtue of its state. For a particle it is a scalar quantity equal to the work done in a
conservative force field in moving the particle from a given position to a reference
position where the potential energy is conventionally taken to be zero. Gravitational
potential energy or strain energy are good examples.
Equilibrium (mechanical) - a state of a body which is at rest or undergoes a uniform rectilinear motion in an inertial space.
Equilibrium equations - equations expressing static equilibrium conditions of a
body (mechanical system, structure). They state that a body (mechanical system,
structure) is in equilibrium if the resultant of all forces and moments is equal to
zero.
Equilibrium of system of forces - the system of forces is in equilibrium if the resultant of all its forces and moments is equal to zero.
Excitation - a time dependent external force whereby energy is imparted to the system.
Finite element method - an approximate numerical method for modelling and solving the engineering problems of mechanics. The system being solved is replaced
by means of small interconnected parts (elements) whose stiffness, damping and
inertia properties are expressed in matrix form. Variational principles applied on
governing equations of continuum mechanics are used for deriving both element
and global matrices. Applying the initial and boundary conditions the statics, steady
A
=
=
+
APPENDIX A. GLOSSARY
480
state, transient and other problems can be solved.
Force - can only be defined by means of its effects. It is the cause of the change of
the velocity of a body and/or the cause of its deformation.
Force (axial) - a force acting along a specified axis.
Force (friction) - a force tangential to the sliding surfaces of bodies opposing the
motion of one body against the other.
Force (normal) - see internal stress resultants.
Force (shear) - a force causing a shear deformation. A body (its element) is deformed in such a way that some of its parts are moved parallel to a fixed plane thus
changing its original shape.
Force (shearing force, sometimes only shear) - see internal stress resultants.
Forced vibration - vibration of a system caused by a sustained excitation.
Forward / Inverse problem - the forward dynamics deals with the determination of
the motion of the system that is subjected to prescribed applied forces and torques.
The inverse dynamics deals with the determination of the applied and constraint
forces and torques for a mechanical system whose motion is prescribed.
Free vibration - vibration over an interval of time during which the system is free
from excitation.
Free body diagram - is a tool for establishing the governing equations of a mechanical system by replacing the effects of other systems and/or constraints by idealized set of forces acting on the system under investigation. The system is usually
sketched in an arbitrarily displaced position. It is advantageous to set the positive
sense of assumed displacements, velocities, accelerations and forces with a positive
sense of the chosen coordinate system.
Frequency (angular) - is the cyclic frequency multiplied by .
Frequency (fundamental) - lowest of the set of frequencies associated with the
harmonic components of a periodic quantity.
Frequency (natural), also eigenfrequency - frequency of free simple harmonic vibration of an undamped linear system.
Frequency (sometimes cyclic frequency) - for a periodic motion, the number of
times the same event occurs per unit of time. The inverse value of frequency is
called period.
Frequency equation - the equation from which natural frequencies could be determined. The frequency equation is a result of the substitution of an assumed solution
into the equation of motion of a body free of external loading.
Frequency ratio - a ratio of the excitation frequency to natural frequency.
General solution - the general solution of a linear differential equation usually consists of a complementary function and a particular integral.
Generalized coordinates - length or angular coordinates (or other quantities)
uniquely defining the position of the system at any time.
Generalized eigenvalue problem, generalized eigenproblem - is a mathemati-
2
APPENDIX A. GLOSSARY
481
cal task that for given matrices K; M and for any non-zero vector x related by
K M x 0 requires to find matrices ; X such that KX MX. If, in the
linear vibration problem, K; M are stiffness and mass matrices respectively, then is a diagonal matrix of eigenvalues i (i.e. the squares of natural frequencies i2 )
and X is a corresponding matrix of eigenvectors x(i ) (also called natural modes of
vibration) stored columnwise.
Generalized forces - may be forces or moments (or other suitable quantities) energetically associated to generalized coordinates.
Group velocity - the velocity at which a wave packet travels in a particular medium.
If the medium exhibits dispersion, the group velocity differs from the phase velocity.
Harmonic motion - an oscillatory motion described by sine or cosine functions of
time.
Hinge - a kinematic pair that allows only rotary motion between two links.
Holonomic constraints a) A relation between coordinates describing position and
orientation of elements. They restrict displacements and rotations of the system. b)
Relations containing time derivatives of coordinates (velocities) are holonomic too,
if they can be integrated and transformed into the form a) without integrating the
whole system.
Ideal constraints - constraint forces caused by ideal constraints do not produce
work. That means, constraint forces are always orthogonal to any admissible displacement.
Inertia - this notion may be best explained by a story of young Richard Feynman
and his dad, as presented when R. Feynman was already a Nobel prize winner.
Young Richard Feynman asked his dad: ’When I pull the wagon the ball rolls to the
back of wagon, and when I am pulling it along and I suddenly stop, the ball rolls to
the front of wagon - why is that?’ And his dad replied: ’That nobody knows. The
general principle is that things that are moving try to keep moving and things that
are standing still tend to stand still unless you push on them hard. This tendency is
called inertia but nobody knows why it’s true.’
Inertial space - a space in which the first Newtont’s law applies.
Initial conditions - conditions (at the instant of time taken as origin) imposed on
the solution of differential equations allowing to determine the arbitrary constants
in the solution. See also initial value problem.
Initial value problem - problem of obtaining the solution of a differential equation
that satisfies prescribed initial condition, e.g. values at the begining of a motion.
This solution represents a time evolution of the system.
Instantaneous center of rotation - is a conceptual point of a moving body, which
is instantaneously at rest. This term is usually associated with a body in planar motion and it provides a means of analyzing for example the velocities of mechanism’s
members.
(
) =
=
APPENDIX A. GLOSSARY
482
Instantaneous motion - a motion happening right now. The term instantaneous applies to other quantities as well. It is the concept based on the limit approach when
t ! . For example the instantaneous velocity is v
t!0 xt as a contrast
to average velocity which is x= t.
Internal (sometimes cross-sectional) stress resultants - when a slender body (bar,
beam, shaft, strut, truss) is cut by a plane and separated into two parts, the crosssectional resultants could be imagined occurring at the centroid of the cut-face (section) of one part such that the resultants represent all the forces acting from the
other part. The cross sectional resultants are the normal force, shearing force, bending moment and torsional moment. The normal force at a given section is equal to
the sum of all forces acting normal to that section. The shearing force at a given
section is equal to the sum of all forces acting parallel to that section. The bending
moment at a given section is the algebraic sum of all moments of normal forces
acting on one side of that section with respect to its centroid. The analogous sum of
moments of tangential forces gives torsional moment.
Internal forces - see internal (sometimes cross-sectional) stress resultants.
Joint, also hinge - the physical realization of a kinematic pair, an idealized connection of structural elements.
Kinematic pair - idealization of a physical joint that is concerned only with the
type of constraint that the joint offers.
Kinematically determined system - system constraints determine the motion of
all members of the system uniquely, the motion of the system is fully prescribed.
The system has zero degrees of freedom.
Kinematically indetermined system - the motion of the system is not fully prescribed. The system members can carry out particular motion compatible with the
imposed constraints. If there are n independent "modes" of motion (fundamental
types of motion in accordance with the constraints) the system has n degrees of
freedom.
Kinematics - is concerned with study of motion of bodies (mechanical systems,
structures) without regard to the forces causing the motion.
Lagrange’s formalism - deriving the equations of motion via Lagrange’s equations
of second kind using the kinetic and potential energy of a system. Ideal constraint
forces do not appear in the equations. See also Lagrange’s equations.
Lagrange’s equations - a tool for deriving equation of motion of a mechanical
system based on kinetic and potential energy balance, and on the concept of generalized coordinates and velocities. For a system with n degrees of freedom they
@T
@V
can be stated in the form ddt @@Tq_j
Qj , j ; ; : : :; n where T is
@qj
@qj
kinetic energy, V is potential energy, Qj represents generalized forces and qj are
generalized coordinates. The time derivative is indicated by a dot.
Linear / Nonlinear system, response - If all basic components of a system behave
linearly, the response of the system is known as linear vibration. This means that
0
= lim
( )
+
=
=12
APPENDIX A. GLOSSARY
483
there is a linear relation between displacements and forces, velocities and damping
forces, etc. Linear systems are based on assumptions of small strains and small displacements. Equations of equilibrium are written for an undeformed configuration
of the structure and the principle of superposition can safely be used. The resulting
system of differential equations is of linear nature as well. On the other hand if any
of the basic components behave nonlinearly, the vibration is nonlinear. See also
linear problems.
Linear problems - unknown variable in governing equations of static or dynamic
problems appears only in linear form. The magnitude of the response of linear
systems is proportional to the magnitude of its excitation and the principle of superposition holds. See also linear/nonlinear system, response.
Linearization - performing series expansion of nonlinear terms in an equation and
neglecting terms of quadratic and higher order. Note: If a linearized variable has to
be differentiated during the subsequent calculations, terms of higher order have to
be regarded in the series expansion.
Link - a connecting element (member) with two joints in a mechanism.
Magnification factor - the ratio of dynamic to static amplitude of motion under the
forced vibration conditions. Also called amplification ratio or amplitude ratio.
Mathematical model - model of reality expressed by equations.
Mechanical model - image of a real system were the real parts are represented by
structural elements with particular idealized properties.
Mechanics - the branch of physics concerned with the analysis of behavior of objects under the action of forces. Historically the subject dealt with rigid bodies only,
since then it is extended and applied to problems of solid and fluid continuum. In
the latter scope it is sometimes called classical mechanics as a contrast to quantum
mechanics.
Mechanism - a constrained mechanical system that has at least one degree of freedom.
Method of joints - a method for the stress analysis in truss structures. Conditions
of equilibrium are written for successively isolated joints.
Method of sections - a method for the stress analysis in planar truss structures.
Conditions of equilibrium are written for a part of the structure, which is cut by an
imaginary section through three truss members.
Minimal form - representation of the governing equations of a constrained system
with n degrees of freedom as a n-dimensional set of dynamical equations called
equations of motion. These equations are obtained by introduction of generalized
coordinates and elimination of constraint forces.
Mode of vibration also natural mode of vibration, eigenmode - any oscillatory
response of a linear mechanical system can be expressed as a superimposition of all
modes of vibration. The modes of vibration can be obtained by solving the generalized eigenvalue problem.
APPENDIX A. GLOSSARY
484
Model, modelling - The phenomena of nature are inherently complex and it is impossible to consider every detail when trying to describe, formulate and predict their
future behaviour or occurrence. Always certain simplifications have to be accepted.
The idea behind modelling is to neglect what seems to have a negligible or small
influence on what is to be grasped. Accepting simplifying assumptions, however,
leads to limited validity of models.
Moment of force - a measure of the ability of a force to rotate a body on which it
acts.
Moment of force about a point - vector product of a radius vector from the point
to the line of action of the force and the force itself.
Moment of force about an axis - component along a given axis of the moment of
a force about any point on the axis.
Multibody dynamics - sub-domain of dynamics that deals with dynamics of multibody systems (see: Multibody system). Classical multibody dynamics deals with
multibody systems of rigid bodies while flexible multibody dynamics deals with the
systems that contain flexible bodies as well.
Multibody system - mechanical system of interconnected rigid or deformable bodies that may undergo large displacements and rotations.
Newton-Euler formalism - procedure to obtain the equations governing the dynamical behaviour of a mechanical system. All coupling and all constraint elements
on each body are replaced by forces and torques to formulate Newton‘s and Euler‘s
equations for each body leading to Newton-Euler equations containing applied and
constraint forces. Elimination of constraint forces and reduction to generalized coordinates lead to equations of motion (synthetic procedure).
Newtonian mechanics - a scope of Newtonian physics. The Newtonian mechanics
admits the notion of absolute time independent of space. The Newtonian description is unchanged under a Galileo transformation which is a mathematical device
relating a single phenomenon recorded by two observers whose frame of reference
differ by virtue of their travelling at different uniform velocities. According to Newton’s concept, while the position of two observers differ by virtue of their relative
motion, both have an identical perception of time, which does not depend on the
frame of reference. Newtonian mechanics provides a vital tool, which still perfectly
works in all manners of ways from the motion of billiard balls to galaxy formation.
The Newtonian concept is deterministic. If positions, velocities and masses of various bodies are given at one time, together with history of excitation effects, then
their positions, velocities and accelerations are mathematically determined for all
later times. And for earlier as well.
Newton’s laws - Isaac Newton was born in Woolsthorpe, on Christmas Day 1642,
the same year Galileo died. Newton’s Principia Mathematica was written between
1684 and 1687. The introduction contains the laws of motion. They read as follows.
Newton’s law - first - A body continues in its state of rest or uniform rectilinear
APPENDIX A. GLOSSARY
485
motion unless it is compelled to change that state by forces impressed on it.
Newton’s law - second - The change of motion is proportional to the motive force
impressed; and is made in the direction of the right line in which that force is impressed.
Newton’s law - third - To every action there is always opposed an equal reaction;
or the mutual actions of two bodies upon each other are always equal, and directed
to opposite parts.
Nonholonomic constraints - constraint relations that are not holonomic. See holonomic constraints.
Nonlinear system, response - see Linear / Nonlinear system, response.
Numerical methods - algorithms to solve equations (algebraical, transcendental,
differential). The solution is not in the form of an analytical function but is given
by a set of discrete numbers.
Numerical solution is a solution of the established mathematical model that is obtained by utilising numerical procedures.
Ordinary differential equations, system of ODE - equations in which derivatives
of unknown variables with respect to a single variable appear. The highest order
of derivative defines the order of the ODE. In dynamics of rigid bodies derivatives
are usually with respect to time. In statics of deformable bodies derivatives with respect to space variables occur. In dynamics of deformable bodies the variables can
depend on time and space simultaneously leading to partial differential equations.
Orthogonal - describing a set of vectors in which every pair of vectors is mutually
perpendicular. The property can readily be extended to matrices, tensors, functions,
etc. For an orthogonal matrix it holds that A 1 AT .
Orthogonal transformation - a transformation procedure which, when applied to
vectors, matrices or tensors, satisfies that the transformed orthogonal entities remain
orthogonal.
Oscillation - a variation of a physical quantity (say displacement, voltage), usually
with time, about its mean value.
Oscillatory motion - mechanical oscillation, see also vibration.
Parallelogram law for addition of vectors - a combined effect of two vectors is
given by the vector defined by the diagonal of the parallelogram with sides representing the two vectors.
Partial differential equations - unknowns depend on more than one independent
variables, the appearing derivatives are partial ones.
Particular integral - a solution to a differential equation for a particular right hand
side. Together with a complementary integral it forms a general solution.
Period - is the time taken to complete one cycle of the periodic motion.
Periodic motion - an oscillatory motion having the same pattern after a definite
time interval (period) or after its integer multiples.
Phase - instantaneous argument of a simple harmonic function.
=
APPENDIX A. GLOSSARY
486
Phase characteristics - the relation describing the phase shift as a function of frequency.
Phase shift or Phase lag - for a vibrating system it is an angular difference of the
displacement with respect to the excitation force.
Phase velocity - the rate at which a point of constant phase in a wave travels through
a medium.
Principle of transmissibility - states that a sliding vector being moved along its
line of action has the same effects concerning the equilibrium considerations. The
principle is valid within the scope of rigid body mechanics only.
Principle of virtual work - states that the mechanical work done by internal forces
(moments, stresses) is equal to mechanical work done by external forces if a body
(mechanical system, structure) is subjected to virtual displacements. Virtual admissible displacements are small conceptual displacements that are in agreement
with constraints. During virtual displacements the time is ’frozen’, i.e., forces and
stresses do not change. In rigid bodies the work done by stresses is equal to zero.
Also, in ideal constraints the work done by reaction forces equals zero.
Pure rolling - a contact motion of two bodies with no sliding. At a given moment
the contact point is the instantaneous center of rotation.
Radius of curvature - radius of a circle going through a point of a curve and having
a contact with that curve at this point at least of second order (three-point contact).
The center of the circle is called center of curvature.
Random vibration - vibration whose magnitude cannot be precisely predicted for
any given instant of time.
Relative motion - a motion related to a ’movable’ frame of reference.
Resonance - is a state of vibration occurring if the frequency of external loading
coincides with one of the natural frequencies. As a result the system undergoes
dangerously large oscillations.
Rheonomic constraints - constraints which are dependent on time, which means
that time appears explicitly in constraints equations.
Rigid body (mechanics) - within the scope of solid continuum it is a model characterized by infinite stiffness of matter. By definition a rigid body perfectly resists
deformation due to the action of forces, as such it cannot be deformed.
Rolling motion - relative angular displacement about a common tangent of two
bodies in contact.
Rolling resistance (friction) - resistance to motion that occurs when one deformable body rolls on another.
Scleronomic constraints - time does not appear explicitly in constraint equations.
Skew-symmetric matrix - is an antisymmetric matrix. For such a matrix A
AT .
Sliding - a contact of two bodies with different velocities at the contact point.
Stabilization procedure - the procedure that is included in dynamic simulation nu-
=
APPENDIX A. GLOSSARY
487
merical algorithms that assures minimization of violation of the system constraints
during integration of governing equations formulated in descriptor form.
Standard eigenvalue problem - is a mathematical task, which for given A and any
non-zero vector x in A I x 0 finds the diagonal matrix of eigenvalues i
and the full matrix X of eigenvectors x(i) such that AX X is satisfied.
State space form - governing equations of mechanical systems usually are differential equations of the second order. For numerical integration they are sometimes
transformed in a set of first order equations introducing the state vector. It contains
positions and velocities of the system.
Static deflection - is a deflection due to the statically applied load.
Statically determined system - Internal : a system for which the distribution of
internal forces is determined by equilibrium equations alone. External : a system
for which the distribution of reaction forces is determined by equilibrium equations
alone.
Statically indetermined system - Internal : a system for which the distribution
of internal forces depends on material properties of the members of the system.
External: a system for which the distribution of reaction forces depends on material properties of the members of the system.
Steady-state vibration - continuing periodic vibration.
Stiff system - different definitions can be found. Stiff equations are equations where
certain implicit methods perform better, usually tremendously better, than explicit
ones. This pragmatic opinion is still valid. A useful indicator for stiffness of linear systems is e.g. if the norms of the eigenvalues of the system are very different.
Mechanical systems can become numerically stiff because of their mechanical properties (the natural frequencies are far apart from each other) or due to discretization
properties of applied numerical procedure (e.g. discretization using finite element
method).
Stiffness - a measure of body’s resistance to deformation.
Stiffness (coefficient, also spring constant) - change of force (or moment) divided
by the corresponding translational (or rotational) displacement of an elastic element.
Structural dynamics - it deals with deformable mechanical structures. Its elements
generally do not carry out large translations or rotations. Properties of mass and
compliance are distributed along the structure.
Structure - a mechanical structure is implied. It is an assemblage of engineering
design elements as bars, trusses, beams, plates, shells, etc.
System - a mechanical system is implied. Assemblage of bodies and/or structural
elements.
Thin beam (slender beam) - a structural element whose length is considerably
larger than its cross sectional dimensions. It is a model usually based on following
assumptions: material is homogeneous and elastic, small deformations and strains
are assumed, central axis of the beam does not undergo any extension during its de-
(
) =
=
APPENDIX A. GLOSSARY
488
formation, cross sections remain planar and perpendicular to the neutral axis during
the deformation. Under these assumptions the model is known as the BernoulliEuler model of the beam.
Torque - a moment of force applied to driving link or supplied by the output link of
a mechanism. See also twist.
Transient vibration - vibratory motion of a system other than steady state.
Twist (also torsion) - relative rotation of two cross-sections of a bar or shaft about
its longitudinal axis. State suffered by a bar or shaft as a result of an axial torque
applied to it.
Undamped / Damped vibration, system - The vibration is known to be undamped
if no energy is lost or dissipated during oscillations. If any energy is lost in this way,
it is called damped vibration. These properties are often attributed to a system as
well.
Unilateral constraint - relation between coordinates restricting the motion of a system described by an inequality, e.g. contact problem with non-penetrating bodies.
Varignon’s theorem - states that the moment of a force about any point is equal to
algebraic sum of moments of the components of the force about the point.
Vector - is a quantity possessing both magnitude and direction.
Vector (bound or fixed) - vector fixed to a point. If the principle of transmissibility
can be applied, then the vector may be applied at any point along its line of action
(sliding vector). External effects of a sliding vector remain the same.
Vector (free) - may be moved anywhere in space provided it maintains the same
direction and magnitude.
Vector (radius) - a vector from origin to a point, particle.
Vector (unit) - is vector one unit in length.
Velocity (angular) - a time derivative of angular displacement.
Velocity (linear) - the rate of change of position vector or displacement vector, the
time derivative of these vectors with respect to time, i.e. v
t!0 xt ddxt
u
d
u . See displacement.
t!0 t dt
Vibration - a subset of dynamics of rigid and deformable bodies dedicated to study
of oscillatory motions. Also an oscillatory motion a body (mechanical system,
structure) about the equilibrium state (position).
Vibration (harmonic) - the quantities describing the vibratory motion of a body
(mechanical system, structure) are sine and/or cosine functions of time.
Vibration (periodic) - a body (mechanical system, structure) is repeatedly in the
same state after the same amount of time called period.
Virtual displacements - see principle of virtual work.
Wave equation - a partial differential equation describing the propagation of a disturbance in space and time.
Wave length - the distance between two successive peaks of a periodic wave (or
more generally, between any two points having the same phase).
lim
=
= lim
=
=
APPENDIX A. GLOSSARY
489
2
Wave number - the reciprocal of the wave length, sometimes multiplied by .
Work, mechanical work - a scalar product of force and displacement vectors.