In theory of
vibration
Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic function, periodic, such as the motion of a pendulum ...
s, Duhamel's integral is a way of calculating the response of
linear system
In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator.
Linear systems typically exhibit features and properties that are much simpler than the nonlinear case.
As a mathematical abstraction o ...
s and
structures
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
to arbitrary time-varying external perturbation.
Introduction
Background
The response of a linear, viscously damped
single-degree of freedom (SDOF) system to a time-varying mechanical excitation ''p''(''t'') is given by the following second-order
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
:
where ''m'' is the (equivalent) mass, ''x'' stands for the amplitude of vibration, ''t'' for time, ''c'' for the viscous damping coefficient, and ''k'' for the
stiffness
Stiffness is the extent to which an object resists deformation in response to an applied force.
The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
Calculations
The stiffness, k, of a b ...
of the system or structure.
If a system initially rests at its
equilibrium position, from where it is acted upon by a unit-impulse at the instance ''t''=0, i.e., ''p''(''t'') in the equation above is a
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
''δ''(''t''),
, then by solving the differential equation one can get a
fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not ad ...
(known as a unit-impulse response function)
:
where
is called the
damping ratio
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples inc ...
of the system,
is the natural
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of the undamped system (when ''c''=0) and
is the
circular frequency when damping effect is taken into account (when
). If the impulse happens at ''t''=''τ'' instead of ''t''=0, i.e.
, the impulse response is
: