Structural dynamics is a type of
structural analysis
Structural analysis is a branch of Solid Mechanics which uses simplified models for solids like bars, beams and shells for engineering decision making. Its main objective is to determine the effect of loads on the physical structures and their ...
which covers the behavior of a
structure
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 a ...
subjected to
dynamic
Dynamics (from Greek δυναμικός ''dynamikos'' "powerful", from δύναμις ''dynamis'' "power") or dynamic may refer to:
Physics and engineering
* Dynamics (mechanics)
** Aerodynamics, the study of the motion of air
** Analytical dyn ...
(actions having high acceleration) loading. Dynamic loads include people, wind, waves, traffic,
earthquake
An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in intensity, fr ...
s, and blasts. Any structure can be subjected to dynamic loading. Dynamic analysis can be used to find dynamic
displacements, time history, and
modal analysis
Modal analysis is the study of the dynamic properties of systems in the frequency domain. Examples would include measuring the vibration of a car's body when it is attached to a shaker, or the noise pattern in a room when excited by a loudspeak ...
.
Structural analysis is mainly concerned with finding out the behavior of a physical structure when subjected to force. This action can be in the form of
load due to the weight of things such as people, furniture, wind, snow, etc. or some other kind of excitation such as an earthquake, shaking of the ground due to a blast nearby, etc. In essence all these loads are dynamic, including the self-weight of the structure because at some point in time these loads were not there. The distinction is made between the dynamic and the static analysis on the basis of whether the applied action has enough acceleration in comparison to the structure's natural frequency. If a load is applied sufficiently slowly, the inertia forces (
Newton's first law of motion) can be ignored and the analysis can be simplified as static analysis.
A
static load is one which varies very slowly. A dynamic load is one which changes with time fairly quickly in comparison to the structure's natural frequency. If it changes slowly, the structure's response may be determined with static analysis, but if it varies quickly (relative to the structure's ability to respond), the response must be determined with a dynamic analysis.
Dynamic analysis for simple structures can be carried out manually, but for complex structures
finite element analysis
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
can be used to calculate the mode shapes and frequencies.
Displacements
A dynamic load can have a significantly larger effect than a static load of the same magnitude due to the structure's inability to respond quickly to the loading (by deflecting). The increase in the effect of a dynamic load is given by the
dynamic amplification factor (DAF) or dynamic load factor (DLF):
:
where ''u'' is the deflection of the structure due to the applied load.
Graphs of dynamic amplification factors vs non-dimensional
rise time (''t''
''r''/''T'') exist for standard loading functions (for an explanation of rise time, see time history analysis below). Hence the DAF for a given loading can be read from the graph, the static deflection can be easily calculated for simple structures and the dynamic deflection found.
Time history analysis
A history will give the response of a structure over time during and after the application of a load. To find the history of a structure's response, you must solve the structure's
equation of motion
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (V ...
.
Example
A simple single
degree of freedom system
A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purpose and express ...
(a
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
, ''M'', on a
spring
Spring(s) may refer to:
Common uses
* Spring (season), a season of the year
* Spring (device), a mechanical device that stores energy
* Spring (hydrology), a natural source of water
* Spring (mathematics), a geometric surface in the shape of a h ...
of
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 ...
''k'', for example) has the following equation of motion:
:
:
where
is the acceleration (the double
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the displacement) and x is the displacement.
If the loading ''F''(''t'') is a
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argum ...
(the sudden application of a constant load), the solution to the equation of motion is:
: