Structural dynamics is a branch of structural analysis which covers the behavior of a structure subjected to dynamic loading. Dynamic loading is any time-varying loading which changes quickly enough that the response of the structure differs from the response to the same loading applied statically. Causes of dynamic loading include people, wind, waves, traffic, earthquakes, and blasts. Dynamic analysis can be used to find dynamic displacements, time history, and natural frequencies and mode shapes. Whether a given load should be treated as static or dynamic depends on how quickly the load varies 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 analytically, but for complex structures finite element analysis is more often used to calculate the mode shapes and frequencies.
Applications Structural dynamics is applied in a number of engineering fields, including
Earthquake engineering Wind engineering Coastal engineering Human-structure interaction in structural engineering.
Dynamic loading Structural analysis is mainly concerned with finding out the behavior (forces or displacements) 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. All loads are dynamic in the literal sense, because at some point in time they were not present. 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. Dynamic loads on a structure can be categorized as periodic or non-periodic. Periodic loads may be simple harmonic, as in the case of a rotating machine with an unbalanced flywheel (a familiar example is a washing machine operating at a steady speed), or they may be more complex but representable by a Fourier series. Non-periodic loads include impulsive (very short duration) loading caused by blasts or impacts, and longer duration loads including earthquakes and wind. In the case of random excitation, the amplitude-time history of the load and the structural response are defined in terms of statistical distributions.
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):
DAF = DLF = u max u static {\displaystyle {\text{DAF}}={\text{DLF}}={\frac {u_{\max }}{u_{\text{static}}}}}
where u is the deflection of the structure due to the applied load. Graphs of dynamic amplification factors vs non-dimensional rise time (tr/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 full time history will give the response of a structure over time during and after the application of a load. To find the full time history of a structure's response, you must solve the structure's equation of motion.
Example
A simple single degree of freedom system (a mass, M, on a spring of stiffness k, for example) has the following equation of motion:
M x ¨ + k x = F ( t ) {\displaystyle M{\ddot {x}}+kx=F(t)}
where x ¨ {\displaystyle {\ddot {x}}} is the acceleration (the double derivative of the displacement) and x is the displacement. If the loading F(t) is a Heaviside step function (the sudden application of a constant load), the solution to the equation of motion is:
x = F 0 k [ 1 − cos ( ω t ) ] {\displaystyle x={\frac {F_{0}}{k}}[1-\cos(\omega t)]}
where ω = k M {\displaystyle \omega ={\sqrt {\frac {k}{M}}}} and the fundamental natural frequency, f = ω 2 π {\displaystyle f={\frac {\omega }{2\pi }}} . The static deflection of a single degree of freedom system is:
x static = F 0 k {\displaystyle x_{\text{static}}={\frac {F_{0}}{k}}}
so we can write, by combining the above formulae:
x = x static [ 1 − cos ( ω t ) ] {\displaystyle x=x_{\text{static}}[1-\cos(\omega t)]}
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