- Structural dynamics
**Structural dynamics**is a subset ofstructural analysis which covers the behaviour ofstructure s subjected to dynamic loading. Dynamic loads include people, wind, waves, traffic,earthquake s, and blasts. Any structure can be subject to dynamic loading. Dynamic analysis can be used to find dynamic displacements, time history, andmodal analysis .A static load is one which does not vary. A dynamic load is one which changes with time. 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 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)::$DAF\; =\; frac$u_{max}u_{static}

where u is the deflection of the structure due to the 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 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 (amass , M, on a spring ofstiffness , k for example) has the following equation of motion::$M\{ddot\{x\; +\; kx\; =\; F(t)$:where $ddot\{x\}$ 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::$x\; =\; frac$F_0k(1 - cos{(wt)}):

where ω = $sqrt\{frac$kM} and the fundamental natural frequency, $f\; =\; frac$w2pi.

The static deflection of a single degree of freedom system is:

:$x\_\{static\}\; =\; frac$F_0k:so you can write, by combining the above formulae:

:$x\; =\; x\_\{static\}(1\; -\; cos(wt))$

This gives the (theoretical) time history of the structure due to a load F(t), where the false assumption is made that there is no

damping .Although this is too simplistic to apply to a real structure, the Heaviside Step Function is a reasonable model for the application of many real loads, such as the sudden addition of a piece of furniture, or the removal of a prop to a newly cast concrete floor. However, in reality loads are never applied instantaneously - they build up over a period of time (this may be very short indeed). This time is called the

rise time .As the number of degrees of freedom of a structure increases it very quickly becomes too difficult to calculate the time history manually - real structures are analysed using

non-linear finite element analysis software.**Damping**Any real structure will dissipate energy (mainly through friction). This can be modelled by modifying the DAF:

:$DAF\; =\; 1\; +\; e^\{-cpi\}$:where $c=frac$ ext {Damping Coefficient} ext{Critical Damping Coefficient} and is typically 2%-10% depending on the type of construction:

* Bolted steel ~6%

* Reinforced concrete ~ 4%

* Welded steel ~ 2%Generally damping would be ignored for transient events (for example, an

impulse load such as a bomb blast), but would be important for non-transient events (such as wind loading or crowd loading).**Modal analysis**A

modal analysis calculates the frequencymodes or natural frequencies of a given system, but not necessarily its full time history response to a given input. The natural frequency of a system is dependent only on thestiffness of the structure and themass which participates with the structure (including self-weight). It is not dependent on the load function.It is useful to know the modal frequencies of a structure as it allows you to ensure that the frequency of any applied periodic loading will not coincide with a modal frequency and hence cause

resonance , which leads to largeoscillations .The method is:

a) Find the natural modes (the shape adopted by a structure) and natural frequencies

b) Calculate the response of each mode

c) Optionally superpose the response of each mode to find the full modal response to a given loading

**Energy method**It is possible to calculate the frequency of different mode shapes of system manually by the energy method. For a given mode shape of a multiple degree of freedom system you can find an "equivalent" mass, stiffness and applied force for a single degree of freedom system. For simple structures the basic mode shapes can be found by inspection, but it is not a conservative method. Rayleigh's principle states:

"The frequency ω of an arbitrary mode of vibration, calculated by the energy method, is always greater than - or equal to - the fundamental frequency ω

_{n}."For an assumed mode shape $ar\{u\}(x)$, of a structural system with mass, M; stiffness, EI (

Young's modulus , E, multiplied by thesecond moment of area , I); and applied force, F(x)::$ext\{Equivalent\; mass,\; \}\; M\_\{eq\}\; =\; int\{Mar\{u\}^2\}\; du$::$ext\{Equivalent\; stiffness,\; \}k\_\{eq\}\; =\; int\{EI(frac$d^2ar{u}dx^2)}dx::$ext\{Equivalent\; force,\; \}F\_\{eq\}\; =\; int\{Far\{udx$:then, as above:

:$w=sqrt\{frac$k_{eq}M_{eq

**Modal response**The complete modal response to a given load F(x,t) is $v(x,t)=sum\{u\_n(x,t)\}$. The summation can be carried out by one of three common methods:

* Superpose complete time histories of each mode (time consuming, but exact)

* Superpose the maximum amplitudes of each mode (quick but conservative)

* Superpose the square root of the sum of squares (good estimate for well-separated frequencies, but unsafe for closely spaced frequencies)To superpose the individual modal responses manually, having calculated them by the energy method:

:$T\; =\; 2pi\; w$

Assuming that the rise time t

_{r}is known it is possible to read the DAF from a standard graph. The static displacement can be calculated with $u\_\{static\}=frac\{F\_\{1,eq\{k\_\{1,eq$. The dynamic displacement for the chosen mode and applied force can then be found from::$u\_\{max\}\; =\; u\_\{static\}DAF$

**Modal participation factor**For real systems there is often mass participating in the

forcing function (such as the mass of ground in anearthquake ) and mass participating ininertia effects (the mass of the structure itself, M_{eq}). Themodal participation factor Γ is a comparison of these two masses. For a single degree of freedom system Γ = 1.:Γ $=\; frac\{sum\{M\_nar\{u\}\_n\{sum\{M\_nar\{u\}\_n^2$

**ee also***

Seismic analysis

*Response spectrum

*Earthquake engineering

*Seismic performance analysis

*Vibration

*Vehicle dynamics

*Modal analysis using FEM

*Vibration control **External links*** [

*http://structdynviblab.mcgill.ca/index.html Structural Dynamics and Vibration Laboratory of McGill University*]

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