Structural dynamics

Single degree of freedom system: simple mass spring model

A simple single degree of freedom system (a mass, M, on a spring of stiffness k, for example) has the following equation of motion:

${\displaystyle M{\ddot {x}}+kx=F(t)}$

where ${\displaystyle {\ddot {x}}}$ 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:

${\displaystyle x={\frac {F_{0}}{k}}[1-\cos(\omega t)]}$

where ${\displaystyle \omega ={\sqrt {\frac {k}{M}}}}$ and the fundamental natural frequency, ${\displaystyle f={\frac {\omega }{2\pi }}}$.

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

${\displaystyle x_{\text{static}}={\frac {F_{0}}{k}}}$

so we can write, by combining the above formulae:

${\displaystyle x=x_{\text{static}}[1-\cos(\omega t)]}$

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.

It is possible to calculate the frequency of different mode shape 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 ${\displaystyle {\bar {u}}(x)}$, of a structural system with mass M; bending stiffness, EI (Young's modulus, E, multiplied by the second moment of area, I); and applied force, F(x):

${\displaystyle {\text{Equivalent mass, }}M_{\text{eq}}=\int M{\bar {u}}^{2}\,du}$

${\displaystyle {\text{Equivalent stiffness, }}k_{\text{eq}}=\int EI\left({\frac {d^{2}{\bar {u}}}{dx^{2}}}\right)^{2}\,dx}$

${\displaystyle {\text{Equivalent force, }}F_{\text{eq}}=\int F{\bar {u}}\,dx}$

then, as above:

${\displaystyle \omega ={\sqrt {\frac {k_{\text{eq}}}{M_{\text{eq}}}}}}$

The complete modal response to a given load F(x,t) is ${\displaystyle v(x,t)=\sum u_{n}(x,t)}$. 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:

Assuming that the rise time t_r is known (T = 2π/ω), it is possible to read the DAF from a standard graph. The static displacement can be calculated with ${\displaystyle u_{\text{static}}={\frac {F_{1,{\text{eq}}}}{k_{1,{\text{eq}}}}}}$. The dynamic displacement for the chosen mode and applied force can then be found from:

${\displaystyle u_{\max }=u_{\text{static}}{\text{DAF}}}$