In foundation soil system damping is always present in one form or another. For this case fig. 1 (a), the equation of motion is:

———-(1)

———-(2)

The solution of equation (2) is done by applying the concept of rotating vector. In the figure 1 the exciting force vector Q _{0} is placed with a phase angle É¸ ahead of the displacement vector A. The equation of displacement may be expressed as:

———-(3)

In figure 1 (b) the position of motion vector is shown. In figure 1 (c) the position of force vector is shown. The force vectors act opposite to that of motion vectors. The force vector Q_{0} is placed with a phase angle É¸ ahead of the displacement vector A Resolving the force in the vertical and horizontal direction, we have,

———-(4)

And ———-(5)

Solving for A and we have,

———-(6)

And ———-(7)

———-(8)

———-(9)

———-(10)

———-(11)

The equations are plotted for various values D as shown in figure 2(a) and (b). These curves are referred to here as response curves for Constant-force-amplitude-excitation. In the figure it is seen that maximum amplitude occurs at a frequency slightly less than the un-damped natural circular frequency, where . The frequency at maximum amplitude will be referred to as resonant frequency f_{m} for constant force Q_{0}.

Resonant frequency at maximum amplitude

———-(12)

Put and , then

Or,

Or,

Or,

Or,

Or, ———-(13)

Taking positive sign, we get,

———-(14)

Or, ———-(15)

———-(16)

Magnification factor at resonant frequency is given by,

———-(17)

———-(18)

When , f_{max} = 0 which means that the maximum response is the static response.

The maximum amplitude at resonance will be,

———-(19)

When damping in the system is neglected, then

———-(20)