# PILEDYN - User documentation - Impedance problem

Impedances

The impedances relate displacements/rotations with forces/moments. In general, this can be written as a matrix relation , where is the impedance matrix, is the displacements/rotations vector, and is the forces/moments vector. Each impedance term is a frequency dependent complex-valued function which can be written as:

where is the real part or stiffness coefficient, is the dimensionless frequency, is the circular frequency (rad/s), is a pile diamater, is a shear wave velocity, and is the damping coefficient.

On the other hand, for pile configurations with planes of symmetry calculations can be made applying symmetry conditions, so the studied geometry is reduced. This way, the results are obtained more quickly and easily: if there are planes of symmetry, the studied geometry is reduced to of the original.

Vibration modes

The impedances recieve different names depending on the displacement/rotation they relate. PILEDYN does not take into account torsion, therefore it considers five vibration modes. Those are:

VM 1: displacement along X axis

To produce this displacement (), a force applied along x axis () is needed. Nevertheless, as a consecuence of and the imposition of null rotation, a resistant momentum appears ().

Figura 1: Vibration mode 1 diagram.

Then, two impedances can be obtained with this vibration mode: Kxx, the horizontal impedance along x axis, and Kryx, the crossed impedance along y axis that appears with .

VM 2: displacement along Y axis

In this case, the situation is similar to the one described in Vibration mode 1, but changing the direction in which the pile vibrates to the y axis. The diagram below represents it:

Figura 2: Vibration mode 2 diagram.

Where is the displacement along y axis, the force applied ncessary to produce and (negative) is the resistant momentum.

Then, two impedances can be obtained with this vibration mode too: Kyy, the horizontal impedance along y axis, and Krxy, the crossed impendance along x axis that appears with .

VM 3: displacement along Z axis

Now, to produce the displacement along z axis () a force applied in the same direction () is needed.

Figura 3: Vibration mode 3 diagram.

Then, only one impedance can be obtained with this vibration mode: Kzz, the vertical impedance.

VM 4: rotation along X axis

To produce a rotation along x axis (), a momentum have to be applied along x axis. However, as a consecuence of , a resistant force appears along z axis (), that produces another momentum. This can be seen in the diagram below:

Figura 4: Vibration mode 4 diagram.

Then, two impedances can be obtained with this vibration mode: Krx, the rocking impedance in x axis, and Kyrx, the crossed impedance that appears with .

VM 5: rotation along Y axis

In this case, another momentum is needed (), applied along y axis. But, as a consecuence of and again the imposition of null desplacement and rotation in other axes, a resistant force appears along z axis (), that applied in x axis produces another momentum. This can be seen in the diagram below:

Figura 5: Vibration mode 5 diagram.

Then, two impedances can be obtained with this vibration mode too: Kry, the rocking impedance along y axis, and Kxry, the crossed impedance along y axis that appears with .