The FRF Calculator employs the residue formulation to estimate the FRF between two given degrees of freedom (DOFs) using the modal parameters:
where the following notation is used:
- FRF between the i-th and
j-th DOFs, where DOF i is the output and DOF
j is the input.
- Number of modes considered in the FRF calculation.
- r-th vibration mode, with 
its real part and 
its imaginary part:
In terms of the damping coefficient ξr and the undamped frequency
, the mode can be expressed as:
- residue of the ij DOF pair for mode
r. 
is calculated as:
where
is the mass normalized mode shape evaluated at DOF
i (a complex magnitude in general).
- complex conjugate of x.
can be understood as the output generalized displacement measured in DOF
i when an input generalized force is imposed in DOF
j as a function of frequency. Thus, depending on the nature of the DOFs
(whether they are displacement or rotation DOFs), 
is measured as Displacement/Force, Rotation/Force, Displacement/Torque or
Rotation/Torque.
The analogous relationships between generalized velocity (translational or rotational) and generalized force, or generalized acceleration (translational or rotational) and generalized force are called Mobility and Accelerance, respectively.
They are calculated as:
For a given pair of input and output nodes, a 3x3 FRF matrix can be computed relating
the 3 input and 3 output DOFs measured at each node in the Global Coordinate System (GCS).
If that matrix is denoted as 
, the FRF matrix 
expressed in the DOFs along two arbitrary coordinate systems centered in
the input and output nodes is:
where 
and 
are the rotation matrices that transform from the GCS to the local output
(O) and input (I) coordinate systems (CS), respectively.