2 Mathematics behind Virtual Points

On this page, we summarize the fundamentals of the Virtual Point Transformation. For a broader overview, please refer to the dissertation of our colleague Maarten van der Seijs: link.

Relations between measured data and Virtual Points

A typical instrumentation around a Virtual Point is shown in the following figure:

The underlying assumption of the Virtual Point Transformation is that the interface behaves rigidly. Based on this, the measured motion $$  \mathbf{u} $$ (e.g., accelerometer channels) can be related to the Virtual Point motion $$  \mathbf{q} $$ (e.g., translational and rotational acceleration). Analogously, a relation between Virtual Point loads $$  \mathbf{m} $$ (i.e., forces and moments) and the measured forces $$  \mathbf{f} $$ (e.g., hammer impacts or shaker excitations) exists:

The 6 rigid body modes at the Virtual Point resulting from the reduction with $$ \mathbf{R} $$ can be visualized as:

Inverting these relations

Since the reduction basis $$ \mathbf{R} $$ is generally not a square matrix, you can not directly invert them. Instead, the left- and right-inverse of the reduction matrices can be used:

Pseudoinverse

If a matrix $$ \mathbf{A} $$ is underdetermined (fewer rows than columns) or rank deficient, then multiple solutions exist for the least-squares problem that minimize the norm $$ \mathbf{(Ax – b)} $$. In such a case, the Moore-Penrose inverse (pseudoinverse) allows to still find a unique solution. This solution is characterized by minimizing the norm $$ \mathbf{(x)} $$. The pseudoinverse of a matrix $$ \mathbf{A} $$ is denoted by the “plus” superscript $$ \mathbf{A^+} $$.
Since we can not guarantee the reduction basis $$ \mathbf{R} $$ to be determined and of full rank, the relations above will be substituted by the pseudoinverse:

Transforming Frequency Response Functions (FRFs) to the Virtual Point

The measured FRF $$ \mathbf{Y}_{\mathrm{uf}} $$ is denoted by the subscript “uf” and relates the measured forces $$ \mathbf{f} $$ (at the impact locations) to the measured motion $$ \mathbf{u} $$ (at accelerometers):

$$ \mathbf{u = Y}_{\mathrm{uf}} \mathbf{f} $$

The Virtual Point FRF $$ \mathbf{Y}_{\mathrm{qm}} $$ is denoted by the subscript “qm” and relates the Virtual Point loads $$ \mathbf{m} $$ to Virtual Point motion $$ \mathbf{q} $$:

$$ \mathbf{q = Y}_{\mathrm{qm}} \mathbf{m} $$

Similar to the previous relations, the FRF $$ \mathbf{Y}_{\mathrm{uf}} $$ can be transformed to the Virtual Point by the following equation:

$$ \mathbf{Y}_{\mathrm{qm}} = \mathbf{T}_{\mathrm{u}} \mathbf{Y}_{\mathrm{uf}} {\mathbf{T}_{\mathrm{f}}}^{T} = \mathbf{Y}_{\mathrm{qf}} {\mathbf{T}_{\mathrm{f}}}^{T} = \mathbf{T}_{\mathrm{u}} \mathbf{Y}_{\mathrm{um}} $$

Summary

An overview on relating the Virtual Point frame to the measurement frame is given in the following table:

Practical benefits:

• The transformation matrices $$ \mathbf{R}_{\mathrm{u}} $$ and $$ \mathbf{R}_{\mathrm{f}}$$ are independent! Therefore, no excitations on the sensors are necessary.
• The transformation can be performed on left $$ (\mathbf{u}) $$, right $$ (\mathbf{f}) $$ or both sides (useful for e.g. TPA).