 # 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).