 # 2 Mathematics behind Blocked Forces

Let’s get back to our well-known assembled system AB.

In this assembly, the responses $$\mathbf{u}_{\mathrm{3}}$$ on the passive subsystem B are caused by unknown loads $$\mathbf{f}_{\mathrm{1}}$$ which are acting on the active subsystem A.

When considering the Blocked Forces method, one is essentially looking for a set of forces $$\mathbf{f}\mathrm{_{2}^{bl}}$$ that, applied to the interface of the system at rest $$(\mathbf{f}\mathrm{_{1} = 0})$$, yields the same responses at $$\mathbf{u}_{\mathrm{3}}$$. This can be directly formulated using the admittance of the assembly $$\mathbf{Y}\mathrm{_{32}^{AB}}$$, or expanded in terms of its subsystem admittances.

Comparing both equations it follows naturally that the Blocked Forces should take the form:

$$\mathbf{f}\mathrm{_{2}^{bl} = (\mathbf{Y}_{22}^{A})^{-1}} \mathbf{Y}\mathrm{_{21}^{A}} \mathbf{f}\mathrm{_{1}}$$

An alternative explanation is shown in the following figure. Here, the Blocked Forces, applied in the opposite direction to the assembly with the source in operation, must cancel out all vibrations “downstream” of $$\mathbf{u}\mathrm{_{2}}$$.

This shows that the Blocked Forces are indeed a property on the active component A only. The existence of such an equivalent source problem offers tremendous potential for practical component-based TPA methods. There is however one crucial limitation:

The Blocked Forces only properly represent the operational source excitations for responses on the receiving structure or on the interface. Responses obtained on the source will be different and therefore of no use.

This limitation can be understood as follows:

• Responses at the passive side B are caused only by forces through or onto the interface,
• whereas the responses at the source side A are a result of both the direct contribution of  $$\mathbf{f}\mathrm{_{1}}$$ and its reflection through the coupled subsystem B.