An accurate estimation of frequency response functions in output-only measurements

Abstract

Frequency response functions (FRFs) can be estimated only from responses of structures without knowledge of input forces. Mode shapes identified from operational modal analysis (OMA) methods are unscaled, and the unscaled mode shapes provide unscaled FRFs. In this paper, mass change and mass–stiffness methods are employed to construct FRFs from the response-only measurements. A numerical case study of a cantilever beam is investigated using the finite element method. It is shown that the mass–stiffness change method provides more accurate results compared to the mass change method in the low-frequency range. A laboratory-scale steel beam is also tested using an OMA method and conventional hammer test. The experimental results show better accuracy of the identified FRFs in the low-frequency range when the mass–stiffness method is used.

Appendix

The transformations from Eqs. (5), (6) are presented here.

By subtracting Eq. (5) from Eq. (4), the following relation is obtained:

$$\begin{aligned} \left[ M \right] \left( {\left\{ {\phi _1 } \right\} \omega _1^2 -\left\{ {\phi _2 } \right\} \omega _2^2 } \right) -\left[ {\Delta M} \right] \left\{ {\phi _2 } \right\} \omega _2^2 =\left[ K \right] \left( {\left\{ {\phi _1 } \right\} -\left\{ {\phi _2 } \right\} } \right) -\left[ {\Delta K} \right] \left\{ {\phi _2 } \right\} \end{aligned}$$

(15)

Considering small mass and stiffness changes, the mode shapes before and after modification change negligibly, i.e.

$$\begin{aligned} \left\{ {\phi _2 } \right\} \cong \left\{ {\phi _1 } \right\} =\left\{ \phi \right\} \end{aligned}$$

(16)

Considering Eqs. (16), (15) can be simplified as:

$$\begin{aligned} \left[ M \right] \left\{ \phi \right\} \left( {\omega _1^2 -\omega _2^2 } \right) =\left( {\left[ {\Delta M} \right] \omega _2^2 -\left[ {\Delta K} \right] } \right) \left\{ \phi \right\} \end{aligned}$$

(17)

Pre-multiplying by $\left\{ \phi \right\} ^{t}$ and considering orthogonality of mode shapes, Eq. (17) becomes:

$$\begin{aligned} \left( {\omega _1^2 -\omega _2^2 } \right) =\left\{ \phi \right\} ^{t}\left( {\left[ {\Delta M} \right] \omega _2^2 -\left[ {\Delta K} \right] } \right) \left\{ \phi \right\} \end{aligned}$$

(18)

By substituting Eq. (2) into Eq. (18), the following equation is obtained:

$$\begin{aligned} \alpha =\sqrt{\frac{\left( {\omega _1^2 -\omega _2^2 } \right) }{\left\{ \psi \right\} ^{t}\left( {\left[ {\Delta M} \right] \omega _2^2 -\left[ {\Delta K} \right] } \right) \left\{ \psi \right\} }} \end{aligned}$$

(19)