**2. Transmissibility in MDOF systems**

very important problem in various areas, such as vibration control, fatigue life prediction

Although the force identification problem may be solved from the dynamic responses by simply reversing the direct problem, this is usually ill-posed and sensitive to perturbations in

Over the past years, the theory of inverse methods has been actively developed in many research areas presenting in common the effects of matrix ill-conditioning, reflecting the illposedness nature of the inverse problem itself. Those problems can often be overcome by methods such as pseudo-inversion for over-determined systems, use of Kalman filters [6, 7],

Various research works in force identification can be found in the literature, such as those related to the identification of impact forces, implementation of prediction models based on reflected waves or simply from the dynamic responses [11-18], prediction of forces in plates for systems with time dependent properties [11] and identification of harmonic forces [13]. These methods to identify operational loads based on response measurements can be clas‐ sified into three main categories: deterministic methods, stochastic methods and methods based on artificial intelligence. Two main classes of identification technique are consid‐ ered in the group of deterministic methods for load identification: frequency-domain methods and time-domain methods. The force identification in time domain has been less studied than its frequency domain equivalent, therefore there are not that many force identification studies in the literature. A review on the state of the art for dynamic load

Although out of the scope of this chapter, some references are here given with respect to recent time-domain force identification developments. One interesting approach based on modal filtering [15] is the Sum of Weighted Accelerations Technique (SWAT), which allows to obtain the time-domain force reconstruction by isolating the rigid body modal accelerations. Another approach for time-domain force reconstruction is the Inverse Structural Filter (ISF) method of Kammer and Steltzner [16] that inverts the discrete-time equations of motion. A variant of this, expected to produce a stable ISF when the standard method fails was recently developed and named as Delayed Multi-step ISF (DMISF). For a more detailed description on these methods (SWAT, ISF and DMISF) see e.g. [17] and for its application to rotordynamics, see [18].

In this chapter, the authors treat the frequency-domain problem from a different perspective, which is based on the MDOF transmissibility concept. As aforementioned, usually the transmissibility of forces is defined in textbooks for SDOF systems, simply as the ratio between the modulus of the transmitted force magnitude to the support and the modulus of the applied force magnitude. For SDOF systems, the expression of either the transmissibility of motion or forces is exactly the same; however, as explained in [1], that is not the case for MDOF systems. On the one hand, the problem of extending the idea of transmissibility of motion to an MDOF system is essentially a problem of how to relate a set of unknown responses to a set of known responses associated to a given set of applied forces; on the other hand, for the transmissibility

of forces the question is how to relate a set of reaction forces to a set of applied ones.

Singular Value Decomposition and Tikhonov regularization [8-10].

and health monitoring.

104 Advances in Vibration Engineering and Structural Dynamics

identification may be found in [3, 14].

the measured data.

The transmissibility concept may be found in any fundamental textbook on mechanical vibrations (e.g. [28]), related to SDOF systems.

The transmissibility of motion is defined as the ratio between the modulus of the response amplitude (output) and the modulus of the imposed base harmonic displacement (input). Depending on the imposed frequency, the result can vary from an amplification to an attenu‐ ation in the response amplitude relatively to the input one.

On the other hand, the transmissibility of force is defined as the ratio between the modulus of the transmitted force magnitude to the support and the modulus of the imposed force magnitude.

It happens that for SDOF systems the expression for calculating the transmissibility is the same, either referring to forces or to motion. This is not the case for MDOF systems.

The generalization of these definitions to MDOFs has been developed in the last decade, as mentioned before. In this section a brief review of these generalizations is given, introducing also the concepts and notation used for the force identification problem.

inverse of the dynamic stiffness matrix *Z*(ω). One may underline that the mass-normalized

= =

**Φ ΦI Φ Φ diag**

*M*

<sup>2</sup> −*ω* <sup>2</sup>

then the force reconstruction (in frequency-domain) would be given by:

Eliminating the external forces *FA* between (5) and (6), one obtains

*T T*

Assuming proportional damping, *C=αK+βM* and therefore,

*Y* =*H F* = *Φ diag*(*ω<sup>r</sup>*

*2.1.1. Transmissibility of motion in terms of FRFs*

relationships:

where

<sup>2</sup> ( ) w

) + *i ω*(*α diag*(*ω<sup>r</sup>*

where *Φ* is the mode shape matrix, *ωr* is the *r*th natural frequency and *α* and *β* are constants.

From (1) it is easy to understand that if the responses *Y* at the discretization points are known,

Based on harmonically applied forces at co-ordinates *A*, one may establish that displacements at co-ordinates *U* and *K* are related to the applied forces at co-ordinates *A* by the following

> ( ) ( ) <sup>+</sup> = = *<sup>A</sup>*

( ) ( )

of the sub-matrix *HKA*. An important property of the transmissibility matrix to be used here is that it does not depend on the magnitude of the involved forces and only requires the

= *<sup>A</sup>*

is the transmissibility matrix relating both sets of displacements. *(HKA)*

+

2

*r*

*<sup>K</sup>* (2)

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*Y HF U UA A* = (5)

*Y HF K KA A* = (6)

*U UA KA K d K UK Y HH Y T Y* (7)

*T HH <sup>d</sup> UK UA KA* (8)

*+*

is the pseudo-inverse

) + *β I*) <sup>−</sup><sup>1</sup> *Φ <sup>T</sup> F* (3)

orthogonality properties are observed here:

## **2.1. Transmissibility of motion in MDOF systems**

To introduce the problem, the authors follow here as near as possible the notation used in [1]. Let *K* be the set of *nK* co-ordinates where the displacement responses *YK* are known (measured or computed), *U* the set of *nU* co-ordinates where the displacement responses *YU* are unknown, and *A* the set of co-ordinates where the forces *FA* may be applied (Fig. 1).

**Figure 1.** Illustration of an elastic body with the three sets of co-ordinates *K*, *U* and *A*.

To obtain the needed transmissibility of motion one may consider two distinct ways. The first is based on the frequency response function (FRF) matrices *H*(ω), known as the fundamental formulation, while the second is based on the dynamic stiffness matrix *Z*(ω) and is named alternative formulation.

The receptance frequency response matrix *H*(ω) relates the dynamic displacement amplitudes *Y* with the external force amplitudes *F* as (using harmonic excitation, in steady-state condi‐ tions):

$$\mathbf{Y} = \mathbf{H} \cdot \mathbf{F} \quad \Leftrightarrow \quad \mathbf{Y} = \left(\mathbf{K} - \alpha^2 \ \mathbf{M} + i\alpha \ \mathbf{C}\right)^{-1} \mathbf{F} \tag{1}$$

where *K, M* and *C*are the stiffness, mass and viscous damping matrices, respectively. *H*(ω) includes all the degrees of freedom in which the system is discretized and corresponds to the inverse of the dynamic stiffness matrix *Z*(ω). One may underline that the mass-normalized orthogonality properties are observed here:

$$\begin{aligned} \boldsymbol{\Phi}^{\top} \; \boldsymbol{M} \; \boldsymbol{\Phi} &= \mathbf{I} \\ \boldsymbol{\Phi}^{\top} \; \boldsymbol{K} \; \boldsymbol{\Phi} &= \text{diag}(\boldsymbol{\phi}\_{r}^{2}) \end{aligned} \tag{2}$$

Assuming proportional damping, *C=αK+βM* and therefore,

$$\mathbf{Y = H \ F \ = \mathbf{0}} \mathbf{T} \begin{bmatrix} \operatorname{diag}(\omega\_r^2 - \omega^2) + \operatorname{i}\omega \left(\operatorname{a}\operatorname{diag}(\omega\_r^2) + \beta \operatorname{I}\right) \end{bmatrix} \mathbf{I}^{-1} \mathbf{O}^T \ \mathbf{F} \tag{3}$$

where *Φ* is the mode shape matrix, *ωr* is the *r*th natural frequency and *α* and *β* are constants.

From (1) it is easy to understand that if the responses *Y* at the discretization points are known, then the force reconstruction (in frequency-domain) would be given by:

$$F = H^{-1}\ Y$$

#### *2.1.1. Transmissibility of motion in terms of FRFs*

Based on harmonically applied forces at co-ordinates *A*, one may establish that displacements at co-ordinates *U* and *K* are related to the applied forces at co-ordinates *A* by the following relationships:

$$\mathbf{Y}\_{\upsilon} = \mathbf{H}\_{\upsilon\mu} \mathbf{F}\_{\mu} \tag{5}$$

$$\mathbf{Y}\_{\mathcal{K}} = \mathbf{H}\_{\mathcal{K}\mathcal{A}} \mathbf{F}\_{\mathcal{A}} \tag{6}$$

Eliminating the external forces *FA* between (5) and (6), one obtains

$$\mathbf{Y}\_{U} = \mathbf{H}\_{U4} \left(\mathbf{H}\_{\text{KA}}\right)^{+} \mathbf{Y}\_{K} = \left(\mathbf{T}\_{d}\right)^{A}\_{U\&} \mathbf{Y}\_{K} \tag{7}$$

where

The generalization of these definitions to MDOFs has been developed in the last decade, as mentioned before. In this section a brief review of these generalizations is given, introducing

To introduce the problem, the authors follow here as near as possible the notation used in [1]. Let *K* be the set of *nK* co-ordinates where the displacement responses *YK* are known (measured or computed), *U* the set of *nU* co-ordinates where the displacement responses *YU* are unknown,

also the concepts and notation used for the force identification problem.

and *A* the set of co-ordinates where the forces *FA* may be applied (Fig. 1).

**Figure 1.** Illustration of an elastic body with the three sets of co-ordinates *K*, *U* and *A*.

alternative formulation.

tions):

To obtain the needed transmissibility of motion one may consider two distinct ways. The first is based on the frequency response function (FRF) matrices *H*(ω), known as the fundamental formulation, while the second is based on the dynamic stiffness matrix *Z*(ω) and is named

The receptance frequency response matrix *H*(ω) relates the dynamic displacement amplitudes *Y* with the external force amplitudes *F* as (using harmonic excitation, in steady-state condi‐

> ( ) <sup>1</sup> 2 w

where *K, M* and *C*are the stiffness, mass and viscous damping matrices, respectively. *H*(ω) includes all the degrees of freedom in which the system is discretized and corresponds to the

 w - *Y HF Y K M C F* = Û =- + *i* (1)

**2.1. Transmissibility of motion in MDOF systems**

106 Advances in Vibration Engineering and Structural Dynamics

$$\left(\left(\boldsymbol{T}\_{d}\right)\_{\ell\mathcal{K}}^{4} = \boldsymbol{H}\_{\cup\mathcal{U}}\left(\boldsymbol{H}\_{\times\mathcal{U}}\right)^{\ast}\tag{8}$$

is the transmissibility matrix relating both sets of displacements. *(HKA) +* is the pseudo-inverse of the sub-matrix *HKA*. An important property of the transmissibility matrix to be used here is that it does not depend on the magnitude of the involved forces and only requires the knowledge of a set of co-ordinates that include all the co-ordinates where the forces are applied. Indeed, it is required that *nK* be greater or equal to *nA*. One important aspect of this definition is that sub-matrices *HUA* and *HKA* may be obtained experimentally.

### *2.1.2. Transmissibility of motion in terms of dynamic stiffness*

There exists an alternative approach to obtain the transmissibility matrix for the displacements, using the dynamic stiffness matrices introduced in (1). Assuming again harmonic loading and defining two subsets, *A* and *B*, *A* being the set where the dynamic loads may be applied and *B* the set formed by the remaining co-ordinates, where no forces are applied (*FB* = **0**), one can obtain (after grouping adequately the degrees of freedom of the problem):

$$
\begin{bmatrix}
\mathbf{Z}\_{\mathcal{A}\mathcal{K}} & \mathbf{Z}\_{\mathcal{A}\mathcal{U}} \\
\mathbf{Z}\_{\mathcal{B}\mathcal{K}} & \mathbf{Z}\_{\mathcal{B}\mathcal{U}}
\end{bmatrix}
\begin{Bmatrix}
\mathbf{Y}\_{\mathcal{K}} \\
\mathbf{Y}\_{\mathcal{U}}
\end{Bmatrix} = \begin{Bmatrix}
\mathbf{F}\_{\mathcal{A}} \\
\mathbf{0}
\end{Bmatrix}
\tag{9}
$$

be other co-ordinates, where neither there are any applied forces nor there are any reactions,

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With the definition of the new sets *K*, *U* and *C*, the problem may be defined in the following

*K*

(13)

*F*

*F*

*U*

*HF HF UK K UU U* + = **0** (14)

*Uf K* <sup>=</sup> ( )*UK FT F* (15)

ì ü é ù ï ï ê ú ì ü í ý <sup>=</sup> ê ú í ý ï ï î þ ê ú î þ ë û *K KK KU*

*U UK UU*

*Y HH*

*Y HH*

*Y HH*

*C CK CU*

that shall constitute the set *C*.

**Figure 2.** Illustration of both sets of co-ordinates *K* and *U*.

*2.2.1. Transmissibility of forces in terms of FRFs*

Imposing *YU* = **0**, it follows that

way:

and so

Developing eq. (9), it follows that

$$\begin{aligned} \mathbf{Z}\_{AK} \ Y\_K + \mathbf{Z}\_{AU} \ Y\_U &= \mathbf{F}\_A & \mathbf{a} \\ \mathbf{Z}\_{BK} \ Y\_K + \mathbf{Z}\_{BU} \ Y\_U &= \mathbf{0} & \mathbf{b} \end{aligned} \tag{10}$$

From (10b) one obtains the transmissibility in terms of the dynamic stiffnesses:

$$\mathbf{Y}\_{U} = -\left(\mathbf{Z}\_{BU}\right)^{\*} \mathbf{Z}\_{\text{RX}} \mathbf{Y}\_{K} = \left(\mathbf{T}\_{d}\right)^{4}\_{\text{UK}} \mathbf{Y}\_{K} \tag{11}$$

where *(ZBU) +* is the pseudo-inverse of *ZBU*.

From (11) it is possible to obtain the response at the unknown co-ordinates, as long as the pseudo-inverse is viable, which requires that *nB* is greater or equal to *nU.*

Indeed, from all this resulted two conditions:

$$\left(\left(\mathbf{Z}\_d\right)\_{\cup K}^{\wedge} = -\left(\mathbf{Z}\_{BU}\right)^{+}\mathbf{Z}\_{BK} = \mathbf{H}\_{\cup 4} \left(\mathbf{H}\_{K4}\right)^{+} \qquad n\_B \ge n\_U \text{ and } n\_K \ge n\_A \tag{12}$$

#### **2.2. Transmissibility of forces in MDOF systems**

To introduce the transmissibility of forces for MDOF systems, the authors follow a similar procedure to the one used in the previous sub-section. The problem consists now of relating the set of known applied forces to a set of unknown reactions (or the other way around), relating the set of known applied forces (set *K*) with a set of unknown reaction forces (set *U*), which are illustrated in Fig.2. At the set *U* it will be assumed that *YU* = **0**. In general, there will be other co-ordinates, where neither there are any applied forces nor there are any reactions, that shall constitute the set *C*.

**Figure 2.** Illustration of both sets of co-ordinates *K* and *U*.

#### *2.2.1. Transmissibility of forces in terms of FRFs*

With the definition of the new sets *K*, *U* and *C*, the problem may be defined in the following way:

$$
\begin{bmatrix}
\mathbf{Y}\_{\mathcal{K}} \\
\mathbf{Y}\_{U} \\
\mathbf{Y}\_{C}
\end{bmatrix} = \begin{bmatrix}
\mathbf{H}\_{\mathcal{K}\mathcal{K}} & \mathbf{H}\_{\mathcal{K}U} \\
\mathbf{H}\_{U\mathcal{K}} & \mathbf{H}\_{\mathcal{U}U} \\
\mathbf{H}\_{\mathcal{K}\mathcal{K}} & \mathbf{H}\_{\mathcal{C}U}
\end{bmatrix} \begin{Bmatrix}
\mathbf{F}\_{\mathcal{K}} \\
\mathbf{F}\_{U}
\end{Bmatrix} \tag{13}
$$

Imposing *YU* = **0**, it follows that

$$\mathbf{H}\_{\text{UK}} \left[ \mathbf{F}\_{\text{K}} + \mathbf{H}\_{\text{UU}} \right] \mathbf{F}\_{\text{U}} = \mathbf{0} \tag{14}$$

and so

knowledge of a set of co-ordinates that include all the co-ordinates where the forces are applied. Indeed, it is required that *nK* be greater or equal to *nA*. One important aspect of this definition

There exists an alternative approach to obtain the transmissibility matrix for the displacements, using the dynamic stiffness matrices introduced in (1). Assuming again harmonic loading and defining two subsets, *A* and *B*, *A* being the set where the dynamic loads may be applied and *B* the set formed by the remaining co-ordinates, where no forces are applied (*FB* = **0**), one can

is that sub-matrices *HUA* and *HKA* may be obtained experimentally.

obtain (after grouping adequately the degrees of freedom of the problem):

é ù ì ü ì ü ê ú í ýí ý <sup>=</sup> ë û î þ î þ **0** *AK AU K A*

*Z Z <sup>Y</sup>* (9)

a b

*ZY ZY* (10)

*<sup>U</sup> BU BK K d K UK Y Z ZY T Y* (11)

*Z Z Y F*

*BK BU U*

+ = + = **0** *AK K AU U A BK K BU U*

From (10b) one obtains the transmissibility in terms of the dynamic stiffnesses:

pseudo-inverse is viable, which requires that *nB* is greater or equal to *nU.*

( ) ( ) ( ) and + + = - ³ ³ *<sup>A</sup>*

( ) ( ) <sup>+</sup> = - = *<sup>A</sup>*

From (11) it is possible to obtain the response at the unknown co-ordinates, as long as the

To introduce the transmissibility of forces for MDOF systems, the authors follow a similar procedure to the one used in the previous sub-section. The problem consists now of relating the set of known applied forces to a set of unknown reactions (or the other way around), relating the set of known applied forces (set *K*) with a set of unknown reaction forces (set *U*), which are illustrated in Fig.2. At the set *U* it will be assumed that *YU* = **0**. In general, there will

*<sup>d</sup> UK BU BK UA KA BU K A T Z Z =H H nn nn* (12)

*Z ZY F Y*

*2.1.2. Transmissibility of motion in terms of dynamic stiffness*

108 Advances in Vibration Engineering and Structural Dynamics

Developing eq. (9), it follows that

where *(ZBU)*

*+*

is the pseudo-inverse of *ZBU*.

Indeed, from all this resulted two conditions:

**2.2. Transmissibility of forces in MDOF systems**

$$\boldsymbol{F}\_{\boldsymbol{U}} = \left(\boldsymbol{T}\_{\boldsymbol{f}}\right)\_{\mathrm{t}\boldsymbol{\mathcal{K}}} \boldsymbol{F}\_{\boldsymbol{K}} \tag{15}$$

$$\left(\boldsymbol{T}\_{\boldsymbol{f}}\right)\_{\boldsymbol{t}\boldsymbol{\&}\boldsymbol{\ell}} = -\left(\boldsymbol{H}\_{\boldsymbol{t}\boldsymbol{U}}\right)^{-1}\boldsymbol{H}\_{\boldsymbol{t}\boldsymbol{\&}\boldsymbol{\ell}}\tag{16}$$

Assuming harmonic loading and the mentioned sets *K, U* and *C*, one can obtain (after grouping

(21)

111

é ù ì üì ü ê ú ï ïï ï í ýí ý <sup>=</sup>

*ZZZY F ZZZY F ZZZY F*

*KK KC KU K K CK CC CU C C UK UC UU U U*

ï ïï ï ë û î þî þ

é ù ì ü ì ü ê ú íýí ý <sup>=</sup> ë û î þ î þ **0** *EE EU E E UE UU U Z Z Y F*

> = = *EE E E UE E U*

( ) ( )

(( ) )

(( ) ) ( ) <sup>+</sup> <sup>+</sup>

+

*ZY F*

It is worthwhile noting that joining together the sets *K* and *C* in a new set *E* makes it easier to

a b


*Z Z <sup>F</sup>* (22)

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*ZY F* (23)

*Uf E* <sup>=</sup> ( )*UE FTF* (24)

*<sup>f</sup>* <sup>=</sup> *UE EE UE T ZZ* (25)

*Ef U* <sup>=</sup> *UE FT F* (26)

*<sup>f</sup>* <sup>=</sup> *EE UE UE T ZZ* (27)

adequately the degrees of freedom of the problem) the following result:

see that imposing *YU* = **0** one obtains the following relationships:

Eliminating *YE* between (23a) and (23b), it turns out that

from which it is clear that:

The inverse problem corresponds to

where

with

is the force transmissibility matrix.

This is the direct force identification method, i.e., one knows the applied forces and calculate the reactions at the supports, where the displacements are assumed as zero. The inverse problem is also possible, if one is able to measure the reaction forces and if their number is higher than the number of applied forces, in order to calculate the pseudo-inverse of *HUK*:

$$\boldsymbol{F}\_{\boldsymbol{K}} = \left( \left( \boldsymbol{T}\_{\boldsymbol{f}} \right)\_{\boldsymbol{t} \in \mathcal{K}} \right)^{\*} \boldsymbol{F}\_{\boldsymbol{U}} \tag{17}$$

where

$$\left(\left(\boldsymbol{T}\_{f}\right)\_{\cup\mathbb{X}}\right)^{\*} = -\left(\boldsymbol{H}\_{\cup\mathbb{X}}\right)^{\*}\boldsymbol{H}\_{\cup\mathbb{U}}\tag{18}$$

Note that in spite of the fact that here the reaction forces are known, the notation *U* (that in principle stands for "unknown") is kept.

In the inverse problem, one may not know how many applied force exist and where they are applied. If that is the case, one must follow a different approach, as it will be explained in section 4.1

If the condition *YU* = **0** is relaxed, from eq. (13) it follows that:

$$\mathbf{Y}\_{\mathcal{U}} = \mathbf{H}\_{\mathcal{U}\mathcal{K}}\mathbf{F}\_{\mathcal{K}} + \mathbf{H}\_{\mathcal{U}\mathcal{U}}\mathbf{F}\_{\mathcal{U}} \tag{19}$$

$$\begin{aligned} \mathbf{F}\_{\upsilon} &= \left(\mathbf{T}\_{f}\right)\_{\upsilon\_{\mathcal{K}}} \mathbf{F}\_{\mathcal{K}} + \left(\mathbf{H}\_{\upsilon\upsilon}\right)^{-1} \mathbf{Y}\_{\upsilon} & \mathbf{a} \\ \text{and} \\ \mathbf{F}\_{\mathcal{K}} &= \left(\left(\mathbf{T}\_{f}\right)\_{\upsilon\mathcal{K}}\right)^{+} \mathbf{F}\_{\upsilon} + \left(\mathbf{H}\_{\upsilon\mathcal{K}}\right)^{+} \mathbf{Y}\_{\upsilon} & \mathbf{b} \end{aligned} \tag{20}$$

#### *2.2.2. Transmissibility of forces in terms of dynamic stiffness*

Again, there is an alternative approach to obtain the force transmissibility matrix, using the dynamic stiffness matrices.

Assuming harmonic loading and the mentioned sets *K, U* and *C*, one can obtain (after grouping adequately the degrees of freedom of the problem) the following result:

$$
\begin{bmatrix}
\mathbf{Z}\_{\mathcal{K}} & \mathbf{Z}\_{\mathcal{K}} & \mathbf{Z}\_{\mathcal{K}U} \\
\mathbf{Z}\_{\mathcal{C}\mathcal{K}} & \mathbf{Z}\_{\mathcal{C}\mathcal{C}} & \mathbf{Z}\_{\mathcal{C}U} \\
\mathbf{Z}\_{\mathcal{UK}} & \mathbf{Z}\_{\mathcal{UC}} & \mathbf{Z}\_{\mathcal{UU}}
\end{bmatrix}
\begin{Bmatrix}
\mathbf{Y}\_{\mathcal{K}} \\
\mathbf{Y}\_{\mathcal{C}} \\
\mathbf{Y}\_{\mathcal{U}}
\end{Bmatrix} = \begin{Bmatrix}
\mathbf{F}\_{\mathcal{K}} \\
\mathbf{F}\_{\mathcal{C}} \\
\mathbf{F}\_{\mathcal{U}}
\end{Bmatrix} \tag{21}
$$

It is worthwhile noting that joining together the sets *K* and *C* in a new set *E* makes it easier to see that imposing *YU* = **0** one obtains the following relationships:

$$
\begin{bmatrix}
\mathbf{Z}\_{EE} & \mathbf{Z}\_{EU} \\
\mathbf{Z}\_{UE} & \mathbf{Z}\_{UU}
\end{bmatrix}
\begin{Bmatrix}
\mathbf{Y}\_{E} \\
\mathbf{0}
\end{Bmatrix} = 
\begin{Bmatrix}
\mathbf{F}\_{E} \\
\mathbf{F}\_{U}
\end{Bmatrix}
\tag{22}
$$

from which it is clear that:

$$\begin{aligned} \mathbf{Z}\_{EE} \ Y\_E &= \mathbf{F}\_E & \mathbf{a} \\ \mathbf{Z}\_{UE} \ Y\_E &= \mathbf{F}\_U & \mathbf{b} \end{aligned} \tag{23}$$

Eliminating *YE* between (23a) and (23b), it turns out that

$$\boldsymbol{F}\_{\boldsymbol{U}} = \left(\boldsymbol{T}\_{\boldsymbol{f}}\right)\_{\text{UE}} \boldsymbol{F}\_{\boldsymbol{E}} \tag{24}$$

where

where

where

section 4.1

is the force transmissibility matrix.

110 Advances in Vibration Engineering and Structural Dynamics

principle stands for "unknown") is kept.

If the condition *YU* = **0** is relaxed, from eq. (13) it follows that:

and

*2.2.2. Transmissibility of forces in terms of dynamic stiffness*

dynamic stiffness matrices.

( ) ( )

*FTFH Y*

*U f K UU U UK*

= +

= +

(( ) ) ( )

*F T FHY*

*K f U UK U UK*

<sup>+</sup> <sup>+</sup>

( ) ( )


This is the direct force identification method, i.e., one knows the applied forces and calculate the reactions at the supports, where the displacements are assumed as zero. The inverse problem is also possible, if one is able to measure the reaction forces and if their number is higher than the number of applied forces, in order to calculate the pseudo-inverse of *HUK*:

+

Note that in spite of the fact that here the reaction forces are known, the notation *U* (that in

In the inverse problem, one may not know how many applied force exist and where they are applied. If that is the case, one must follow a different approach, as it will be explained in

1


Again, there is an alternative approach to obtain the force transmissibility matrix, using the

(( ) )

(( ) ) ( ) <sup>+</sup> <sup>+</sup>

*<sup>f</sup>* = - *UU UK UK T HH* (16)

*Kf U* <sup>=</sup> *UK FT F* (17)

*<sup>f</sup>* = - *UK UU UK T HH* (18)

*Y HF HF U UK K UU U* = + (19)

a

b

(20)

$$\mathbf{Z}\left(\mathbf{Z}\_f\right)\_{UE} = \mathbf{Z}\_{UE}\left(\mathbf{Z}\_{EE}\right)^{-1} \tag{25}$$

The inverse problem corresponds to

$$\boldsymbol{F}\_{E} = \left( \left( \boldsymbol{T}\_{f} \right)\_{\mathrm{UE}} \right)^{\*} \boldsymbol{F}\_{U} \tag{26}$$

with

$$\mathbb{E}\left(\left(\mathbf{Z}\_f\right)\_{\text{UE}}\right)^\* = \mathbf{Z}\_{\text{EE}}\left(\mathbf{Z}\_{\text{UE}}\right)^\* \tag{27}$$

It is important to note that only some of the co-ordinates of the set *E* have applied forces. This means that in (23) some rows of *F<sup>E</sup>* are zero and only the columns (in *ZEE*) whose co-ordinates have applied forces (set *K*) are needed for the transmissibility matrix. In other words, from the set *E* only the co-ordinates corresponding to the *K* set are used.

#### **2.3. Summary**

From sections 2.2.1 and 2.2.2, one can conclude that for the direct problem of transmissibility of forces there is no restrictions in the number of co-ordinates used:

$$\begin{aligned} \left(\mathbf{T}\_f\right)\_{\iota\chi} &= -\left(\mathbf{H}\_{\iota\iota\iota}\right)^{-1}\mathbf{H}\_{\iota\chi} \\ \left(\mathbf{T}\_f\right)\_{\iota\iota\varepsilon} &= \mathbf{Z}\_{\iota\varepsilon}\left(\mathbf{Z}\_{EE}\right)^{-1} \end{aligned} \tag{28}$$

where *C* represents the viscous damping matrix, often of the proportional type, i.e., *C=αK+βM*,

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To build the dynamic stiffness matrix, a specific structural finite element is chosen according to the approximation considered. For example, in the case of a reasonably long and slender beam one can use the Euler-Bernoulli beam element (instead of a shell or solid structural element). Then, the global matrices are assembled for the chosen discretization of the structure.

In order to improve the accuracy of the numerical model when simulating what is obtained experimentally, concentrated masses are often added at the corresponding nodes to model the

Although the receptance matrix *H*(ω) is the inverse of the corresponding dynamic stiffness matrix, one should avoid such direct numerical inversion (frequency by frequency). Instead,

*<sup>A</sup>* or *(Tƒ)UK.*

where *α* and *β* are constants to be evaluated experimentally.

effect of the accelerometers used in the testing positions.

*H*(ω) is calculated from eq. (3), after a modal analysis in free vibration.

**3.2. Transmissibility in terms of experimental measurements**

transmissibility of forces is always a simply supported beam.

For the experimental setup, the following equipment is used:

**•** Force transducers(PCB PIEZOTRONICS Model 208C01);

**•** Data acquisition equipment (Brüel & Kjær Type 3560-C).

suspended (free-free) beam is always used.

**•** Vibration exciter (Brüel & Kjær Type 4809);

**•** Power amplifier (Brüel & Kjær Type 2706);

structure.

Then, using (8) or (16), one can calculate the needed transmissibility matrices *(Td)UK*

Of course, alternatively one may use the equivalent expressions (11) or (25), respectively.

Depending on the type of transmissibility to obtain, the corresponding experimental setup should be established. Essentially, it is important to observe that for the transmissibility of motion one measures the FRFs relating co-ordinates *U* and *K* with co-ordinates A, normally using accelerometers and force transducers. For validation purposes, one may also measure the applied forces. In the examples presented next for the transmissibility of motion, a

For the transmissibility of forces, in the direct problem, one measures the applied forces at coordinates *K* (in the inverse problem, one measures the reaction forces at co-ordinates *U*). For validation purposes, one also measures the ones to be estimated. The test specimen for the

In Fig. 3, a schematic representation of the experimental setup used for the force transmissi‐ bility tests is presented, in this case a simply supported beam with a single applied force.

The excitation signal used was a multi-sine transmitted to the exciter, with constant amplitude in the frequency. In reality the signal measured by the force transducer does not exhibits a constant amplitude along the frequency, as it depends on the dynamic response of the

whereas in the inverse problem of transmissibility of forces there are some restrictions that can make this option not very useful in practice, especially when using the dynamic stiffnesses, since one needs to calculate the pseudo-inverse matrices:

$$\begin{aligned} \left( \left( \mathbf{T}\_{f} \right)\_{\mathrm{UE}} \right)^{+} &= -\left( \mathbf{H}\_{\mathrm{UE}} \right)^{+} \mathbf{H}\_{\mathrm{UE}} & \qquad \mathbf{n}\_{U} \ge \mathbf{n}\_{K} \\ \left( \left( \mathbf{T}\_{f} \right)\_{\mathrm{UE}} \right)^{+} &= \mathbf{Z}\_{EE} \left( \mathbf{Z}\_{U\mathbb{E}} \right)^{+} & \qquad \mathbf{n}\_{U} \ge \mathbf{n}\_{E} \end{aligned} \tag{29}$$

## **3. Numerical and experimental applications**

As explained before, the transmissibility matrices may be obtained from a numerical model (which should be updated for the range of frequencies involved) or from results obtained experimentally. In this section, the methodology used in each case is described and illustrated through a comparison example.

#### **3.1. Transmissibility in terms of the numerical model**

For the numerical model, one needs the knowledge of the structure within the discretization chosen, to create the receptance matrix *H*(ω), which is the inverse of the corresponding dynamic stiffness matrix *Z*(ω). Here, the numerical model is created using the Finite Element Method (FEM), although other alternatives may also be used. As seen before, the dynamic stiffness matrix is defined as:

$$\mathbf{Z}(\alpha) = \mathbf{K} - \alpha^2 \mathbf{M} + i\alpha \mathbf{C} \tag{30}$$

where *C* represents the viscous damping matrix, often of the proportional type, i.e., *C=αK+βM*, where *α* and *β* are constants to be evaluated experimentally.

To build the dynamic stiffness matrix, a specific structural finite element is chosen according to the approximation considered. For example, in the case of a reasonably long and slender beam one can use the Euler-Bernoulli beam element (instead of a shell or solid structural element). Then, the global matrices are assembled for the chosen discretization of the structure.

In order to improve the accuracy of the numerical model when simulating what is obtained experimentally, concentrated masses are often added at the corresponding nodes to model the effect of the accelerometers used in the testing positions.

Although the receptance matrix *H*(ω) is the inverse of the corresponding dynamic stiffness matrix, one should avoid such direct numerical inversion (frequency by frequency). Instead, *H*(ω) is calculated from eq. (3), after a modal analysis in free vibration.

Then, using (8) or (16), one can calculate the needed transmissibility matrices *(Td)UK <sup>A</sup>* or *(Tƒ)UK.* Of course, alternatively one may use the equivalent expressions (11) or (25), respectively.

## **3.2. Transmissibility in terms of experimental measurements**

It is important to note that only some of the co-ordinates of the set *E* have applied forces. This means that in (23) some rows of *F<sup>E</sup>* are zero and only the columns (in *ZEE*) whose co-ordinates have applied forces (set *K*) are needed for the transmissibility matrix. In other words, from the

From sections 2.2.1 and 2.2.2, one can conclude that for the direct problem of transmissibility

1


whereas in the inverse problem of transmissibility of forces there are some restrictions that can make this option not very useful in practice, especially when using the dynamic stiffnesses,

1

(28)

(29)


*n n*

*n n*

=- + *i* (30)

As explained before, the transmissibility matrices may be obtained from a numerical model (which should be updated for the range of frequencies involved) or from results obtained experimentally. In this section, the methodology used in each case is described and illustrated

For the numerical model, one needs the knowledge of the structure within the discretization chosen, to create the receptance matrix *H*(ω), which is the inverse of the corresponding dynamic stiffness matrix *Z*(ω). Here, the numerical model is created using the Finite Element Method (FEM), although other alternatives may also be used. As seen before, the dynamic

<sup>2</sup> *Z KM C* ( )

 ww

w

= ³

= - ³

*<sup>f</sup> UK UU U K UK*

*<sup>f</sup> EE UE U E UE*

set *E* only the co-ordinates corresponding to the *K* set are used.

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of forces there is no restrictions in the number of co-ordinates used:

since one needs to calculate the pseudo-inverse matrices:

**3. Numerical and experimental applications**

**3.1. Transmissibility in terms of the numerical model**

through a comparison example.

stiffness matrix is defined as:

(( ) ) ( )

(( ) ) ( )

*T ZZ*

( ) ( )

= -

*<sup>f</sup> UU UK UK*

*T HH*

( ) ( )

*<sup>f</sup> UE EE UE*

=

<sup>+</sup> <sup>+</sup>

*T HH*

<sup>+</sup> <sup>+</sup>

*T ZZ*

**2.3. Summary**

Depending on the type of transmissibility to obtain, the corresponding experimental setup should be established. Essentially, it is important to observe that for the transmissibility of motion one measures the FRFs relating co-ordinates *U* and *K* with co-ordinates A, normally using accelerometers and force transducers. For validation purposes, one may also measure the applied forces. In the examples presented next for the transmissibility of motion, a suspended (free-free) beam is always used.

For the transmissibility of forces, in the direct problem, one measures the applied forces at coordinates *K* (in the inverse problem, one measures the reaction forces at co-ordinates *U*). For validation purposes, one also measures the ones to be estimated. The test specimen for the transmissibility of forces is always a simply supported beam.

For the experimental setup, the following equipment is used:


In Fig. 3, a schematic representation of the experimental setup used for the force transmissi‐ bility tests is presented, in this case a simply supported beam with a single applied force.

The excitation signal used was a multi-sine transmitted to the exciter, with constant amplitude in the frequency. In reality the signal measured by the force transducer does not exhibits a constant amplitude along the frequency, as it depends on the dynamic response of the structure.

*3.3.1. Numerical model*

may be expressed as

as they match reasonably well.

needed numerical model of the beam.

bering of nodes and co-ordinates *y* coincide.

*3.3.2. Example 1 — Transmissibility of motion*

direction, denoted as Y3, Y7, Y13, and Y17 (Fig. 4).

model is included using *α* and *β* as updating parameters (Table 1).

**Figure 4.** Schematic representation of the accelerometers and force transducer positions.

**T** *<sup>A</sup> Ud K UK*

The standard two-node Euler-Bernouli bidimensional finite element is used here to build the

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The beam was discretized into sixteen finite elements, which correspond to *N* = 17 nodes, ordered from 1 up to 17. As the analysis and the model are limited to the plane xOy, each node has three degrees of freedom (which are *ux*, *uy*, *θ).* Hence, the matrices of the numerical model have an order of *3xN* for the free-free beam. In what the measurements are concerned, only the displacements and applied forces along the *y* direction are used and therefore the num‐

The model was updated using *E*, *ρ* and *I* as updating parameters and a proportional damping

In this example, the free-free beam has only one applied force at node 11 along the *y* direction, denoted as F11, and the measurements are taken at nodes 3, 7, 13 and 17 also along the *y*

The transmissibility of motion between the pair of nodes (3,7) and the pair of nodes (13,17)

( ) <sup>13</sup> 13,3 13,7 <sup>3</sup>

= Û= í ý ê ú í ý î þ ë û î þ

These transmissibilities were obtained from eq. (8), using the FRFs calculated from the numerical model and measured experimentally. Two of them are plotted in Fig. 5, where one can observe that both ways are able to produce the transmissibility response of the structure,

17 17,3 17,7 7

*Y Y T T Y Y T T Y Y* (31)

ì ü é ù ì ü

**Figure 3.** Example of the experimental setup developed for the transmissibility of forces.

For the transmissibility of motion, a beam suspended by nylon strings is used, to simulate freefree conditions. In order to facilitate the interchange of the available accelerometers between the measure positions without affecting the dynamics of the structure, it is important to add equivalent masses (dummies) to model the effect of the sensors.

After obtaining experimentally the needed receptances, by using (8) or (16) one can establish the transmissibility matrices *(Td)UK <sup>A</sup>* or *(Tƒ)UK*.

## **3.3. Examples**

The same steel beam was used in all the examples. With the purpose of illustrating the applicability of the presented formulations to obtain the transmissibility plots, the authors used the geometric and material parameters presented in Table 1. Note that these data correspond to the values obtained after updating the FE model.


**Table 1.** Beam properties (after updating).

## *3.3.1. Numerical model*

**Figure 3.** Example of the experimental setup developed for the transmissibility of forces.

equivalent masses (dummies) to model the effect of the sensors.

to the values obtained after updating the FE model.

the transmissibility matrices *(Td)UK*

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**Table 1.** Beam properties (after updating).

**3.3. Examples**

For the transmissibility of motion, a beam suspended by nylon strings is used, to simulate freefree conditions. In order to facilitate the interchange of the available accelerometers between the measure positions without affecting the dynamics of the structure, it is important to add

After obtaining experimentally the needed receptances, by using (8) or (16) one can establish

The same steel beam was used in all the examples. With the purpose of illustrating the applicability of the presented formulations to obtain the transmissibility plots, the authors used the geometric and material parameters presented in Table 1. Note that these data correspond

*m*

*m*

*m*<sup>2</sup>

*s* <sup>−</sup><sup>1</sup>

*<sup>A</sup>* or *(Tƒ)UK*.

Young's modulus – E 208 GPa Density – ρ 7840 kg/m3 Length – L 0.8 m Section width - b 5.0×10−<sup>3</sup>

Section height - h 20.0×10−<sup>3</sup>

Second moment of area - I 2.0883×10−10*m*<sup>4</sup> proportional damping - α 4 *s* proportional damping - β 2.0×10−<sup>6</sup>

Section area - A 1×10−<sup>4</sup>

The standard two-node Euler-Bernouli bidimensional finite element is used here to build the needed numerical model of the beam.

The beam was discretized into sixteen finite elements, which correspond to *N* = 17 nodes, ordered from 1 up to 17. As the analysis and the model are limited to the plane xOy, each node has three degrees of freedom (which are *ux*, *uy*, *θ).* Hence, the matrices of the numerical model have an order of *3xN* for the free-free beam. In what the measurements are concerned, only the displacements and applied forces along the *y* direction are used and therefore the num‐ bering of nodes and co-ordinates *y* coincide.

The model was updated using *E*, *ρ* and *I* as updating parameters and a proportional damping model is included using *α* and *β* as updating parameters (Table 1).

## *3.3.2. Example 1 — Transmissibility of motion*

In this example, the free-free beam has only one applied force at node 11 along the *y* direction, denoted as F11, and the measurements are taken at nodes 3, 7, 13 and 17 also along the *y* direction, denoted as Y3, Y7, Y13, and Y17 (Fig. 4).

**Figure 4.** Schematic representation of the accelerometers and force transducer positions.

The transmissibility of motion between the pair of nodes (3,7) and the pair of nodes (13,17) may be expressed as

$$\begin{aligned} \mathbf{Y}\_U = \left(\mathbf{T}\_d\right)^d\_{\ell \mathcal{K}} \mathbf{Y}\_K \quad \Longleftrightarrow \quad \begin{Bmatrix} Y\_{13} \\ Y\_{17} \end{Bmatrix} = \begin{bmatrix} T\_{13,3} & T\_{13,7} \\ T\_{17,3} & T\_{17,7} \end{bmatrix} \begin{Bmatrix} Y\_3 \\ Y\_7 \end{Bmatrix} \end{aligned} \tag{31}$$

These transmissibilities were obtained from eq. (8), using the FRFs calculated from the numerical model and measured experimentally. Two of them are plotted in Fig. 5, where one can observe that both ways are able to produce the transmissibility response of the structure, as they match reasonably well.

*Y Y* **T**

 *<sup>A</sup> Ud K UK*

Figure 4. Schematic representation of the accelerometers and force transducer positions.

17 17,3 17,7 7

*Y Y T T Y Y T T* 

<sup>13</sup> 13,3 13,7 <sup>3</sup>

response of the structure, as they match reasonably well.

The transmissibility of motion between the pair of nodes (3,7) and the pair of nodes (13,17) may be expressed as

experimentally. Two of them are plotted in Fig. 5, where one can observe that both ways are able to produce the transmissibility

*3.3.3. Example 2 — Transmissibility of forces*

The experimental setup is illustrated in Fig. 6.

{ } <sup>1</sup> 1,7

<sup>=</sup> (32)

17 17,7 *F T*

*F T*



Amplitude (dB)

Amplitude (dB)

7

*F*

**4. Force identification** 

**Figure 7.** Numerical and experimental transmissibilities (upper plot, T1,7; bottom plot, T17,7)

expressed as

In this case, a simply supported beam is considered with one applied force at node 7 and reactions at nodes 1 and 17. Only the magnitude of the forces is measured, and the transmis‐ sibility is obtained directly from the measurements and compared with the numerical results.

numerical results. The experimental setup is illustrated in Fig. 6.

Figure 6. Schematic representation of the positions of the force transducers.

**3.3.3. Example 2 — Transmissibility of forces** 

The force transmissibility relation between node 7 and the pair of nodes (1,17) may be

{ } <sup>1</sup> 1,7

17 17,7 ì ü é ù í ý <sup>=</sup> ê ú î þ ë û *F T*

7

model because it was not important, as these results are only of an illustrative type.

Numerical force transmissibility Experimental force transmissibility

Numerical force transmissibility Experimental force transmissibility

Figure 7. Numerical and experimental transmissibilities (upper plot, T1,7; bottom plot, T17,7)

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> -50

Frequency (Hz)

*<sup>F</sup> <sup>T</sup>* (32)

The force transmissibility relation between node 7 and the pair of nodes (1,17) may be expressed as

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In this case, a simply supported beam is considered with one applied force at node 7 and reactions at nodes 1 and 17. Only the magnitude of the forces is measured, and the transmissibility is obtained directly from the measurements and compared with the

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The force transmissibilities were obtained using eq. (16) and are plotted in Fig. 7, where it is clear that both numerical and experimental FRFs are able to produce the transmissibility response of the structure. Note that around 100 Hz there is a "bump" in the experimental curve, due to the effect of the supports of the beam themselves; this effect has not been included in the numerical

*F*

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> -50

Frequency (Hz)

(31)

Figure 5. Numerical and experimental transmissibilities (upper plot, T13,3; bottom plot, T13,7) **Figure 5.** Numerical and experimental transmissibilities (upper plot, T13,3; bottom plot, T13,7)

**Figure 6.** Schematic representation of the positions of the force transducers.

### *3.3.3. Example 2 — Transmissibility of forces* **3.3.3. Example 2 — Transmissibility of forces**

17 17,7

Figure 4. Schematic representation of the accelerometers and force transducer positions.

Figure 5. Numerical and experimental transmissibilities (upper plot, T13,3; bottom plot, T13,7)

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> -80

**Figure 5.** Numerical and experimental transmissibilities (upper plot, T13,3; bottom plot, T13,7)

**Figure 6.** Schematic representation of the positions of the force transducers.

Frequency (Hz)

17 17,3 17,7 7

Numerical motion trasmissibility Experimental motion trasmissibility

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> -80

Numerical motion trasmissibility Experimental motion trasmissibility

Frequency (Hz)

*Y Y T T Y Y T T* 

<sup>13</sup> 13,3 13,7 <sup>3</sup>

response of the structure, as they match reasonably well.

 *<sup>A</sup> Ud K UK*

*Y Y* **T**

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Amplitude (dB)

0

20



Amplitude (dB)

0

20

The transmissibility of motion between the pair of nodes (3,7) and the pair of nodes (13,17) may be expressed as

These transmissibilities were obtained from eq. (8), using the FRFs calculated from the numerical model and measured experimentally. Two of them are plotted in Fig. 5, where one can observe that both ways are able to produce the transmissibility

(31)

In this case, a simply supported beam is considered with one applied force at node 7 and reactions at nodes 1 and 17. Only the magnitude of the forces is measured, and the transmis‐ sibility is obtained directly from the measurements and compared with the numerical results. The experimental setup is illustrated in Fig. 6. In this case, a simply supported beam is considered with one applied force at node 7 and reactions at nodes 1 and 17. Only the magnitude of the forces is measured, and the transmissibility is obtained directly from the measurements and compared with the numerical results. The experimental setup is illustrated in Fig. 6. The force transmissibility relation between node 7 and the pair of nodes (1,17) may be expressed as

The force transmissibility relation between node 7 and the pair of nodes (1,17) may be expressed as { } <sup>1</sup> 1,7 7 *F T F F T* <sup>=</sup> (32)

7

model because it was not important, as these results are only of an illustrative type.

*<sup>F</sup> <sup>T</sup>* (32)

The force transmissibilities were obtained using eq. (16) and are plotted in Fig. 7, where it is clear that both numerical and experimental FRFs are able to produce the transmissibility response of the structure. Note that around 100 Hz there is a "bump" in the experimental curve, due to the effect of the supports of the beam themselves; this effect has not been included in the numerical

*F*

{ } <sup>1</sup> 1,7

17 17,7 ì ü é ù í ý <sup>=</sup> ê ú î þ ë û *F T*

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> -50 -40 -30 -20 -10 0 10 20 30 Frequency (Hz) Amplitude (dB) Numerical force transmissibility Experimental force transmissibility <sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> -50 -40 -30 -20 -10 0 10 20 30 Frequency (Hz) Amplitude (dB) Numerical force transmissibility Experimental force transmissibility

$$\mathsf{Fdgure 7.}\text{ Numerical and experimental transmission}\\
\text{bilites (upper plot, }\mathsf{T}\_{1\rangle};\text{ bottom plot, }\mathsf{T}\_{1\rangle\rangle}$$

**4. Force identification** 

The force transmissibilities were obtained using eq. (16) and are plotted in Fig. 7, where it is clear that both numerical and experimental FRFs are able to produce the transmissibility response of the structure. Note that around 100 Hz there is a "bump" in the experimental curve, due to the effect of the supports of the beam themselves; this effect has not been included in the numerical model because it was not important, as these results are only of an illustrative type.

The maximum number of forces must be less or equal to the dimension of the known dynamic

The successive combinations of the tested nodes are obtained according to the following

The error in each combination is kept in a vector to identify the combination with the least associated error (in absolute value). Firstly, the algorithm scrolls through the possible combi‐ nations of position and number of forces. For each combination, the associated error between the calculated vector *Y<sup>U</sup>* and the measured response vector *Ỹ* is calculated; this is carried out over a frequency range defined by the user. The error between the predicted and the measured

( ( ( )) ( ( )))

For each combination, the calculated error is kept in an entry of the error vector and analyzed

The accumulated error for a given combination of co-ordinates where *F* can be located is the norm of *ε*. The calculations are repeated for sucessive combinations of number and position of forces. The combination of the force locations that gives the lowest error leads to the number and position of the forces applied to the structure. As already mentioned, the maximum number of forces that can be found is equal to the dimension of the known dynamic response

As one does not know *a priori* how many forces exist, one has to follow a trial and error procedure that consists basically in assuming an increasing number of forces and the corre‐

sponding number of measurements; if the right number of forces is *Nf*

error log abs ( ) log abs ( )

w

2

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**ε** = {errori} (34)

, one has a minimum

 w- *i i iU U Y Y* (33)

dynamic response at each co-ordinate *i* can be defined as:

w

<sup>=</sup> å %

response vector *Y*.

scheme:

later on:

vector.
