**4. Force identification**

This section shall be divided into (i) part one for the force localization algorithm based on the transmissibility of motion and reconstruction using the measured responses and the updated numerical model, and (ii) part two for the force reconstruction using the transmissibility of forces.

## **4.1. Force localization based on the transmissibility of motion and force reconstruction**

The force identification problem is a difficult matter, as one has a limited knowledge of the measured responses, due to the complexity of the structure, lack of access to some locations, etc. In other words, there are difficulties due to the incompleteness of the model.

Due to this difficulty in calculating the load vector directly, the authors propose to divide the process into two distinct steps:


For the first step, a search for the number and position of forces using the transmissibility of motion is performed. Essentially, this step consists of searching for the transmissibility matrix correspondent to the dynamics of the system and using the available measured data and the numerical model involved.

Once the corresponding transmissibility matrix is found, one has a solution for the number and position of the forces applied to the structure.

The second step consists of reconstructing the load vector with the results obtained in the first step. A more detailed description about this methodology is given in the following sections.

## *4.1.1. Force localization*

In a first stage, to apply the method proposed in the previous section, one finds the transmis‐ sibility matrix that converts the dynamic responses *YK* into *YU*. As one does not know the position of the applied forces, it was decided to cover all the possibilities until the calculated responses (*YU*) match the measured ones *Ỹ*, over a range of frequencies. To calculate the vector *YU* one may use either eq. (7) or (11).

The maximum number of forces must be less or equal to the dimension of the known dynamic response vector *Y*.

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

$$\begin{array}{ll} \text{For one force}: & \{ (1), \ldots \quad (N) \}; \\\\ \text{For two forces}: & \begin{cases} (1,2), \ldots \quad (1,N); \\ (2,3), \ldots \quad (2,N); \\ (3,4), \ldots \quad (3,N); \\ \vdots \\ (N-1,N); \\ \text{For three forces}: & \begin{cases} (1,2,3), \ldots \quad (1,2,N) \\ \vdots \end{cases} \end{array} \\\\ \text{For three forces}: & \begin{cases} (1,2,3), \ldots \quad (1,2,N) \\ \vdots \end{cases} \end{array}$$

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

This section shall be divided into (i) part one for the force localization algorithm based on the transmissibility of motion and reconstruction using the measured responses and the updated numerical model, and (ii) part two for the force reconstruction using the transmissibility of

**4.1. Force localization based on the transmissibility of motion and force reconstruction**

etc. In other words, there are difficulties due to the incompleteness of the model.

applied forces using the concept of transmissibility of motion;

The force identification problem is a difficult matter, as one has a limited knowledge of the measured responses, due to the complexity of the structure, lack of access to some locations,

Due to this difficulty in calculating the load vector directly, the authors propose to divide the

**1.** the localization of the forces, i.e. the identification of the number and position of the

For the first step, a search for the number and position of forces using the transmissibility of motion is performed. Essentially, this step consists of searching for the transmissibility matrix correspondent to the dynamics of the system and using the available measured data and the

Once the corresponding transmissibility matrix is found, one has a solution for the number

The second step consists of reconstructing the load vector with the results obtained in the first step. A more detailed description about this methodology is given in the following sections.

In a first stage, to apply the method proposed in the previous section, one finds the transmis‐ sibility matrix that converts the dynamic responses *YK* into *YU*. As one does not know the position of the applied forces, it was decided to cover all the possibilities until the calculated responses (*YU*) match the measured ones *Ỹ*, over a range of frequencies. To calculate the vector

type.

forces.

**4. Force identification**

118 Advances in Vibration Engineering and Structural Dynamics

process into two distinct steps:

**2.** the load vector reconstruction.

and position of the forces applied to the structure.

numerical model involved.

*4.1.1. Force localization*

*YU* one may use either eq. (7) or (11).

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

$$\text{error}\_{i} = \sum\_{o} \left( \log \left( \text{abs} \left( \tilde{Y}\_{U\_{i}}(oo) \right) \right) - \log \left( \text{abs} \left( Y\_{U\_{i}}(oo) \right) \right) \right)^{2} \tag{33}$$

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

$$\mathbf{z} = \begin{pmatrix} \mathbf{error}\_i \end{pmatrix} \tag{34}$$

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 vector.

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* , one has a minimum error **ε** for a certain set of co-ordinates. When one proceeds and assumes *Nf +1* forces and measurements, the error will be higher then **ε**, telling us that the right answer was effectively *Nf* at a certain set of co-ordinates.

It is clear that all the combinations of the *Nf +1* forces that contain the right combination of the *Nf* forces should exhibit a local minimum, though not the absolute one.

The method was implemented computationally (in MatLab®).

#### *4.1.2. Force reconstruction*

In a second step, the reconstruction of the force amplitudes consists of solving an inverse problem using the measured dynamic responses *Y*K:

$$\boldsymbol{F}\_{\boldsymbol{A}} = \left(\boldsymbol{H}\_{\boldsymbol{K}\boldsymbol{A}}\right)^{\*}\boldsymbol{Y}\_{\boldsymbol{K}} \tag{35}$$

Considering these responses, the maximum number of identifiable applied forces is three, as explained before. The forces are uncorrelated and applied to the structure at co-ordinates 1 and 5. A series of force combinations have to be systematically generated as follows. In this

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Applying the localization method described in the previous subsection 4.1.1, one obtains the

<sup>0</sup> <sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>600</sup> <sup>700</sup> <sup>800</sup> <sup>10</sup>-30

It is clear that there are several situations (combinations) where the error is close to zero and

The minimum error happens with the combination number 21, corresponding to two forces applied at co-ordinates 1 and 5, thus identifying the correct positions and number of forces

combination number

case, all combinations up to three forces have been considered:

plot of the error defined in eq. (33), as shown in Fig.9.

**Figure 9.** Accumulated error in frequency for each combination of forces.

10-20

other where is not.

(Table 2).

10-10

accumulated error

10<sup>0</sup>

10<sup>10</sup>

Note that for the given system to be invertible, the number of dynamic responses to be used (set K) must be higher or equal than the number of applied forces (set A). However, this is always verified, as in the first step one has already imposed it.

#### *4.1.3. Example 3 — Localization of the applied forces*

This is a numerical example, illustrated in Fig.8, where a set of uncorrelated forces is applied at co-ordinates 1 and 5 (set *A*), and one uses the three known responses (set *K*) to identify the number and location of forces.

A set of simulated results (to mimic the experimental measurements) are obtained at nodes 1, 3, 5, 11 and 17 (see Fig. 8); they define the following sets:

$$\mathbf{Y}\_K = \begin{pmatrix} Y\_3 & Y\_5 & Y\_{17} \end{pmatrix}^T \quad \text{and} \quad \mathbf{Y}\_U = \begin{pmatrix} Y\_1 & Y\_{11} \end{pmatrix}^T \tag{36}$$

**Figure 8.** Illustration with the responses and applied force locations for example 3.

Considering these responses, the maximum number of identifiable applied forces is three, as explained before. The forces are uncorrelated and applied to the structure at co-ordinates 1 and 5. A series of force combinations have to be systematically generated as follows. In this case, all combinations up to three forces have been considered:

error **ε** for a certain set of co-ordinates. When one proceeds and assumes *Nf +1* forces and measurements, the error will be higher then **ε**, telling us that the right answer was effectively

In a second step, the reconstruction of the force amplitudes consists of solving an inverse

Note that for the given system to be invertible, the number of dynamic responses to be used (set K) must be higher or equal than the number of applied forces (set A). However, this is

This is a numerical example, illustrated in Fig.8, where a set of uncorrelated forces is applied at co-ordinates 1 and 5 (set *A*), and one uses the three known responses (set *K*) to identify the

A set of simulated results (to mimic the experimental measurements) are obtained at nodes 1,

*K U YYY Y Y* (36)

**Y Y** = = { 3 5 17} and { 1 11} *T T*

( ) +

forces should exhibit a local minimum, though not the absolute one.

The method was implemented computationally (in MatLab®).

always verified, as in the first step one has already imposed it.

problem using the measured dynamic responses *Y*K:

*4.1.3. Example 3 — Localization of the applied forces*

3, 5, 11 and 17 (see Fig. 8); they define the following sets:

**Figure 8.** Illustration with the responses and applied force locations for example 3.

number and location of forces.

 *+1* forces that contain the right combination of the

*FHY A KA K* = (35)

*Nf*

*Nf*

at a certain set of co-ordinates.

*4.1.2. Force reconstruction*

It is clear that all the combinations of the *Nf*

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Applying the localization method described in the previous subsection 4.1.1, one obtains the plot of the error defined in eq. (33), as shown in Fig.9.

**Figure 9.** Accumulated error in frequency for each combination of forces.

It is clear that there are several situations (combinations) where the error is close to zero and other where is not.

The minimum error happens with the combination number 21, corresponding to two forces applied at co-ordinates 1 and 5, thus identifying the correct positions and number of forces (Table 2).


Considering these responses, the maximum number of identifiable applied forces is two, as

This is an experimental example, where a multisine signal is fed into the shaker, attached to the beam at co-ordinate 13. Later on, the

A series of force combinations was systematically generated as described before. Applying the

A series of force combinations was systematically generated as described before. Applying the localization method, one obtains the

error

As one can see, the absolute minimum corresponds to combination no. 13, which is right because this combination represents the

As one can see, the absolute minimum corresponds to combination no. 13, which is right because this combination represents the force applied at co-ordinate 13. So, the method could

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>10</sup><sup>0</sup>

combination number

Once the localization of the force is accomplished, its reconstruction is a simple calculation, using the measured displacements relating those co-ordinates to the force location. As the force is located at node 13, taking the measurement at co-ordinates 5, 7, 11

> 5 5 5,13 5,13 7 7 7,13 7,13 13 13 11,13 11,13 11 11 15,13 15,13 <sup>15</sup> <sup>15</sup>

 

5 5 5,13 5,13 7 7 7,13 7,13 13 13 11,13 11,13 11 11 15,13 15,13 <sup>15</sup> <sup>15</sup>

% % % % % % % %

*Y Y H H Y Y H H F F Y Y H H H H Y Y*

<sup>+</sup> ìü ìü ìüìü ïï ïï ïïïï ïï ïï íýíýíýíý =Û=

îþîþ îþ îþ

Using the information from the measured responses, the reconstruction is now immediate. To validate this methodology, the result was ploted against its experimentally measured curve, as in Fig. 11. One may affirm that the method is able to predict the applied

*Y Y H H Y Y H H F F Y Y H H H H Y Y*

Once the localization of the force is accomplished, its reconstruction is a simple calculation, using the measured displacements relating those co-ordinates to the force location. As the force is located at node 13, taking the measurement at co-ordinates 5, 7, 11 and 15, for instance, it

 

force applied at co-ordinate 13. So, the method could localize correctly the position of the force at co-ordinate 13.

force. It is possible that a better matching of the curves may be obtained with a finer updated FE model.

ïïïïïïïï

7 15 5 11 and *T T* **Y Y** *K U Y Y Y Y* (37)

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more combinations including co-ordinate 13

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explained before.

10<sup>1</sup>

10<sup>2</sup>

accumulated error

10<sup>3</sup>

10<sup>4</sup>

graph of Fig. 10.

applied force is compared with the reconstructed one.

localization method, one obtains the graph of Fig. 10.

The experimental measurements are obtained at nodes 5, 7, 11 and 15. The measured vectors are as follows:

Considering these responses, the maximum number of identifiable applied forces is two, as explained before.

**4.1.4 Example 4 – localization and reconstruction 1** 

Figure 10. Accumulated error in frequency for each force combination.

**Figure 10.** Accumulated error in frequency for each force combination.

localize correctly the position of the force at co-ordinate 13.

and 15, for instance, it follows that:

follows that:

**Table 2.** Data of the combination with minimum error.

To better understand why there exist more combinations with small errors, Table 3 shows these combinations with its corresponding error value. All of them have a common group of coordinates, corresponding to the correct combination of number and positions of the forces. In this case the correct positions are obtained with success through the minimum error.


**Table 3.** Some combinations and their respective error.

This illustrates the localization step performed with two forces, whose number and location were not known at the beginning. From these results, it can be stated that the transmissibility of motion can be considered as adequate to perform this task. Note that, in spite of the high numbers of combinations that exist, the computations are relatively quick, as they involve only sub-matrices, and for the first permutations they are of a small order.

#### *4.1.4. Example 4 — Localization and reconstruction 1*

This is an experimental example, where a multisine signal is fed into the shaker, attached to the beam at co-ordinate 13. Later on, the applied force is compared with the reconstructed one.

The experimental measurements are obtained at nodes 5, 7, 11 and 15. The measured vectors are as follows:

$$\mathbf{Y}\_K = \begin{pmatrix} Y\_\uparrow & Y\_{1\uparrow} \end{pmatrix}^T \quad \text{and} \quad \mathbf{Y}\_U = \begin{pmatrix} Y\_\uparrow & Y\_{11} \end{pmatrix}^T \tag{37}$$

Considering these responses, the maximum number of identifiable applied forces is two, as explained before. applied force is compared with the reconstructed one. The experimental measurements are obtained at nodes 5, 7, 11 and 15. The measured vectors are as follows:

This is an experimental example, where a multisine signal is fed into the shaker, attached to the beam at co-ordinate 13. Later on, the

A series of force combinations was systematically generated as described before. Applying the localization method, one obtains the graph of Fig. 10. 7 15 5 11 and *T T* **Y Y** *K U Y Y Y Y* (37) Considering these responses, the maximum number of identifiable applied forces is two, as explained before.

A series of force combinations was systematically generated as described before. Applying the localization method, one obtains the

**Figure 10.** Accumulated error in frequency for each force combination.

Figure 10. Accumulated error in frequency for each force combination.

**Combination Number of forces Position of the forces**

**Table 2.** Data of the combination with minimum error.

122 Advances in Vibration Engineering and Structural Dynamics

**Table 3.** Some combinations and their respective error.

*4.1.4. Example 4 — Localization and reconstruction 1*

are as follows:

**Number of identified forces**

21 2 1, 5 2 1, 5 2,69e-29

To better understand why there exist more combinations with small errors, Table 3 shows these combinations with its corresponding error value. All of them have a common group of coordinates, corresponding to the correct combination of number and positions of the forces. In

**Combination Real position of the forces Absolute error** 1,5 2,69e-29 1,2,5 7,99e-26 1,3,5 2,85e-27 1,4,5 6,99e-28 1,5,7 8,70e-29 1,5,8 1,86e-28 1,5,9 1,82e-28 1,5,10 5,54e-28

This illustrates the localization step performed with two forces, whose number and location were not known at the beginning. From these results, it can be stated that the transmissibility of motion can be considered as adequate to perform this task. Note that, in spite of the high numbers of combinations that exist, the computations are relatively quick, as they involve only

This is an experimental example, where a multisine signal is fed into the shaker, attached to the beam at co-ordinate 13. Later on, the applied force is compared with the reconstructed one.

The experimental measurements are obtained at nodes 5, 7, 11 and 15. The measured vectors

*K U Y Y Y Y* (37)

**Y Y** = = { 7 15} and { 5 11} *T T*

sub-matrices, and for the first permutations they are of a small order.

this case the correct positions are obtained with success through the minimum error.

**Identified positions**

**Absolute error**

graph of Fig. 10.

As one can see, the absolute minimum corresponds to combination no. 13, which is right because this combination represents the force applied at co-ordinate 13. So, the method could localize correctly the position of the force at co-ordinate 13. Once the localization of the force is accomplished, its reconstruction is a simple calculation, using the measured displacements As one can see, the absolute minimum corresponds to combination no. 13, which is right because this combination represents the force applied at co-ordinate 13. So, the method could localize correctly the position of the force at co-ordinate 13.

relating those co-ordinates to the force location. As the force is located at node 13, taking the measurement at co-ordinates 5, 7, 11 and 15, for instance, it follows that: 5 5 5,13 5,13 *Y Y H H Y Y H H* Once the localization of the force is accomplished, its reconstruction is a simple calculation, using the measured displacements relating those co-ordinates to the force location. As the force is located at node 13, taking the measurement at co-ordinates 5, 7, 11 and 15, for instance, it follows that:

7 7 7,13 7,13 13 13

was ploted against its experimentally measured curve, as in Fig. 11. One may affirm that the method is able to predict the applied

force. It is possible that a better matching of the curves may be obtained with a finer updated FE model.

*F F*

$$\begin{aligned} \begin{bmatrix} \tilde{Y}\_{5} \\ \tilde{Y}\_{7} \\ \tilde{Y}\_{11} \\ \tilde{Y}\_{13} \end{bmatrix} = \begin{bmatrix} H\_{5,13} \\ H\_{7,13} \\ H\_{11,13} \\ H\_{13,13} \end{bmatrix} \end{aligned} \left\{ F\_{13} \oplus F\_{13} = \begin{bmatrix} H\_{5,13} \\ H\_{7,13} \\ H\_{11,13} \\ H\_{13,13} \end{bmatrix}^{\*} \begin{bmatrix} \tilde{Y}\_{5} \\ \tilde{Y}\_{7} \\ \tilde{Y}\_{11} \\ \tilde{Y}\_{15} \end{bmatrix} \right\} \tag{38}$$

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Using the information from the measured responses, the reconstruction is now immediate. To validate this methodology, the result was ploted against its experimentally measured curve, as in Fig. 11. One may affirm that the method is able to predict the applied force. It is possible that a better matching of the curves may be obtained with a finer updated FE model.

Figure 11. Comparison between the experimental and the reconstructed forces at co-ordinate 13.

The experimental measures are obtained at nodes 3, 7, 17 and 13. The measured vectors are as follows.

Here, the same multisine signal is fed into two shakers, attached to the free-free beam at the co-ordinates 1 and 11. Later on, the

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

Force reconstruction at co-ordinate 13 Measured force at co-ordinate 13

Frequency (Hz)

3 7 17 <sup>13</sup> , , and *<sup>T</sup>*

Considering these responses, the maximum number of identifiable applied forces is three. Applying the localization method, one

As one can see, the absolute minimum corresponds to the right combination (no. 27), representing the forces applied at co-ordinates 1 and 11. So, the method located correctly the position of the forces. Note that, as there are two forces, a high number of combinations with small errors appear; observing the co-ordinates of those combinations, the best of them have in common the correct co-ordinates where the forces are applied and the others include co-ordinates physically close to them. Again, the force reconstruction is obtained

As one can see, the absolute minimum corresponds to the right combination (no. 27), repre‐ senting the forces applied at co-ordinates 1 and 11. So, the method located correctly the position of the forces. Note that, as there are two forces, a high number of combinations with small errors appear; observing the co-ordinates of those combinations, the best of them have in common the correct co-ordinates where the forces are applied and the others include coordinates physically close to them. Again, the force reconstruction is obtained using the

<sup>0</sup> <sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> <sup>600</sup> <sup>700</sup> <sup>800</sup> <sup>10</sup><sup>0</sup>

combination number

 3,1 3,11 3,1 3,11 3 7,1 7,11 1 1 7,1 7,11 7 13,1 13,11 11 11 13,1 13,11 13 17,1 17,11 17,1 17,11 17 17

 3,1 3,11 3,1 3,11 3 7,1 7,11 1 1 7,1 7,11 7 13,1 13,11 11 11 13,1 13,11 13 17,1 17,11 17,1 17,11 17 17

% % % % % % % %

*Y Y H H H H Y Y H H F F H H Y Y HH F F HH Y Y H H H H*

<sup>+</sup> ìü éùéù ìü ïï êúêú ïï ïï ìüìü ïï íý <sup>=</sup> íýíý Û= íý êúêú ïï îþîþ ïï ïï ïï îþ ëûëû îþ

Figs. 13 and 14 present the reconstructed forces *versus* the measured ones. Again, one can state that the method is able to predict the

Figs. 13 and 14 present the reconstructed forces versus the measured ones. Again, one can state

*Y Y H H H H Y Y H H F F H H Y Y HH F F HH Y Y H H H H*

 

**Y Y** *K U YYY Y* (39)

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(40)

(40)

**4.1.5 Example 5 – localization and reconstruction 2** 

Figure 12. Accumulated error in frequency for each force combination.

measured displacements:

**Figure 12.** Accumulated error in frequency for each force combination.

that the method is able to predict the applied force.

10<sup>1</sup>

10<sup>2</sup>

accumulated error

10<sup>3</sup>

10<sup>4</sup>

applied forces are compared with the reconstructed ones.

0.05

Amplitude of force (N)

0.1

obtains the plot shown in Fig.12.

using the measured displacements:

applied force.

**Figure 11.** Comparison between the experimental and the reconstructed forces at co-ordinate 13.

#### *4.1.5. Example 5 — Localization and reconstruction 2*

F11

Here, the same multisine signal is fed into two shakers, attached to the free-free beam at the co-ordinates 1 and 11. Later on, the applied forces are compared with the reconstructed ones.

The experimental measures are obtained at nodes 3, 7, 17 and 13. The measured vectors are as follows.

$$\mathbf{Y}\_{\mathcal{K}} = \begin{Bmatrix} Y\_3, Y\_7, Y\_{17} \end{Bmatrix}^T \quad \text{and} \quad \mathbf{Y}\_U = \begin{Bmatrix} Y\_{13} \end{Bmatrix} \tag{39}$$

Considering these responses, the maximum number of identifiable applied forces is three. Applying the localization method, one obtains the plot shown in Fig.12.

**Y Y** *K U YYY Y* (39)

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Here, the same multisine signal is fed into two shakers, attached to the free-free beam at the co-ordinates 1 and 11. Later on, the

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

Force reconstruction at co-ordinate 13 Measured force at co-ordinate 13

Frequency (Hz)

3 7 17 <sup>13</sup> , , and *<sup>T</sup>*

Considering these responses, the maximum number of identifiable applied forces is three. Applying the localization method, one

**Figure 12.** Accumulated error in frequency for each force combination.

Figure 12. Accumulated error in frequency for each force combination.

Figure 11. Comparison between the experimental and the reconstructed forces at co-ordinate 13.

**4.1.5 Example 5 – localization and reconstruction 2** 

applied forces are compared with the reconstructed ones.

0.05

Amplitude of force (N)

0.1

obtains the plot shown in Fig.12.

applied force.

Using the information from the measured responses, the reconstruction is now immediate. To validate this methodology, the result was ploted against its experimentally measured curve, as in Fig. 11. One may affirm that the method is able to predict the applied force. It is possible

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

Frequency (Hz)

Here, the same multisine signal is fed into two shakers, attached to the free-free beam at the co-ordinates 1 and 11. Later on, the applied forces are compared with the reconstructed ones.

The experimental measures are obtained at nodes 3, 7, 17 and 13. The measured vectors are as

Considering these responses, the maximum number of identifiable applied forces is three.

*K U YYY Y* (39)

**Y Y** = = { 3 7 17 , , and } { <sup>13</sup>} *<sup>T</sup>*

Applying the localization method, one obtains the plot shown in Fig.12.

**Figure 11.** Comparison between the experimental and the reconstructed forces at co-ordinate 13.

*4.1.5. Example 5 — Localization and reconstruction 2*

124 Advances in Vibration Engineering and Structural Dynamics

that a better matching of the curves may be obtained with a finer updated FE model.

Force reconstruction at co-ordinate 13 Measured force at co-ordinate 13

0.05

follows.

Amplitude of force (N)

F11

0.1

As one can see, the absolute minimum corresponds to the right combination (no. 27), representing the forces applied at co-ordinates 1 and 11. So, the method located correctly the position of the forces. Note that, as there are two forces, a high number of combinations with small errors appear; observing the co-ordinates of those combinations, the best of them have in common the correct co-ordinates where the forces are applied and the others include co-ordinates physically close to them. Again, the force reconstruction is obtained using the measured displacements: 3 3,1 3,11 3,1 3,11 3 7 7,1 7,11 1 1 7,1 7,11 7 *Y Y H H H H Y Y H H F F H H* As one can see, the absolute minimum corresponds to the right combination (no. 27), repre‐ senting the forces applied at co-ordinates 1 and 11. So, the method located correctly the position of the forces. Note that, as there are two forces, a high number of combinations with small errors appear; observing the co-ordinates of those combinations, the best of them have in common the correct co-ordinates where the forces are applied and the others include coordinates physically close to them. Again, the force reconstruction is obtained using the measured displacements:

$$
\begin{bmatrix} \tilde{Y}\_{1} \\ \tilde{Y}\_{1} \\ \tilde{Y}\_{13} \\ \tilde{Y}\_{13} \\ \tilde{Y}\_{17} \end{bmatrix} = \begin{bmatrix} H\_{3,1} & H\_{3,11} \\ H\_{7,1} & H\_{7,11} \\ H\_{13,1} & H\_{13,11} \\ H\_{13,1} & H\_{7,11} \end{bmatrix} \begin{Bmatrix} F\_{1} \\ F\_{1} \\ \end{Bmatrix} \Leftrightarrow \begin{Bmatrix} F\_{1} \\ F\_{1} \\ F\_{11} \end{Bmatrix} = \begin{bmatrix} H\_{3,1} & H\_{3,11} \\ H\_{7,1} & H\_{7,11} \\ H\_{13,1} & H\_{13,11} \\ H\_{7,1} & H\_{7,11} \end{bmatrix} \begin{Bmatrix} \tilde{Y}\_{1} \\ \tilde{Y}\_{1} \\ \tilde{Y}\_{1} \\ \tilde{Y}\_{1} \end{Bmatrix} \tag{40}
$$

Figs. 13 and 14 present the reconstructed forces versus the measured ones. Again, one can state that the method is able to predict the applied force.

**4.2. Force reconstruction based on the transmissibility of forces**

problems involving estimation of forces are here considered:

set of known reactions, as expressed by equation (17).

the number of reactions. So, it is a required condition that*nK* ≤*nU* .

*4.2.1. Example 6 — Reaction forces estimation knowing the applied ones*

Reconstructed reaction at node 1 Measured reaction at node 1

**Figure 15.** Comparison between the experimental and estimated force reaction F1

reactions at nodes 1 and 17 (set *U*).

0.005

0.01

0.015

0.02

0.025

Amplitude (N)

F15

0.03

0.035

0.04

0.045

from a set of known applied loads, as expressed by equation (15);

The main objective of this section is the estimation of the existing forces (reactions or applied forces) in the structure using the MDOF concept of transmissibility of forces. Two types of

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**1.** Reaction forces estimation, with the objective of calculating a set of unknown reactions

**2.** Applied forces estimation, with the objective of calculating a set of applied forces from a

The method to estimate the applied forces is limited by the number of reactions, as it is not possible to perform the needed pseudo-inverse if the number of applied forces is greater than

The first experimental reconstruction case was carried out with the configuration presented in Fig. 6 (simple supported beam). One has a single applied force at node 7 (set *K*) and two

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

Frequency (Hz)

In this case, with one applied force and two reactions, the transmissibility has a dimension of 2x1 and can be obtained either from the receptance matrix or from the dynamic stiffness matrix,

F13 **Figure 13.** Comparison between the experimental and the reconstructed forces at co-ordinate1

F14 **Figure 14.** Comparison between the experimental and the reconstructed forces at co-ordinate11

## **4.2. Force reconstruction based on the transmissibility of forces**

The main objective of this section is the estimation of the existing forces (reactions or applied forces) in the structure using the MDOF concept of transmissibility of forces. Two types of problems involving estimation of forces are here considered:


The method to estimate the applied forces is limited by the number of reactions, as it is not possible to perform the needed pseudo-inverse if the number of applied forces is greater than the number of reactions. So, it is a required condition that*nK* ≤*nU* .

### *4.2.1. Example 6 — Reaction forces estimation knowing the applied ones*

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

Force reconstructed at co-ordinate 1 Measured force at co-ordinate 1

Frequency (Hz)

Force reconstructed at co-ordinate 11 Measured force at co-ordinate 11

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

Frequency (Hz)

F13 **Figure 13.** Comparison between the experimental and the reconstructed forces at co-ordinate1

F14 **Figure 14.** Comparison between the experimental and the reconstructed forces at co-ordinate11

0.02

0.02

0.04

0.06

Amplitude of force (N)

0.08

0.1

0.04

0.06

Amplitude of force (N)

0.08

0.1

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The first experimental reconstruction case was carried out with the configuration presented in Fig. 6 (simple supported beam). One has a single applied force at node 7 (set *K*) and two reactions at nodes 1 and 17 (set *U*).

F15 **Figure 15.** Comparison between the experimental and estimated force reaction F1

In this case, with one applied force and two reactions, the transmissibility has a dimension of 2x1 and can be obtained either from the receptance matrix or from the dynamic stiffness matrix, as proposed in this work. The two different formulations are equivalent and are very close to the experimental results.

As the objective is to estimate the reaction forces, one needs the numerical model for the transmissibility matrix and to know the experimental vector of applied forces, which in this case has only one component. The calculation of the reactions is then reduced to the following form:

$$
\begin{Bmatrix} F\_1 \\ F\_{1\gamma} \end{Bmatrix} = \begin{bmatrix} T\_{1,\gamma} \\ T\_{1\gamma,\tau} \end{bmatrix} \begin{Bmatrix} F\_\tau \end{Bmatrix} \tag{41}
$$

Knowing the reaction forces, the reconstruction of the applied force follows, in this case, the

17,7 17

The reconstructed values are compared with the experimentally measured ones, in Fig. 17.

Reconstructed applied force at node 7 Measured applied force at node 7

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

Frequency (Hz)

In all the tested cases a good approach of the reconstructed forces was verified, as the values obtained by the direct and inverse problems are close enough to the experimentally measured

In this work, the authors reviewed recent advances in the application of MDOF transmissibil‐

From these developments, one can draw the following main conclusions:

F17 **Figure 17.** Comparison between the experimental and estimated applied load F7

ity-based methods for the identification of forces.

*<sup>T</sup> <sup>F</sup>* (42)

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

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

7

*F*

following expression:

0.005

ones.

**5. Conclusions**

0.01

0.015

0.02

0.025

Amplitude (N)

0.03

0.035

0.04

0.045

F16 **Figure 16.** Comparison between the experimental and estimated force reaction F17

From Figs. 15 and 16, it is clear that the reconstructed reactions match reasonably well the experimentally measured ones. Better results may even be possible if a finer updating procedure on the FE model is achieved.

#### *4.2.2. Example 7 — Applied force reconstruction knowing the reaction forces*

For the reconstruction of the applied forces (the inverse problem), one needs to known the vector of the reaction forces {*FU*} and the inverse transmissibility matrix that can be obtained from the numerical model.

In this case the same configuration presented in Fig. 6 was used (simple supported beam), with one applied force at node 7 (set *K*) and two reaction forces at nodes 1 and 17 (set *U*).

Knowing the reaction forces, the reconstruction of the applied force follows, in this case, the following expression:

$$\{F\_{\tau}\} = \begin{bmatrix} T\_{1,\tau} \\ T\_{1\tau,\tau} \end{bmatrix}^{\*} \begin{Bmatrix} F\_{1} \\ F\_{1\tau} \end{Bmatrix} \tag{42}$$

The reconstructed values are compared with the experimentally measured ones, in Fig. 17.

F17 **Figure 17.** Comparison between the experimental and estimated applied load F7

In all the tested cases a good approach of the reconstructed forces was verified, as the values obtained by the direct and inverse problems are close enough to the experimentally measured ones.

## **5. Conclusions**

as proposed in this work. The two different formulations are equivalent and are very close to

As the objective is to estimate the reaction forces, one needs the numerical model for the transmissibility matrix and to know the experimental vector of applied forces, which in this case has only one component. The calculation of the reactions is then reduced to the following

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

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

Reconstructed reaction at node 17 Measured reaction at node 17

F16 **Figure 16.** Comparison between the experimental and estimated force reaction F17

*4.2.2. Example 7 — Applied force reconstruction knowing the reaction forces*

procedure on the FE model is achieved.

from the numerical model.

7

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

Frequency (Hz)

From Figs. 15 and 16, it is clear that the reconstructed reactions match reasonably well the experimentally measured ones. Better results may even be possible if a finer updating

For the reconstruction of the applied forces (the inverse problem), one needs to known the vector of the reaction forces {*FU*} and the inverse transmissibility matrix that can be obtained

In this case the same configuration presented in Fig. 6 was used (simple supported beam), with

one applied force at node 7 (set *K*) and two reaction forces at nodes 1 and 17 (set *U*).

*<sup>F</sup> <sup>T</sup>* (41)

*F*

the experimental results.

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form:

0.005

0.01

0.015

0.02

0.025

Amplitude (N)

0.03

0.035

0.04

0.045

In this work, the authors reviewed recent advances in the application of MDOF transmissibil‐ ity-based methods for the identification of forces.

From these developments, one can draw the following main conclusions:

**i.** it is possible to localize forces acting on a structure, based on the motion transmissi‐ bility matrix, comparing the expected responses with the ones measured along the structure;

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