**3. Experimental Set-Up**

To investigate the impact force reconstruction experimentally, the structure shown in Figure 2 is used with the parameters given in Table 1. The structure is a hollow rectangular steel beam, fixed at the bottom and free at the top, on which eight lumped masses are clamped at equally distributed distances. In the following, these masses are called level 1 to 8 with level 1 at the bottom and level 8 at the top.

Four sensors, namely, two DC-response MEMS accelerometers (Measurement Specialties 4000A-005) at level 3 and level 8, a laser Doppler vibrometer (Polytec PDV-100) at level 3, and a laser triangulation sensor (Micro-Epsilon optoNCDT 1302, ILD 1302-50) at level 3 were employed to gauge system responses, as shown in Figure 2.

The impact forces were applied by a modally tuned instrumented impact hammer that provided the measurement of the actual dynamic load. This hammer was used with different tips (i.e., steel, hard rubber, medium rubber, and soft rubber tips) in order to simulate various hit modes.

**Figure 2.** Experimentalset-up showing the multi-storey tower and primary response transducers including (1) laser Doppler vibrometer, (2) laser triangulation sensor, and (3) accelerometer.


**Table 1.** Experimental set-up parameters.

#### **4. Results and Discussion**

#### *4.1. Effect of Regularization*

As stated previously, identification of transfer functions as well as reconstruction of impact forces are both ill-posed problems. Solving (6) using the least-square technique without applying regularization, given in (8), led to transfer functions with magnitudes in the order of 103, while solving the problem considering regularization, defined in (9), gave the transfer functions with magnitudes in the order of 10−5. Figure 3 depicts the reconstructed impact force implementing the transfer function obtained with and without regularization. In this figure, the impact force was applied at level 5, and the vibration response was measured in level 8. As seen in Figure 3, the reconstruction process utilizing the transfer function obtained without regularization yields a reconstructed impact force with virtually zero amplitude. This is because the transfer function has large singular values. The large singular values are inverted through the inverse algorithm, causing the reconstructed force to approach zero.

**Figure 3.** The effect of regularization in establishing the transfer function and reconstructing the impact force.

#### *4.2. Establishing the Transfer Function*

Different hammer tips can produce different half-sine shapes of impact force, with different rising patterns and time durations. As shown in Figure 4, harder tips produce sharper graphs of impact force, i.e., closer to the graph of Dirac delta function. The sharpest graph shows the results of the hammer with the hardest tip (steel tip), which, theoretically, should result in the most accurate transfer function. This result is also shown illustratively in Figure 5 in which the impact force applied by the soft rubber tip at level 4 is reconstructed by using the measured acceleration at level 8 and different transfer functions established, namely, by the steel tip, the hard rubber tip, the medium rubber tip, and the soft rubber tip itself. As can be seen, the transfer function obtained by the steel tip gives the most accurate force reconstruction result. Quantitatively, the correlation coefficient between the measured impact force and the corresponding reconstructed forces, shown in Figure 5, are 0.9917, 0.9829, 0.9917, and 0.9752, respectively, for the steel tip, the hard rubber tip, the medium rubber tip, and the soft rubber tip. Additionally, the percentage of peak

errors are, respectively, −1.12%, 7.50%, 11.29%, and −1.58%. Interestingly, the best force reconstruction is not necessarily achieved when the transfer function is obtained with the same tip as the tip generating force. Even if that were the case, it would not be applicable as the material of the object impacting the structure is usually not known or predictable in practice.

To show the effectiveness of the force reconstruction, two quantities, correlation coefficient and peak error, should be considered simultaneously. In other words, the force reconstruction with a higher correlation coefficient (i.e., closer to 1), and concurrently, lower peak error (i.e., closer to 0%) is more desirable. Therefore, we introduce the following reconstruction accuracy error in order to use only one variable:

$$e = \sqrt{(correlation\ coefficient - 1)^2 + (peak\ error)^2}.\tag{12}$$

In the worst case scenario, the maximum value of the accuracy error is <sup>√</sup>2. On the other hand, when *e* is closer to zero, the reconstruction is more precise. For instance, in Figure 5, the reconstruction accuracy errors for steel tip, hard rubber tip, and medium rubber tip are 0.0139, 0.0769, and 0.1132, respectively, which shows the reconstruction precision in the case of steel tip as its corresponding accuracy error is closer to zero. Similar results were observed for impact at other levels. It is worth mentioning that neither the correlation coefficient nor the peak error can singly lead to this conclusion. As concluded from the above discussion, from now on, the transfer functions are established by using the steel tip hammer.

**Figure 4.** Impact force graphs produced using different hammer tips.

**Figure 5.** Impact force reconstruction using transfer functions (TF) established by different hammer tips.

## *4.3. Influence of Sensor Type and Location*

As previously pointed out in Section 3, four transducers are mounted on the experimental set-up, measuring the displacement, velocity, and acceleration at level 3, as well as the acceleration at level 8. These measurements can be utilized in combination or individually for the force reconstruction.

Figure 6 shows the reconstruction of the impact forces applied by steel tip hammer implementing different system responses (i.e., the velocity at level 3, acceleration at level 3, and acceleration at level 8). Since the results of using displacement at level 3 were not satisfactory at all, these are not shown for better clarity. To investigate this comparison quantitatively, Table 2 shows the values of the accuracy error for each condition, as well.

As shown in Figure 6 and Table 2, the distance between the impact location and the sensor location is a dominant factor in impact force reconstruction. More specifically, employing velocity measurement at level 3 leads to better reconstruction results for impacts at lower levels (1, 2, 3, and 4). On the other hand, to reconstruct impact forces applied at higher levels (5, 6, 7, and 8), using the acceleration measurement at level 8 is more effective. Note that the minimum accuracy error in each row is colored in Table 2.

It is observed that making the problem over-determined (e.g., employing a combination of velocity at level 3 and acceleration at level 8) does not necessarily improve the reconstruction. Therefore, in this paper, the problem is kept even-determined in order not to use extra ineffective computation costs. In this regard, the impact forces will be reconstructed by using the velocity measurement at level 3 when the impact location is in the lower half of the structure and employing the acceleration measurement at level 8, otherwise.


**Table 2.** Accuracy errors of impact force reconstruction using different system responses.

Figure 7 illustrates the reconstruction accuracy errors for impact forces applied on different levels of the structure. As shown in Figure 7a, when using the velocity measurement at level 3, the minimum of the accuracy error occurs when the impact force is applied at level 3. Similarly, when it comes to using the acceleration measurement at level 8, the accuracy error is the minimum if the impact location is also at level 8, as illustrated in Figure 7c. Concerning the acceleration measurement at level 3, it gives its most accurate result for mid-levels (not end-levels), as can be seen in Figure 7b. Figure 7d shows the accuracy error when using the velocity at level 3 for the lower half of the structure (i.e., level 1 to 4), and employing the acceleration at level 8 for the upper half (i.e., level 5 to 8). As can be seen both in Figure 7d and Table 2, the reconstruction is poorer when the impact force is applied at level 1. The reason is that this level is very close to the fixed support, which prevents the proper stimulation of vibration modes and hence the signal can not satisfactorily be captured by sensors placed distant from this location. On the other hand, the most accurate results are obtained at levels 3 and 8, exactly where the transducers are placed, which demonstrates the effect of proximity of impact location and sensor location.

**Figure 6.** Using different transducers for impact force reconstruction at different locations.

(**a**) Using velocity measurement at level 3.

(**c**) Using acceleration measurement at level 8.

(**b**) Using acceleration measurement at level 3.

(**d**) Minimum accuracy error accessible using either of the sensor measurements.

**Figure 7.** Relation between the sensor location and impact force reconstruction accuracy error.

In order to complement the above discussion, the impact forces applied by different rubber tip hammers (i.e., soft, medium, and hard rubber tips) are reconstructed in the following. Note that based on the conclusion made in Section 4.2, the transfer functions are obtained by using the steel tip hammer. Additionally, based on the conclusion made earlier in the current subsection, the velocity measurement at level 3 and the acceleration measurement at level 8 are employed for lower half and upper half of the structure, respectively. Table 3 shows the accuracy errors of the reconstruction for each rubber hammer tip at different levels. These errors could be reduced by manually changing the regularization parameter, however, it was not the purpose of the current paper. As shown, the accuracy error is acceptable in most of the cases, which demonstrates the efficacy of the transfer function obtained and the responses used.

**Table 3.** Accuracy errors of the reconstruction of impact forces applied by rubber tip hammers at different levels.


#### *4.4. Impact Force Location*

In the following results, it is assumed that eight impact forces are applied concurrently at levels 1 to 8, while the magnitude of only one impact force is non-zero. Herein, the superposition approach introduced in Section 2.3 is employed for location identification. Different scenarios were tested, namely, (i) using each of the available measurements singly, and (ii) different combinations of two system responses. Among all, the combination of

the acceleration at level 3 and the acceleration at level 8 leads to the most satisfactory impact localization. This is shown quantitatively in Table 4, where the accuracy errors corresponding to the reconstruction of the actual impact forces are presented for all abovementioned scenarios. The minimum possible accuracy error for each impact location is colored in the table. As demonstrated, for most levels, the minimum occurs when the combination of acceleration at level 3 and acceleration at level 8 is employed. It is concluded that reducing the degree of under-determinacy can improve the localization accuracy. Moreover, it seems that when the two measurements, selected in combination, are of the same type, the impact force can be localized more accurately. Therefore, the actual impact location can be detected through the following relation:

$$
\begin{bmatrix} \mathbf{r}\_3 \\ \mathbf{r}\_8 \end{bmatrix} = \begin{bmatrix} \mathbf{T}\_3^1 & \mathbf{T}\_3^2 & \dots & \mathbf{T}\_3^8 \\ \mathbf{T}\_8^1 & \mathbf{T}\_8^2 & \dots & \mathbf{T}\_8^8 \end{bmatrix} \begin{bmatrix} \mathbf{f}\_1 \\ \mathbf{f}\_2 \\ \vdots \\ \mathbf{f}\_8 \end{bmatrix} \tag{13}
$$

where **r**<sup>3</sup> and **r**<sup>8</sup> are the acceleration response at level 3 and 8, respectively, and **T***<sup>i</sup> <sup>j</sup>* is the transfer function between the impact location *i*, *i* = 1, ..., 8, and measurement location *j*, *j* = 3, 8. As presented in Section 2.3, (13) is solved for **f***i*, *i* = 1, ..., 8, where the magnitude of one of these reconstructed forces is significantly greater that others which specifies the actual impact location.

Figure 8 shows the reconstruction of impact forces at all possible locations when the actual impact force is applied at levels 1 to 8, individually, and a combination of the acceleration at level 3 and the acceleration at level 8 is considered as the system response. It demonstrates the efficacy of the approach as the reconstructed impact force associated to the true impact location has a smooth half-sine shape with a higher peak amplitude than other possible locations, as expected.

**Table 4.** Accuracy errors of the reconstruction of actual impact force using different traducers at each individual level.


(**a**) True impact location is level 1.

(**b**) True impact location is level 2.

**Figure 8.** *Cont*.

**Figure 8.** Identification of the impact location.

#### **5. Conclusions**

Inverse identification of an impact force acting on a multi-storey tower structure was studied experimentally using dynamic signals measured by different transducers. Herein, both the magnitude and location of the impact force were investigated. It was shown that using the hammer with the hardest tip can lead to a more accurate transfer function, where an accuracy error function was proposed to evaluate the reconstruction precision as a function of the correlation coefficient and the peak error. Moreover, it was observed that the proximity between the impact and sensor locations is a dominant factor in impact force reconstruction. Therefore, the velocity measurement at level 3 was used for the lower half of the structure and the acceleration measurement at level 8 was employed for the upper half and the effectiveness of this idea for impact force reconstruction at all positions was demonstrated both for steel tip hammer and rubber tip hammers. For impact localization, the superposition method was exploited, where the effect of different transducers was studied. It was concluded that reducing the degree of under-determinacy by using a combination of system responses of the same type can improve the localization accuracy. Therefore, a combination of the acceleration at level 3 and the acceleration at level 8 was employed for the localization.

As a potential real-world application of this study, identification of impact forces on bridge structures can be of great interest to the bridge owners and engineers. The bridge can be modelled as a multi-degree of freedom system with the expansion joints of the bridge deck taken as the potential impact locations. Measurement of the vibration response generated by the impact of heavy trucks can be carried out using accelerometers or contactless sensors such LDVs installed distant from the impact location. Another possible application of the current study is in oil-well drilling industry. During the whirling motion, the rotating drill string strikes the borehole wall, generating shocks from lateral vibrations. The location and magnitude of these impact forces are unknown as it is indeed impossible to place sensors on the string. However, using top-side measurements and inverse algorithms, the impact force can be identified, which helps in stability analysis and controller design for such structures.

**Author Contributions:** Conceptualization, H.K.; Formal analysis, S.T.; methodology, H.K. and S.T.; software, H.K. and B.H.; validation, H.K. and S.T.; supervision, H.K.; writing—original draft preparation, H.K. and S.T.; writing—review and editing, H.K. and S.T.; project administration, H.K. and B.H.; funding acquisition, B.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

