**1. Introduction**

Many structures are subjected to impact forces, which can be a matter of serious concern in terms of structural integrity. Measurement of these accidental impact forces is of great importance since it can help prevent system failure through evaluating the system stress and comparing it to its tolerance threshold or fatigue limit. Direct measurements of impact forces are difficult, expensive, and tedious, especially for large structures due to the difficulty of sensor installation and dynamic characteristic altering, while beforehand, localization of the impact area can make the examinations more efficient. Using system dynamic responses, captured by sensors placed distant from the impact location, the impact forces can be estimated by inverse algorithms.

The basis of inverse algorithms is to indirectly identify the impact force using responses measured at given points of the body subjected to impact. Inverse algorithms exploited in the literature can be categorized into two main classifications, namely, modelbased techniques [1,2] and neural networks [3–6]. The superiority of neural networks emerges when the underlying dynamics is infeasibly complicated or inaccessible. However, as the accuracy of these techniques relies on massive training data, which is usually impractical, the model-based methods are more widely used. In model-based methods, a transfer function is found by utilizing the input and output of the system. Some examples of these methods are as follows: deconvolution technique [7–14], state variable formulation [15–20], and sum of weighted accelerations [21,22]. In [23], the inverse structural filter

**Citation:** Kalhori, H.; Tashakori, S.; Halkon, B. Experimental Study on Impact Force Identification on a Multi-Storey Tower Structure Using Different Transducers. *Vibration* **2021**, *4*, 101–116. https://doi.org/ 10.3390/vibration4010009

Received: 2 December 2020 Accepted: 25 January 2021 Published: 29 January 2021

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method, which leans on the dynamics state-space model, and the sum of the weighted accelerations technique are compared. Therein, deficiencies of the mentioned strategies are discussed and some modifications are proposed in order to enhance their performance. Among the model-based methods introduced, the deconvolution method has received significant attention in the literature. Two main attitudes of the deconvolution method are the time-domain [1,2,24,25] and the frequency-domain approach [26]. In [23], a comparison is made between the results of two time-domain strategies and those of a frequency-domain approach in order to determine the pros and cons of each method. Generally speaking, frequency-domain methods need lower computational efforts while they are usually infeasible for transient phenomena such as impact events. Solving a deconvolution problem might not result in a sufficiently good outcome since the force reconstruction problem is intrinsically ill-posed due to the ill-conditioned nature of the transfer function, i.e., the condition number of the transfer function matrix is very large, making the problem sensitive to small perturbations such as measurement errors or noise. To avoid divergent or inaccurate results, it is usually necessary to exploit a regularization method.

Several regularization techniques have been proposed in the literature. The most popular ones are Tikhonov regularization [27–31] and Singular Value Decomposition (SVD) based methods, including truncated SVD (TSVD) [27,32,33]. These two methods are compared in [34]. The theoretical backgrounds of five regularization methods, namely, generalized cross-validation, singular value decomposition, iterative method, data filtering approach, and Tikhonov regularization are introduced and main restrictions of each method are discussed in [35]. Some other exploited methods in the literature are QR factorization [36], explicit block inversion algorithms [37], Bayesian regularization [38], and the least-square QR (LSQR) iterative regularization method [39]. A combination of *l*<sup>1</sup> regularization and sparse reconstruction is proposed in [40]. In [11], a primal-dual interior point method is exploited and compared to the Tikhonov method. More recently, nonconvex sparse regularization based on generalized minimax-concave (GMC) and non-negative Bayesian learning are used in [25,41], respectively. In [42], Bayesian sparse regularization is exploited for identification and localization of multiple forces in time domain, and compared with Tikhonov regularization associated with the Generalized Cross Validation (GCV) criterion. Existing regularization methods which are proposed for force reconstruction are vector-based, while for large-scale inverse problems, matrix-based regularization has several privileges. Matrix-based regularization was recently introduced in [43] where the parameter of regularization was chosen with the Bayesian Information Criterion (BIC). Another issue that has been raised in recent years is that of moving force identification. In [44], a comparison is made between four regularization methods, i.e., (i) truncated generalized singular value decomposition (TGSVD), (ii) piecewise polynomial truncated singular value decomposition (PP-TSVD), (iii) modified preconditioned conjugate gradient (M-PCG) method, and (iv) preconditioned least-square QR-factorization (PLSQR) method, all used for reconstruction of moving forces, where it is concluded that the TGSVD method is preferred on the issue of identification accuracy. On the other hand, the M-PCG method is recommended in regard to identification efficiency.

To perform a comprehensive identification of an impact force, both its magnitude (force history) and location should be assessed. The location of the impact force is obscure in numerous cases in practice, which violates the fundamental presumption of the above mentioned methods. Various methods are introduced in the literature to localize the impact force. In [45], an experimental method is used in which an objective function is defined based on transfer functions and minimized in order to find the impact force location and in [46], a pseudo-inverse direct method is utilized to identify both the magnitude and location of the impact force. More recently, [12] pursued a similarity searching technique, and [14] introduces a superposition approach to estimate the impact location and magnitude simultaneously.

In the current paper, the identification of (i) the impact force history, and (ii) the impact location is presented. The impact force is applied on a scaled eight-storey tower structure in the laboratory. The identification is performed using recorded system outputs, i.e., the displacement, velocity, and acceleration measurements at level 3, as well as the acceleration measurement at level 8. The impact force reconstruction consists of two procedures, namely, (i) obtaining a transfer function between a reference impact force and its resulting response captured by a specific sensor, and (ii) identifying an unknown impact force using the transfer function obtained and the responses. Herein, the deconvolution technique is exploited to solve these inverse problems and the Tikhonov regularization method is used in order to deal with the ill-conditioned nature of the transfer function. To identify the impact location, the superposition approach is exploited where it is assumed that impact forces are concurrently applied on all 8 potential locations, while only one of them has a non-zero magnitude. This expresses the condition when only one impact is exerted at one of the possible locations. The actual impact location is then detected among all potential locations through an extended matrix form of the convolution equation.

The contributions of this paper are, firstly, investigating the influence of the hammer tip material on the effectiveness of the transfer function obtained, secondly, proposing an accuracy error function to evaluate the reconstruction precision, thirdly, studying the effect of sensor type and location on the accuracy of the impact force reconstruction, fourthly, using distinct sensors for the force reconstruction of different levels (i.e., using recorded signals at level 3 for the lower half of the structure and employing measurements at level 8 for the upper half), and fifthly, studying the localization accuracy based on the system responses used individually or in combination. The effectiveness of the method used for impact force reconstruction is demonstrated for all positions, with steel, soft rubber, medium rubber, and hard rubber tip hammers. The paper is organized as follows. The problem formulation is presented in Section 2. The experimental set-up is introduced in Section 3. Section 4 presents the results and discussion. Finally, the conclusions are presented in Section 5.

#### **2. Problem Formulation**

#### *2.1. Single Impact Force Reconstruction*

The impact force reconstruction consists of two procedures, namely, (i) obtaining a transfer function between a reference impact force and its resulting response captured by a specific sensor, and (ii) identifying an unknown impact force using this transfer function and the collected vibration responses. Suppose *n* sensors are deployed on a structure subjected to impacts to measure impact responses (e.g., displacement, velocity or acceleration) and the following assumptions hold:


Then, the relation between the impact force *f* applied at point *x* and the response *r* measured at point *y* at time *t* is given by a convolution integral as follows:

$$r(y\_\prime t) = \int\_0^t T\_s(\mathbf{x}, y\_\prime t - \zeta) f(\mathbf{x}, \zeta) d\zeta,\tag{1}$$

where *Ts*(*x*, *y*, *t* − *ζ*), *s* = 1, ...*n*, is the transfer function between the impact force at point *x* and the *sth* sensor at point *y* at time *t* = *ζ*. The discretized form of the forward model (1), which is more applicable in practice, can be written as follows:

$$\mathbf{r} = \mathbf{T}\_s \mathbf{f}\_\prime \tag{2}$$

with **<sup>r</sup>** ∈ *<sup>R</sup>m*, **<sup>T</sup>***<sup>s</sup>* ∈ *<sup>R</sup>m*×*m*,**<sup>f</sup>** ∈ *<sup>R</sup>m*, where **<sup>r</sup>** is the recorded response vector, **<sup>f</sup>** is the vector of impact force which is to be reconstructed, and **T***s* is the impulse response matrix, which is a lower triangular toeplitz matrix, given by

$$\mathbf{r} = \begin{bmatrix} r(\Delta t) \\ r(2\Delta t) \\ \vdots \\ r((m-1)\Delta t) \\ r(m\Delta t) \end{bmatrix}, \mathbf{f} = \begin{bmatrix} f(\Delta t) \\ f(2\Delta t) \\ \vdots \\ f((m-1)\Delta t) \\ f(m\Delta t) \end{bmatrix}'$$

$$\mathbf{T}\_s = \begin{bmatrix} T\_s(\Delta t) & 0 & \dots & 0 \\ T\_s(2\Delta t) & T\_s(\Delta t) & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ T\_s((m-1)\Delta t) & T\_s((m-2)\Delta t) & \dots & 0 \\ T\_s(m\Delta t) & T\_s((m-1)\Delta t) & \dots & T\_s(\Delta t) \end{bmatrix}. \tag{3}$$

In (3), *m* is the number of samples and Δ*t* is the time interval, which should be small enough since the above discretization assumes that the impact force *f* is constant within each time interval. In other words, with a higher sampling frequency, the results given by (2) are theoretically more accurate.

The solution of (2) can be theoretically obtained by using the following least squares problem:

$$\min \|\mathbf{r} - \mathbf{T}\_s \mathbf{f}\|\_{2'}^2 \tag{4}$$

where **r** is contaminated by experimental errors in practice. Moreover, **T***<sup>s</sup>* is a matrix with a very large condition number and hence is ill-conditioned. Therefore, the problem must be regularized. The Tikhonov regularization method alternatively searches for an approximation of **f** through the following penalized least-squares problem:

$$\min \{ \|\mathbf{r} - \mathbf{T}\_s \mathbf{f}\|\|\_2^2 + \delta \|\|I\mathbf{f}\|\|\_2^2 \},\tag{5}$$

where *δ* ≥ 0 is the regularization parameter, determined by L-curve method, and *I* is the identity matrix.

#### *2.2. Transfer Function*

In order to solve (2), the transfer function **T***<sup>s</sup>* should be obtained in advance. This is achieved by using a reference impact force, its corresponding measured response, and the following relation:

$$\mathbf{r} = \mathbf{F}\mathbf{t}\_{\mathbf{s}} \tag{6}$$

with **<sup>t</sup>***<sup>s</sup>* ∈ *<sup>R</sup>m*, **<sup>F</sup>** ∈ *<sup>R</sup>m*×*m*, where **<sup>F</sup>** is a lower triangular toeplitz matrix, and **<sup>t</sup>***<sup>s</sup>* is the vector of transfer function, as follows:

$$\mathbf{F} = \begin{bmatrix} f(\Delta t) & 0 & \dots & 0 & 0 \\ f(2\Delta t) & f(\Delta t) & \dots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ f((m-1)\Delta t) & f((m-2)\Delta t) & \dots & f(\Delta t) & 0 \\ f(m\Delta t) & f((m-1)\Delta t) & \dots & f(2\Delta t) & f(\Delta t) \end{bmatrix}, \mathbf{t}\_s = \begin{bmatrix} T\_s(\Delta t) \\ T\_s(2\Delta t) \\ \vdots \\ T\_s((m-1)\Delta t) \\ T\_s(m\Delta t) \end{bmatrix}. \tag{7}$$

The solution of (6) can be obtained by using the following least squares problem:

$$\min \|\mathbf{r} - \mathbf{F}\mathbf{t}\_{\delta}\|\_{2}^{2}.\tag{8}$$

However, in practice, the collected impact force and the measured dynamic response are associated with noise, equivalent to high-frequency components of signals. This causes matrix **F** to, potentially, have a large condition number, making it ill-conditioned. The large condition number of **F** together with presence of noise in **r** results in deviated transfer functions. Therefore, applying regularization is deemed necessary. Employing Tikhonov regularization method, an approximation of **t***<sup>s</sup>* can be found instead, by the following penalized least-squares problem:

$$\min \{ \|\mathbf{r} - \mathbf{F}\mathbf{t}\_{\delta}\|\|\_{2}^{2} + \beta \|\|I\mathbf{t}\_{\delta}\|\|\_{2}^{2} \},\tag{9}$$

with *β* ≥ 0 the regularization parameter, determined by the L-curve method. Summarizing, impact force reconstruction consists of two steps:


As discussed, both above problems are ill-posed. Therefore, in this paper, the Tikhonov regularization method is exploited in order to avoid the sensitivity to perturbations, which can potentially make the solution unstable.

#### *2.3. Impact Force Location*

Two approaches have been employed in the literature for impact force localization: the one-to-one approach and the superposition approach [14]. In the one-to-one approach, the impact reconstruction is performed for each pair of impact and response location, while in the superposition approach, the impact forces at all possible locations are reconstructed concurrently. Generally speaking, the superposition approach considers a superposition of responses corresponding to each impact force exerted at different locations. In the following, the superposition approach is presented more in detail.

Assuming several impact forces at various points (*i* = 1, . . . , *p*) concurrently applied to a structure, the vibration response collected by a single sensor installed at position *s* is, therefore, a superposition of the responses generated by each individual impact force.

$$\mathbf{r} = \sum\_{i=1}^{p} \mathbf{T}\_{s}^{i} \mathbf{f}\_{i\prime} \tag{10}$$

where **f***<sup>i</sup>* is the impact force applied on the location *i*, *i* = 1, ..., *p*, and **T***<sup>i</sup> <sup>s</sup>* is the transfer function between the location *i* and the *sth* sensor location. Equation (10) can be written in matrix-vector form as follows:

$$\mathbf{r} = \begin{bmatrix} \mathbf{T}\_s^1 & \mathbf{T}\_s^2 & \dots & \mathbf{T}\_s^p \end{bmatrix} \begin{bmatrix} \mathbf{f}\_1 \\ \mathbf{f}\_2 \\ \vdots \\ \mathbf{f}\_p \end{bmatrix}. \tag{11}$$

The procedure for creating the transfer functions were already discussed in Section 2.2. For brevity, (11) is presented by **r** = **T***s***f**. As previously pointed out, matrix **T***<sup>s</sup>* is ill-conditioned and vector **r** is contaminated with noise, necessitating applying regularization to solve for **f**. Similar to (5), Tikhonov regularization is implemented. It is worth mentioning that (11) is severly under-determined, as there is one equation with *p* unknown forces. Now, let us make an important assumption that is the magnitude of all impact forces but one is actually equal to zero. This condition entails that an impact occurs at only one location. The purpose is, therefore, to detect the actual impact location among all potential locations, together with its force history. Using this approach, a reconstructed impact force is obtained for each potential impact location. In other words, *p* impact forces, i.e., **f**1,**f**2,...**f***p*, are reconstructed, keeping in mind that there is only one actual non-zero impact force. The reconstructed impact forces at spurious locations are expected to have zero magnitude as no impact has actually occurred at these locations. However, there might be some non-zero reconstructed impact forces at spurious locations. The reconstructed force at each location is qualitatively assessed, addressing key characteristics of a normal impact force such as the shape and the maximum amplitude of the first peak if applicable. A normal impact force

has typically a smooth half-sine shape. More comprehensive description of this method as well as various case studies can be found in previous works of the authors [10,12,14]. Figure 1 shows a schematic of the problem.

**Figure 1.** Schematic of impact force localization using the superposition approach.
