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, model-based techniques [1
] and neural networks [3
]. 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
], state variable formulation [15
], and sum of weighted accelerations [21
]. In [23
], the inverse structural filter 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
] 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
] and Singular Value Decomposition (SVD) based methods, including truncated SVD (TSVD) [27
]. 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
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
], 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
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.