This paper proposes a novel approach for identifying distributed dynamic loads in the time domain. Using polynomial and modal analysis, the load is transformed into modal space for coefficient identification. This allows the distributed dynamic load with a two-dimensional form in terms of time and space to be simultaneously identified in the form of modal force, thereby achieving dimensionality reduction. The Impulse-based Force Estimation Algorithm is proposed to identify dynamic loads in the time domain. Firstly, the algorithm establishes a recursion scheme based on convolution integral, enabling it to identify loads with a long history and rapidly changing forms over time. Secondly, the algorithm introduces moving mean and polynomial fitting to detrend, enhancing its applicability in load estimation. The aforementioned methodology successfully accomplishes the reconstruction of distributed, instead of centralized, dynamic loads on the continuum in the time domain by utilizing acceleration response. To validate the effectiveness of the method, computational and experimental verification were conducted.

Accurately reconstructing the dynamic forces acting on a structure holds significant importance in various engineering applications. Frequently, direct measurements of the input are not feasible, while limitations in practicality and cost prevent measurements of the structure’s response at all physical locations. In such cases, the forces must be indirectly determined from dynamic response measurements using system inversion techniques.

A distributed dynamic load, such as the air pressure load experienced by a building or the water pressure load on ships, plays a critical role in the analysis of dynamic loads. Therefore, the recognition of distributed dynamic loads has considerable practical significance. While numerous methods and techniques [

Two primary avenues have emerged for studying distributed dynamic loads: one involves the technology of spatial fitting, while the other emphasizes establishing load relationships in either the time or frequency domain. It is comparatively straightforward to identify the load with the frequency response function [

This paper presents the Impulse-based Force Estimation Algorithm (IFEA), in which the recursive algorithm based on convolution integral combined with the moving mean and polynomial fitting methods can avoid the inversion of super-high dimensional matrices and, as a result, can accurately identify loads with a long history and fast-changing characteristics over time. For distributed dynamic load, the utilization of polynomial function and modal analysis has proven successful in converting a dynamic load, which encompasses two dimensions of space and time, into coefficient identification for a single-degree-of-freedom (SDOF) system. Ultimately, the successful identification of distributed dynamic loads with a sufficiently long duration and diverse forms on a continuum with an infinite number of degrees of freedom is achieved in the time domain, in contrast to the frequency domain proposed in [

The theoretical framework of this study commences with

A conventional point-load excitation is inherently constrained in terms of spatial degrees of freedom, with a mathematical form of

For deterministic structures, modal parameters can be obtained with modal testing or finite-element modal-analysis technique [

where

For decoupled SDOF systems, the distributed dynamic load

Here,

Considering a SDOF damped system subjected to an excitation

Theoretically,

Among these,

Upon substituting

From the perspective of impulse,

Within each discrete time interval, the integrand function of the convolution integral comprises known quantities, including the constant basis, basis

where

The calculated velocity is shown in

The calculated acceleration is given by

At the end of the time interval

Here,

The convolution integration takes the form of a parametric integral in mathematics, namely the form

The method reconstructs information regarding the force within the interval and a recursion relationship was established for the load between the two discrete points. This approach enables efficient handling of a large number of sampling points, as each step only requires solving the equation at the current moment. Specifically, for 10,000 sampling points, a matrix with 10,000 rows and 10,000 columns needs to be constructed according to references [

Through experiments and simulations, we have observed that multiple iterations can lead to cumulative errors. To visualize these errors, we introduced the concept of Direct Current (DC) in the circuit, which we refer to as DC trends. According to the process shown in

To address the issue of DC trend, we incorporate a moving mean filtering algorithm. Let us consider a set of time-series discrete signals denoted as

The schematic diagram is depicted in

In the absence of a DC trend, the moving mean remains relatively constant. However, when a DC trend is present, the moving mean also changes, reflecting the trend value. The sliding window should be appropriately sized according to the sampling rate and frequency of the response. As the moving mean is calculated using a sliding window, any change in the input signal’s characteristics will cause fluctuations at the junctions of the two segments. To address this issue, we employ a polynomial fitting method to fit the DC trend of each segment. By using a polynomial, we perform fitting based on the least square index to approximate the DC trend from the sampled signal. Once the DC trend is fitted, it can be removed from the recognized excitation signal.

where

In accordance with

This study assumes that the spatial distribution function of the distributed dynamic load displays spatial smoothness, indicating a low spatial frequency. Consequently, multi-order polynomials are utilized to approximate it. The modal load of each order at discrete time

where

where

The matrix

Hence, the distributed dynamic load at discrete time

In order to objectively and quantitatively describe the obtained dynamic load, commonly used signal evaluation methods should be adopted. Let the theoretical load signal at time

where

Here,

The IFEA is validated for the SDOF system depicted in

The acceleration response of the example is illustrated in

Fixed frequency plus random excitation is a common condition in engineering, and the time-domain algorithm has the potential to identify the excitation after fixed frequency and random superposition simultaneously. However, this requires a smaller sampling interval and more sampling points. It can be observed from

In the field of structural dynamics, low-frequency loads induce low-order modes in a structure, which typically exhibit larger amplitudes and have the potential to cause more damage. On the other hand, high-frequency loads are often treated as random loads acting upon the structure in engineering scenarios. Consequently, a simply supported beam model with a remarkably low first-order mode was specifically designed, as depicted in

The distributed dynamic load, acting perpendicular to the beam, is described by the equation

The form of the load is shown in

The estimation of each order of modal force is depicted in

Due to the linear nature of the first-order and third-order modal forces, the correlation coefficient index yields a poor result and loses its functionality. However, by observing the NMES index and the modal force curve, it is evident that the identified force fluctuates around a straight line with minimal error. Overall, both the modal force curves and the index curves indicate a highly accurate recognition of each order of modal force.

The errors between the reconstructed and original loads are shown in

As depicted in

Furthermore, to validate the method’s estimation capability for rapidly changing and long time-history loads, we devised an excitation that undergoes a mutation. The mutation of this excitation alternates in both spatial distribution and frequency, thereby testing the method’s recognition ability. The form of excitation is shown in

The excitation is continuous in the time domain. All other conditions remain unchanged from the previous example, with the only alteration being the extension of sampling time to three seconds. The form of excitation is illustrated in

As evident from

The third-order modal force is represented by a straight line with a value of 0, and it is evident from the figure that the identification is highly accurate. Hence, calculating its index is not possible as the denominator of the index becomes extremely small, resulting in an infinite value. Similarly, this applies to the first 0.5 s of the first-order modal force, so the index for the first-order modal force is calculated starting from 0.5 s. The modal force diagram and all indices demonstrate the accuracy of the estimated force.

The errors between the reconstructed and original loads are shown in

The example demonstrates that the proposed method can effectively identify long time-history distributed dynamic loads undergone a mutation, with highly accurate identification results and minimal error.

To assess the method’s robustness against noise, we introduced 2% Gaussian white noise to the response stimulated in

The first and third order modal force indices can also be elucidated by the analysis in

The modal force for single-point concentrated loads is given by

where

About contributions of single-point concentrated to the response, it can be obtained similarly from

Given the proposed nature of IFEA as a time-domain method in this study, it is feasible to employ experimental data comprising concentrated excitations to verify the algorithm’s performance in the time domain. Since concentrated dynamic loads are employed, the multi-order polynomial fitting is unnecessary during the computation process. Merely utilizing zero-order polynomial fitting is adequate. Among the various forms of concentrated excitations, impulsive loads, characterized by high amplitude and rapid variations, pose more stringent demands on numerical computations, particularly in terms of load tracking in the time domain and accuracy of amplitude reconstruction [

Considering the challenges associated with controlling and measuring spatially distributed dynamic loads, including difficulties in accurately controlling the amplitude of distributed loads in the spatial domain and challenges in measuring the amplitude of continuously distributed dynamic loads. These challenges can lead to significant errors in experimental data, thereby impeding the validation of the algorithm itself. Therefore, the validation procedure uses the mature and easily controllable concentrated dynamic load to emphasize the performance of the time domain of the newly proposed IFEA. If the experiment incorporates impulse loads, it would provide an opportunity to evaluate the performance of the IFEA under an impulse load. Additionally, this would serve as a supplementary demonstration of simulations. Thus, a pulsive load is the excitation source during the experimental validation process.

The configuration of the simply supported beam model is depicted in

Length/m | Width/m | Height/m | Elasticity modulus/GPa | Density/kg/m^{3} |
---|---|---|---|---|

0.7 | 0.04 | 0.008 | 210 | 7800 |

A modal experiment was performed on the simply supported beam employing the hammer impact technique, wherein the natural frequencies and modal damping ratios of the simply supported beam were measured and are presented in

First order | Second order | Third order | |
---|---|---|---|

Modal frequency/Hz | 39.37 | 153.41 | 346.74 |

Modal damping ratio | 0.026 | 0.013 | 0.009 |

Given the instantaneous characteristics and brief duration of impact loads, there are stringent demands for dynamic-load identification algorithms. To assess the accuracy of amplitude recognition, the peak relative error is introduced,

An impact load was imparted using a force hammer at a distance of 0.37 m. Accelerometers were strategically positioned at response point 1 (located at 0.28 m) and response point 2 (located at 0.49 m) to capture the acceleration responses, as depicted in

Notably, in the magnified section of

The reconstructed load is shown in

Similarly, in the identified impact-load graph, the zoomed-in segment illustrates the load attaining its peak at 0.492 s and commencing its effects between 0.49 and 0.492 s. These findings affirm the experiment’s success in verifying the algorithm’s precision in reconstructing the load duration. In

In this study, a novel approach for distributed load identification in the time domain is introduced. The novelty of the methodology can be summarized as follows:

(1) By employing polynomials and modal analysis, the task of recognizing a dynamic distribution load which has two dimensions of space and time is converted into the identification of coefficients within a single-degree-of-freedom system. This method can effectively reduce the dimension of the system and recognize the spatial and temporal distribution simultaneously in the form of model force. In coefficient recognition, each step of the reconstruction is solely based on the modal force at the current time. This prevents the accumulation of spatial reconstruction errors and has no impact on the errors in the time domain. Through this approach, the distributed dynamic loads on a continuum are effectively established, not for discrete systems with multiple degrees of freedom, nor for concentrated loads.

(2) Based on the convolution integral, a recursive algorithm is proposed. The recursive scheme possesses the ability to calculate long time-history loads undergone a mutation in the time domain. This is an ability that ultra-high-dimensional matrices, which require regularization, do not possess. Utilizing the response of the sampling point, the algorithm reconstructs the force in the sampling interval while considering the pulse energy. This enhancement obtains the analytical solution of the convolution integral of the reconstructed force in discrete time, instead of assuming a constant interval force. Additionally, the method employs acceleration response instead of strain, making it easier to measure in engineering applications.

(3) The strategy of moving mean and polynomial fitting to eliminate trend is introduced, providing IFEA with the capability to accurately estimate the dynamic loads of structures in the time domain.

As demonstrated by the examples, IFEA possesses the capability to estimate rapidly changing loads in the time domain after converting the loads into modal space, which is also an advantage of this algorithm. However, when identifying the spatial distribution, this paper assumes a smooth spatial distribution of load. For loads that exhibit significant spatial variations, higher-order polynomials are necessary for fitting, but correspondingly more measurement points are required, potentially leading to overfitting. However, as the spatial distribution recognition is confined to each individual step, any inaccuracy in spatial recognition does not impact the accuracy of the modal forces in the time domain.

The invaluable advice of Tian Tang during the preparation of this experiment is greatly appreciated. We are also grateful to the editors and reviewers for their helpful suggestions.

The authors received no specific funding for this study.

The authors confirm contribution to the paper as follows: study conception and design: Yuantian Qin, Yucheng Zhang; data collection: Yucheng Zhang; analysis and interpretation of results: Yuantian Qin, Yucheng Zhang; draft manuscript preparation: Yucheng Zhang, Vadim Silberschmidtb. All authors reviewed the results and approved the final version of the manuscript.

The data used in this paper has been made available within the paper. Moreover, the data supporting the findings can be obtained from the corresponding author upon a reasonable request.

The authors declare that they have no conflicts of interest to report regarding the present study.