Theoretical foundation, methods, and criteria for calibrating human vibration models using frequency response functions

Highlights

• This study examined the theory and methods for calibrating human vibration models.

• It proposed a set of criteria for assessing the validation of the model calibration.

• It identified the requirements for achieving a unique solution of the calibration.

• It demonstrated the applications of the proposed theory, methods, and criteria.

• They can be used to guide the development and evaluation of human vibration models.

Abstract

While simulations of the measured biodynamic responses of the whole human body or body segments to vibration are conventionally interpreted as summaries of biodynamic measurements, and the resulting models are considered quantitative, this study looked at these simulations from a different angle: model calibration. The specific aims of this study are to review and clarify the theoretical basis for model calibration, to help formulate the criteria for calibration validation, and to help appropriately select and apply calibration methods. In addition to established vibration theory, a novel theorem of mechanical vibration is also used to enhance the understanding of the mathematical and physical principles of the calibration. Based on this enhanced understanding, a set of criteria was proposed and used to systematically examine the calibration methods. Besides theoretical analyses, a numerical testing method is also used in the examination. This study identified the basic requirements for each calibration method to obtain a unique calibration solution. This study also confirmed that the solution becomes more robust if more than sufficient calibration references are provided. Practically, however, as more references are used, more inconsistencies can arise among the measured data for representing the biodynamic properties. To help account for the relative reliabilities of the references, a baseline weighting scheme is proposed. The analyses suggest that the best choice of calibration method depends on the modeling purpose, the model structure, and the availability and reliability of representative reference data.

Introduction

The characterization of human vibration biodynamics is important for understanding vibration effects and for assessing the risks of vibration exposures [1], [2], [3], [4], [5]. The biodynamic responses are basically a passive mechanical process, similar to the responses of many engineering structures to vibration. In engineering, the model of a system is usually constructed using a forward dynamic approach, in which the dynamic properties of the system are directly measured and used in the model to predict the dynamic responses. Because it is difficult to directly measure the biodynamic properties of living human subjects, human vibration models are most frequently constructed using an inverse dynamic approach or a hybrid forward and inverse approach [6], [7], [8], [9], [10], [11], [12]. The inverse approach estimates the unknown biodynamic properties of a model using the measured time-history responses (forces and motions) and/or frequency response functions (apparent mass, mechanical impedance, and vibration transmissibility) of the system. Mathematically, the property parameters are determined by imposing matches between modeled responses/functions and the corresponding measured responses/functions. In principle, this imposed matching process passes the dynamic properties included in the reference data to the model, similar to the calibration of an accelerometer. In engineering modeling, this process is usually termed as parameter identification, as it is only used to identify the unknown values of one or a few well-defined parameters of a model that provides a close simulation of an engineering system [13], [14]. In contrast, the vast majority of human vibration models are lumped-parameter models that only provide a very crude simulation of the complex human body structure. The exact physical representations of some model components are not known or predetermined before their values are identified. Their representations are actually interpreted according to their values. As a result, the components in the same model structure may represent different parts or properties of the human body or segment, depending on the characteristics of the reference functions used in the modeling, as observed in our previous studies [9], [15]. Therefore, this process not only calibrates the values of the model parameters, but it may also calibrate the physical meanings of some model components. It may be more appropriate to refer to this process as model calibration in human vibration modeling.

Such model calibration makes it possible for many researchers to efficiently construct human vibration models for understanding the basic characteristics and motion mechanisms of human biodynamic responses using inexpensive modeling techniques. This further explains why the vast majority of human vibration models have been developed using this approach. Such models are also adequate and efficient for enhancing the designs, analyses, and evaluations of tools, seats, and anti-vibration devices [16], [17], [18], and/or for constructing test apparatuses or human test dummies [19], [20]. To help further understand vibration effects and to help assess the risks of vibration exposures, it is required to improve such models for better predictions and understanding of the biodynamic responses in the human body. A major deficiency of the reported lumped-parameter models is that they may not be used to predict the dynamic forces at many joints and the vibration power absorption distributed in different tissues, as these models do not provide a reasonable simulation of the relationships among the bones, joints and soft tissues. Although vibration is likely to be primarily transmitted through the bones and joints, these structures only account for less than 20 percent of the body mass [21]; the soft tissues account for the vast majority of the body mass and are thus likely to play a major role in determining the overall responses, according to a recently derived theorem [22]. While it is not difficult to configure a model that separately simulates the bones, joints, and soft tissues, it remains a formidable research task to determine how such models can be reliably calibrated and reasonably validated. Enhancing the understanding of model calibration may accelerate the development of such models.

Based on function, human vibration models are conventionally classified into three categories: mechanistic, quantitative, and effects models [23]. A set of validation criteria and a checklist for the models in each category have also been proposed [23]. Because these criteria are not based on the concept of model calibration, they do not systemically and directly address the technical issues and requirements of the model development using the inverse dynamic approach. Instead, the human vibration modeling society generally interprets model calibrations only as quantitative summaries of biodynamic measurements [23], similar to the statistical regressions used in the analyses of experimental data. According to such limited understanding, the majority of the reported human vibration models are classified as quantitative models [23]. Such a classification actually incorrectly considers the quantitative summaries as the final function of these models, as the endpoints of these models are not the quantitative summaries but they are primarily for the above-mentioned applications. This classification also contradicts the major purposes of such computer modeling − to predict the responses or parameters that cannot be directly measured and/or to understand the phenomena that cannot be easily understood from the experimental data themselves. Another questionable practice is that, according to how quantitative models are defined, these models are often considered to be validated as long as they can provide a good curve fit to a set of measured data. Because there is no uniform specification for judging the goodness of curve fitting in human vibration modeling, many studies have portrayed their models as being validated simply because those models reflect the study’s best curve fits; there is often insufficient assessment of the model’s ability to represent the targeted biodynamic properties or response functions. Such a practice is problematic for the following reasons: (i) the best curve fitting can always be achieved for any experimental data on any model, even if its equations of motion include serious errors, as observed in a recently reported study [24]; (ii) the best curve fitting can also be achieved even if the reference functions include significant errors, and/or the transmissibility measured on one substructure is used to represent the vibrations of other substructures, which are also observed in the reported study [24]. Obviously, more effective validation criteria that directly address these issues are required for the further development and evaluation of human vibration models.

Also because there is lack of understanding of model calibration, it is not clear how to appropriately select and apply the specific calibration schemes. While apparent mass or impedance has been used to calibrate some models [7], [8], [9], vibration transmissibility has been used to calibrate some other models [10], [25], [26]. However, more researchers have used both transmissibility and driving-point response functions [6], [24], [27]. Some of these researchers hypothesized that such a combination would provide a more reliable calibration [24], [27]. The current study hypothesized that the best choice of the calibration method depends on many factors, which may include the specific application of the model, the model structure, and the availability and reliability of the reference functions. Further studies are required to test these hypotheses.

In many cases, it is difficult to clearly differentiate or categorize these models, especially the mechanistic and quantitative models, as they usually both include mechanistic and quantitative components. Even though some finite element models were originally developed using the forward modeling approach, some of the element properties have to be adjusted through model calibration so that the modeling predictions can be more reliable [11], [28]; the biodynamic properties are usually measured with cadaver tissues that may not be fully representative of live subject properties, and the individual differences may further increase the complexities and difficulties. On the other hand, some models were originally developed to simulate the biodynamic measurements, but they can be considered as mechanistic models after their parameters are understood and can be physically associated with the specific substructures of the human body or segments [9], [15]. As such merging marks the advancement of modeling knowledge [23], the hybrid forward and inverse dynamic approach is likely to be more frequently used in the future. It is important to formulate the calibration validation criteria that are generally applicable to models in every category.

Based on this background, the specific aims of this study are to review and enhance the theoretical foundation for human model calibration, to systematically examine the calibration methods, to identify their basic criteria and requirements, and to explore the improvements of the calibration methods. This study also proposed some new interpretations of the reported models.