Every day, the human body is exposed to vibrations, whether during work, transport or sports. Two modes of exposition can be distinguished: whole body vibration and local vibration. Whole body vibrations may have some beneficial effects including ossification [1
] and treatment of child muscular disabilities [3
]. However, their negative effects have also been proven. Vibration could lead to an accelerated degeneration of joints [4
] and vascular diseases [5
]. Likewise, local vibrations produce beneficial effects, in sports training for example [6
], as well as significant drawbacks, especially in the hand-arm system. Related symptoms can be split into four groups: vascular effects, muscular effects, skeletal effects and joints effects [7
], including Raynaud’s disease [8
]. All of them are generally referred to as Hand Arm Vibration Syndrome (HAVS) [9
Investigating the effects of vibrations leads us to consider the human body as an extremely complex system, if only because of its responses to vibrations. It is a non-uniform, nonlinear and anisotropic structure. Nevertheless, simplified numerical models, like Finite Elements (FE) models, have been used to study the human body on various aspects: evaluate the injury risk of the lumbar vertebra of a seated human body submitted to vibrations [10
] or predict the strains and strength of human femora [11
]. A recent study featuring a two-dimensional FE model of the human hand-arm system was also used to estimate significant modal frequencies [12
]. These frequencies were then compared to experimental results in order to update the mechanical properties of the model. The study concluded that FE models could be used to obtain reliable resonance frequencies of the human hand-arm system.
In order to update and validate numerical models, it is necessary to carry out experimental measurements. In this study, Operational Modal Analysis (OMA) is used. The OMA represents the set of modal identification methods based on the sole measurement of the system’s response. It has been successfully used in various studies on the human body, mostly for whole body vibrations. Indeed, one of the main fields of interest of modal analysis in this scope is related to transport, including the effects of cabin vibrations [13
] or the resonance behavior of the seated human body [14
]. OMA and finite elements have already been used jointly to determine modal frequencies, damping and mode shapes, but only on a given part of the human body, e.g., the tibia [15
]. Moreover, most of the studies about vibrations and the human body only focus on the hand-arm system: the upper limbs are indeed frequently exposed, for example during biking [16
]. However, lower limbs should be given proper attention, as well, since many everyday activities expose legs to vibrations. Solicitations could be similar to white noise in public transport [17
] or similar to shock in walking and running activities [18
]. Therefore, we have chosen to focus our study on the lower limbs, combining OMA and finite elements on a different scale, i.e., the half-body.
The aim of this study is to identify the experimental modal parameters of the human lower limbs and to propose an updated two-dimensional finite element model. Experimental modal parameters were extracted by operational modal analysis using poly-reference implementation of a Least Squares Complex Frequency domain (p-LSCF) method.
The present study identified eight modes in the range of 0–150 . Six modes are identified in 80% of the tests. The fifth mode seems difficult to exploit given its low repeatability in the identification. This mode is most certainly due to a measurement artifact. Two of these eight modes ( and ) are in agreement with the numerical modes, and the identification is supported by an analysis of modal deformations. The parameters used in the updated model clearly match both physical and anatomical realities. This proves that an FE model, once updated, is a possible approach to simulate the mechanical behavior of human lower limbs. It should be noted that these two modes are differentiated by the sensor located in the middle of the tibia. The tibia actually contains a vibration node for solicitations around . On the other hand, the thigh does not have a vibration node in the frequency range of interest.
In order to validate the numerical model, this study proposes a methodology that allows describing the modal behavior of the leg with no direct measures. This methodology can come in addition to those used in the literature that focus on modal parameters of a particular bone. One can distinguish two main in vivo methods that are compared in the work of Cornelissen in the late 1980s [35
]. The Impulse Response (IFR) method involves generating impact with a shock hammer where the leg is relaxed and left hanging. The Bone Resonance Frequency (BRA) method consists of generating a sinusoidal signal over a frequency range with a vibrating shaker. These methods involve the measurement of the forces and their responses on the bone: it is therefore impossible to carry out measurements under real conditions. An OMA offers an interesting alternative, being an application in real conditions where the input forces are not known. Recent developments in OMA make it possible to identify the modal parameters, with possible applications in various sports with stationary impacts or noise. In terms of instrumentation, the proposed setup (featuring four sensors) extracts two modes in the 0–150-
range. The sensor in the middle of the tibia is essential to differentiate modes. It might be wise to deploy a fifth sensor in the middle of the thigh to observe a possible modal deformation displaying a vibration node on the femur. However, properly attaching the sensor would be difficult in view of the presence of the rectus femoris. Experimental studies propose a mathematical modeling of the phenomenon and could be integrated directly into the FE model [36
Concerning the identified frequencies, the frequency of
is very similar to that found in previous studies [37
]. Both resonance frequencies were determined by using a vibratory excitation at different frequencies between 10 and
. The resonance frequency of the ankle was found to lie in the 40–50-
range. The modal deformation of the second mode at
matches the modal deformations of the tibia, which is identified experimentally in the work of Van der Perre [39
] and numerically in that of Taylor [40
]. The latter highlights the torsional modes obtained for higher values (
). The protocol of the present study unfortunately prevents the measurement of this deformation.
Considering results in a qualitative way, the frequency values lie in wide ranges. The tibia study shows that the bending resonance frequencies are of the order of magnitude
]. The modal frequencies for the femur are quite controversial. In free-free boundary conditions, Kumar [41
] identified modal frequencies at 449, 524 and
, while Gupta [42
] determined frequencies of
for bending mode and
for double bending. In fixed-fixed conditions, the extracted frequencies are 1211, 1269 and
]. These studies identified modes on a particular bone with boundary conditions that are completely different from those depicted in the present paper. Our methodology is dependent on the mass of the subject and the whole anatomical environment, which accounts for the lower values and corroborates previous studies on the influence of anatomical parts. Tsuchikane [43
] shows from cadaverous specimens that the resonance frequencies are uninfluenced by the skin, increased with the dissection of the muscles and foot and decreased with the dissection of the femur and the fibula. This is confirmed by the work of Cornelissen [35
], who adds that the joint has no influence on the modal frequencies of the tibia, but alters modal damping by 10–16%. These results are in agreement with the present work, where the values of frequencies are much lower than the values determined in vitro. In addition, the results of the sensitivity of the parameters show that a variation of 50% in the parameters of soft tissues, including the trunk and hard tissues, leads to a variation of about 10% in the frequencies.
The proposed FE model is a preliminary model that requires many developments, but the values identified experimentally are very similar to the numerical ones, even without updates. Several perspectives are possible. The model does not integrate the fibula, which may actually be necessary: the works of Cornelissen and Tsuchikane show that the fibula makes the lower limb more rigid. Moreover, the work of Tseng [44
] proves that the fibula has modal properties close to those of the tibia (
): there is no denying that it has a given influence on the dynamic behavior of the limb. Joint cartilage has also not been modeled: its influence, especially on damping, may be investigated in future works. A possible source of error is also the choice of a linear elastic constitutive law for soft tissues. Switching to hybrid models featuring Ogden-type hyperelasticity and exponential anisotropic behaviors [45
] may improve the model. Then, it would be interesting to calculate the resonance modes at different knee angles. The work of Munera [8
] shows that the resonance frequencies are a function of the knee angle during a squat. The identified frequencies could be located in the excited frequency range 6–8 or 9–20
, matching that of running activities [46
]. Finally, upgrading to a 3D model would most certainly provide a greater number of modes in the range 0–150
, two-dimensional modeling only allowing extraction of the modes in the mid-lateral plane.
Reading the standards on human exposure to vibrations, the frequency of
seemed to be the most harmful frequency for lower limbs. Indeed standard ISO 2631-1 [23
] defines a frequency weighting filter: the gain at
is 0.87, but drops below 0.10 beyond
. In the field of public transport and sports, the development of specific materials able to absorb these specific frequencies will decrease the amount of vibrations received by the human lower limbs. For example, a ground made of this material in public transport or in industries could prevent or at least significantly decrease diseases such as arthritis among workers. It could also drastically reduce the tiredness of lower limbs for passengers and make travel far more comfortable. We can note that Taiar [47
] determined the vibration responses of anti-fatigue mats to reduce the pain associated with vibratory exposure by considering these standards and defined the optimal geometry and design for risk prevention.