# FE Model and Operational Modal Analysis of Lower Limbs

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## Abstract

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## Featured Application

**In the field of public transport and sports, the development of specific materials able to absorb the modal frequencies identified in this paper will help decrease the amount of vibrations received by the human lower limbs. For example, shoes made of this material in running activities could prevent, or at least significantly decrease, injuries such as tibial stress syndrome. Moreover, the proposed model could be used to simulate dynamic solicitations stemming from different foot-strike patterns during running activities and, thus, to decrease stresses in the lower limb.**

## Abstract

## 1. Introduction

## 2. Method

#### 2.1. Measurement

#### 2.2. Modal Parameters

#### 2.3. FE Model

^{®}software (Version 6.12, Dassault Systèmes, Vélizy-Villacoublay, France) and features the foot (calcaneus, talus, metatarsus lumped together, tarsus lumped together), leg (tibia bone), thigh (femur bone), soft tissues, ankle, knee, hip joints and trunk (Figure 2). The femur and tibia bones were made of a central part of cortical bone and trabecular bones at the ends. The bones were held together by soft tissues, which represent muscles, skin, tendons and adipose tissue. Bones are in contact at joints; however, joint cartilage has not been modeled. The anthropometric dimensions used to create the model have been retrieved from several references (Table 2) and calculated for a virtual subject $175\phantom{\rule{0.277778em}{0ex}}\mathrm{c}\mathrm{m}$ tall. Obviously, the natural modes and frequencies are strongly dependent on these anthropometric dimensions, yet focusing on a single virtual subject showed promising results and allowed us to validate the methodology.

^{®}on a standard workstation. Modal frequencies’ sensitivity to the mesh density was investigated.

## 3. Results

#### 3.1. Modal Identification

#### 3.2. FE Model Results

#### 3.3. Model Update

## 4. Discussions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(Left) Accelerometers locations and orientations; (right) geometry of the lower limb model in the Modal Analysis software. Points 1, 2, 3, 4 are the location of the accelerometers and the number of node for the numerical model.

**Figure 3.**Mode shapes: (top) Frequency 1 ($12.8\phantom{\rule{0.277778em}{0ex}}\mathrm{Hz}$); (middle) Frequency 2 ($54.5\phantom{\rule{0.277778em}{0ex}}\mathrm{Hz}$); (bottom) Frequency 3 ($126.9\phantom{\rule{0.277778em}{0ex}}\mathrm{Hz}$).

Parameters | Ranges | Mean | STD |
---|---|---|---|

Age (years) | 22–34 | 25.90 | 2.81 |

Height ($\mathrm{m}$) | 1.68–1.84 | 1.78 | 0.04 |

Weight ($\mathrm{k}\mathrm{g}$) | 65.30–84.70 | 73.52 | 4.93 |

Body Mass Index (BMI) | 20.73–27.04 | 23.34 | 2.09 |

Hip-knee ($\mathrm{c}\mathrm{m}$) | 50.00–60.00 | 53.73 | 3.88 |

Knee-tibia ($\mathrm{c}\mathrm{m}$) | 15.00–21.07 | 17.29 | 1.09 |

Tibia-ankle ($\mathrm{c}\mathrm{m}$) | 20.00–27.93 | 22.89 | 1.52 |

Total length ($\mathrm{c}\mathrm{m}$) | 80.00–107.00 | 93.91 | 5.72 |

**Table 2.**Anthropometric table for a virtual subject $175\phantom{\rule{0.277778em}{0ex}}\mathrm{c}\mathrm{m}$ tall.

Segment | Parameters | Dimension ($\mathbf{m}\mathbf{m}$) |
---|---|---|

Thigh | Femoral length [25] | 482.0 |

Femoral head diameter [26] | 46.1 | |

Neck width of the femoral head [26] | 24.1 | |

Femoral diaphysiswidth [27] | 30.0 | |

Femoral condyles width [28] | 81.4 | |

Thigh diameter [29] | 191.0 | |

Knee radius [29] | 125.4 | |

Leg | Tibia length [28] | 383.4 |

Tibia width ($13.7\phantom{\rule{0.277778em}{0ex}}\mathrm{c}\mathrm{m}$ above ankle) [30] | 28.2 | |

Tibia proximal epiphysis width [31] | 60.1 | |

Tibia distal epiphysis width [31] | 43.6 | |

Calf radius [29] | 119.7 | |

Foot | Foot length [32] | 260.0 |

Foot height [33] | 81.2 | |

Ankle height [33] | 127.9 |

Body Zones | Initial | Updated | ||
---|---|---|---|---|

E $\left(\mathbf{M}\mathbf{Pa}\right)$ | D $\left({10}^{3}\phantom{\rule{0.277778em}{0ex}}\mathbf{k}\mathbf{g}/{\mathbf{m}}^{3}\right)$ | E $\left(\mathbf{M}\mathbf{Pa}\right)$ | D $\left({10}^{3}\phantom{\rule{0.277778em}{0ex}}\mathbf{k}\mathbf{g}/{\mathbf{m}}^{3}\right)$ | |

Trunk | 600 | 1.00 | 600 | 1.30 |

Cortical | 17,500 | 2.10 | 17,500 | 2.10 |

Trabecular | 450 | 1.80 | 450 | 1.80 |

Soft tissues | 345 | 1.20 | 615 | 1.11 |

**Table 4.**Identification results: repeatability (Protocol 1). m: mean, std: standart deviation, CV: coefficient of variation.

Mode | Score | Frequency ($\mathbf{Hz}$) m ± std (CV%) | Damping % m ± std (CV%) |
---|---|---|---|

1 | 100% | 36.82 ± 0.98 (2.7) | 10.03 ± 1.73 (17.2) |

2 | 60% | 42.26 ± 2.02 (4.8) | 7.5 4± 3.29 (43.7) |

3 | 100% | 49.48 ± 1.23 (2.5) | 7.04 ± 1.74 (24.7) |

4 | 90% | 62.51 ± 2.68 (4.3) | 7.78 ± 2.19 (28.2) |

5 | 30% | 71.30 ± 1.26 (1.8) | 4.81 ± 2.40 (49.9) |

6 | 80% | 82.14 ± 2.60 (3.2) | 7.82 ± 1.26 (16.2) |

7 | 100% | 98.26 ± 3.16 (3.2) | 7.53 ± 2.33 (31.0) |

8 | 90% | 122.89 ± 1.07 (0.9) | 2.58 ± 0.76 (29.3) |

Mode | Frequency ($\mathbf{Hz}$) m ± std (CV%) | Damping % m ± std (CV%) |
---|---|---|

1 | 35.66 ± 1.00 (2.8) | 1.39 ± 0.97 (69.8) |

2 | 43.24 ± 2.09 (4.8) | 7.28 ± 2.36 (32.4) |

3 | 52.54 ± 2.05 (3.9) | 6.41 ± 2.35 (36.8) |

4 | 61.99 ± 1.67 (2.7) | 5.16 ± 2.28 (44.2) |

5 | 70.40 ± 1.60 (2.3) | 4.73 ± 1.57 (33.1) |

6 | 84.06 ± 3.71 (4.4) | 4.46 ± 1.34 (30.1) |

7 | 100.38 ± 3.71 (3.7) | 3.52 ± 2.12 (60.1) |

8 | 118.94 ± 2.70 (2.3) | 1.60 ± 0.76 (47.7) |

**Table 6.**Comparison between the confidence interval and the frequencies calculated by the numerical model.

Frequencies | Mode 3 | Mode 8 |
---|---|---|

Mean experimental frequencies ($\mathrm{Hz}$) | 52.54 | 118.94 |

Confidence interval ($\mathrm{Hz}$) | 51.17–53.91 | 117.13–120.75 |

Initial model ($\mathrm{Hz}$) | 54.55 | 126.85 |

Updated model ($\mathrm{Hz}$) | 51.17 | 120.70 |

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**MDPI and ACS Style**

Pionteck, A.; Chiementin, X.; Munera, M.; Murer, S.; Chadefaux, D.; Rao, G.
FE Model and Operational Modal Analysis of Lower Limbs. *Appl. Sci.* **2017**, *7*, 853.
https://doi.org/10.3390/app7080853

**AMA Style**

Pionteck A, Chiementin X, Munera M, Murer S, Chadefaux D, Rao G.
FE Model and Operational Modal Analysis of Lower Limbs. *Applied Sciences*. 2017; 7(8):853.
https://doi.org/10.3390/app7080853

**Chicago/Turabian Style**

Pionteck, Aymeric, Xavier Chiementin, Marcela Munera, Sébastien Murer, Delphine Chadefaux, and Guillaume Rao.
2017. "FE Model and Operational Modal Analysis of Lower Limbs" *Applied Sciences* 7, no. 8: 853.
https://doi.org/10.3390/app7080853