# Relationship between Joint Stiffness, Limb Stiffness and Whole–Body Center of Mass Mechanical Work across Running Speeds

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Participants

#### 2.2. Experimental Protocol and Data Collection

#### 2.3. Data Analysis

^{2}), and instantaneous COM height (${h}_{i}$) [28], expressed as

## 3. Results

#### 3.1. Stiffness

#### 3.2. Mechanical Work and Power

#### 3.3. Multiple and Simple Linear Regression

^{2}= 0.384, p = 0.046), and ${K}_{knee}$ had the strongest unique association with ${K}_{vert}$ at this speed ($\beta $ = 0.509, p = 0.022). At 2.2 m/s, the model accounted for 49.8% of the variance in ${K}_{vert}$ (R

^{2}= 0.498, p = 0.014), and ${K}_{knee}$ again had the strongest unique association with ${K}_{vert}$ at this speed ($\beta $ = 0.553, p = 0.011).

^{2}= 0.424, p = 0.028), and ${K}_{knee}$ had the strongest unique association with ${K}_{leg}$ ($\beta $ = 0.532, p = 0.014). At 2.2 m/s, the model accounted for 79.3% of the variance in ${K}_{leg}$ (R

^{2}= 0.793, p < 0.0001). For this speed, however, ${K}_{knee}$ ($\beta $ = 0.553, p = 0.0004) and ${K}_{hip}$ ($\beta $ = 0.526, p = 0.001) both had a strong unique association with ${K}_{leg}$. At 2.6 m/s, the model accounted for 39.9% of the variance in ${K}_{leg}$ (R

^{2}= 0.399, p = 0.039), and ${K}_{knee}$ had a unique association with ${K}_{leg}$ ($\beta $ = 0.456, p = 0.04). At 3.4 m/s, the model accounted for 47.4% of the variance in ${K}_{leg}$ (R

^{2}= 0.474, p = 0.026), and ${K}_{hip}$ had a strong unique association with ${K}_{leg}$ ($\beta $ = 0.721, p = 0.009).

^{2}= 0.133, p = 0.477). However, ${K}_{vert}$ was positively associated with ${W}_{coms}^{+}$ across the speeds (R

^{2}= 0.902, r = 0.95, p = 0.004) (Table 4).

#### 3.4. Interpretation of Graph Patterns

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic representative of a spring–mass model in the running stance phase. The model consists of a point mass (COM) equivalent to the body mass and the leg as a massless linear spring. The leg spring is compressed, and reaches maximum compression ($\u2206L$) at mid–stance. The COM displacement in the vertical direction is denoted as $\u2206y$. The half swept angle of the leg spring is denoted as $\theta $. IC: initial–contact. MS: mid–stance. TO: toe–off. Point O is the ground contact location.

**Figure 2.**Group average (n = 20) leg–spring force–length curves at three representative speeds. GRF: vertical ground reaction force normalized to body weight. Virtual leg length: instantaneous leg length/${L}_{0}$.

**Figure 3.**Group average (n = 20), (

**a**) whole–body COM gravitational potential energy (${E}_{pot}$) and mechanical kinetic energy (${E}_{kin}$) in the stance phase of three representative speeds; (

**b**) sagittal plane whole–body COM instantaneous mechanical power (${P}_{coms}$) in the stance phase of three representative speeds.

**Table 1.**Vertical stiffness (kN/m), leg stiffness (kN/m) and joint stiffness (Nm/kg/deg) across running speeds. Sample mean (standard deviation); n = 20.

Stiffness | Running Speed (m/s) | |||||
---|---|---|---|---|---|---|

1.8 | 2.2 | 2.6 | 3.0 | 3.4 | 3.8 | |

${K}_{vert}$ | 23.03 (5.19) ^{a} | 24.98 (4.77) ^{b} | 27.10 (4.50) ^{a,c} | 29.79 (4.70) ^{a,b,d} | 32.84 (6.40) ^{a,b,c} | 40.29 (9.16) ^{a,b,c,d} |

${K}_{leg}$ | 13.49 (3.40) | 13.39 (3.85) | 13.22 (3.28) | 13.07 (2.76) | 12.96 (3.65) | 13.45 (4.17) |

${K}_{ankle}$ | 0.18 (0.08) | 0.18 (0.05) | 0.19 (0.06) | 0.19 (0.09) | 0.21 (0.07) | 0.23 (0.09) |

${K}_{knee}$ | 0.10 (0.02) ^{e} | 0.11 (0.02) ^{f} | 0.12 (0.03) | 0.14 (0.04) ^{e,f} | 0.15 (0.06) | 0.18 (0.08) ^{e,f} |

${K}_{hip}$ | 0.25 (0.14) | 0.22 (0.11) | 0.26 (0.12) | 0.24 (0.07) | 0.27 (0.10) | 0.27 (0.10) |

^{a}: Statistically significant differences of ${K}_{vert}$ between 1.8 m/s and all speeds between 2.6 and 3.8 m/s, respectively (p < 0.0001);

^{b}: differences of ${K}_{vert}$ between 2.2 m/s and all speeds between 3.0 and 3.8 m/s, respectively (p $\le $ 0.0001);

^{c}: differences of ${K}_{vert}$ between 2.6 m/s and 3.4 m/s (p = 0.001), and 2.6 m/s and 3.8 m/s (p = 0.0002);

^{d}: differences of ${K}_{vert}$ between 3.0 m/s and 3.8 m/s (p = 0.0032);

^{e}: differences of ${K}_{knee}$ between 1.8 m/s and 3.0 m/s (p = 0.002), and 1.8 m/s and 3.8 m/s (p = 0.001);

^{f}: differences of ${K}_{knee}$ between 2.2 m/s and 3.0 m/s (p = 0.001), and 2.2 m/s and 3.8 m/s (p = 0.003).

**Table 2.**Whole–body COM positive and negative mechanical work (J/kg) and sagittal plane COM peak positive and negative power (W/kg) across the speeds. Sample mean (standard deviation); n = 20.

Running Speed (m/s) | ||||||
---|---|---|---|---|---|---|

1.8 | 2.2 | 2.6 | 3.0 | 3.4 | 3.8 | |

Work | ||||||

${W}_{coms}^{+}$ | 1.03 (0.14) ^{a} | 1.06 (0.23) | 1.16 (0.14) | 1.21 (0.20) ^{a} | 1.22 (0.31) ^{a} | 1.31 (0.29) ^{a} |

${W}_{coms}^{-}$ | −0.85 (0.11) ^{b} | −0.90 (0.12) | −0.96 (0.11) | −0.96 (0.13) | −0.98 (0.15) ^{b} | −0.94 (0.19) |

${W}_{comh}^{+}$ | 0.21 (0.05) ^{c} | 0.26 (0.08) ^{d} | 0.33 (0.09) ^{c,e} | 0.39 (0.12) ^{c,d,f} | 0.43 (0.17) ^{c,d,e} | 0.54 (0.17) ^{c,d,e,f} |

${W}_{comh}^{-}$ | −0.17 (0.05) ^{g} | −0.22 (0.05) ^{h} | −0.30 (0.08) ^{g} | −0.33 (0.08) ^{g} | −0.37 (0.09) ^{g,h} | −0.39 (0.12) ^{g} |

${W}_{comv}^{+}$ | 0.83 (0.14) | 0.81 (0.17) | 0.85 (0.12) | 0.84 (0.13) | 0.81 (0.16) | 0.79 (0.18) |

${W}_{comv}^{-}$ | −0.69 (0.11) | −0.69 (0.11) | −0.68 (0.10) | −0.65 (0.11) | −0.62 (0.10) | −0.56 (0.11) |

Power | ||||||

${P}_{coms}^{+}$ | 10.80 (2.63) ^{i} | 12.42 (2.30) ^{i,j} | 13.99 (2.74) ^{i,j,k} | 16.43 (3.46) ^{i} | 17.55 (2.75) ^{i,j,k} | 18.80 (4.92) ^{i,j.k} |

${P}_{coms}^{-}$ | −11.39 (1.97) ^{l} | −12.69 (2.08) ^{l,m} | −14.70 (3.01) | −15.48 (2.15) ^{l,m} | −16.56 (2.57) ^{l,m} | −17.75 (4.62) ^{l} |

^{a}: Statistically significant differences of ${W}_{coms}^{+}$ between 1.8 m/s and 3.0 m/s (p = 0.002), 1.8 m/s and 3.4 m/s (p < 0.0001), and 1.8 m/s and 3.8 m/s (p = 0.003);

^{b}: differences of ${W}_{coms}^{-}$ between 1.8 m/s and 3.4 m/s (p = 0.002);

^{c}: differences of ${W}_{comh}^{+}$ between 1.8 m/s and all speeds between 2.6 and 3.8 m/s, respectively (p < 0.0003);

^{d}: differences of ${W}_{comh}^{+}$ between 2.2 m/s and all speeds between 3.0 and 3.8 m/s, respectively (p < 0.002);

^{e}: differences of ${W}_{comh}^{+}$ between 2.6 m/s and all speeds between 3.4 and 3.8 m/s, respectively (p < 0.001);

^{f}: differences of ${W}_{comh}^{+}$ between 3.0 m/s and 3.8 m/s (p = 0.0009);

^{g}: differences of ${W}_{comh}^{-}$ between 1.8 m/s and all speeds between 2.6 and 3.8 m/s, respectively (p < 0.0001);

^{h}: differences of ${W}_{comh}^{-}$ between 2.2 m/s and 3.4 m/s (p = 0.0004);

^{i}: differences of ${P}_{coms}^{+}$ between 1.8 m/s and all speeds between 2.2 and 3.8 m/s, respectively (p < 0.001);

^{j}: differences of ${P}_{coms}^{+}$ between 2.2 m/s and 2.6 m/s, 2.2 m/s and 3.4 m/s, 2.2 m/s and 3.8 m/s (p < 0.001);

^{k}: differences of ${P}_{coms}^{+}$ between 2.6 m/s and 3.4 m/s, and 2.6 m/s and 3.8 m/s (p $\le $ 0.001);

^{l}: differences of ${P}_{coms}^{-}$ between 1.8 m/s and 2.2 m/s, and 1.8 m/s at all speeds between 3.0 and 3.8 m/s, respectively (p $\le $ 0.002);

^{m}: differences of ${P}_{coms}^{-}$ between 2.2 m/s and 3.0 m/s, and 2.2 m/s and 3.4 m/s (p < 0.001).

**Table 3.**Multiple linear regression models between joint stiffness and vertical stiffness (first two rows), and leg stiffness (lower four rows). Only the speed conditions with statistically significant associations are shown; n = 20.

Variable | Speed (m/s) | ${\mathit{\beta}}_{\mathit{K}\mathit{a}\mathit{n}\mathit{k}\mathit{l}\mathit{e}}$ | ${\mathit{\beta}}_{\mathit{K}\mathit{k}\mathit{n}\mathit{e}\mathit{e}}$ | ${\mathit{\beta}}_{\mathit{K}\mathit{h}\mathit{i}\mathit{p}}$ | Model Summary |
---|---|---|---|---|---|

${K}_{vert}$ | 1.8 | 0.246 | 0.509 * | 0.142 | ${\beta}_{0}$ = 8.298, R^{2} = 0.384, p = 0.046 |

${K}_{vert}$ | 2.2 | 0.040 | 0.553 * | 0.338 | ${\beta}_{0}$ = 9.289, R^{2} = 0.498, p = 0.014 |

${K}_{leg}$ | 1.8 | −0.076 | 0.532 * | 0.323 | ${\beta}_{0}$ = 4.815, R^{2} = 0.424, p = 0.028 |

${K}_{leg}$ | 2.2 | −0.237 | 0.553 * | 0.526 * | ${\beta}_{0}$ = 3.210, R^{2} = 0.793, p < 0.0001 |

${K}_{leg}$ | 2.6 | 0.048 | 0.456 * | 0.404 | ${\beta}_{0}$ = 4.512, R^{2} = 0.399, p = 0.039 |

${K}_{leg}$ | 3.4 | −0.353 | 0.046 | 0.721 * | ${\beta}_{0}$ = 9.760, R^{2} = 0.474, p = 0.026 |

**Table 4.**Simple linear regression model between vertical stiffness and whole–body COM sagittal plane positive work across the speeds; n = 20.

Variable | ${\mathit{\beta}}_{{W}_{coms}^{+}}$ | Model Summary |
---|---|---|

${K}_{vert}$ | 0.950 | ${\beta}_{0}$ = 0.677, R^{2} = 0.902, p = 0.004 |

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**MDPI and ACS Style**

Jin, L.; Hahn, M.E.
Relationship between Joint Stiffness, Limb Stiffness and Whole–Body Center of Mass Mechanical Work across Running Speeds. *Biomechanics* **2022**, *2*, 441-452.
https://doi.org/10.3390/biomechanics2030034

**AMA Style**

Jin L, Hahn ME.
Relationship between Joint Stiffness, Limb Stiffness and Whole–Body Center of Mass Mechanical Work across Running Speeds. *Biomechanics*. 2022; 2(3):441-452.
https://doi.org/10.3390/biomechanics2030034

**Chicago/Turabian Style**

Jin, Li, and Michael E. Hahn.
2022. "Relationship between Joint Stiffness, Limb Stiffness and Whole–Body Center of Mass Mechanical Work across Running Speeds" *Biomechanics* 2, no. 3: 441-452.
https://doi.org/10.3390/biomechanics2030034