6.4 The musculoskeletal dynamics problem

Credit: Chyn Wey Lee/Western Herald 2009

  • Inertial effects cannot always be ignored
    • Automotive accident
    • Collisions between people
    • Falls
  • Equations become:
    • \(\displaystyle\sum_j F_{ij} = m_i \ddot{r}_i\)
    • \(\displaystyle\sum_j r_{ij} \times F_{ij} + \displaystyle\sum_k M_{ik} = \dot{H}_i = I \ddot{\theta}\)
  • The RHS of these equations is the rate of change of inertia

6.4.1 Methods to solve the dynamics problem

  • Direct solution
    • Internal and external forces are known as a function of time
    • Directly integrate the equations of motion
      • \(\displaystyle\iint \displaystyle\sum_j F_{ij} \, dt \, dt = \displaystyle\iint m_i \ddot{r}_i \, dt \, dt\)
      • \(\displaystyle\iint \displaystyle\sum_j r_{ij} \times F_{ij} \, dt \, dt + \displaystyle\iint \displaystyle\sum_k M_{ik} \, dt \, dt = \displaystyle\iint I \ddot{\theta} \, dt \, dt\)
    • Note: the mathematics of the discrete calculation are interesting… covered in my advanced finite element class
  • Inverse solution
    • External forces are known, internal unknown
    • Motion has been measured
      • \(\displaystyle\sum_j \left(F_{ij}\right)_\mathrm{internal} = m_i \ddot{r}_i -\displaystyle\sum_j \left(F_{ij}\right)_\mathrm{external}\)
      • \(\displaystyle\sum_k \left(M_{ik}\right)_\mathrm{internal} = I \ddot{\theta}-\displaystyle\sum_k \left(M_{ik}\right)_\mathrm{external}\)
  • The inverse method is far more common in biomechanics – internal forces are rarely known
  • Unfortunately, redundancy remains and we must make assumptions
  • Further, typically substantial error associated with calculation of position!

6.4.2 Body segment mass and geometric properties

CCSA4.0 BruceBlaus 2015

  • Accuracy requires:
    • Accurate measurement of anatomical segments and mass distributions
    • Lines of action of muscles, tendons, ligaments
      • Can be determined approximately by origins and insertions of tendon and ligament
  • Empirical data is often used

6.4.3 Anthropometric Data

@Bartel2006 @Bartel2006

  • Approximate properties can be generalized from living subjects and cadavers
  • These properties are averages from a small number of samples
  • Significant variability should be expected from the published data
  • You must recognize and acknowledge the limitations