A Hybrid Approach to Stress Analysis in Skeletal Systems Alan Barhorst Department of Mechanical Engineering Texas Tech University alan.barhorst@ttu.edu Lawrence Schovanec Department of Mathematics & Statistics Texas Tech University schov@math.ttu.edu Abstract This paper provides a continuum analysis of skeletal elas- tic structures in which loading conditions are derived from neural-musculotendon dynamics. Forward dynamic simu- lations of human motion are based on an ensemble of ar- ticulating segments controlled by Hill-type musculotendon actuators. The joint torques and reaction forces as predicted by this analysis determine loading conditions for the stress analysis of the segmental links which are modeled as hy- brid parameter systems. This approach accounts for both the rigid body motions of the articulating links and the elas- tic deformations that represent the continuum effects in the bone. Although the methods in this paper are readily ex- tended to general multi-link segmental models, simulations for the arm-shoulder complex are presented in order to il- lustrate the method. 1 Musculotendon Dynamics Direct dynamic, or forward, models of human movement are utilized in this work in order to simulate human move- ment as a consequence of the applied muscle forces that are generated in response to a given neural stimulation. These muscle forces and joint passive effects determine the load- ing conditions that are incorporated into a stress analysis of elastic skeletal elements. The muscle models utilized in this investigation are referred to as a Hill-type models. They have been shown to incorporate realistic physiology and complexity while remaining computationally practical (see [2, 3] and references therein). In Figure 1, a phemeno- logical model of muscle-tendon of length l tm is depicted. The muscle of length l m is in series and off-axis by a pen- nation angle α with the tendon of length l t . The muscle is assumed to consist of two components: an active force generator and a parallel passive component. The passive component includes a parallel elastic element (F pe ) that de- scribes the passive muscle elasticity and a linear damping component which corresponds to the passive muscle viscos- ity (B m ). l m l tm l t Passive Component Active Contractile Component M m F pe α l w B m K t (F t ) Figure 1. Hill Type Model of the musculotendon complex. The model for the active contractile component is based on the generally accepted notion ([5]) that the active muscle force is the product of three factors: (1) a length-tension relation f l (l m ), (2) a velocity-tension relation f v ( ˙ l m ), and (3) the activation level a(t). The length at which the max- imum active muscle force, F o , is developed is called the optimal muscle length, l o . Consequently, it is convenient to visualize active force generation as a collections of sur- faces obtained as a product of the nondimensionalized force velocity and force length curves scaled by activation, F act = a(t)F o f l ( ˜ l m )f v (f v ( ˙ l m )). where ˜ l m = l m /l o . Several analytical models have been proposed to describe the curves F pe , f l (l m ) and f v ( ˙ l m ). The approach in this paper follows that previously em- ployed in [2, 3] in which the active and passive force length curves and the force velocity curve are constructed from an- alytical models or as natural cubic splines fitted to physio- logical data. Muscle activation, a(t), is related to the neural input, u(t), by a process known as contraction dynamics. The process through which neural input is transformed into activation is known to be mediated through a calcium diffusion process and is represented by the first order differential equation da(t) dt +  1 τ act (β + (1 − β)u(t))  a(t)= 1 τ act u(t) (1.1) where 0 <β< 1, β = τ act /τ deact , and τ act ,τ deact are an activation and deactivation time constants that vary with fast and slow muscle.