ME 176 Final Exam, Fall 1997
Page 1 of 10
Tuesday, December 16, 5:00–8:00 PM, 1997.
Answer all questions for a maximum of 100 points. Please write all answers in the space provided. If you
need additional space, write on the back sides. Indicate your answer as clearly as possible for each
question. Write your name at the top of each page as indicated.
Read each question very carefully!
(30 points total) Dynamic Planar (2D) Analysis of the Skeleton
During a “giant circle” routine (Figure 1), a gymnast rotates in the sagittal plane around a fixed bar
which they grip in an approximately frictionless manner. In some routines, the gymnast will actively
move their legs from the hyperextended position (shown below) to a more flexed position,
will move their legs closer to the front of the body, thereby decreasing the angle between the
anterior aspect of the torso and the legs. As with most kinetic problems in musculoskeletal
biomechanics, we can use the inverse-dynamics approach to solve for the internal loads in the body,
where some of the motions and external loads are directly measured and the unknown internal loads
are then solved for with the appropriate dynamics analysis.
In analyzing this situation, assume that the body can be modeled as a two-link rigid bar system
(Figure 1b), one rigid bar (termed here the “torso”) representing the torso, head, arms, and hands,
and the other rigid link (termed here the “legs”) representing the legs and feet. The links are
connected at the hip joint. The masses and mass moments of inertia (about the mass center) are m
for the torso, and m
for the legs. Assume for this problem that the only known kinematic data
are the angular position, velocity, and accelerations of the torso, denoted
The resultant (vector) force exerted by the bar on the gymnast’s hands,
, is also known. The
appropriate dimensions are known and are given in Figure 1.
Two-bar linkage model (left) of gymnast performing a “giant circle” routine (right).
Draw a fully labeled free-body diagram for each link. Add also the appropriate linear and angular