ME 577 – Human Motion Kinetics Homework #4 1. Assume that a person is lifting a weight in a quasi-static manner using the muscles shown in the diagram below. The hand holds a 15 kg mass a distance of 30 cm (the length of the forearm) from the elbow joint. The forearm weighs 15 N. 1a. For the single force – no moment model, determine the muscle load (and joint contact forces) if we assume only the biceps is acting. Then, assume only the brachialis is acting. Then, assume only the brachioradialus is acting. 1c. Determine the muscle forces for the multi-force-no moment model (assume each muscle develops the same stress). This is a common technique for solving these kinds of problems, but not necessarily the best one. Also calculate the joint contact forces. Which model yields the lowest joint contact forces? Adapted from Orthopaedic Biomechanics by Mow and Hayes. 2 Muscle Angle θ (degrees) Moment Arm (cm) PCSA (cm ) Biceps brachii (BIC) 80.3 4.6 4.6 Brachialis (BRA) 68.7 3.4 7.0 Brachioradialis (BRR) 23.0 3.0 1.5 PCSA = Physiological Cross-Sectional Area 2. Repeat the problem above (only for the SFNM model of the biceps muscle) assuming that the elbow makes an angle of φ = πt2/2 – π/4. At what time will the forearm be horizontal? At that time, determine the force in the biceps muscle assuming it acts alone. 3. A weight lifter starts with his hands at his sides and raises his arms 90o in T=1 second, π ⎛ ⎛ πt ⎞ ⎞ θ(t) = 1− cos ⎜ ⎝⎜ ⎠⎟ ⎟ 4 ⎝ T ⎠ . His arm is 80 cm long and rotates about the center of the humeral head. The deltoid attaches at a point approximately 9 cm from the center of the humeral head. For simplicity, assume that it always makes an angle of 10o with the long axis of the humerus. The weight lifter raises a weight with a mass of 30 kg. 3a. Draw a free body diagram. 3b. Write out the equations of motion that govern the system. 3c. Calculate the acceleration of the center of mass. 3d. If the mass of the arm is 8 kg, determine the system’s moment of inertia relative to the center of the humeral head. 3e. Determine the contact forces in the shoulder and the muscle forces in the deltoid. 3f. Use Cotter’s method to determine the most important parameters for this model. Use the deltoid force as your output. 4. Consider a static analysis of an adult picking up a 30 lbs. child and holding the child at arm’s length (approximately 2.5 ft. from the shoulder). Assuming the shoulder lines up with the vertical axis of the spine, and the distance from the vertebral body to the erector spinae muscle is 1.5 inches behind the spine, calculate the compressive and shear loads on the spine as well as the force in the erector spinae muscle. Assume that the head and torso weight 40 lbs. (2 inches in front of the spine) and the two arms weight 10 lbs. What happens if you rotate 45o at the pelvis? .