Biomechanics BME 315
Rod Lakes
University of Wisconsin

Assignments are taken from problems given in I. P. Herman, "Physics of the human body" or from the following set (provisional).

1. Consider the mandible, as shown in the diagram (adapted and modified from Smith D. M., McLachlan, K. R., and McCall, W. D., "A numerical model of temporomandibular joint loading", J. Dental Research, 65, 1046-1052, (1986)). Some people experience pain in the temporomandibular joint (TMJ).
(a) Calculate the force Fcondyle in the joint based on the following assumptions. The biting force Fbite = 750 N and is a vertical force. Assume X1 = 11 cm and Xm1 = 3.5 cm. Assume only muscle m1 is active (other muscles are relaxed) and that its force is vertical. Assume, moreover that the points of application of joint, muscle, and tooth forces lie in a plane. Discuss briefly the implications. If you were designing an implant for the joint, how would your design be influenced by the result of your calculation?
(b) In reality, the person can tense any combination of the jaw muscles. In that case, is the problem statically determinate? What are the implications?

2. Measure the horizontal and vertical distance corresponding to five flights of stairs in the Engineering Research Building. If you prefer, use another building. Go up the stairs at a brisk pace, and measure the time it takes. Calculate your vertical and horizontal components of velocity in miles per hour. How does this compare with walking speed on a level surface? How much work was done against gravity? How much metabolic energy (in Joules) was expended, assuming an efficiency of 25%? Convert this energy to calories, keeping in mind the distinction between cal and kcal. How many flights of stairs could you ascend with the energy in one bag of potato chips (or your favorite snack)? Remark: if you calculate the power as energy divided by time, consider for comparison that the resting metabolism of an adult human is about 100 watts.

3. We have considered scaling of walking speed. Now determine how maximum running speed depends on organism size L. Assume the limitation upon speed is the strength of the leg muscles. Hint. Recall that the torque M which accelerates the leg is M = I d2q/dt2 with I as mass moment of inertia, and t as time, and q as angular position. I depends on mass distribution of the leg via a proportionality constant K. We have I = K m L 2 with L as the length of the leg. As with walking, assume the angle is oscillatory in time. The moment M is muscle force F times a moment arm times an angle.

4. Consider strain in the skin. Straighten your leg. Make two small marks with ink on the skin over your knee. Measure the distance. Bend your leg. Measure the distance again. Calculate the strain. Squeeze the skin until just before creases form. Again determine the strain. How do these strains compare with the strain (0.05) for onset of damage in ligament? We remark that bone undergoes damage at a strain of about 0.006. Discuss briefly.

5. Determine the axial load necessary to cause a strain of 400 micro-strain in the femoral diaphysis. Assume a circular section, outer diameter 25 mm, inner diameter 12.5 mm. For body weight, calculate the weight in newtons from a mass of 70 kg. A further question: do you think the stress in the femur is purely compressional as assumed in the problem statement?

6. Consider again the jaw bone in Problem 1. If a person under age 40 complains of pain in the temporo-mandibular joint (TMJ), should an implant be considered? If so, how might you incorporate the biomechanics of the joint in the design? For example, is a porous polymer suitable as a joint bearing surface? What about Teflon (polytetrafluoroethylene)? After you have written some preliminary thoughts, read the following case study, Jaw joint implant pdf , and comment further.

7. Consider stress relaxation in bone as shown in the graph in the on-line web notes (Bone viscoelasticity: bone relaxation and damping plot). Owing to the relationship between relaxation and dynamic (sinusoidal) response, the relaxation curve may be continued, overlapping the G' curve, over an effective time scale 10-3 sec to 105 sec. Evaluate the appropriateness of the function G(t) = A + B e-t/T to model the relaxation. Here t is time and T is a relaxation time constant. This exponential function arises from a simple differential equation involving strain and strain rate. Specifically, make a formal plot of the experimental data on the same log time scale as the model and discuss. Can one use a three-element spring-dashpot model for the relaxation behavior? If a static component of stress (superposed on time varying components in normal activities) were maintained for a longer period, say one year, what do you think would happen?

8. Determine the scaling of strength (maximum limb force) to body weight ratio assuming the scaling law is governed by bending of bones in the body. Hint: begin with the analysis done in class for three-point bending. Does the scaling differ from results obtained from consideration of strength of muscle or tendon in tension?

Study questions (not assigned)

Suppose that the femur mid shaft is approximately tubular with a 30 mm outside diameter and a 6 mm wall thickness. Consider the relative bending strength of the bone and of "nails" used for fracture fixation. Suppose that the bone has a yield strength of 120 MPa in tension and a Young's modulus of 15 GPa and that the steel in the "nail" has a yield strength of 690 MPa and a Young's modulus of 200 GPa. Remark: this is a fairly high strength steel, much stronger than mild structural steel. (a) How much bending moment can an intact femur withstand? (b) How much moment can a solid cylindrical stainless steel "nail" 9 mm in diameter withstand? (c) It is claimed a cloverleaf nail can withstand 23 Nm. A patient with a broken femur was treated with such a nail. One day he stood on one leg to put on his pants. He felt the nail bend, and returned to the doctor. Discuss. Was the nail at fault?

Find the force exerted by the pliers jaw (sketched in class) in terms of the force applied by the hand. Diagram to be given. Discuss briefly the ergonomic aspects of the design of hand tools. How will your choice of parameters influence how easy it is to use the tool?

A sprinter runs 100 yards in 10 seconds. (i) Show that this is equivalent to an average speed of 20.5 miles per hour. (ii) The sprinter has thicker legs than the cheetah, but cannot run as fast as the cheetah (70 miles per hour). Why not? (iii) The same sprinter can approach 40 miles per hour in a bicycle sprint. Why the difference in speed? Remark: the four minute mile corresponds to 15 miles per hour.

What is the density of the dumbbells in the Natatorium? Are they solid?

Suppose that the femur mid shaft is approximately tubular with a 25 mm outside diameter and a 6 mm wall thickness. (a) How much torsional moment can an intact femur withstand? (b) Suppose a skier strikes a tree stump with a ski, so that the ski receives a twisting action of 200 pounds of force component orthogonal to the ski at a distance of 4 feet along the ski from the ankle. Compare the twisting moment of the ski with the result obtained in (a). What do you think will happen?

Referring to the femur mid shaft problem, How much bending moment can a tubular cylindrical stainless steel nail 9 mm outside diameter and 1 mm wall thickness withstand?

Consider scaling of muscle strength. How does the strength to weight ratio of a human scale with height L? Based on your answer, do you think a tall athlete will be able to perform more pull - up exercises than a shorter athlete who has the same proportions? Assume that the maximum stress s in the muscles is independent of L.