Illustrative exercises for the lectures

Transcription

1 Biomechanics Illustrative exercises for the lectures Ingrid Svensson

2 1. To practise the use of free-body diagram, consider the problem of analyzing the stress in man s back muscles when performing hard work. The figure below shows a man shovelling snow. If the snow and the shovel weigh 10 kg and have a centre of gravity located at a distance of 1 m from the lumbar vertebra of his backbone, what is the moment about that vertebra? The construction of a human spine is sketched in the figure above. The disks between the vertebrae serve as the pivots of rotation. It may be assumed that the intervertebral disks cannot resist rotation. Hence the weight of the snow and shovel has to be resisted by the vertebral column and the back extensor muscle. Estimate the loads in his back muscle, vertebrae, and disks. Lower back pain is such a common disorder that the loads acting on the disks of patients were measured with strain gauges in some cases. It was found that no agreement can be obtained if we do not take into account the fact that when one lifts a heavy weight, one tense up the abdominal muscles so that the pressure in the abdomen is increased. A free-body diagram of the upper body of a man is shown in the figure. Show that it helps to have a large abdomen and strong abdominal muscles. 2

3 2. Compare the bending moment acting on the spinal column at the level of a lumbar vertebra for the following cases: a) A student at LTH when buying beer, lifting the case with (i) the knees straight and (ii) the knees bent. b) A water skier skis with (i) the arms straight and (ii) with elbows hugging the sides. Discuss these cases quantitatively with proper free-body diagrams. 3. A Rusell s contraction mechanism for clinically loading is shown in the figure. The mass of the lower leg is about 4.5 kg. Derive the required force W too keep the leg in equilibrium and the tensile force in the femur under these conditions. 4. Many biological materials are built up of more than one constituent, e.g. bone can be considered as a mixture of collagen and mineral. The individual stiffness and structural arrangement of the constituents will give the elastic modulus of the mixture. Since the structural arrangement is quite complex, two different mathematical models known as the Voigt and Reuss models, are often used to derive the upper and lower limit for the mixture. The Voigt model assumes that the strain is equal in the two constituents whereas the Reuss model assumes equal stresses. Express the total elastic modulus for the mixture if the elastic moduli for the constituents are E 1 and E 2 and the volume fractions are V 1 and V 2. Assume the volume fractions of apatite in bone to be 45 % (elastic modulus 114 GPa) and consequently the volume fraction of collagen to 55 % (elastic modulus 1.3 GPa). Derive the upper and lower limits for the stiffness in bone and compare these values with experimental data. 3

4 5. Consider the question of the strength of human tissues and their margin of safety when we exercise. Take the example of the tension in the Achilles tendon in our foot when we walk and when we jump. The bone structure of the foot is shown the figure. To calculate the tension in the Achilles tendon we may consider the equilibrium of forces that act on the foot. The joint between the tibia and the talus bones may be considered as a pivot. 6. The head in the picture below is considered as a rigid body for the present purpose. It rocks on the occipital condyles C where an axial force F A, a shear force, V, and a neck torque T o, resist motion. An uppercut (treated as an instantaneous load B at an angle of 63 o to the horizontal) is applied to the chin, and the initial linear acceleration, a, of the mass centre, G, is photographically determined to be 140g. For a head mass of 3.5 kg, a moment of inertia about an axis perpendicular to the sagittal plane of kg m 2, and the dimensions shown, what are the reactions at the occipital condyles if torque T 0, which requires muscle activation, is temporarily neglected? 4

5 7. Resilin is a protein found in e.g. insects where it functions as a very effective energy storage. Studies presented in literature show that it is able to store about J/m 3 when loaded near the failure point. Compare this ability with steel with the stress of 425 MPa and elastic modulus 200 GPa. (Assume that the steel is loaded to the yield point and has approximate linear elastic behaviour.) Consider a flea with the mass kg having dual resilin springs containing a total of m 3 resilin. How high can it jump? 8. The cervical disc can be considered as a Kelvin-Voigt solid. In the cervical region, the average section of the fist six discs is 150 mm2 and the average thickness is 5 mm. The Young s modulus (spring constant) of the disk is 30 MPa and the viscous constant is 25 Ns/m. A mass of 20 kg is suddenly placed and maintained for some time on the head. Assume that the entire load is carried by the discs and determine: a) the strain in the discs after 10 s b) the total change in length of the neck (cervical region) after 10 s 9. A serious dental problem is bruxism, i.e. the unconscious sideways gnashing of teeth that produces major material damage by flattening out sharp teeth as well as producing erosion to the root. In order to quantify this problem, consider the digestion of food by biting involving (i) vertical forces only, fig. 1, and (ii) sideways chewing involving horizontal loads which act cyclically on the tooth creating this horizontal motion, fig. 2. Assume that the two force components required to chew the food are the same, 50 N. The root of the idealized tooth has a width of 4 mm and a depth of 3 mm. The root is located 10 mm below the surface where the load is applied. a) What the direct stress acting on the root of the tooth for case (i)? b) What is the maximum bending stress acting on the root of the tooth for case (ii)? The area is the same as in (a). c) What is the factor of safety with sideways chewing after 1 year and 50 years if 200 such motions occur per day? Figure 1. Simple model of a tooth. Figure 2. Fatigue curve. 5

6 10. The safety factor for a total hip replacement is to be determined. 1. (inches) The problem is to estimate the actual load on the prosthesis. For this purpose, a study of an ordinary hip is performed. To get a worst static case the situation of standing on one leg is considered. The free-body diagram is presented below. Make an estimation of the size and direction of the joint reaction force using the freebody diagram in the hip in order to get the loading to use in the calculation of the safety factor. 6

7 11. A jumper executes a standing jump from a platform that is moving upwards with constant speed v p. - Derive an expression for the maximum elevation of the jumper s centre of gravity in terms of the crouch depth c, the equivalent force to weight ratio F equiv /W, the platform speed and other relevant parameters. The elevation of centre of gravity is to be measured with respect to a stationary frame of reference (i.e. one not attached to the platform). - If the crouch depth is 45 cm, the ratio F equiv /W is 2 and the platform speed is 1.5 m/s, compute the elevation of the centre of gravity. 12. A 75 kg stuntwoman executes a standing jump with the aid of a harness and a support wire. In addition to the constant 1100 N force that her legs exert during the push-off phase, the wire has a lift mechanism that applies a force given by 550e -s/l (in Newtons) where L=0.4m and s is the distance travelled. The lift mechanism is engaged at the bottom of the crouch (where s=0) and a safety catch detaches the wire when the stuntwoman leaves the ground. - For a crouch depth of 0.4m, compute the maximal elevation of her centre of gravity. - Compute also the maximal elevation of her centre of gravity if the safety catch fails to disengage. 7

8 Sources: Biomechanics, Mechanical Properties of Living Tissues, Y.C. Fung, 1993 Introduction to Bioengineering, S.A. Berger, W.Goldsmith, E.R. Lewis, 2000 Fundamentals of Machine Elements, B.J. Hamrock, B. Jacobson, S.R. Schmid, 1999 Introductory Biomechanics, From cells to organisms, C. R. Either, C. A. Simmons,