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The Rotational Equations of Motion H O I O dH O M O rG maO O dT angular velocity M O can be an arbitrary point on the body However, if O=G or aO 0 or O accelerates in the direction of G, than: dH O MO dT rG / O G aO O M O I xx dH d I yx dt dt I zx I xy I yy I zy I xz x I yz y I zz z Euler’s Equations I dH O d xx M I yx dt dt I zx I xy I yy I zy I xz x I yz y I zz z Condition: O is the center of mass G Or is a non accelerating point and for a simplified case, when all products of inertia are zero and rotation is about a single axis (planar case): I xx x M O I yy y I zz z Kinetics of Rigid Bodies - Example A weight lifter raises a barbell to his chest. Determine the torque developed by the back and the hip extensor muscles (Mj) when the barbell is about the knee height (as shown in the illustration). Weight of barbell: Wb=1003 N, a=38 cm, b=32 cm, d=64 cm IG=7.43 kg-m2, =8.7 rad/s2, aGx=0.2 m/s2, aGy=-0.1 m/s2. Weight of upper body: Fw,u=525 N Y Fj G d Fwb -Fwb Fwu 600 Mj b a Wb X G Fj G ma Fwb Fwu I G Mj b a The effect of forces • Compression or tension (a) • Bending (b) b • Torsion a C Inside the material • In compression or tension Normal Stresses P Stress=P/A Internally P A Externally P/A P P Inside the material • In Bending My I I- is moment of inertia compression Tension Inside the material • In torsion Shearing stresses Tr Jp T Response of the material Normal strain (linear) Shearing strain (angular) • Failure Yield (limit of reversibility) Ultimate (complete failure) Force (stress) • Deformation U y Ductile materials Displacement (strain) Elastic material Force (stress) • Linear materials (Elastic) • Non linear materials Force (stress) Stress Strain Relations Non elastic material – Non linear elastic – Non elastic (plastic) – Viscoelastic Displacement (strain) Displacement (strain) The Mechanical Analysis • Determine forces through modeling – Identify which muscles are active • Determine: – Forces in the muscles – Forces in the joints and the bones – Forces in the ligaments (if possible) • Check stresses in the system • Do the stresses exceed the material strength Properties of Materials • • • • • Elastic (Young’s) modulus Shear modulus Poisson’s ratio Yield point (stress) Ultimate stress Materials can be • Isotropic: respond equally in all directions • Non Isotropic: respond differently to loading in different direction • Examples: – Metals are mostly isotropic – Wood is non isotropic and so is human skin and most biological tissues What is the meaning of those parameters • Stress strain relations FA l l E E • If ductile material deforms plastically it may become ‘harder’ • Strain hardening produces more brittle properties • Strain hardening can contribute to material Fatigue Force (stress) Strain Hardening & Fatigue y U Ductile materials Displacement (strain) • If an object is subjected to repetitive loading, the maximum stress that can be applied depends on the number of cycles Maximum Stress Material Fatigue Fatigue curve Fatigue limit 1 Number of cycles Properties of some materials • Surgical stainless steel (the strength changes depending on treatment) – Yield stress (2.0 to 8.0)x108 N/m2 – UTS up to 10x108 N/m2 – Ductile elongation from 7% up to 65% depending on type – Young’s Modulus 2.0x1011 N/m2 – Fatigue limit 3.0x108 N/m2 – Hardness (VPN)from 175 up to 300 Properties of some materials (cont.) • Some Cobalt Chromium alloys – Yield stress (4.9 to 10.5)x108 N/m2 – UTS up to 15.4x108 N/m2 – Ductile elongation from 8% up to 60% depending on treatment – Young’s Modulus 2.3x1011 N/m2 – Fatigue limit up to 4.9x108 N/m2 – Hardness (VPN)from 240 up to 450 Properties of some materials (Cont.) • Some Titanium alloys – Yield stress (1.6 to 4.7)x108 N/m2 – UTS up to (4 to 7)x108 N/m2 – Ductile elongation from 15% up to 30% depending on treatment and composition – Young’s Modulus ~1.1x1011 N/m2 – Fatigue limit up to 4.9x108 N/m2 – Hardness (VPN)from 240 up to 450 Biological materials • • • • • Bone Cartilage Tendon Ligament Skin Bone • Read about bone Morphology & Histology • Wolfs Law of Functional Adaptation (1870) – “The shape of bone is determined only by static stressing…” – “Only static usefulness and necessity determines the existence of every bony element, and consequently of the overall shape of the bone”. Some Physical Properties of Bone Variable Comment Value Unit DENSITY Cortical Lumbar Water Bone Kg/m3 Kg/m3 Kg/m3 % Kg/m3 Gpa (109N/m3) MPa MINERAL CONTENT WATER CONTENT ELASTIC MODULUS (tension) TENSILE STRENGTH COMPRESION STRENGTH Femur (cortical) Femur Tibia Fibula Femur Tibia Wood Granite Steel 1700-2000 600-1000 1000 60-70 150-200 5-28 80-150 95-140 93 131-224 106-200 40-80 160-300 370 MPa (106N/m3) Head and Neck of the Femur Bone Structure Bone Diseases- Osteoporosis • Normal Bone Osteoporosis Osteoporosis Cont. • Normal Hip Osteoporotic Hip See you next time