1 / 29

The Rotational Equations of Motion

O can be an arbitrary point on the body However, if O=G or or O accelerates in the direction of G, than:. angular velocity. G. Angular acceleration. O. The Rotational Equations of Motion. Euler’s Equations. Condition: O is the center of mass G Or is a non accelerating point.

yamka
Download Presentation

The Rotational Equations of Motion

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. O can be an arbitrary point on the body However, if O=G or or O accelerates in the direction of G, than: angular velocity G Angular acceleration O The Rotational Equations of Motion

  2. Euler’s Equations 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):

  3. Kinetics of Rigid Bodies- Example Y Fj G  d Fwb 600 Fwu -Fwb Mj X b a Wb 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, Weight of upper body: Fw,u=525 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.

  4. Fj G  Fwb Fwb Fwu Mj b a IG G ma 

  5. C b a The effect of forces • Compression or tension (a) • Bending (b) • Torsion

  6. P P/A P P Inside the material Normal Stresses Stress=P/A • In compression or tension Externally Internally

  7. compression Tension Inside the material • In Bending I- is moment of inertia

  8. T Inside the material • In torsion Shearing stresses

  9. U Force (stress) y Ductile materials Displacement (strain) Response of the material • Deformation • Normal strain (linear) • Shearing strain (angular) • Failure • Yield (limit of reversibility) • Ultimate (complete failure)

  10. Elastic material Force (stress) Displacement (strain) Stress Strain Relations • Linear materials (Elastic) • Non linear materials • Non linear elastic • Non elastic (plastic) • Viscoelastic Force (stress) Non elastic material Displacement (strain)

  11. 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

  12. Properties of Materials • Elastic (Young’s) modulus • Shear modulus • Poisson’s ratio • Yield point (stress) • Ultimate stress

  13. 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

  14. What is the meaning of those parameters • Stress strain relations

  15. U y Ductile materials Displacement (strain) Strain Hardening & Fatigue • If ductile material deforms plastically it may become ‘harder’ • Strain hardening produces more brittle properties • Strain hardening can contribute to material Fatigue Force (stress)

  16. Material Fatigue • If an object is subjected to repetitive loading, the maximum stress that can be applied depends on the number of cycles Maximum Stress Fatigue curve Fatigue limit 1 Number of cycles

  17. 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

  18. 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

  19. 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

  20. Biological materials • Bone • Cartilage • Tendon • Ligament • Skin

  21. 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”.

  22. Some Physical Properties of Bone

  23. Head and Neck of the Femur

  24. Bone Structure

  25. Osteoporosis Bone Diseases- Osteoporosis • Normal Bone

  26. Osteoporotic Hip Osteoporosis Cont. • Normal Hip

  27. See you next time

More Related