second semester

• the statical moment about plane (pi) is equal to the product

  • of the total mass of the body
  • distance from the mass center from the plane
• Moment of inertia is calculated with respect to a straight line passing through the centre of mass.
• Product of inertia is calculated with respect to two perpendicular oriented planes
• Moments of inertia relative to the principal axes of inertia are extreme.
• Principal moments of inertia I1*, I2*, I3* and principal axes of inertia 1*, 2*, 3* of a rigid body determined at its mass centre 0* are referred to as centroidal principal moments of inertia and centroidal principal axes of inertia.
• Products of inertia relative to the planes determined by principal axes of inertia are equal to zero.
• Any material system possesses minimum three principal axes; exactly three when I1 ≠ I2 ≠ I3, one and whole plane principal axes perpendicular to this axis when I1 ≠ I2 = I3, the whole space principal axes when I1 = I2 = I3.
• Kinetic energy is a sum of a kinetic energy of the mass of the body applied in the mass center and kinetic

  energy of rotation about the z axis passing through the mass center
• Eigenvalues of the inertia tensor (roots of its characteristic equation I1, I2, I3) are called principal

  moments of inertia at the chosen point 0.

  The corresponding axes determined by the eigenvectors of the inertia tensor at point 0 are called

  principal axes of inertia at this point.
• Principal moments of inertia I1*, I2*, I3* and principal axes of inertia 1*, 2*, 3* of a rigid body are determined at its mass centre 0*
• These are called centroidal principal moments of inertia and centroidal principal axes of inertia
• Any material system possesses minimum three principal axes
• Exactly three principal axes exist when I1 ≠ I2 ≠ I3
• One principal axis and whole plane principal axes are perpendicular to this axis when I1 ≠ I2 = I3
• The whole space principal axes exist when I1 = I2 = I3
• Moments of inertia relative to the principal axes of inertia are extreme
• Products of inertia relative to the planes determined by principal axes of inertia are equal to zero
• Mass center of a particle system 0*

  • is a center of the parallel system of vectors.
  • The measures of these vectors are masses of the particles
• Stainer


  moment of inertia of a body with respect to any given axis is equal to the moment of inertia of the body with respect to a centroidal axis parallel to the axis plus the product

  of the total body mass and the square of the distance between these two axes.
• Euler’s first law:

  If the sum of external forces acting on the particle system or the rigid body equals zero the linear momentum is CONSTANT.

  The linear momentum then is conserved
• A rigid body -6 degrees of freedom.

  translational motion of an arbitrary point - 3 degrees

  rotational motion - 3 degrees

  Consequently, two balance laws, one for translation and another for rotation are required to specify completely the motion of the rigid body.
• Koenig’s theorem

  kinetic energy of a rigid body is the

  • sum of the translational kinetic energy of the center of mass of the rigid body
  • the rotational kinetic energy about the center of mass of the rigid body.
• Theorem 1.

  For the material system acted upon by the potential forces the equilibrium state is characterised by

  the extremal values of the potential energy of the system.
• Friction - force preventing or resisting the relative motion of bodies

  which are in contact
• Lyapnov equilibrium stability

  The system is in stable equilibrium if when knocked out of balance by the small impulse it remains

  in the vicinity of its initial position
• EQUATIONS OF MOTION OF RIGID BODIES.

  6 degrees of freedom

  • 3 degrees in translational
  • 3 rotational motion

  two balance laws, one for translation and another for rotation are

  required to specify completely the motion of the rigid body
• These equations, called Euler’s equations of motion, may be used to analyse the motion of a rigid

  body about its mass center.

  Three differential equations of motion of the mass center together with above Euler’s equations

  form a system of six differential equations. Given appropriate initial conditions, these differential

  equations have a unique solution.

  Thus, the motion of a rigid body is completely defined by the sum and the total moment about the

  mass center of the external force system acting on it.
• principal moments of inertia

  eigenvalues of inertia tensor

  -The corresponding axes determined by the eigenvectors of the inertia tensor at point 0 are called principal axes of inertia at this point