Mechanics- FORCE SYSTEMS

• 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. The position of the mass center depends on the

  masses of the particles and their position
• Consider a body of the total mass M. Let l be an any arbitrary axis parallel to the axis Lo passing

  through , the center of mass of the body. The distance between these two axes is equal to a.
• The moment of force F

  applied at point A about point B

  is equal to the cross product of the force vector

  AND the vector AB
• Properties of the moment

  • Magnitude of the moment is equal to the magnitude of the force multiplied by the moment arm.
  • Line of action of the moment is perpendicular to both F and AB.
  • Sense of the moment is established by the right hand rule.
• r - moment arm - the perpendicular

  distance from the line of action of the force

  to the moment center.
• The physical meaning of the moment:

  • The moment of a force about a point quantifies its tendency to rotate a body about that point.

  • The magnitude of the moment specifies the magnitude of the rotational force.

  • The direction of a moment shows the axis of rotation
• The moment of a force about a line is

  equal to the moment of a projection of

  the force onto a plane perpendicular

  to this line about the point of

  intersection of the plane and the line
• Moment of a force about a line is parallel to this line.

  When the moment of a nonzero force about a line equals zero?

  Moment of a non-zero force about a line equals zero if, and only if, the force and the line are coplanar
• system of forces is characterized by

  • The sum of the system (the net force)
  • The total moment of a force system about an arbitrary point B
• Moment transport theorem:

  The moment of a force system about the new center is equal to the moment about the

  old center plus the cross product of the sum of the system and the vector connecting

  the old center to the new one.
• Conclusions:

  • If the sum of any given force system equals zero then the total moment of this system is constant (independent of the center).
  • If the moments of a force system about three non-collinear points are equal, then the sum of this system equals zero
  • For an arbitrary force system the dot product of its sum and its moment about any point is constant and called THE PARAMETER OF THE SYSTEM
• Zero force system

  If a force system has a sum equal zero and a total moment about any point equal zero then it is called a zero system
• A couple is defined as two parallel non-zero forces with equal magnitudes, opposite directions and different lines of action (separated by a perpendicular distance d)

  The sum of the couple equals zero

  • The total moment of the couple is the same about any point
• Two force systems are equivalent if their sums are equal and their total moments about any arbitrary point are EQUAL

  1. Two force systems are equivalent if their sums and total Moments about one point are equal.
  2. Two force systems are equivalent if their total moments about three non-collinear points are equal respectively.
  3. Two force systems are equivalent if their total moments about two points are equal and projections of their sums on the line connecting these points are equal.

  Conclusion:

  After adding to any given force system a zero force system we obtain an equivalent force system.
• Concurrent system of forces:

  Let F1 and F2 be a concurrent system of forces whose lines of action intersect at point P then

  -> The sum of the moments of concurrent forces about a given point equals the moment of their sum about this point.

  CONCLUSIONS:

  • The parameter of each concurrent force systems is zero
  • The moment of a concurrent force system about any point is always perpendicular to the sum vector
• To reduce a system of forces means to replace it with the equivalent, simpler system
• Types of reduction:

  • Reduction of a system of forces to one force and one couple

  (reduction at a chosen poit)

  • Reduction of a system of forces to the simplest equivalent system

  (finding a simplest equivalent force system)
• If SUM=0 and Moment about a point=0 of a system then the simplest equivalent system is a zero force system
• if sum=0 and moment is different from zero then the simplest equivalent system is a couple with the moment like before
• if sum is different from zero and parameter ,,k" is equal to 0 then the system can be further reduced to a single force: a resultant force
• All centers of reduction in which the total moment equals zero form a line called the line of action of the resultant force ( or the central axis of the system).

  The line of action of a resultant force passes through O’ and is parallel

  to the sum vector
• Force systems are equivalent if the sums of the forces are equal

  and the total moments of the systems about a chosen point are equal.
• Every force system can be reduced to a force-couple system comprising one force, equal to the sum of the original system, applied to the center of reduction, and one couple, with a moment

  equal to the total moment of the original system about the center of reduction .
• Planar system of forces

  In a planar force system the lines of action of all forces belong to one

  plane.
• The function r ( t ) is used to define other kinematic quantities describing motion
• Kinematic quantities describing motion:

  • Path

  • Displacement vector

  • Distance travelled

  • Velocity vector

  • Acceleration vector

  • Components of the acceleration vector tangential and normal to the path
• Tangential acceleration - the projection of the acceleration vector onto the velocity direction
• •The magnitude of tangential acceleration reflects the change in the speed of the particle.

  •The tangential acceleration equals zero if, and only if, the speed of the particle is constant .
• Normal acceleration - the projection of the acceleration vector onto the plane perpendicular to the velocity vector.

  •The normal acceleration reflects the change in the direction of motion.

  •The normal acceleration of the moving particle equals zero if, and only if, the path is a straight line (or a particle passes through a point of inflection of the path).
• r = r (t ) - the vector function defining the motion of the particle
• Circular motion- motion of a particle on a circular path

  Corollaries:

  • the path is a circle of given radius R

  • the circular motion is a planar motion
• circular motion

  
• A rigid body – an object in which the distance between every pair of points remains constant during motion

  • A rigid body – an idealized model of a solid body whose deformations during motion are negligible.
• Theorem 2

  If the points of the rigid body belong to a line then the heads of their

  velocity vectors form a line
• Number of degrees of freedom (NDOF) is the number of independent variables which should be specified to determine the position of all points in the system

  (the number of independent movements the system has).
• Description of motion of the rigid body by motion of its three non-collinear points
• Translation of the reference frame doesn’t alter the components of the vector
• Particular types of rigid body motion

  • translation

  • rotation about a fixed point

  • rotation about a fixed axis

  • planar motion
• Translation – motion of a rigid body in which a segment connecting two arbitrary points of the body keeps the same direction during motion.
• Rotation about a fixed point – motion in which the distance from the fixed point to an arbitrary particle of a rigid body remains constant during motion