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8/4/2019 Force System Force Systems Are the Starting Point of Engineering Analysis
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Force systemForce systems are the starting point of engineering analysis.
Force Vectors
A force vector is a force defined in two or more dimensions with a
component vector in each dimension which may all be summed to equalthe force vector. Similarly, the magnitude of each component vector,
which is a scalar quantity, may be multiplied by theunit vectorin that
dimension to equal the component vector.
Moment
For a system wherein a rigid body experiences a force F at a orthogonal
distance L from a fixed point, the momentM is the quantity (oddly
enough of the same units as energy) defined by the force multiplied by
the length of distance between the fixed point and the point where the
force is applied. The direction of the moment is perpendicular to the
force ecotro and the length, using theright hand rule.
In the event that a force impacts the rigid body at an angle other than a
right angle , the moment is determined by the
component of the force vector that is orthogonal to the length L.
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The general case in three dimensions can be calculated with thecross
product. Do note that the order of the distance vector and the
force Fdoes matter in cross products as opposite order will change
signs.
The components of a moment vector is the moment around the
respective axis, following theright hand rule.
Example:
M = Force * Length = 100 Newtons * 10 Meters = 1,000 Newton-meters (N-m)
Example: Force F is incident on the end of a rigid body of length L at an
angle A degrees from the central axis of the body x (Hint: draw a free
body diagram).
Then and
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Couple
A couple is a pair of equal and opposite force vectors that are some
distance apart and that act upon the same body, thus causing a rotation.
Imagine that force and force are incident at two locations along arigid body of total length at positions and , where . (Hint:
draw a free body diagram)
Then
In 3D the same rule applies, using
that which
means that the moment will be the same around any point in the system.
Resultant
Any system of forces may be reduced to a system of components and a
resulting moment.
That is to say, and about the point
and and and then
with
...then is magnitude in the direction of
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Mechanical equilibrium
Apendulumin a stable equilibrium (left) and unstable equilibrium (right).
A standard definition of static equilibrium is:
A system of particles is in static equilibrium when all the particles of
the system are at rest and the total force on each particle is
permanently zero.
This is a strict definition, and often the term "static equilibrium" is used
in a more relaxed manner interchangeably with "mechanical
equilibrium", as defined next.
A standard definition of mechanical equilibrium for a particle is:
The necessary and sufficient conditions for a particle to be inmechanical equilibrium is that thenet forceacting upon the particle is
zero.
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The necessary conditions for mechanical equilibrium for a system
of particles are:
(i)The vector sum of all external forcesis zero;(ii) The sum of the moments of all external forcesabout any line is
zero.
As applied to a rigid body, the necessary and sufficient
conditions become:
Arigid bodyis in mechanical equilibrium when the sum of
allforceson all particles of the system is zero, and also the sum of
alltorqueson all particles of the system is zero.
A rigid body in mechanical equilibrium is undergoing neither linear norrotational acceleration; however it could be translating or rotating at a
constant velocity.
However, this definition is of little use incontinuum mechanics, for which
the idea of a particle is foreign. In addition, this definition gives no
information as to one of the most important and interesting aspects of
equilibrium states theirstability.
An alternative definition of equilibrium that applies toconservative
systemsand often proves more useful is:[6]
A system is in mechanical equilibrium if its position inconfiguration spaceis
a point at which thegradientwith respect to thegeneralized coordinatesof
thepotential energyis zero.
Because of the fundamental relationship between force and energy, this
definition is equivalent to the first definition. However, the definition
involving energy can be readily extended to yield information about the
stability of the equilibrium state.
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For example, from elementarycalculus, we know that a necessary
condition for alocal minimumoramaximumof a differentiable function is a
vanishing first derivative (that is, the first derivative is becoming zero). To
determine whether a point is a minimum or maximum, one may be able to
use thesecond derivative test. The consequences to the stability of the
equilibrium state are as follows:
This is an unstable equilibrium.
Second derivative< 0 : The potential energy is at a local maximum, which
means that the system is in an unstable equilibrium state. If the system is
displaced an arbitrarily small distance from the equilibrium state, the forces
of the system cause it to move even farther away.
This is a stable equilibrium.
Second derivative > 0 : The potential energy is at a local minimum. Thisis a stable equilibrium. The response to a small perturbation is forces
that tend to restore the equilibrium. If more than one stable equilibrium
state is possible for a system, any equilibria whose potential energy is
higher than the absolute minimum represent metastable states.
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This is an indifferent equilibrium.
Second derivative = 0 or does not exist: The second derivative test fails,
and one must typically resort to using thefirst derivative test. Both of the
previous results are still possible, as is a third: this could be a region in
which the energy does not vary, in which case the equilibrium is called
neutral or indifferent or marginally stable. To lowest order, if the systemis displaced a small amount, it will stay in the new state.
In more than one dimension, it is possible to get different results in different
directions, for example stability with respect to displacements in the x-
direction but instability in the y-direction, a case known as asaddle point.
Without further qualification, an equilibrium is stable only if it is stable in all
directions.
The special case of mechanical equilibrium of a stationary object is static
equilibrium. A paperweight on a desk would be in static equilibrium. Theminimal number of static equilibria of homogeneous, convex bodies (when
resting under gravity on a horizontal surface) is of special interest. In the
planar case, the minimal number is 4, while in three dimensions one can
build an object with just one stable and one unstable balance point, this is
calledGomboc. A child sliding down aslideat constant speed would be in
mechanical equilibrium, but not in static equilibrium.
An example of mechanical equilibrium will be a person trying to press a
spring, he can push it up to a point after which it reaches a state where theforce trying to compress it and the resistive force from the spring are equal,
so the person can not further press it, at this state the system will be in
mechanical equilibrium. When the pressing force is removed the spring
attains its original state.
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