force system force systems are the starting point of engineering analysis

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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