Ligature (Theoretical mechanics)

Link or ligature. In Theoretical Mechanics , it is everything that restricts the movements of a body in space .

Summary

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• 1 Degree of freedom of a body
• 2 Types of links
  • 1 Contact between bodies
  • 2 Flexible link
  • 3 Rigid embedment
  • 4 Force diagrams
• 3 Example of force diagram
• 4 References
• 5 Sources

Degree of freedom of a body

The number of independent coordinates that determine the position of any body or material system is called a body’s degree of freedom.

The concept of degree of freedom can be analyzed by taking any body found in space and using a rectangular coordinate system as a reference.

This body has six degrees of freedom, that is, that the rectilinear movements with respect to the X , Y , and Z coordinate axes constitute three degrees of freedom, and the rotational movements with respect to these axes also constitute three degrees of freedom.

This same body if placed on the X or Y plane . Only three degrees of freedom will remain: two displacements and one rotational movement with respect to the Z axis .

Degree of freedom

Any rigid body can be in space in a free or dependent (bound) state. If a rigid body is able to move in any direction, this body is called free. If a rigid body meets any obstacle in its path, preventing its movement, this body is called dependent (bound). From this it follows that everything that restricts the movements of a body in space is called a ligature. In the example it will be linked to the X or Y plane .

In the solution of most of the problems of Mechanics it is necessary to deal with non-free bodies, that is, with bodies that have contact or are subject to other bodies, by virtue of which some or other translations of the body become impossible. If the body exerts pressure on the link, under the action of the applied forces, the link in turn will act on the given body, that is, the force with which a link acts on the body, hindering its translation in one u Another direction is called the reaction force or simply reaction of this bond. ^[1] .

According to the law of equality of action and reaction ( Newton’s third law ), the reaction force of a bond is equal in modulus to the pressure force on the bond and is oriented in the opposite direction to this force, that is, in the opposite direction to the sense in which the link hinders the translation of the body. Most of the technical problems of the Statics consist precisely in the determination of the reaction forces of the bonds. Knowing these forces, it is also possible to know the pressure forces on the links, that is, the necessary data will be obtained to calculate the resistance of the constructions.

Types of links

Some types of ligatures, assuming they are made of rigid materials without friction at the points of contact with the bodies examined.

Contact between bodies

1. The body rests on point Aon a smooth surface. The surface is called smooth if you can do without rubbing a body against it. Since the smooth bearing surface does not hinder the sliding of the surface of a body on it, the reaction R of the smooth surface is always oriented according to the normal common to the surface of the body and to the surface of the bond at the point of contact of the same.
2. This type of ligatures can be referred to as the so-called mobile roller. Here the beam or reinforcement, with the help of a rocker, has its end Asupported on a cylindrical roller. Reaction RA is applied to the beam at point A and is normally directed at the bearing surface , on which the rollers can roll.

Moving roller

Flexible link

In this case, the ligatures or bonds are the threads that retain the body . If flexible bodies ( wire , cable , chain ) are used as ligatures , the reaction forces TA and TB are applied to the body at the fastening points and are directed along the wires.

Rigid embedment

The AB beam has one end rigidly embedded in a wall, the other end of which serves as a support for a construction. In embedment there are reactions made up of a force which is the reaction of embedment RA and its torque, the embedment moment MA .

Force diagrams

The balance of the linked bodies, is studied in the Statics based on the following axiom: “every linked body can be considered as free if we suppress the ligatures and substitute their actions for the reactions of these ligatures (including their own weight)”, ( this is called a force diagram or free body). ^[2] .

Embedment

When solving a problem involving the equilibrium of a rigid solid, it is essential to consider all the forces acting on it; it is equally important to exclude any force that is not directly applied to the solid. Omitting a force or adding a foreign one destroys equilibrium conditions. Therefore, the first step in solving the problem should be to draw a free diagram of the rigid solid under study. Due to its importance, we will summarize the steps that must be followed when drawing a free-body diagram. ^[3]

1. Make a clear decision regarding the choice of the free solid to use. This body is then separated from its base of support, as well as from any other body. Below is a sketch of the contour of the isolated body.
2. All the external forces that represent the action exerted on the rigid body by the support base and by the bodies that have been separated are indicated; These forces must be applied at the various points by which the free body was supported at its base of support or by which it was connected to the other bodies. The weight of the free solid must also be included among the external forces, since it represents the attraction exerted by the earth on the various particles that make up the free body. It should be applied at the body’s center of gravity.
3. The modulus, direction, and direction of known external forces should be clearly marked on the diagram. Care must be taken to indicate the direction of the force exerted on the free body. Known external forces generally comprise the weight of the free body and the forces applied for a specific purpose.
4. The free-body diagram must also include dimensions, since they may be required for calculating force moments.

Example of force diagram

Force diagram

The smooth bar that rests at point A on the floor and the wall and at B on a pole (Fig. 5a). When one of the contacting surfaces is a point, the reaction is directed by the normal to the other surface. It can be considered as a free body (Fig. 5b) that is in balance under the action of the known force G (beam weight) and the reactions of the ligatures: RAX , RAY and RB