Guide Dubbel Handbook of Mechanical Engineering

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Industrial Robot Control Systems. Main Applications and Selection of Robots. Moments or Couples. These consist of two equal and opposite forces with parallel lines of action Fig. Ib or. In this situation. Couples can be displaced at random in their planes of effect and perpendicular to it; in other words, the moment vector is a free vector, determined by the vector product. Equilibrium prevails if a body is at rest or is in uniform motion in a straight line. Rigid bodies as understood in statics are bodies of which the deformations are so small that the points at which f rce is applied undergo negligible displacement.

These are vectors of varying direction and displaceable in their lines of action, which cause changes in the motions or shapes of bodies. The determinant factors of forces are magnitude. External forces are all the forces that exert an effect on a body capable of free motion from the outside see AI. S loads and supporting forces. Intenlal forces are all the cutting forces and binding forces occurring inside a system. M signifies the magnitude or amount of the moment, and provides a graphic representation of the area of the parallelogram formed by rand F. In this situation, h is the moment ann perpendicular to F.

For the direction cosines Fig. Ic , the following applies:. The effect of an individual force with a random point of contact in relation to a point 0 becomes clear if a null vector is added; i. A single force F is derived at the point 0, as well as a pair of forces or moment M displacement moment. If rand F are given in components x, Fy , F, Fig. ResolutioR of a Force in the Plane. For a graphical solution, see Fig. Sa, b. Calculated Solution. Sc components: The following applies to the components, the value of the moment vector, and the direction cosines:.


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If the force vector is located in the x, y plane, i. These formulae correspond to the sine law for a scalene triangle Fig.

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Forces are added together geometrically vectolially , for two forces with the parallelogram or triangle of forces Fig. The calculated solution is. Forces are added geometrically vectorially , as the polygon of force is plotted in space Fig. For this purpose, the projection is made on two planes, i. The projection of the F; in the z-direction produces the true component F R, of the resultant force. This is. Combination oJ two Figure 4. Combination of sevplane forces: a paraJlelognun of eral plane forces: a location plan, b polygon of forces. Figure S.

Resolution of a plane force: a in two directions, single value ; b in three directions multiple value ; c calculated. Combination of forces in space: a location plan, b polygon of forces, c overall resultant force. The size and direction of the resultant force are.

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The Resolution of a Force in a plane is uniquelv possible only in three directions; in four and more directions. In the graphical solution Fig.

It meets the line of intersection C' of plane B' 7' D' formed by 2 and 3 in the horizontal projection at the projected intersection point of line G'. The projection of ;' into the vertical projection provides G" and therefore efl.

Force P is now resolved in the horizontal and vertical projections in directions J and c, which are located in two different planes, namely ABD and BCD. The final value of PI is. The same applies correspondingly to F, and F, Fig. As in Fig. These three linear equations for the three unknown forces 1'" 1', and 1', have a unique solution only if their system determinant does not equal zero, i.

According to Fig. The solution is single-value if the parallelipiped product is e,e,e, ". Figure Composition of several plane forces: a location plan funicular polygon , b polygon of forces solid angle. The forces are added geometricaUy to the result-.

The paraUeis to these are transferred to the location plan Fig. The point of intersection ofthe first and last rays of the funicular polygon provides the contact point of the resultant force, the value and direction of which are derived from the polygon of forces. Calculation Procedure. By reference to the zero point, the plane group of forces provides a resultant force and a resultant displacement moment Fig.

For a random point, the effect of the group of forces is the same as that of their resultant force. If the resultant force is displaced parallel from the zero point so far that M. Resolution of a Force. The resolution of force in a plane is possible as a single value, in three given directions that do not intersect at a point, and of which a maximum of two may be paraUe!.

A force is graphicaUy resolved with the aid of Culmann's auxiliary vectors Fig. In addition to this, the force F is made to intersect one of the three lines of application, and the other two lines of application are made to intersect each other. After resolution of the force F in the polygon of forces in the directions 3 and c, F. The force Fc is then again resolved in directions 1 and 2, giving F, and F z. Figure Resolution of a plane force: a location plan with mann lines c, b polygon of forces, c calculated solution.

The calculated solution follows from the condition that the application of force and moment of the individual forces F; and the force F must be equal in relation to the zero point Fig. There are three equations for the three unknowns F" Fz, F,. Their denominator determinants may not be zero; i. Reduction of forces in space: a location plan, b force and moment resultant. Force Combination Reduction.


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  7. The calculated solution Fig. Force Resolution in Space. A force can be resolved in space as a single value in six given directions. If the directions are given by their direction cosines, and if the forces are designated F, '. F o, then.

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    From these six linear equations a unique solution may be derived, if the denominator determinant is not equal to zero. Parallelogram of Forces. A further simplification of the reduced system of forces is possible if there is an axis that has a specific position in which the force vector and the moment vector are parallel Fig. This axis is called the central axis.

    The vector that pertains to this is.. Since all acceleration values are zero, it follows from the basic principles of dynamics that no resultant force and no resultant moment are exerted on the body. Each of the three conditions of equilibrium for the forces can be replaced by another for the moments about another random axis, which may not pass through the origin Six unknown values forces or moments can be calculated from the six conditions of equilibrium.

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    If more than six unknowns exist, the problem is described as statically indeterminate. Its solution is possible only by invoking deformation values see B2. If forces with common.