6.8: Drawing the Free-Body Diagram
The free-body diagram (FBD) is a drawing of an object (=body) or a set of objects, isolated from their surroundings and all force vectors and moment vectors that act on it. Its main purpose is to determine the sum of the force vectors \(\sum \overrightarrow{\boldsymbol{F}}\). The word free indicates that the object is cut free from surrounding objects, and all effects of the surroundings are represented by force and moment vectors. A key purpose of an FBD is to define the difference between internal and external forces, since all objects drawn in the FBD can be considered to be internal and their forces should not be drawn, whereas all drawn forces are external forces generated by objects that are external and not drawn. Although we will focus on the FBD for single point masses in this chapter, we will later also consider FBDs of objects and rigid bodies consisting of many point masses.
If a good sketch has been made in the previous steps, drawing the FBD is straightforward: draw the object, and draw all force vectors that act on the object just like they were drawn in the sketch. An example is given in Figure 6.4. Also provide the unit vectors, which are used to obtain the force components by projection.
6.8.1 Drawing the FBD
To draw the FBD we follow the methodology as described in Vallery and Schwab [9]. This description already includes the elements in the FBD needed for rigid bodies in the presence of moments that will be introduced in Ch. 10.
- Draw the system in a free state, i.e. "cut" the system at convenient locations. Don’t draw objects that do not belong to the system. Draw outlined shapes of the separate pieces belonging to the system. Each cut at a contact point can introduce new external action-reaction forces at the system boundaries. Some helpful guidelines:
a) Always draw the system in a generic state. So, for example, if you draw the FBD of a pendulum, do not draw it in the vertical position, since in that position the horizontal component of the force in the rope is zero, while it is not always zero.
b) Whenever possible, choose your system boundaries such that you expose only action-reaction forces that you are actually interested in calculating, or that are easily determined from the information you have. Otherwise, extra equations and unknowns are introduced, complicating calculations.
- Draw a 3D coordinate system with a clearly defined position for the origin \(O\). Check if rotating and/or translating the coordinate system can make things easier, and if Cartesian or cylindrical coordinate systems makes the analysis simplest. Consider drawing the three unit vectors. If the coordinate system is already clear from the sketch it may be omitted in the FBD.
- Some rules for drawing forces and moments in a FBD:
a) Do not show internal forces or moments.
b) If a connection prevents movement or rotation in a particular direction, then forces/moments are drawn to represent that restriction.
c) If a rigid segment is split in two, the forces and moments acting on the two segments in the separated FBDs are equal in magnitude and opposite in direction.
d) Forces acting on a rigid body may be shifted along their lines of action (see sliding vectors in Ch. 3).
e) Couple moments acting on a rigid body may be placed anywhere (they are free vectors).
f) Only draw moment vectors if the forces that generate them are not drawn to prevent double counting of forces or moments.
g) Projections of forces on the coordinate axes can be drawn in a FBD, but should be uniquely labelled as such (e.g. \(F_{g, x}\) and \(F_{g, y}\) ), and use dashed arrows to distinguish them from the original force.
h) Draw the arrows of the force vector in the expected direction of the force.
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Indicate and label all known and unknown external force vectors and moment vectors that act on the system uniquely, and place dots at the
correct points of action and reference points. Decide on the preferred vector notation method Sec. 3.2.4. - Label relevant dimensions, distances, angles and relative position vectors.
The main goal of these guidelines is that an FBD should include all relevant information needed to correctly establish the sums of forces and moments that will later be used to determine the equations of motion.
6.8.2 Common errors in drawing FBDs
To help you generate proper FBDs, we list errors that often occur in drawing FBDs.
- Drawing internal forces.
- Drawing forces generated by the free body (e.g. on the constraints). Only the external forces acting on the free body should be drawn.
- Drawing the same force twice.
Both drawing the force and its moment (can lead to double counting).
Drawing both the vector and its projection on the axes without clearly differentiating them, e.g. with dashed lines (can lead to double counting).
- Drawing velocity or acceleration vectors in an FBD without clearly distinguishing them from the forces.
- Drawing objects that do not belong to the free body in the FBD.
- Forgetting to draw forces generated by constraints on the body.
- Forgetting gravity or other forces.
- Forgetting to draw or indicate a coordinate system or unit vector directions.
- Drawing the FBD in a non-generic (=trivial) state or equilibrium position (see previous subsection), such that certain force components or angles become zero, while they are not always zero.
- Using scalar unsigned magnitude notation (e.g. \(F_{A}\) see Ch. 3), for a force of which the direction is unknown or time-dependent.
- Drawing resultant or sums of vectors like \(\overrightarrow{\boldsymbol{F}}_{\text {tot }}\) (leads to double counting). Each individual physical force vector needs to be drawn as a separate vector.
- Already using Newton’s laws to determine the value of forces, e.g. writing \(m g \hat{\boldsymbol{\jmath}}\) for the normal force, instead of keeping it as an unknown and determining it from Newton’s laws.
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Drawing a vector \(m \overrightarrow{\boldsymbol{a}}\). Acceleration is a result of force, but is not the force itself. An FBD should only contain the physical forces acting on
the free body. - Drawing virtual or pseudo force vectors, based on assumed motions like the centrifugal (or centripetal) force in rotations. Only forces resulting from physical interactions between objects can be drawn in the FBD, since it is the purpose of the FBD to determine the sum of forces and use that to determine the motion.