6.5: Sketch of the dynamic system
Before analysing the dynamics, it is important to make a sketch for a good overview of the dynamic system and the interactions between the different objects. Such a sketch, like shown in Figure 6.3, should include the following elements:
- Constraining objects
- Massless mechanisms
- Objects with mass
- Forces vectors with points of action
- Labels
- Direction of the gravitational acceleration vector \(\overrightarrow{\boldsymbol{g}}\) (if present)
- Distances, angles and position vectors
- Coordinate systems and unit vectors
We will now discuss these different elements, starting with three different types of objects: constraining objects (like walls), objects with mass (like point masses) and massless mechanisms.
6.5.1 Constraining objects
In the sketch, first draw all the constraining objects. These are the objects or structures that constrain the motion of other objects in the system. Constraining objects can either have a constant position, like walls, ground, ceiling and rails, but can also move, like elevators, drive shafts and carousels. In Figure 6.3, the grey sloped surface is a constraining object. Since the time dependent motion of these constraining objects is fully known, Newton’s second law doesn’t have to be applied to them. This is normally also not possible since the mass of the constraining objects is usually not given. The effect of constraining objects can be described by constraint equations (Sec. 5.2).
6.5.2 Objects with mass
Secondly draw the masses at an arbitrarily chosen position and angle that satisfies the constraints.
Important: when choosing such an arbitrary position, make sure to select a generic, non-trivial position, for which all relevant forces are non-zero and have, if possible, non-zero components along the coordinate axes. For example, when drawing a pendulum, a mass suspended by a string, do not draw it in
the lowest position, since then there will be no horizontal component of the string force.
6.5.3 Massless mechanisms
Finally draw the massless mechanisms, these are mechanisms with zero or negligible mass, which can generate non-zero forces on point masses and constraining objects via mechanical connections. Examples include massless ropes, springs, rods, wheels, pulleys, gears and crankshafts and will be discussed in Sec. 6.13.6.
6.5.4 Force vectors
After all objects and mechanisms have been sketched, the force vectors can be sketched according to the guidelines in Sec. 3.2.4. We distinguish 3 types of force vectors:
- Force vectors at contact points. Newton’s third law holds for all forces, such that 2 force vectors should be drawn at each contact point between 2 objects and/or mechanisms: The force vector exerted by object 1 on object 2, and the (equal and opposite) force vector from object 2 on object 1. In Figure 6.3 the forces \(\overrightarrow{\boldsymbol{F}}_{A, N}\) and \(\overrightarrow{\boldsymbol{F}}_{A, f}\) are force vectors that act on \(A\) at contact point \(C\). Note that we often only draw the force vectors acting on objects for which we want to apply Newton’s second law.
- Contactless forces acting at a distance. Certain forces, like the gravitational force and the Coulomb force can act between 2 objects over a distance, without requiring mechanical contact. In Figure 6.3 the gravitational force \(\overrightarrow{\boldsymbol{F}}_{A, g}\) is a force with as point-of-action \(G\), the center-of-mass of \(A\).
6.5.5 Labelling objects and vectors
After drawing all objects and vectors, it is important to label them uniquely, since these labels link the sketch to the mathematics that you will use to solve the dynamics. Either letters, numbers or combinations of letters and numbers can be used as labels (see Sec. 10.7.2). For instance in Figure 6.3 you have point mass \(A\) with mass \(m_{A}\). Force vectors can be labelled with two subscripts, the first indicates the object on which the force acts and the second indicates the object or type of force which generates the force, e.g. \(\overrightarrow{\boldsymbol{F}}_{A, N}\) is the normal force on \(A, \overrightarrow{\boldsymbol{F}}_{A, g}\) the gravitational force and \(\overrightarrow{\boldsymbol{F}}_{A, f}\) the friction force.