5.2: Constrained and predetermined kinematics
In the next chapter on kinetics we will discuss how the motion of objects in time and space can be described based on the knowledge of forces and masses. However, often there is also different information about the motion available. For example when a train is constrained by a rail-track, when a measurement of the motion of a bird’s trajectory was made, or when it is assumed that a rolling ball always touches a curved surface. This information can be cast in the form of constraint equations.
5.2.1 Constraint equations
Although all dynamics is governed by Newton’s laws, rigorously applying these laws for every atom can be very inefficient. For instance, when a hockey puck is sliding on ice, modelling the forces of all ice molecules would be a tremendous task. Instead, we know that the ice molecules generate forces on the puck that prevent it from entering the ice, while gravity prevents the puck from
Thus in such cases, where we know the effective properties of a number of forces, it is much more efficient to describe their combined effect on the motion by constraint equations instead of applying Newton’s laws. Remember however, that such constraint equations always are the result of forces, and the accuracy of the resulting dynamics will depend on the accuracy by which the constraint equations represent these forces. In the next chapter we will also discuss how constraint equations can be combined with Newton’s laws to determine the forces generated by the constraints.
In other cases, information about the motion of an object has been obtained from a measurement, or is assumed to be known, and a constraint equation that describes the measured trajectory can be used to analyse the kinematics, and determine velocities and accelerations.
Equation that provides information on the motion and trajectories of objects in time and space, without providing direct information on the forces and mass. Constraint equations describe the kinematic constraints, which are the constraints that limit the potential motion of a mechanical system.
A constraint equation \(f\) can often be written as a function of the position vectors of the objects \(\overrightarrow{\boldsymbol{r}}_{i}\), and their time derivatives as:
\[f\left(\overrightarrow{\boldsymbol{r}}_{i}, \dot{\overrightarrow{\boldsymbol{r}}}_{i}, \ddot{\overrightarrow{\boldsymbol{r}}}_{i}, \ldots, t\right)=0 \tag{5.1} \label{5.1}\]
Where \(\overrightarrow{\boldsymbol{r}}_{i}\), with \(i=1,2, \ldots, N\) (or \(i=A, B, \ldots\) ) are the position vectors of the objects in the system. By taking the time-derivative or time-integral of a constraint equation, additional constraint equations can be derived. We will now discuss several examples of constraint equations.
5.2.2 Spatially constrained kinematics
Although space is always three-dimensional, in some cases there are forces on the object that constrain the motion to a certain part of space. The analysis of the dynamics in these cases can be significantly simplified. For instance, when a hockey puck \(A\) slides over a flat ice surface, and we know or assume that it will not lift from the surface, this can be described by the constraint equation:
\[z_{A}(t)=0 \tag{5.2} \label{5.2}\]
Since in this case the puck will not move along the \(z_{A}\) coordinate, its dynamics can be fully dealt with in 2 dimensions (2D) using \((x, y)\) coordinates or polar
\((\rho, \phi)\) coordinates, simply applying the Cartesian and cylindrical coordinate systems from the previous sections with \(z=0\). In fact this textbook deals with many examples of this \(2 \mathrm{D}\) in-plane dynamics situation, the derived equations and techniques are valid in 3D unless explicitly stated otherwise.
5.2.3 Path curve and path coordinate
An even more constrained situation is when the motion of an object is limited to a one-dimensional (1D) curvilinear path. This situation arises for instance when a train, marble or other point mass \(i\) moves along a predefined track, guided by a rail or tube.
As shown in by the blue curve in Figure 5.2 such a curvilinear path traces out a 1D path in 3D space, and this path can be described by a parametrised path curve \(\overrightarrow{\boldsymbol{r}}_{s}(s)\), such that for every value of the scalar path coordinate \(s\) there is 1 vector \(\overrightarrow{\boldsymbol{r}}_{s}(s)\) that defines the position on the track. It is convenient (although not always easy), to choose this function such that \(s\) is a measure of the distance along the track with respect to a certain origin \(\overrightarrow{\boldsymbol{r}}_{s}(s=0)\), for instance this can be done by taking a flexible tape measure, and measuring the distance \(s\) along the track. Since it is known that the point mass \(i\) always resides on the path curve, we have the following constraint equation:
\[\overrightarrow{\boldsymbol{r}}_{i}(t)=\overrightarrow{\boldsymbol{r}}_{s}(s) \tag{5.3} \label{5.3}\]
At every time \(t\), the path coordinate of object \(i\) has to obey equation (5.3), such that: \(\overrightarrow{\boldsymbol{r}}_{i}(t)=\overrightarrow{\boldsymbol{r}}_{s}\left(s_{i}(t)\right)\). This has the advantage that, because the path curve \(\overrightarrow{\boldsymbol{r}}_{s}(s)\) is known, the motion of \(i\) in \(3 \mathrm{D}\) can be described by a scalar function \(s_{i}(t)\), instead of requiring a more complex \(3 \mathrm{D}\) vector function \(\overrightarrow{\boldsymbol{r}}_{i}(t)\). Equation (5.3) and also the equation \(z_{i}(t)=0\) from the previous subsection are examples of constraint equations.
5.2.4 Relatively constrained kinematics and pulleys
Besides spatial constraints that limit the motion of a point mass to a certain plane or curve in space discussed in the previous subsection 5.2, it is also possible to have constraints that limit motion of a point mass with respect to one or more other point masses. Consider for example two point masses \(A\) and \(B\) with their motion described by separate path curves and path coordinates \(s_{A}(t)\) and \(s_{B}(t)\), see Figure 5.3. When making such a drawing, you are free to choose for each path coordinate the positive direction of the path curve and also the point at which the path coordinate is zero, which is called the datum. It is important to indicate those in the drawing.
We can write down a relative constraint equation for the length \(L_{\text {rope }}\) of the rope. From Figure 5.3 we see that always:
\[s_{A}(t)+s_{B}(t)+R \pi=L_{\mathrm{rope}} \tag{5.4} \label{5.4}\]
The term \(R \pi\) comes from the part of the rope that touches the pulley with radius \(R\) over half of its perimeter. By taking the time derivatives of this equation and using that the length of the rope \(L_{\text {rope }}\) and the radius of the pulley \(R\) is constant in time we find:
\[\begin{align} s_{A}(t)+s_{B}(t) & =c_{1}=\text { constant } \tag{5.5} \label{5.5}\\[4pt] \dot{s}_{A}(t)+\dot{s}_{B}(t) & =0 \tag{5.6} \label{5.6}\\[4pt] \ddot{s}_{A}(t)+\ddot{s}_{B}(t) & =0 \tag{5.7} \label{5.7}\end{align}\]
This shows that at all times the velocity and acceleration of the masses are equal and in opposite directions \(\left(\ddot{s}_{A}=-\ddot{s}_{B}\right)\). These relative constraint equations facilitate analysis of dynamic problems, since one does not need to determine the force in the rope. Since one is usually only interested in the constraint equation for the velocities, it is often not necessary to determine the exact value of the constant \(c_{1}=L_{\text {rope }}-R \pi\).
The shown procedure for obtaining the constraint equations is also quite generally applicable:
- Determine the path coordinates of all moving objects and point masses based on the geometry.
- Determine the constraint equations for the positions of the objects and simplify them.
- Take two times the time derivative of the constraint equation to obtain the constraint equations for the velocities, and for the accelerations.
Figure 5.4 shows a more complicated pulley system. The question is: find the relative constraint equation that relates the velocities \(\dot{s}_{A}, \dot{s}_{B}\) of point masses \(A\) and \(B\).
Exemplary solution
Since pulleys \(C\) and \(D\) are also moving, it is useful to also monitor their path coordinates \(s_{C}\) and \(s_{D}\), which have been indicated in Figure 5.4. Now we write down all constraint equations and combine them to obtain a single equation relating \(s_{A}\) and \(s_{B}\) :
\[\begin{align} L_{\mathrm{rope}} & =s_{A}+2 s_{C}+s_{D}+L_{C D}+c_{1} \tag{5.8} \label{5.8}\\[4pt] s_{C} & =s_{B}-\left(L_{C D}+L_{B}\right) \tag{5.9} \label{5.9}\\[4pt] s_{D} & =s_{B}-L_{B} \tag{5.10} \label{5.10}\\[4pt] L_{\mathrm{rope}} & =s_{A}+3 s_{B}+c_{2} \tag{5.11} \label{5.11}\end{align}\]
The actual value of the constant \(c_{2}\) is not needed to solve the problem, since we now take the time derivative to obtain the relative constraint equation for the velocities:
\[\begin{align} s_{A}+3 s_{B} & =c_{3}=\text { constant } \tag{5.12} \label{5.12}\\[4pt] \dot{s}_{A}+3 \dot{s}_{B} & =0 \tag{5.13} \label{5.13}\\[4pt] \dot{s}_{A} & =-3 \dot{s}_{B} \tag{5.14} \label{5.14}\end{align}\]
The three constraint equations (5.8-5.10) represent the effect of the three constraining elements in Figure 5.4: the rope, the rigid rod between pulleys \(C\) and \(D\) and the short rope between pulley \(D\) and point mass \(B\). Initially there are four unknown scalar variables: \(s_{A}, s_{B}, s_{C}\) and \(s_{D}\). Generally each (scalar) constraint equation eliminates one unknown, such that in the end we are left with one constraint equation (with one unknown). Besides pulleys and ropes there are of course many other elements that constrain motion, which will be discussed later.