Epsilon co-function for planar mechanisms

One of the most important and most challenging tasks of a mechanism analysis is the problem of mechanism kinematics. This paper shows the way to determine unknown angular accelerations of joint-bar mechanism components, by applying so-called (by authors) epsilon co-function. Using this method, the problem is reduced onto an analysis of relative angular accelerations of neighboring members within the mechanism and determination of a moment of all those vectors with respect to a point or an axis. The main contribution of this paper is that it shows the novel method how to calculate angular accelerations of mechanism members using analog form of equations that are similar to the moment balance equations in statics. Considering that the relative angular velocity vectors play role of forces in statics, this paper shows how to form a system of kinematic equations similar to moment equations in statics, which are sufficient to solve for all angular accelerations of a mechanism.


Introduction
*Several methods exist in the theory of planar mechanisms to determine particular kinematic quantities (such as angular velocity and angular acceleration) of the mechanism members or particular points of the mechanism, such as: using a particle kinematics, using planar kinematics equations, using the method of complex numbers and using the method of mechanism reduction. Each of these methods has some advantages with respect to others, depending on the type of the problem that should be solved.
Particularly, there are methods that use relative motion of neighboring members to determine unknown angular velocities of a hinge-lever mechanism (Ilic, 1966;Hufnagl, 1974;Voloder, 2005).

Important facts about the ⃗⃗⃗ co-function
Let us assume that a planar mechanical system is consisted of n rigid bodies (members) that are mutually interconnected. We will analyze relative angular velocities and relative angular accelerations of all pairs out of n observed members in a cyclic order (Fig. 1).
If we add all relative angular velocities, according to (1), we obtain the following vector equation where the index = + 1 denotes the same vector as for = 1, due to the periodicity of the index.
The main vector of the moment of ⃗ ⃗ co-function for an arbitrary point as a pole is equal zero Equations (1) and (2) show that the vectors of the ⃗ ⃗ co-function have same properties as an arbitrary balanced system of forces. In this way, the same laws, as for an arbitrary balanced system of forces, can be applied for the ⃗ ⃗ co-function.

Definition of ⃗ co-function
Let us observe a set of relative angular accelerations of sequential members (Fig. 2), given by the following equations 2,1 = 2 − 1 , , −1 = − −1 , Such a set of relative angular acceleration vectors we will call co-function, similarly to the previous ⃗ ⃗ co-function. Now, we will prove some properties of the co-function. (2), we obtain the following vector equation 2,1 + 3,2 + ⋯ + , −1 + 1, = 0 ⃗ , which yields If the first member is denoted as 0, then the sum in (4) starts with 0, e.g. if the ground member is assigned with 0.

Moment of ⃗ co-function for an arbitrary pole
Equation (4) shows that the principal vector (the vector sum) of the angular accelerations in cofunction is equal zero, hence this property is similar to the first property of the ⃗ ⃗ co-function.
However, when the moment of the co-function vector is considered with respect to an arbitrary point O, then significant difference can be noticed when compared with the corresponding property of the ⃗ ⃗ co-function. The following analysis will show this difference.
Then, relative acceleration of points +1, and

Fig. 3: With the proof of the principal moment of the co-function
Due to the following 2,1 − 1, + 3,2 − 2,1 + ⋯ and using (5), we obtain If we choose an arbitrary point as a pole O, then Using (7) and (8) Concise form of (11) is the following and considering that: a) the first term under the sum represents the moment of the relative angular acceleration vector with respect to the arbitrary point O, and b) the second term represents the normal relative acceleration of the point P i+1,i with respect to the previous point in the sequence P i,i−1 , then (12) can be written in the following form Equation (13) shows that the vector addition of the principal moment for an arbitrary pole O and all relative normal accelerations between sequential connection points of the mechanism.

Example
Bar 1 of the mechanism shown in Fig. 4 has a constant angular velocity: 1 = 20 rad s , and the bar 4 has a constant angular velocity: 4 = 40 rad s . Using the described method of co-function, angular accelerations of members 2 and 3 need to be determined. The following is given: AB ̅̅̅̅ = DE ̅̅̅̅ = EJ ̅ = EH ̅̅̅̅ = , at the given instant the angles are BAD = 60 0 and ABD = 90 0 .

Fig. 4: Example: mechanism configuration
Solution can be given using the classic approach and the described method using the co-function.

Solution using classic method
Using kinematics theory of planar motion, we can establish vector equations for velocities of mechanism connection points, as shown in Fig. 5.
where the shown velocity magnitudes in [ Angular accelerations can be obtained similarly, by using the vector loop equation, i.e., by expressing accelerations using accelerations of other points in the kinematic chain over connection points of the mechanism, as shown in Fig. 6. The acceleration of the same connection point, e.g. the joint D, can be obtained using two different chains of points. In this way, we obtain = + , + , = + , + , ,

Conclusion
It is shown that kinematic quantities, such as angular velocity and angular acceleration can be obtained using an alternative way, which shows an analogy to static balance equations. Using relative angular velocities and relative angular accelerations, and position vectors of connection points of a mechanism, two vector equations can be established. Using analogy between statics equations and the equations derived here, the first equation, the sum of relative angular velocities equal zero is analog to the sum of forces equal zero in statics. The second equation, the sum of moments of angular velocity vectors with respect to an arbitrary point-pole corresponds to the sum of moments of all forces for an arbitrary point-pole in statics. Besides the equations related to ⃗ ⃗ , it is shown here that almost the same analogy can be establish for relative angular acceleration vectors . The equation of relative angular accelerations is identically equal to zero and therefore it does not represent an independent equation. Particularly, it is shown that the sum of moments of angular accelerations along with additional terms that represent the sum of normal relative acceleration denoted as the "axp" components (the components perpendicular to and oriented toward the axis of relative rotation, i.e. toward the connection point) is equal to zero. This equation is analog to the moment equation in statics for an arbitrary point. There is no any restriction for the chosen pole, but it is convenient to choose that point such that the position vector calculation is simplified.
Finally, application of the derived equations was demonstrated through an example of a mechanism for a given instant of motion with given angles at the instant, but the method is valid for an arbitrary instant of a joint-bar mechanism.