2021 | Book

# Problems of Locus Solved by Mechanisms Theory

Authors: Prof. Iulian Popescu, Prof. Xenia Calbureanu, Assoc. Prof. Alina Duta

Publisher: Springer International Publishing

Book Series : Springer Tracts in Mechanical Engineering

2021 | Book

Authors: Prof. Iulian Popescu, Prof. Xenia Calbureanu, Assoc. Prof. Alina Duta

Publisher: Springer International Publishing

Book Series : Springer Tracts in Mechanical Engineering

This book reports on an original approach to problems of loci. It shows how the theory of mechanisms can be used to address the locus problem. It describes the study of different loci, with an emphasis on those of triangle and quadrilateral, but not limited to them. Thanks to a number of original drawings, the book helps to visualize different type of loci, which can be treated as curves, and shows how to create new ones, including some aesthetic ones, by changing some parameters of the equivalent mechanisms. Further, the book includes a theoretical discussion on the synthesis of mechanisms, giving some important insights into the correlation between the generation of trajectories by mechanisms and the synthesis of those mechanisms when the trajectory is given, and presenting approximate solutions to this problem. Based on the authors’ many years of research and on their extensive knowledge concerning the theory of mechanisms, and bridging between geometry and mechanics, this book offers a unique guide to mechanical engineers and engineering designers, mathematicians, as well as industrial and graphic designers, and students in the above-mentioned fields alike.

Advertisement

Abstract

The locus is defined and correlated with the notions of generated trajectories and curves. It is considered a geometric Figure in which one side becomes fixed, and the others are moving, so that certain specified points describe the trajectories, i.e. curves assimilated to geometric places, such as conical curves: ellipse, parabola, hyperbola. Drawing these curves starting from geometric problems is easy by using the Theory of Mechanisms. The equivalent mechanisms are built and analyzed. For those who are not familiar with the Theory of Mechanisms, the strictly necessary elements are given in order to be able to construct the equivalent mechanisms and then to determine the trajectories of some points. The diagrams of the kinematic elements with rotational motion R, and translational motion (sliding), P are indicated, showing how the elements can be connected, also the cases when the lengths of some elements are equal to zero are presented too. The method of contours developed by Tchebichev is shown and examples of mechanisms and generated curves are given [1, 2, 3] (Cebâşev in Izobrannâe trud, Izd, Nauka, Moskva, 1953; Popescu in Mecanisme, vol. I, II. Tipografia Universităţii din Craiova, 1995; Popescu in Proiectarea mecanismelor plane, Craiova, Editura Scrisul Românesc, 1977).

Abstract

We start from a simple geometric problem, when a straight line slides with one end on a circle and the other on a straight line (both fixed) and we look for the trajectory of a point on this straight line. We construct the equivalent mechanism which is the rod-crank mechanism, we write the relationships for the positions, we draw the mechanism in several positions and then we obtaine a succession of the curves generated by the points on this line. The case is generalized by replacing the fixed line with a mobile one, resulting a lot of curves, variable depending on the correlation of the angles of the two leading elements. The resulting curves are different from one case to another, being open curves and some having several branches [1, 2] (Luca et al. in Studies regarding of aesthetics surfaces with mechnanisms. In: Mathematical Methods for Information Science and Economics. Proceedings of the 17th WSEAS International Conference on Applied Mathematics (AMATH `12), Montreux, Switzerland, pp. 249–254, 2012; Sass et al. in J Ind Design Eng Graphics. Papers of the International Conference on Engineering Graphics and Design, Sect. 3: Engineering Computer Graphics, Nr. 12, (1):41–46, 2017).

Abstract

We consider two straight lines (cranks) with one end fixed at the base (frame) and we are looking for the trajectory of their intersection point when both straight lines rotate. The mechanism is constructed for the case when a line has a finite length and shows the successive positions and the trajectory of their intersection point [1], which is an incomplete circle. Both straight lines with infinite lengths are considered, resulting another equivalent mechanism with two leading elements. Depending on the correlation between the angles of rotation of these lines, many trajectories result, some being known curves, others unknown [3], with several incomplete branches [2]. The successive positions of the equivalent mechanism are also given. In another case, one of the lines is no longer connected to the base by a R coupling, but by a P coupling, so it slides on a fixed line. Numerous successive curves and positions are also obtained. Another case has in B only two couplings, the trajectory being a straight line.

Abstract

We consider a straight line that can have a rotation motion around a fixed point and another one which slides with one end on the first one and the other end slides on a fixed line, we try to obtain the trajectories of several points situated on the mobile line. Numerous trajectories are obtained depending to the angle of the fixed line with the abscissa [1]. Successive positions are also given. The curves have two branches. When the fixed line have rotation motion too, there are two leading elements and the trajectories depend on the correlation between their movements [3].

Abstract

A straight-line slide with its heads on the axes of the system. The points on this line generate ellipses (example point C), composed of two halves, one for the + sign and the other for the—sign in front of a radical. Successive positions generate asteroids [1, 2]. Diagrams are given with the variations of some coordinates of the tracer point. From the ends of the sliding line we draw parallel lines to the axes of the reference system, establishing the trajectory of their intersection, the point D. The result is a circle containing the ellipse generated by the first point. In another case, the length of the line AB that has the ends on the two fixed lines was considered to be variable depending to the length of x_{B}, i.e. the stroke of the leading element, so AB = q * x_{B}, where q is conveniently adopted. It was found that in this case the trajectory of C point is a line parallel to the abscissa, and that of D point is an inclined line. The case where the fixed lines are inclined to the axes of the system was also studied, resulting in many curves dependent on the inclinations of these lines.

Abstract

It is considered a straight line that rotates around a fixed point, and at the free end it connects with another straight line that rotates around the same point. The mechanism has two leading elements [1], obtaining the trajectories of the end of the second line, with shapes dependent on the correlation of the two movements. Successive positions form aesthetic surfaces [2]. In another case, the first straight line slides on the abscissa, resulting in numerous trajectories and associated positions, also these one being aesthetic surfaces. In another case, the first line slides on the abscissa and the second on the first line, the trajectories being straight lines.

Abstract

A straight line rotates around a fixed point, and at the free end it connects with another straight line that rotates around the same point, and the other end slides on the abscissa. Another line FB rotates around F point, the end—C point sliding to the line DE, requiring the trajectory of point C. Many different curves are obtained, with quite complicated shapes [2], depending on the correlation of the movements of the two leading elements. Another range of curves resulted for the trajectory of the point of intersection of the lines AD with BC, where two slides were introduced that slide on the elements AD and BC and have a torque of rotation between them. In both situations, both directions of rotation of the conductive elements were taken into account: in the same direction or in opposite directions.

Abstract

A crooked line with an angle of 90°, has one end that goes on the abscissa, and the other rotates around the origin of the system and slides through that point. The trajectory of points B and D on the bent line is required. The result is the “Kappa” curve, in several variants depending on the length of the long arm of the bent line. Next we also start from a crooked line but with the bending angle of different values, resulting in a parallelogram and generating curves similar to the “Kappa” curve but with deformed branches. Successive positions are also given, which have the shapes of curves. We study also the “Kieroid” curve obtained when EC line has the ends moving on two fixed lines parallel to the ordinate. Point E is also in rotation around B moving on the conducting element AC. The trajectories of B and D are required, i.e. the “Kieroid” curve, which has two branches, one drawn by D and the other by B. By changing the distance between the fixed vertical lines, other curves result, some of the same kind, others different.

Abstract

Two concurrent fixed lines have at their point of intersection a torque of rotation, so that another line rotates around this point, this line intersecting with another line that goes with its heads on the fixed lines, looking for the geometric place of this point of intersection. This generates the “butterfly” curve. Many similar curves result in changing the angle between a fixed line and an abscissa.

Abstract

On two intersected lines in C, slide the slides 2 and 5, having the line BG = constant. The trajectories of points E and F on these slides are required. The trajectory of E is a woody, and of F a half branch of nephroid. Taking the symmetrical curves, the complete nephroid results, on which the woodcuts of E are also given. By modifying the distance GF, other curves are obtained. Another nephroid generating mechanism is given. Two intersected straight lines in A fixed, rotate around A, both being conductive elements with movements correlated by the coefficient q, and the line BC = const. slides with point C on AC. Point C will draw a rhodonea. By changing the value of q, variants of rhodoneas with several branches are obtained. The plotted curve has the number of sides equal to q − 2. If the sides AB and BC are equal, other types of rhodoneas are obtained, in which the branches pass through the origin of the axis system.

Abstract

We start from the problem in the previous chapter but point A is moving vertically while AB and AC rotate. Sequences of rhodoneas are formed which form aesthetic surfaces, which differ according to the coefficient “c”, where “c” is the correlation coefficient between the angles \(\varphi\) and \(\psi\) (\(\psi = c \cdot \varphi\)). Numerous surfaces with special aesthetics have been obtained. Decimal values for c were also used. Negative values were also taken for c, i.e. the conductive elements rotate in opposite directions. Next, other aesthetic surfaces were obtained considering that point A moves on the x-axis.

Abstract

It is known that bisectors, medians, mediators, heights can be drawn in a triangle. If two heights are taken, for example, they intersect at a point if the triangle is static. However, if the triangle moves, having a single fixed side, several variants of equivalent mechanisms are obtained, resulting in the trajectories of the points of intersection of different shapes. The resulting curves and successive positions are given. Some mechanisms are very complicated, only that they are used only as calculation artifices, not being necessary to build them and only in real cases when it is necessary to draw these trajectories on pieces. The resulting curves are of a great diversity of shapes.

Abstract

If in a triangle there are few usual lines that can intersect, in a quadrilateral there are many more possibilities. Thus, heights intersecting perpendicular from each corner on the opposite sides, as well as bisectors, medians, non-curves, diagonals, can intersect here as well. For each case the synthesis of the equivalent mechanism is made and then the intersection curve of the considered lines is determined. You get lots of curves, of a great diversity of shapes. By changing some lengths of some sides, other curves are obtained. In many cases, two curves plotted by two points on the mechanism were represented. The successive positions of some generating mechanisms are also given. Particular cases of the quadrilateral are also analyzed: square, rectangle, rhombus, parallelogram. The trajectory of the point of intersection of the diagonals of a quadrilateral is searched, when one side of the quadrilateral moves, dragging them on two other sides. The trajectory of a point on the connecting rod is also given for comparison. Woody curves were obtained. Next, two lengths of the mechanism were modified, resulting in other curves, some quite interesting.

Abstract

Here we study the trajectories drawn by the points of intersection of the diagonals in a pentagon. When one side is fixed, and the remaining are mobile, there are two conducting elements. Taking two intersecting diagonals, the result is 5 trajectory-generating points. Mechanisms are built for all these cases and the trajectories found are drawn. Many resulting curves are given for each value of q. Successive positions of the created mechanisms are also given. The resulting curves are generally open, with several branches [1]. The length of the sides of the mechanisms did not correlate with conditions like Grashof, so that in general the mechanisms do not work throughout the cycle, but only in certain subintervals of the cycle [2]. The discontinuities that appear are indicated below in the diagrams where the jumps are seen.

Abstract

A correlation is made between the generation of trajectories by mechanisms and the synthesis of those mechanisms when the trajectory is given. It is exemplified with the articulated quadrilateral mechanism, finding a nonlinear algebraic system with 9 equations and 9 constant unknowns, which are the parameters that define the mechanism. It is shown that this system cannot be solved with numerical analysis methods to provide exact solutions, so that approximate methods are used, given that the measurement of coordinates on the drawing introduces errors, so that the measured points are not sure on that curve. The methods used to roughly solve the system are indicated, but the solutions found cannot be applied because some negative or zero values result. This leads to the solution of the optimal synthesis, i.e. to optimization problems with nonlinear constraints, indicating approximate methods of solving. An example is also given of finding the mechanism that draws a straight-line segment imposed by 6 points, finding several mechanisms with different precisions. The Newton–Raphson algorithm was used, applicable to a nonlinear algebraic system with only 6 equations and 6 unknowns.