Elliptical Trammel Mechanism Final 582b6z

ELLIPTICAL TRAMMEL MECHANISM Abdel hamed Ibrahim abdel hamed For Dr. Rania Mostafa

ELLIPTICAL TRAMMEL MECHANISM

ABSTRACT Instrument used for drawing ellipses it's a mechanism obtained by having two sliding pairs and two turning pairs. The Elliptical Trammel (also known as the Elliptic Trammel, or the Trammel of Archimedes) is a simple mechanism which can trace an exact elliptical path. An ellipsograph is a trammel of Archimedes intended to draw, cut, or machine ellipses, e.g. in wood or other sheet materials They can be used to draw smaller ellipses but only draw one half at a time, having to be reversed to draw the complete ellipse.

INTRODUCTION A Elliptical Trammel is a mechanism that traces out an ellipse. It consists of two shuttles which are confined ('trammeled') to perpendicular channels or rails, and a rod which is attached to the shuttles by pivots at fixed positions along the rod. As the shuttles move back and forth, each along its channel, the end of the rod moves in an elliptical path. The semiaxes a, and b of the ellipse are the distances between the end of the rod and the two pivots. An ellipsograph is a trammel of Archimedes intended to draw, cut, or machine ellipses, e.g. in wood or other sheet materials. An ellipsograph has the appropriate instrument (pencil, knife, router, etc.) attached to the rod. Usually the distances a and b are adjustable, so that the size and shape of the ellipse can be varied. By the application of elliptical trammel, Ellipses are closed curves and are the bounded case of the conic sections, the curves that result from the intersection of a circular cone and a plane that does not through its apex; the other two (open and unbounded) cases are parabolas and hyperbolas.^ A diagram of conic sections including a circle, an ellipse, a parabola, and a hyperbola. Point that Ellipse es through. The ellipse is one of the conic sections , the intersection of a right circular cone with a cutting plane, as shown in the diagram at the right. Ellipses also arise as images of a circle under parallel projection and some cases of perspective projection. The special case of a circle's eccentricity A circle is a special case of an ellipse. The area of the ellipse is (7rab), where a, b are the semi-axes; this result may be deduced by regarding the ellipse as the orthogonal projection of a circle, or by means of the calculus. An isometric image has no perspective and all parallel straight lines are at the same angle. .It is also the simplest Lissajous figure, formed when the horizontal and vertical motions are sinusoids with the same frequency.

Abdel hamed Ibrahim abdel hamed MTE LVL 200

Main Section

Geometry of elliptical trammel Let A and B denote the points on the moving rigid body that coincide with the revolute t axes. Let C denote that point on the moving body that traces a path. Define the following distances: a = |AC| b = |BC| c = |AB|.

(1)

For convenience, assign a reference frame with origin at the intersection of the two prismatic t axes, and with basis vectors collinear with the t axes Let (θ) denote the angle between the line A~C and the x-axis. In this coordinate system, the x and y-coordinates of the point C are given by: x = b cos (θ)

; y = a sin (θ).

(2)

Consequently : (X/b)2 + (y/a)2 = 1

(3)

Which is the equation for an ellipse with major and minor axes having dimensions a, and b. At each instant, the central body moves as if it was rotating about a pole (also known as Instantaneous center of rotation, or ICR). At each location of the mechanism, each sliding block in the prismatic ts moves with a velocity that is equivalent to a rotation about any point on The line which es through the center of the block and which is perpendicular to the axis of sliding. Both of these lines intersect at a unique point . Hence, at each instant, the central body moves as if it were rotating about the ICR located at the center of these two lines.

Abdel hamed Ibrahim abdel hamed MTE LVL 200

Recall that the fixed cent rode consists of the set of instantaneous pole locations (or ICRs) in the fixed reference frame, while the moving cent rode consists of the set of pole locations as Described in the moving frame of the central body. The mechanism moves as if the moving cent rode rolls without slipping on the fixed cent rode. The geometry of the fixed cent rode can be found as follows. For each feasible orientation of the central body (denoted by θ), The x and y-coordinates of the ICR P is: xP = −c cos(θ)

yP = c sin (θ) .

(4)

Consequently, the moving cent rode traces out the circle xp2+ yp2 = c2 .

(5)

The motion generated when a circle of radius R rolls (without slipping) on the inside of another circle with radius 2R. Points on the moving circle trace out elliptical paths. Thus, the Elliptic Trammel implements Cardanic motion. Application · · ·

It is used in automatic tool changer in a machining. Elliptical Trammels are used for drawing large ellipses. They can be used to draw smaller ellipses but only draw one half at a time, having to be reversed to draw the complete ellipse.

Conclusion Elliptical trammel is inversion of four bar mechanism by fixing a link which to provide rotary motion to other link and by using elliptical trammel, we can draw an ellipse. For construction of elliptical trammel we can used a wood as a material.

Abdel hamed Ibrahim abdel hamed MTE LVL 200