# Motions

MACHINE ELEMENTS 2 Cams ? Cam is a link having an irregular surface or groove that imparts motion to a follower ? Cams are very important and frequently occurring elements in many types of machines – especially AUTOMATIC MACHINES ? Cams are the heart of such automatic devices as automatic devices as automatic machine tools, record changers, mechanical calculators, cash registers, and many other devices. Types of Cams: Motions Used for Cam Followers: ? The motion of the follower is of primary interest in the analysis of existing cams or in the design of new cams. It is easier to analyze the motion of cam followers if their motion is plotted as a graph often referred to as DISPLACEMENT DIAGRAM A. Displacement Diagram B. Motions that are most commonly used: 1. Uniform Velocity (straight line) motion – UVM 2. Simple Harmonic Motion – SHM 3. Uniformly Accelerated motion (Parabolic Motion) – UAM or PM 4. Modified Uniform-Velocity Motion – MUVM a. Arc method – MUVM-Arc b. Uniform Acceleration Method – MUVM-UAM 5. Cycloidal Motion – CM A. Uniform Velocity Motion (Straight Line Motion)

If the follower is to move with uniform velocity, its displacement must be the same for equal units of time. Its curve in the displacement diagram, therefore, is a STRAIGHT LINE. Example 1: Lay out the displacement diagram for a cam follower that is to have the following motions: Dwell 30O (at rest) Rise 2 inches in 90O (uniform velocity) Dwell 30O Fall 2 inches in 60O (uniform velocity) Dwell 150O B. Simple Harmonic Motion (SHM) As a point moves around the circumference of a circle with a constant velocity, its projection on the diameter of the circle moves with simple harmonic motion.

How to plot in the displacement diagram? Example 2: Lay out the displacement diagram for a cam follower that is to have the following motions: Rise 2 inches in 120O (SHM) Dwell 30O Fall 1 inch in 90O (SHM) Dwell 30O Fall 1 inch in 60O (SHM) Dwell 30O C. Uniformly Accelerated Motion (Parabolic Motion) ? A motion of uniform or constant acceleration, often called parabolic motion because a graph of its equation ( s = at2 / 2 ) is a parabola. ? It is motion in which the displacement taking place in each successive interval of time proportional to the square of the time.

How to plot in the displacement diagram? Example 3: Layout the displacement diagram for a cam follower that is to have the following motions: Rise 2 inches in 120O (uniform acceleration and deceleration motion) Dwell 30O Fall 1 inch in 90O (uniform acceleration and deceleration motion) Dwell 30O Fall 1 inch in 90O (uniform acceleration and deceleration motion) D. Modified Uniform Velocity Motion a) Arc Method: This method consists merely in introducing arcs at the beginning and at the end of the displacement period.

The size of the arcs is arbitrary, but they are usually drawn with a radius equal to one half the displacement. The arcs are drawn first to an indefinite length then a straight line is drawn tangent to both arcs. b) Uniform Acceleration Method: This method consists in introducing short periods of uniform acceleration or deceleration at both ends of the displacement period. D. Cycloidal Motion ? If a circle rolls along a straight line without slipping, a point on its circumference traces a curve that is called a cycloid. How to plot in the displacement diagram?

In the figure below shows how such motion is laid out in a displacement diagram. Line AB is drawn and is extended to some point such as C. A circle is drawn at C whose circumference is equal to the displacement s or whose diameter is equal to s over constant pi. The circumference of this circle is divided into a number of parts corresponding to the number of divisions along the horizontal scale. The points around the circle are projected to the vertical center line of the circle and then parallel to the line AB to the corresponding vertical lines in the displacement diagram. ———————- [pic] [pic] [pic] [pic] [pic] [pic] In this construction, it should be noted that the diameter of the semicircle is equal to the follower rise and that the number of divisions around the semicircle agrees with the number of divisions along the time axis (angle of cam rotation). [pic] [pic] The figure at the left shows the displacement, s of a falling object compares with arbitrary time units [pic] a) First method (Figure @ the right)

In laying out this type of motion in a displacement diagram, a given displacement is divided into two halves – the first half is uniformly accelerated, and the second half is uniformly retarded motion. It is therefore, necessary that the horizontal distance involved in the total displacement be divided into an even number of divisions. [pic] b) Second Method (Figure @ the left) Figure as shown, shows an alternative method of constructing this type of motion. In this case, the half displacement is divided into equal sized divisions corresponding to the number of horizontal divisions. [pic] [pic] [pic] [pic]

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