:*Linear motion*
Equation describing a linear motion with respect to time is:
The motion curve and velocity and acceleration curves are as shown. Note that the acceleration is zero for the entire motion (a=0) but is infinite at the ends. Due to infinite accelerations, high inertia forces will be created at the start and at the end even at moderate speeds. The cam profile will be discontinuous.
2.
Note that even though the velocity and acceleration is finite, the maximum acceleration is discontinuous at the start and end of the rise period. Hence the third derivative, jerk, will be infinite at the start and end of the rise portion. This curve will not be suitable for high or moderate speeds. 3. Parabolic or Constant Acceleration Motion Curve:Noting that the velocity must be zero at the two ends, we can assume a constant acceleration for the first half and a constant deceleration in the second half of the cycle. The resulting motion curve will be two parabolas. This curve can be graphically drawn by dividing each half displacement into equal number of divisions corresponding to the divisions on the horizontal axis and joining these points with O and O’ for the first and second halves respectively. Point of intersection of these lines with the corresponding vertical lines yield points on the desired curve as shown
*Cycloidal Motion Curve:*
If a circle rolls along a straight without slipping, a point on the circumference traces a curve that is called a
Within the curves we have thus far seen, cycloidal motion curve has the best dynamic characteristics. The acceleration is finite at all times and the starting and ending acceleration is zero. It will yield a cam mechanism with the lowest vibration, stress, noise and shock characteristics. Hence for high speed applications this motion curve is recommended. The maximum velocity and acceleration values are:
This curve is an improvement to the linear motion curve. To avoid infinite acceleration at the ends of the rise motion, circles are drawn as shown. Although the acceleration is finite, it will be of a high magnitude.
^{0} crank rotation. The follower must then move by constant decelaration till it has a rise of 60 mm and dwell.
For the deceleration period q
The general expression for a polynomial is given by: s= c _{0} + c_{1}q + c_{2}q^{2} + c_{3}q^{3} +…….+c_{n}q^{n} where s= displacement of the follower, q = cam rotation angle c _{i} = constants (i= 0,,,n)n= order of the polynomial. For a polynomial of order n we have n+1 unknown constant coefficients. These constant can be determined by considering the end conditions. For cam motion we at least want to have continuity in displacement velocity and acceleration which results with the boundary conditions:,
0=2 c
If we also ask for the third derivative to be zero at the ends (i.e. when q = 0 and when q = b: ) since there are 8 boundary conditions a seventh order polynomial will be required and we obtain 4-5-6-7 polynomial as:
In the previous case we were able to fit a single polynomial for the rise. This procedure does not give us flexibility on the motion itself. In order to have some control over the motion the rise period is divided into a number of parts. The end points of these parts are called knots. The first and the last knot is the start and end of the rise and the boundary conditions must be satisfied. At the interior knots, in order to obtain continuity, the two adjacent polynomials must have the same value, slope (first derivative) and curvature (second derivative) etc. For each part a polynomial of order n is written. Using the boundary conditions at the knots the coefficients are solved. Since the mathematics involved is the topic on numerical analysis, this will be beyond the topic of this book.
In order to compare the motion curves that were discussed we let: w=1 rad/s
s=3q
Now, let us transform these equations to the correct cam angles to obtain the equation
In a similar fashion the derivative ds/dq and for the motion curves can be evaluated at every cam angle and the required formula can be entered into cells in Columns D and E respectively (w=1 s
Depending on the application, manufacturing capabilities, etc. only one type of motion curve is usually used for both rise and return portions. In this example different curves are used just as an example. |