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In the foregoing example we have assigned one or more of the displacement derivatives to be zero at the boundary In the dwell-rise-return-dwell cam, other modes of control may be required to optimize the cam curve We can impose the following three modes of control: 1 no control 2 interior control 3 assignment of nite quantities to terminal point displacement derivations The last category, the assignment of nite quantities to terminal point displacement derivations, will not be discussed here For more see Stoddart (1953) and Chen (1982) In no control, at the ends of a symmetrical DRRD cam (Fig 45), the displacement, the velocity, and the acceleration are to be zero However, at the midstation, displacement, of course, is 1, velocity is zero, and acceleration is left loose (no control) In addition, there is zero jerk at the midpoint Thus the boundary conditions for the rise portion of the motion are when q = 0, y = 0, y = 0; y = 0 when q = 1 ( midpoint ), y = 1, y = 0, y = no control, and y = 0 A polynomial y = C0 + C1q + C2q 2 + C3q 3 + C4q 4 + C5q 5 is assumed which reduces to y = C3q 3 + C4q 4 + C5q 5

Method public static double Sqrt(double v) public static double Tan(double v) public static double Tanh(double v) public static double Truncate(double v) public static decimal Truncate(decimal v)

TABLE 21-1 Methods De ned by Math (continued)

y=1 y, y''

Meaning Returns the square root of v Returns the tangent of v Returns the hyperbolic tangent of v Returns the whole number portion of v Returns the whole number portion of v

1 x(1 x)1/2

Here is an example that uses Sqrt( ) to help implement the Pythagorean theorem It computes the length of the hypotenuse given the lengths of the two opposing sides of a right triangle

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Substituting the end conditions yields C3 + C4 + C5 = 1 3C3 + 4C4 + 5C5 = 0 6C3 + 24C4 + 60C5 = 0 Solving gives y= 20 3 25 4 8 5 q - q + q 3 3 3 100 3 40 4 q + q 3 3 160 3 q 3 (47)

// Implement the Pythagorean Theorem using System; class Pythagorean { static void Main() { double s1; double s2; double hypot; string str; ConsoleWriteLine("Enter length of first side: "); str = ConsoleReadLine(); s1 = DoubleParse(str); ConsoleWriteLine("Enter length of second side: "); str = ConsoleReadLine(); s2 = DoubleParse(str); hypot = MathSqrt(s1*s1 + s2*s2); ConsoleWriteLine("Hypotenuse is " + hypot); } }

Here is a sample run:

SOLUTION The key idea is that we can only handle one singularity at a time This integrand is singular at both endpoints 0 and 1 Therefore we divide the domain of integration somewhere in the middle at 1/2 say (it does not really matter where we divide) and treat the two singularities separately First we treat the integral

y = 20q 2 -

Next is an example that uses the Pow( ) method to compute the initial investment required to achieve a desired future value given the annual rate of return and the number of years The formula to compute the initial investment is shown here: InitialInvestment = FutureValue / (1 + InterestRate)Years

Part II:

The return portion is symmetrical (see Fig 45) In this gure we see the characteristic curves for the fth-degree polynomial applicable as a DRRD cam Interior control may be employed to shift the characteristic curves by manipulation of high-order polynomial equations Chen (1982) elaborated on these different controls We see from the foregoing that controls at the beginning and terminal condition produce an asymmetrical acceleration curve Interior control can elevate this situation Let s assume boundary conditions q = 0, y = 0, y = 0, y = 0, y = 0, q = 1, y = 1, y = 0, y = 0, y = 0, d4 =0 dq 4 (48)

Because Pow( ) requires double arguments, the interest rate and the number of years are held in double values The future value and initial investment use the decimal type

1/2 0

/* Compute the initial investment needed to attain a known future value given annual rate of return and the time period in years */ using System; class InitialInvestment { static void Main() { decimal initInvest; // initial investment decimal futVal; // future value double numYears; double intRate; string str; ConsoleWrite("Enter future value: "); str = ConsoleReadLine(); try { futVal = DecimalParse(str); } catch(FormatException exc) { ConsoleWriteLine(excMessage); return; } ConsoleWrite("Enter interest rate (such as 0085): "); str = ConsoleReadLine(); try { intRate = DoubleParse(str); } catch(FormatException exc) { ConsoleWriteLine(excMessage); return; } ConsoleWrite("Enter number of years: "); str = ConsoleReadLine(); try { numYears = DoubleParse(str); } catch(FormatException exc) { ConsoleWriteLine(excMessage); return; } initInvest = futVal / (decimal) MathPow(intRate+10, numYears); ConsoleWriteLine("Initial investment required: {0:C}", initInvest); } } // number of years // annual rate of return as a decimal

(a) Function generator (cam-driven typewriter mechanism)

21:

Here is a sample run:

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