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In Chap 2 we discussed the basic parabolic, sinusoidal, and cycloidal curves for cam-follower DRD action We also showed in Chap 3 how these curves and others can be modi ed and combined to form a variety of cam pro les Ignoring the torsional aspect of the cam drive (assuming the cam runs at a constant speed), the two fundamental dynamic objectives in designing cam pro les are: 1 to have the smallest possible maximum acceleration and thus the minimum dynamic load on the high-speed follower; 2 To avoid resonant vibratory response in the follower caused by high harmonic content in the designed cam pro le These fundamental objectives are not related Let us compare the basic cam pro les as a means of describing the harmonic content that is addressed in the later paragraphs The parabolic curve has the smallest maximum acceleration but has a high harmonic content, since the Fourier expansion for a parabolic curve has an in nite number of terms Because each term is associated with a certain forcing frequency, it may induce resonance vibrations in the follower at some of those frequencies at some speed as the system is brought up to speed The cycloidal cam, on the other hand, having but a fundamental frequency in the acceleration curve, has a minimum harmonic content The cycloidal pro le is the least likely to excite vibration in the follower, but this occurs at the expense of high acceleration Furthermore, it has a more than 50 percent higher maximum acceleration than does the parabolic pro le In design, it is necessary to nd a cam pro le that has the fewest and lowest possible harmonics, though, in the end, nature calls for a compromised balance of all factors, requiring judicial trade-offs There is no perfect solution A number of acceptable harmonic curves have been developed in consideration of these thoughts Note that for a DRD motion event, the Fourier series used must be an odd harmonic curve, which has polar symmetry with respect to the midpoint of the curve Gutman 1-3 Harmonic (Gutman, 1961) 15 2pq 1 6pq q sin sin y = h b 96p b b 32p y = h b 2pq 1 6pq 15 1 - 16 cos b - 16 cos b hp 8b 3 hp 2 4b 3 2pq 6pq 15 sin b + 3 sin b 2pq 6pq 15 cos b + 9 cos b (411).

The structure corresponding to the char type is Char It is quite useful because it supplies a large number of methods that allow you to process and categorize characters For example, you can convert a lowercase character to uppercase by calling ToUpper( ) You can determine if a character is a digit by calling IsDigit( ) The methods defined by Char are shown in Table 21-9 Notice that several, such as ConvertFromUtf32( ) and ConvertToUtf32( ), give you the ability to work with both UTF-16 and UTF-32 Unicode characters In the past, all Unicode characters could be represented by 16 bits, which is the size of a char However, a few years ago the Unicode character set was expanded and more than 16 bits are required Each Unicode character is represented by a code point The way that a code point is encoded depends on the Unicode Transformation Format (UTF) being used In UTF-16, the most common code points require one 16-bit value, but some need two 16-bit values When two 16-bit values are needed, two char

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21:

y =

Field public static readonly decimal MaxValue public static readonly decimal MinusOne public static readonly decimal MinValue public static readonly decimal One public static readonly decimal Zero

1/2 0

Meaning The largest value that a decimal can hold The decimal representation of 1 The smallest value that a decimal can hold The decimal representation of 1 The decimal representation of 0

y =

values are used to represent it The first character is called the high surrogate and the second is called the low surrogate In UTF-32, each code point uses one 32-bit value Char provides the necessary conversions between UTF-16 and UTF-32 Char defines the following fields: public const char MaxValue public const char MinValue These represent the largest and smallest values that a char variable can hold Char implements the following interfaces: IComparable, IComparable<char>, IConvertible, and IEquatable<char>

Method public int CompareTo(char v) Meaning Compares the character in the invoking object with that of v Returns zero if the characters are equal Returns a negative value if the invoking object has a lower value Returns a positive value if the invoking object has a greater value Compares the character in the invoking object with that of v Returns zero if the characters are equal Returns a negative value if the invoking object has a lower value Returns a positive value if the invoking object has a greater value Converts the Unicode UTF-32 code point in utf32Ch into a UTF-16 string and returns the result Converts the high and low UTF-16 surrogates specified by highSurrogate and lowSurrogate into a UTF-32 code point The result is returned Converts the UTF-16 surrogate pair at str[idx] into its UTF-32 code point The result is returned Returns true if the value of the invoking object equals the value of v Returns true if the value of the invoking object equals the value of v

Gutman s 1-3 harmonic curve can be obtained from the Fourier series expansion of the displacement of the parabolic curve by retaining the rst two terms of the series The resulting curve has a harmonic content triple the frequency of the cycloidal curve and a maximum acceleration of about 130 percent of the parabolic curve, or about 80 percent of the cycloidal curve (Fig 410)

For clarity we let (x) = ln x, (x) = 1/x Then the (inde nite) integral becomes (x) 3 dx = 1/3 4/3 (x) (x) Thus

public int CompareTo(object v)

public static string ConvertFromUtf32(int utf32Ch) pubic static int ConvertToUtf32(char highSurrogate, char lowSurrogate) pubic static int ConvertToUtf32(string str, int idx) public bool Equals(char v) public override bool Equals(object v)

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