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Engineering modeling is the art of reducing a physical system to a mathematical description in order to analyze system performance Unlike many familiar problems in mathematics and engineering, the problem of producing a mathematical model for a given system does not possess a unique solution Many models of various degrees of complexity and detail can be derived for any physical system, and the selection of the right model for a given purpose can at times seem a bit overwhelming The concepts of usefulness and simplicity outlined in the quotes above prove to be helpful guides when faced with such a challenge As the quotes above suggest, the goal is rarely to create the most detailed or realistic model conceivable, but instead to produce a representation that captures the details of interest in a simple, concise manner Transforming a system from physical reality to a set of equations invariably involves some simpli cation in the description and some uncertainty in the measurement or calculation of model parameters Since it is therefore impossible to reproduce the physical system perfectly in the mathematical world, the proper evaluation of a model s quality is really an evaluation of its usefulness and not its detail The operative question thus becomes: can the model explain the behavior it was developed to explain to the degree of accuracy desired Given a choice among models that pass this test of usefulness, there are often advantages to using the simplest possible representation These advantages stem not from computational ef ciency (after all, many commercially available computer codes allow the engineer to build multibody dynamic models of cam-follower systems that are more complex than those discussed here), but rather from the engineering insight provided by simple models Often, simple models enable the engineer to develop analytical results that save time in the design process Instead of running a battery of simulations on a complex model to see how the frequency of an oscillation in the system varies with the length of one element, a simple model could provide an algebraic relationship indicating that the frequency changes with the square root of the length Simple models also enable the engineer to apply intuition from very simple systems (for instance, the idealized mass-springdamper) to more complex systems (such as an automotive valve train) The danger is that the model will be too simple and some relevant behavior will be ignored In the design of engineering systems, therefore, different models with varying complexity each have a role to play Complex models offer the comfort of additional completeness and a closer match to experiment while simple models provide intuition and a means of separating primary and secondary effects Knowing which model to use when is indeed a challenge, but one that can be greatly reduced with the help of a few concepts discussed in later sections This chapter provides the basic tools necessary to produce dynamic models of camfollower systems (and many other mechanical systems), simplify the models through some physical arguments, and roughly assess whether or not such models are useful or too simple The models developed may be used to analyze dynamic phenomena (Rothbart, 1956) such as cam hop (where the cam and follower actually separate), the dynamic forces that occur in each component, and the noise generated by the system while in operation The particular modeling approach taken is known as lumped parameter modeling since the physical characteristics of components are combined, or lumped, to produce simpler representations All components of the cam-follower system possess some mass or inertia, exhibit some compliance or spring-like behavior, and dissipate some energy from the system For modeling purposes, it is convenient to separate these effects into a series of.

Although the implicit type conversions are helpful, they will not fulfill all programming needs because they apply only to widening conversions between compatible types For all other cases you must employ a cast A cast is an instruction to the compiler to convert the outcome of an expression into a specified type Thus, it requests an explicit type conversion A cast has this general form: (target-type) expression Here, target-type specifies the desired type to convert the specified expression to For example, given

As we observed in Section 1, the function f takes on all real values We already have noticed that, since d 1 ln x = > 0, dx x the function ln x is increasing As a result, ln : {x : x > 0} R is one-to-one and onto Hence the natural logarithm function has an inverse

double x, y;

Acceleration (cm/rad/rad)

(int) (x / y)

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Here, even though x and y are of type double, the cast converts the outcome of the expression to int The parentheses surrounding x / y are necessary Otherwise, the cast to int would apply only to the x and not to the outcome of the division The cast is necessary here because there is no implicit conversion from double to int When a cast involves a narrowing conversion, information might be lost For example, when casting a long into an int, information will be lost if the long s value is greater than the range of an int because its high-order bits are removed When a floating-point value is cast to an integer type, the fractional component will also be lost due to truncation For

Splines Polynomials Constraint 8 0 90 180 Cam rotation angle (deg) 270

3:

The inverse function to the natural logarithm function is called the exponential function and is written exp(x) The domain of exp is the entire real line The range is the set of positive real numbers EXAMPLE 67

example, if the value 123 is assigned to an integer, the resulting value will simply be 1 The 023 is lost The following program demonstrates some type conversions that require casts It also shows some situations in which the casts cause data to be lost

FIGURE 510 Comparison of the accelerations obtained by both the polynomial and the spline synthesis techniques in Example 4

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