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1122 Model Suf ciency and Model Reduction Natural frequencies provide some insight into the applicability or suf ciency of the model For mechanical systems, if the system is forced at the natural frequency, resonance (the maximum ampli cation of the forcing input) will occur Intuitively, since the system naturally tends to vibrate at certain frequencies and in certain modes, motion at these frequencies is, in a sense, easier to initiate and maintain Resonance is an important feature to capture in a mathematical model since it is not uncommon for the ratio of system motion to input magnitude to increase by an order of magnitude or more at resonance if the system is lightly damped In a cam follower system, therefore, if the cam rotates at speeds up to 3000 rpm (or 50 Hz), the dynamic model of the follower system must, as a bare minimum, include all natural frequencies up to 50 Hz to capture the basic motion of the follower with reasonable delity Similarly, if experiments show a resonance at 1000 Hz but the highest natural frequency predicted by the model is 100 Hz, the model requires additional re nement It is too simple to predict the observed behavior Continuing with the example of the suspension in Fig 112, this two-mass model is suf cient for simulating the vehicle response to road excitations in the range of around 0 to 15 Hz If a test driver complains about road vibrations being ampli ed around 50 or 60 Hz, the model is too simple, since the mass responsible for this vibration has been lumped into either the body or the tire in the two-mass model Conversely, if the goal of the modeling process is to capture the fundamental response of the vehicle to vibrations in the range of 1 Hz, information about the resonance at 11 Hz is extraneous and the model is more complex than needed Of course the more complex model could still be used Instead of retaining the additional complexity in the model, however, it is often bene cial to look for a simpler model that retains the important characteristics of the original model This process is known as model reduction In this case, the objective of the reduction is to create a model with a single natural frequency that predicts the resonance associated with the low-frequency mode (the sprung mass bouncing on the tire) The task then is to choose an equivalent mass and stiffness for a simpli ed model with a single natural frequency that resembles Fig 113a It cannot be emphasized enough that since any model will be an approximation of the real system, there is no unique or correct way to simplify Simpli cation is again part of the art of engineering modeling The idea of natural frequencies can provide some guidance, however Since natural frequencies are associated with masses and stiffnesses (more formally, with mass and stiffness matrices) and the goal is to remove the higher natural frequencies, the model can be simpli ed by ignoring masses or replacing certain spring elements with rigid links (thus assuming in nite stiffness) These assumptions remove degrees of freedom from the system and reduce the number of natural frequencies and the complexity of the resulting equations of motion It is also possible, in the manner suggested by Chen (1982), to lump masses and stiffnesses together without explicitly assuming any of these to be zero or in nite Three possible ways to apply these concepts to the suspension system are: Approach 1: Assume In nite Tire Stiffness Assume that the stiffer spring (the tire) is rigid and use the mass of the car and the stiffness of the suspension spring to determine natural frequency In this model, the mass of the tire and the degree of freedom associated with the tire are neglected since the rigid spring prevents the tire from moving relative to the road This simpli cation is illustrated in Fig 113b The natural frequency in this case is given by.

Method public void Sort(int startIdx, int count, IComparer<T> comp) public T[ ] ToArray( ) public void TrimExcess( )

Description Sorts a portion of the collection using the specified comparison object The sort begins at startIdx and runs for count elements If comp is null, the default comparer for each object is used Returns an array that contains copies of the elements of the invoking object Reduces the capacity of the invoking list so that it is no more than 10 percent greater than the number of elements that it currently holds

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We see that there are two ways to treat a de nite integral using integration by parts One is to carry the limits of integration along with the parts calculation The other is to do the parts calculation rst (with an inde nite integral) and then plug in the limits of integration at the end Either method will lead to the same solution You Try It: Calculate the integral

(743)

TABLE 24-14 A Sampling of Methods De ned by List<T> (continued)

In addition to the properties defined by the interfaces that it implements, List<T> adds Capacity, shown here: public int Capacity { get; set; } Capacity gets or sets the capacity of the invoking list The capacity is the number of elements that can be held before the list must be enlarged Because a list grows automatically, it is not necessary to set the capacity manually However, for efficiency reasons, you might want to set the capacity when you know in advance how many elements the list will contain This prevents the overhead associated with the allocation of more memory The following indexer, defined by IList<T>, is implemented by List<T>, as shown here: public T this[int idx] { get; set; } It sets or gets the value of the element at the index specified by idx Here is a program the demonstrates List<T> It reworks the first ArrayList program shown earlier in this chapter The only changes necessary are to substitute the name List for ArrayList and to use the generic type parameters

where Gi denotes the ith side of the polygon The aforementioned formulas, thus, can be approximated as A qO 1 n r n i d Gi 2 1 Gi (744a) (744b)

// Demonstrate List<T> using System; using SystemCollectionsGeneric; class GenListDemo { static void Main() { // Create a list List<char> lst = new List<char>(); ConsoleWriteLine("Initial number of elements: " + lstCount); ConsoleWriteLine(); ConsoleWriteLine("Adding 6 elements"); // Add elements to the list lstAdd('C'); lstAdd('A');

The method of partial fractions is used to integrate rational functions, or quotients of polynomials We shall treat here some of the basic aspects of the technique The rst fundamental observation is that there are some elementary rational functions whose integrals we already know I Integrals of Reciprocals of Linear Functions 1 dx ax + b with a = 0 is always a logarithmic function In fact we can calculate 1 1 dx = ax + b a 1 1 dx = log |x + b/a| x + b/a a An integral An integral

Part II:

(744c)

4:

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