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where C and C are constants of integration We thus obtain ln |B(t)| = Kt + D, where we have amalgamated the two constants into a single constant D Exponentiating both sides gives |B(t)| = eKt+D or B(t) = eD eKt = P eKt ( ) Notice that we have omitted the absolute value signs since the number of bacteria is always positive Also we have renamed the constant eD with the simpler symbol P Equation ( ) will be our key to solving exponential growth and decay problems We motivated our calculation by discussing bacteria, but in fact the calculation applies to any function which grows at a rate proportional to the size of the function Next we turn to some examples

in units of radians/second Note that this formula works directly when the stiffness and mass are in SI units (N m and kg, respectively), but when English units of lbf/in and lbm are used a conversion is necessary w n = 322 12 k m (117)

using SystemThreading;

The multithreading system is built upon the Thread class, which encapsulates a thread of execution The Thread class is sealed, which means that it cannot be inherited Thread defines several methods and properties that help manage threads Throughout this chapter, several of its most commonly used members will be examined

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The amplitude and phase can be determined from the initial conditions Since the initial conditions assumed here are x( 0) = A cos( 0 + f ) = x0 the amplitude and phase are given by A = x 0 f = 0 Unlike the amplitude and phase, the frequency of vibration is a property of the system and depends only upon the mass and stiffness, not the initial conditions For this reason, it is called x( 0) = - A sin( 0 + f ) = 0

There are a number of ways to create and start a thread This section describes the basic mechanism Various options are described later in this chapter To create a thread, instantiate an object of type Thread, which is a class defined in SystemThreading The simplest Thread constructor is shown here: public Thread(ThreadStart entryPoint) Here, entryPoint is the name of the method that will be called to begin execution of the thread ThreadStart is a delegate defined by the NET Framework as shown here: public delegate void ThreadStart( )

A population of bacteria tends to double every four hours If there are 5000 bacteria at 9:00 am, then how many will there be at noon

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Thus, your entry point method must have a void return type and take no arguments Once created, the new thread will not start running until you call its Start( ) method, which is defined by Thread The Start( ) method has two forms The one used here is public void Start( ) Once started, the thread will run until the method specified by entryPoint returns Thus, when entryPoint returns, the thread automatically stops If you try to call Start( ) on a thread that has already been started, a ThreadStateException will be thrown Here is an example that creates a new thread and starts it running:

FIGURE 530 Cam displacements for both spline (k = 10) and optimized polydyne with damping ratio = 02 (Cs = Cf) in Example 8

using System; using SystemThreading; class MyThread { public int Count; string thrdName; public MyThread(string name) { Count = 0; thrdName = name; } // Entry point of thread public void Run() { ConsoleWriteLine(thrdName + " starting"); do { ThreadSleep(500); ConsoleWriteLine("In " + thrdName + ", Count is " + Count); Count++; } while(Count < 10); ConsoleWriteLine(thrdName + " terminating"); } } class MultiThread { static void Main() { ConsoleWriteLine("Main thread starting"); // First, construct a MyThread object MyThread mt = new MyThread("Child #1"); // Next, construct a thread from that object Thread newThrd = new Thread(mtRun); // Finally, start execution of the thread newThrdStart(); do { ConsoleWrite("");

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