Investigacion De Operaciones

CAS Project 8A Laplace Transforms: Piecewise and Periodic Drivers In this project you will use your CAS to obtain solutions to mass-spring systems with drivers that are discontinuous and/or periodic. The key tool in this analysis is the Laplace Transform. Using Maple the relevant procedures are laplace and invlaplace in the inttrans package.

inttrans The package that contains integraltransforms and their inverses. with(inttrans).

Load it using

laplace The entry laplace( f(t), t, s ) outputs the Laplace transform of the function f (t). invlaplace The entry invlaplace( F(s), s, t ) outputs the inverse Laplace transform of the function F (s). Heaviside Maple uses Heaviside to denote the step function as defined in Section 8.3 with one small difference: Heaviside(0) is left undefined.To give a value to Heaviside at t = 0 simply make the assignment. For example, Heaviside(0) := 1 would make Heaviside exactly the same as “step”.

Example 1. Plot the solution and the driver for the mass-spring system x + 4x = f (t) if the mass starts from rest at the equilibrium position—the so-called “zero state”—and the driver is f (t) = step(t) − step(t − 10). We begin by defining the aliasstep for Heaviside. Name the left side of the system DE and name the driver f. > alias( step=Heaviside): DE := diff(x(t),t,t) + 4*x(t): f := step(t) - step(t-10): ZeroState := x(0)=0, D(x)(0)=0: #Define the zero-state inits. Now use dsolve/laplace to solve the IVP. Pay attention to the syntax. This yields the solution in a form that is similar to what you would obtain using the Laplace transformalgorithm “by hand”. > soln1 := dsolve( {DE = f, ZeroState} , x(t), method=laplace); plot( [rhs(soln1),f], t=0..20, color=[red,blue]); #rhs means ‘‘right hand side’’. 1 1 1 soln1 := x(t) = − cos(2t) − step(t − 10) sin(t − 10)2 4 4 2
1.0

0.5

0.0 0 5 10 t 15 20

Fig. 1: The solution curve (red) for the driver (blue): step(t) − step(t − 10). Example 2. Drive the same mass-spring system with a10 second “ramp function”, t step(t) − step(t − 10) . 10 Comment on the effect this driver has on the behavior of the solution. g(t) =

> g := t/10*(step(t) - step(t-10)): soln2 := dsolve( {DE = g, ZeroState}, x(t), method=laplace ): plot( [rhs(soln2),g], t=0..20, color=[red,blue]);
1.0

0.5

0.0 0 5 10 t 15 20

Fig 2: The solution curve (red) for the ramp driver (blue):

t 10step(t) − step(t − 10) .

Note that the short term output for the ramp driver is entirely different from the output for the square wave driver. However, the long term behaviors of the two solutions are quite similar. The amplitudes are roughly the same, but they are slightly out of phase with one another. Periodic Drivers. A driver f (t) of period T can be expressed as f (t) = g(t) + step(t − T )g(t −T ) + step(t − 2T )g(t − 2T ) + · · · , where g(t) = f (t) for 0 < t < T and g(t) = 0 otherwise. Consequently, its Laplace transform F (s) has the form F (s) = G(s) + e−T s G(s) + e−2T s G(s) + · · · where G(s) is the Laplace transform of g(t). From this it follows that if φ(t) is the zero-state solution for G(s) mx + bx + kx = g(t): φ(t) = L−1 ms2 +bs+k , then x(t) = φ(t) + step(t − T )φ(t − T )+ step(t − 2T )φ(t − 2T ) + · · · is the zero-state solution for mx + bx + kx = f (t). Although the sum formula for x(t) is infinite, adding the first n terms yields the complete solution curve on the interval [ 0, nT ]. Example 3. Graph the solution curve over several periods when the system in Example 1 is driven by the periodic function f (t) = 1 , 0 < t < 10 0 , 10 < t < 20 , f (t + 20) = f (t).Observe that we have already found the solution when the driver acts only for one period (Example 1). The first entry uses soln1 to define the function φ. The next entry defines the driver in Example 1 as the function g. The plot shows the driver and the solution curve over 4 periods using the sum procedure. > phi := unapply(rhs(soln1),t): g := t -> step(t) - step(t-10): plot( [sum(step(t -...