Project Help Notebook by Keith Stroyan

Help NoteBook Copyright 1998 by Academic Press, Inc. - All rights reserved.
Modify this Notebook to prepare your project report.

Project 36: Second Order Linear Equations with Forcing

by A. Student and B. Pupil

with technical help from Da LabMonitor and Help Notebook

NOTE: Computing plays only a small role in this project.  This project is your own extension of the textbook theory of linear second order equations to account for forced systems.  The project is needed to understand Projects 37 and 38.

My Project Introduction

Write this last.  Begin with the basic question and state your main finding clearly, but with few technical details.  You might even modify the title of this section to make it more interesting.

Derivation of the Equation of Motion of an Un-Forced (Spring) (Circuit)

F = mA for a spring, shock absorber and mass (HINT 36.1)

The forces on our system...

Sample Solutions (HINT 36.2)

The program SpringEq from the main course materials gives the animated sample solutions, while the program SecondOrder gives the exact analytic solutions.

Give samples constructed from those programs to illustrate the three main types of solutions.  We have included a very brief start.

The point is that when the characteristic roots are complex, the real solutions oscillate, when they are real, they do not and the condition between real and complex roots is when we have only one root to the characteristic equation.

Purely Oscillatory Solutions

Clear[x, t, a, b, c]; a = 1; b = 0; c = 1;
Print["The general solution to ", a, "x''[t] + ", b, 
        "x'[t] + ", c, "x[t] = 0"];
DSolve[{a x''[t]+b x'[t]+c x[t]==0},x[t],t]

Typical graphs of the solutions look as follows.

Damped Oscillatory Solutions

Clear[x, t, a, b, c]; a = 1; b = 0.1; c = 1;
Print["The general solution to ", a, "x''[t] + ", b, 
        "x'[t] + ", c, "x[t] = 0"];
DSolve[{a x''[t]+b x'[t]+c x[t]==0},x[t],t]

Typical graphs of the solutions look as follows.

Critically Damped Solutions

Clear[x, t, a, b, c]; a = 1; b = 2; c = 1;
Print["The general solution to ", a, "x''[t] + ", b, 
        "x'[t] + ", c, "x[t] = 0"];
DSolve[{a x''[t]+b x'[t]+c x[t]==0},x[t],t]

Typical graphs of the solutions look as follows.

Over Damped Solutions

Clear[x, t, a, b, c]; a = 1; b = 3; c = 1;
Print["The general solution to ", a, "x''[t] + ", b, 
        "x'[t] + ", c, "x[t] = 0"];
DSolve[{a x''[t]+b x'[t]+c x[t]==0},x[t],t]

Typical graphs of the solutions look as follows.

Transients Leave (HINT 36.3): (lim_(t -> ∞))x[t]  =  0

Negative Real Parts in the Quadratic Formula

The coefficient of x'[t]  in a real system is positive because ...

This makes the real parts of the roots of the characteristic equation negative because  ...  in the various cases of Theorem 23.5 of the main text ...

The General Solutions Tend to Zero

Every solution of  a x''[t]+b x'[t]+c x[t]=0  can be written in the form

x[t]  =  (Subscript[k, 1 ]) ??  + (Subscript[k, 2]) ??

In the various cases described above x[t]  tends to zero as  t  tends to +∞, because ...

Case 1:

Case 2:

Case 3:

Superimposing Forces

Work the hints in the print project and put your solutions here.  The overall point is that a solution with force 1 plus a solution with force 2 is a solution with the sum of the forces - FOR LINEAR EQUATIONS and NOT for nonlinear ones.

Hint 36.5

Hint 36.6

Hint 36.7

Constant Forces (Hints 36.8,9 10)

If we apply a constant force to a linear equation, say by suspending a spring - shock absorber - mass system vertically in gravity,

m y''[t]+c y'[t]+s y[t] = m g

then a solution with any initial condition tends toward a constant solution,  (Subscript[y, SS])[t]  =  k, known as the static solution.

We can find  k  since (Subscript[y, SS'])[t]  =  0, (Subscript[y, SS])''[t]  =  0 and

m (Subscript[y, SS])''[t] + c (Subscript[y, SS])'[t] + s (Subscript[y, SS])[t] = m g

k = ???

Why does any solution tend to the static solution?

Sinusoidal Forcing (Hints 36.11, 12)

If we apply a sinusoidal force to a linear equation, say a spring - shock absorber - mass system,

m y''[t] + c y'[t] + s y[t]=(Subscript[f, C]) Cos[ω t]+ (Subscript[f, S]) Sin[ω t]

then a solution with any initial condition tends toward the sinusoidal "stead state" solution,

\(\(\(	
    \)\(y\_SS[t]\  = \ h\ Cos[ω\ t]\  + \ k\ Sin[ω\ t]\)\)\)

where the constants  h  and  k  are given by

h = ((det|({{(Subscript[f, C]), c ω}, {(Subscript[f, S]), s  -  m (ω^2)}})|)/(det|({{s  -  m ω^2, c ω}, {-c ω, s  -  m (ω^2)}})|))  and  k = ((det|({{s  -  m (ω^2), (Subscript[f, C])}, {-c ω, (Subscript[f, S])}})|)/(det|({{s  -  m ω^2, c ω}, {-c ω, s  -  m (ω^2)}})|))

Why does any solution tend to the steady state oscillation?