Free Oscillations: Damped Systems

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Going back to our spring-mass system, if damping forces that are proportional to the velocity are present, the equation of the motion is

my''(t) + cy'(t)+ky(t)=0, y(0)= , y'(0) = .

Recall that m is the mass of the body attached to the spring, k the spring modulus, c the nonnegative coefficient of proportionality corresponding to the damping force. We shall assume for simplicity that m=1, k=1/4 are given, and study the effects of the damping constant c on the system.

Example 1. Consider the system

y''(t) + cy'(t) + y(t)=0 , y(0)= , y'(0)= .

(a) Find the value of c that results in critical damping for this system.

(b) Plot a few solutions corresponding to various values of c resulting in underdamping, overdamping and critical damping.

Finding the critical damping is very easy. We are looking for c that yields the discriminant of the characteristic equation to be 0. The characteristic equation and its discriminant are, respectively:

, .

Hence, the value of c resulting in critical damping is c = 1. For c < 1 the system is underdamped, for c > 1 overdamped.

We shall solve the system and plot solutions for the values c = , c=1, and c=2. They correspond to underdamping, critical damping, and overdamping, respectively.

We shall not attempt to find a formula for the solutions with an arbitrary c as a parameter. Although Maple can handle parameters, in more complicated equations it may give you a solution that is far from its simplest form. Shortly speaking, we shall be better off solving the initial value problem for different, but always specified, values of c .

We solve the system for the three values of c using the command subs under the dsolve command. We define ODE1 as an equation with a parameter c, and then use the command subs to substitute values for c. You should use the same method in several of your homework problems containing equations with parameters. Hence, pay attention to the syntax below.

Remember that the output of dsolve is neither a function nor an expression. We make it into an expression by assigning the name to the right-hand side ( rhs ) of the output.

> ODE1:=diff(y(t),t$2)+c*diff(y(t),t)+(1/4)*y(t)=0;

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> dsolve({subs(c=1/5,ODE1),y(0)=1/10,D(y)(0)=-1/10},y(t)); expr1:=rhs(%);

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> dsolve({subs(c=1,ODE1),y(0)=1/10,D(y)(0)=-1/10},y(t)); expr2:=rhs(%);

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> dsolve({subs(c=3,ODE1),y(0)=1/10,D(y)(0)=-1/10},y(t)); expr3:=rhs(%);

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> plot([expr1,expr2,expr3],t=0..10*Pi,color=[blue,magenta,green]);

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