A Spring with Friction: Damped Oscillations...The equation for damped oscillations is an example of a linear second-order differential equation with constant coefficients. This section

4/24/14, 7:11 AMLinear Second-Order Differential Equations

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11.11 LINEAR SECOND-ORDER DIFFERENTIALEQUATIONS

A Spring with Friction: Damped Oscillations

The differential equation , which we used to describe the motion of a spring,disregards friction. But there is friction in every real system. For a mass on a spring, the frictional force fromair resistance increases with the velocity of the mass. The frictional force is often approximately proportionalto velocity, and so we introduce a damping term of the form , where a is a constant called thedamping coefficient and is the velocity of the mass.

Remember that without damping, the differential equation was obtained from

With damping, the spring force is replaced by , where a is positive and the term is subtracted because the frictional force is in the direction opposite to the motion. The new differentialequation is therefore

which is equivalent to the following differential equation:

Equation for Damped Oscillations of a Spring

We expect the solution to this equation to die away with time, as friction brings the motion to a stop.

The General Solution to a Linear Differential Equation

The equation for damped oscillations is an example of a linear second-order differential equation withconstant coefficients. This section gives an analytic method of solving the equation,

for constant b and c. As we have seen for the spring equation, if and satisfy the differential



equation, then the principle of superposition says that, for any constants and , the function

is also a solution. It can be shown that the general solution is of this form, provided is not a multiple of.

Finding Solutions: The Characteristic Equation

We now use complex numbers to solve the differential equation

The method is a form of guess-and-check. We ask what kind of function might satisfy a differential equationin which the second derivative is a sum of multiples of and y. One possibility is anexponential function, so we try to find a solution of the form:

where r may be a complex number.22 To find r, we substitute into the differential equation:

We can divide by provided , because the exponential function is never zero. If , then , which is not a very interesting solution (though it is a solution). So we assume . Then is

a solution to the differential equation if

This quadratic is called the characteristic equation of the differential equation. Its solutions are

There are three different types of solutions to the differential equation, depending on whether the solutions tothe characteristic equation are real and distinct, complex, or repeated. The sign of determines the typeof solutions.

The Case with

There are two real solutions and to the characteristic equation, and the following two functions satisfythe differential equation:

The sum of these two solutions is the general solution to the differential equation:



If , the general solution to

is

where and are the solutions to the characteristic equation. If and , the motion is calledoverdamped.

A physical system satisfying a differential equation of this type is said to be overdamped because it occurswhen there is a lot of friction in the system. For example, a spring moving in a thick fluid such as oil ormolasses is overdamped: it will not oscillate.

Example 1

A spring is placed in oil, where it satisfies the differential equation

Solve this equation with the initial conditions and when .

Solution

The characteristic equation is

with solutions and , so the general solution to the differential equation is

We use the initial conditions to find and . At , we have

Furthermore, since , we have

Solving these equations simultaneously, we find and , so that the solution is

The graph of this function is in Figure 11.73. The mass is so slowed by the oil that it passes through theequilibrium point only once (when ) and for all practical purposes, it comes to rest after a short time.The motion has been “damped out” by the oil.



Figure 11.73:

Solution to overdamped equation

The Case with

In this case, the characteristic equation has only one solution, . By substitution, we can check thatboth and are solutions.

If ,

has general solution

If , the system is said to be critically damped.

The Case with

In this case, the characteristic equation has complex roots. Using Euler's formula,23 these complex roots leadto trigonometric functions which represent oscillations.

Example 2

An object of mass is attached to a spring with spring constant , and the objectexperiences a frictional force proportional to the velocity, with constant of proportionality .



At time , the object is released from rest 2 meters above the equilibrium position. Write the differentialequation that describes the motion.

Solution

The differential equation that describes the motion can be obtained from the following general expressionfor the damped motion of a spring:

Substituting , , and , we obtain the differential equation:

At , the object is at rest 2 meters above equilibrium, so the initial conditions are and ,where s is in meters and t in seconds.

Notice that this is the same differential equation as in Example 1 except that the coefficient of hasdecreased from 3 to 2, which means that the frictional force has been reduced. This time, the roots of thecharacteristic equation have imaginary parts which lead to oscillations.

Example 3

Solve the differential equation

subject to , .

Solution

The characteristic equation is

The solution to the differential equation is

where and are arbitrary complex numbers. The initial condition gives

Also,



so gives

Solving the simultaneous equations for and gives (after some algebra)

The solution is therefore

Using Euler's formula, and , we get

Multiplying out and simplifying, all the complex terms drop out, giving

The and terms tell us that the solution oscillates; the factor of tells us that the oscillations aredamped. See Figure 11.74. However, the period of the oscillations does not change as the amplitudedecreases. This is why a spring-driven clock can keep accurate time even as it is running down.

Figure 11.74: Solution to underdamped equation

In Example 3, the coefficients and are complex, but the solution, , is real. (We expect this, since represents a real displacement.) In general, provided the coefficients b and c in the original differential

equation and the initial values are real, the solution is always real. The coefficients and are alwayscomplex conjugates (that is, of the form ).

If , to solve



• Find the solutions to the characteristic equation .

• The general solution to the differential equation is, for some real and ,

If , such oscillations are called underdamped. If , the oscillations are undamped.

Example 4

Find the general solution of the equations

(a).

(b). .

Solution

(a). The characteristic equation is , so . Thus the general solution is

(b). The characteristic equation is , so . The general solution is

Notice that we have seen the solution to this equation in Section 11.10; it's the equation ofundamped simple harmonic motion.

Example 5

Solve the initial value problem



Solution

We solve the characteristic equation

The general solution to the differential equation is

Substituting gives

Differentiating gives

Substituting gives

The solution is therefore

Summary of Solutions to

If , then

If , then

If , then

Exercises and Problems for Section 11.11

Click here to open Student Solutions Manual: Ch 11 Section 11

Click here to open Web Quiz Ch 11 Section 11

Exercises

For Exercises 1–12, find the general solution to the given differential equation.



1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

For Exercises 13–20, solve the initial value problem.

13. , , .

14. , , .

15. , , .

16. , , .

17. , ,

18. , ,

19. , ,

20. , ,

For Exercises 21–24, solve the boundary value problem.



21. , , .

22. , , .

23. , ,

24. , ,

Problems

25. Match the graphs of solutions in Figure 11.75 with the differential equations below.

(a).

(b).

(c).

(d).

Figure 11.75

26. Match the differential equations to the solution graphs (I)–(IV). Use each graph only once.

(a).



(b).

(c).

(d).

(I).

(II).

(III).

(IV).



27. If is a solution to the differential equation

find the value of the constant k and the general solution to this equation.

28. Assuming b, , explain how you know that the solutions of an underdamped differential equationmust go to 0 as .

For each of the differential equations in Problems 29–30, find the values of b that make the general solution:

(a). overdamped,

(b). underdamped,

(c). critically damped.

29.

30.

Each of the differential equations (i)–(iv) represents the position of a 1 gram mass oscillating on the end of adamped spring. For Problems 31–35, pick the differential equation representing the system which answers thequestion.

(a).

(b).

(c).

(d).

31. Which spring has the largest coefficient of damping?

32. Which spring exerts the smallest restoring force for a given displacement?



33. In which system does the mass experience the frictional force of smallest magnitude for a givenvelocity?

34. Which oscillation has the longest period?

35. Which spring is the stiffest?

Hint: You need to determine what it means for a spring to be stiff. Think of an industrial strength spring anda slinky.]

36. Find a solution to the following equation which satisfies and does not tend to infinity as :

37. Consider an overdamped differential equation with b, .

(a). Show that both roots of the characteristic equation are negative.

(b). Show that any solution to the differential equation goes to 0 as .

38. Could the graph in Figure 11.76 show the position of a mass oscillating at the end of an overdampedspring? Why or why not?

Figure 11.76

39. Consider the system of differential equations

(a). Convert this system to a second order differential equation in y by differentiating the second equation



with respect to t and substituting for x from the first equation.

(b). Solve the equation you obtained for y as a function of t; hence find x as a function of t.

Recall the discussion of electric circuits. Just as a spring can have a damping force which affects its motion,so can a circuit. Problems 40–43 involve a damping force caused by the resistor in Figure 11.77. The chargeQ on a capacitor in a circuit with inductance L, capacitance C, and resistance R, in ohms, satisfies thedifferential equation

Figure 11.77

40. If henry, ohms, and farads, find a formula for the charge when

(a). , .

(b). , .

41. If henry, ohm, and farads, find a formula for the charge when

(a). , .

(b). , .

(c). How did reducing the resistance affect the charge? Compare with your solution to Problem 40.



42. If henry, ohm, and farads, find a formula for the charge when

(a). , .

(b). ,

(c). How did increasing the inductance affect the charge? Compare with your solution to Problem 40.

43. Given any positive values for R, L and C, what happens to the charge as t goes to infinity?

44.

(a). If and satisfy show that

is a solution to .

(b). If , show that the solution in part (a) can be written

(c). Using the Taylor series, show that

(d). Use the result of part (c) in the solution from part (b) to show that .

(e). If there is a double root . By direct substitution, show that satisfies .



Copyright © 2006 John Wiley & Sons, Inc. All rights reserved.

A Spring with Friction: Damped Oscillations...The equation for damped oscillations is an example of a linear second-order differential equation with constant coefficients. This section

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