Higher Order Differential Equations

Differential equations of higher order can also be solved using the methods described in this chapter, since a differential equation of order n can be converted into a set of n first-order differential equations. For example, consider the following second-order differential equation (equation 10-30) that describes the damped vibration of a mass m connected to a rigid support by a linear spring with coefficient ks and a vibration damper with coefficient kd, illustrated in Figure 1015.

Figure 10-15. A damped vibration system.

d2x dt2

Equation 10-30 can be rearranged to d2x _kd dx ks dt2 m dt m

The values of the mass, spring coefficient and damper coefficient are shown in Figure 10-16. We want to calculate the position x of the mass at time intervals from t = 0, when the mass has been given an initial displacement of 10 cm from its rest position.

G

H

1

3

mass, kg

coefficient of spring, N/cm

coefficient of damper, N/cm

4

5

5

0.33

5

(ml

(ks)

(kd)

Figure 10-16. Parameters used in the damped vibration calculation in Figure 10-17. (folder 'Chapter 10 Examples', workbook 'ODE Examples', worksheet '2nd Order ODE')

Figure 10-16. Parameters used in the damped vibration calculation in Figure 10-17. (folder 'Chapter 10 Examples', workbook 'ODE Examples', worksheet '2nd Order ODE')

We define x as the displacement of the mass from its rest position at any time t, and x' = dx/dt. Then, since d2x/dt2 - d /dt(dx/dt), equation 10-30 can be written as the two equations dx

You can now use the methods described previously for systems of first-order differential equations to solve the problem.

Figure 10-17 shows part of a spreadsheet describing the displacement x of the damped system as a function of time. The formula for the second derivative, in cell E6, is

(The mass m is multiplied by 0.01 to convert it from kg to N s2 cm-1, in order to obtain the displacement in cm.) The custom function Runge3 is used in columns B and C to calculate x (in column B) and x' (in column C); the array formula entered in cells B7 and C7 is

The value of x' is in both columns C and D, since the same value is both the x value (in column C) and the derivative (in column D); the formula in cell D6 is =C6.

A [ B C J D

E

5

t x x' x' = dx/dt x" = dx'/dt

6

0.000

5

0

0.000

-500

7

0.025

4.853

-11.404

-11.404

-410

B

0.050

4.450

-20.419

-20.419

-310

S

0.075

3.853

-26.893

-26.893

-208

10

0.100

3 126

-30 841

-30.841

-109

11

0.125

2.331

-32.421 -32.421

-19

12

0.150

1,523

-31 903 ! -31.903

58

13

0.175

0.750

-29.634

-29.634

121

14

0.200

0,052

-26.013

-26.013

166

15

0.225

-0.542

-21.451

-21.451

196

16

0 250

-1.016

-16.353

-16.353

209 209

17

0.275

-1.359

-11.094

-11.094

18

0.300

-1.572 -5.998

-5.998 197

Figure 10-17. Portion of the spreadsheet for damped vibration calculation. The initial values for the calculation are in bold, (folder 'Chapter 10 Examples', workbook 'ODE Examples', worksheet '2nd Order ODE')

Figure 10-17. Portion of the spreadsheet for damped vibration calculation. The initial values for the calculation are in bold, (folder 'Chapter 10 Examples', workbook 'ODE Examples', worksheet '2nd Order ODE')

The displacement as a function of time, from 0 to 1 second, is shown in Figure 10-18.

Figure 10-18. Damped vibration, (folder 'Chapter 10 Examples', workbook 'ODE Examples', worksheet '2nd Order ODE')

time (t), seconds

Figure 10-18. Damped vibration, (folder 'Chapter 10 Examples', workbook 'ODE Examples', worksheet '2nd Order ODE')

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