Problem

The motion of a damped spring-mass system (Fig. P20.15) is described by the following ordi...

The motion of a damped spring-mass system (Fig. P20.15) is described by the following ordinary differential equation:

FIGURE P20.15

Year

Moose

Wolves

Year

Moose

Wolves

Year

Moose

Wolves

1959

563

20

1975

1355

41

1991

1313

12

1960

610

22

1976

1282

44

1992

1590

12

1961

628

22

1977

1143

34

1993

1879

13

1962

639

23

1978

1001

40

1994

1770

17

1963

663

20

1979

1028

43

1995

2422

16

1964

707

26

1980

910

50

1996

1163

22

1965

733

28

1981

863

30

1997

500

24

1966

765

26

1982

872

14

1998

699

14

1967

912

22

1983

932

23

1999

750

25

1968

1042

22

1984

1038

24

2000

850

29

1969

1268

17

1985

1115

22

2001

900

19

1970

1295

18

1986

1192

20

2002

1100

17

1971

1439

20

1987

1268

16

2003

900

19

1972

1493

23

1988

1335

12

2004

750

29

1973

1435

24

1989

1397

12

2005

540

30

1974

1467

31

1990

1216

15

2006

450

30

where x = displacement from equilibrium position (m), t = time (s), m = 20-kg mass, and c = the damping coefficient (N. s/m). The damping coefficient c takes on three values of 5 (underdamped), 40 (critically damped), and 200 (over- damped). The spring constant k = 20 N/m. The initial velocity is zero, and the initial displacement x = 1 m. Solve this equation using a numerical method over the time period 0 ≤ t ≤ 15 s. Plot the displacement versus time for each of the three values of the damping coefficient on the same plot.

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Solutions For Problems in Chapter 20