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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