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This system obeys the equation
To set this up on a spreadsheet, I leave a place to enter the constants and
initial conditions in the first few rows; these are t0, x(t0),
x'(t0), m, k, f, c and w.
Then I would devote a column to each of t, x, x' and x", first entering
the initial conditions and then using the formulae above to obtain each new value
from the previous one.
I like to set the second t value to t0 + d, and then all subsequent
ones to twice the previous value minus the value two before  which
means that the intervals in t all have the same size.
The following chart shows how the setup might look on a spreadsheet:
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Column A
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Column B
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Column C |
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M=
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1
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k=
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1
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x0=
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f=
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0.3
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u0=
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d=
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0.01
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u'0=
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c=
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1
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w=
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1.5
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x
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U
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u'
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=D3
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=D4
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=D5
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=A10+B5
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=B10+(A11-A10)*(C10+C11)/2
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=C10+(A11-A10)*(D10+D11)/2
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=2*A11-A10
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=B11+(A12-A11)*(C11+C12)/2
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=C11+(A12-A11)*(D11+D12)/2
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=2*A12-A11
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=B12+(A13-A12)*(C12+C13)/2
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=C12+(A13-A12)*(D12+D13)/2
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=2*A13-A12
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=B13+(A14-A13)*(C13+C14)/2
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=C13+(A14-A13)*(D13+D14)/2
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=2*A14-A13
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=B14+(A15-A14)*(C14+C15)/2
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=C14+(A15-A14)*(D14+D15)/2
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=2*A15-A14
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=B15+(A16-A15)*(C15+C16)/2
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=C15+(A16-A15)*(D15+D16)/2
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=2*A16-A15
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=B16+(A17-A16)*(C16+C17)/2
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=C16+(A17-A16)*(D16+D17)/2
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=2*A17-A16
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=B17+(A18-A17)*(C17+C18)/2
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=C17+(A18-A17)*(D17+D18)/2
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=2*A18-A17
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=B18+(A19-A18)*(C18+C19)/2
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=C18+(A19-A18)*(D18+D19)/2
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=2*A19-A18
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=B19+(A20-A19)*(C19+C20)/2
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=C19+(A20-A19)*(D19+D20)/2
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=2*A20-A19
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=B20+(A21-A20)*(C20+C21)/2
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=C20+(A21-A20)*(D20+D21)/2
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=2*A21-A20
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=B21+(A22-A21)*(C21+C22)/2
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=C21+(A22-A21)*(D21+D22)/2
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=2*A22-A21
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=B22+(A23-A22)*(C22+C23)/2
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=C22+(A23-A22)*(D22+D23)/2
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=2*A23-A22
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=B23+(A24-A23)*(C23+C24)/2
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=C23+(A24-A23)*(D23+D24)/2
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=2*A24-A23
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=B24+(A25-A24)*(C24+C25)/2
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=C24+(A25-A24)*(D24+D25)/2
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=2*A25-A24
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=B25+(A26-A25)*(C25+C26)/2
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=C25+(A26-A25)*(D25+D26)/2
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=2*A26-A25
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=B26+(A27-A26)*(C26+C27)/2
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=C26+(A27-A26)*(D26+D27)/2
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Column D is here:
| Column D |
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0
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1
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0
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=MIN(B1000:B2000)
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u"
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=(-$B$3*B10-$B$4*C10+$B$6*SIN($B$7*A10))/$B$2
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=(-$B$3*(B10+(A11-A10)*C10)-$B$4*(C10+(A11-A10)*D10)-$B$6*SIN($B$7*A11))/$B$2
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=(-$B$3*(B11+(A12-A11)*C11)-$B$4*(C11+(A12-A11)*D11)-$B$6*SIN($B$7*A12))/$B$2
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=(-$B$3*(B12+(A13-A12)*C12)-$B$4*(C12+(A13-A12)*D12)-$B$6*SIN($B$7*A13))/$B$2
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=(-$B$3*(B13+(A14-A13)*C13)-$B$4*(C13+(A14-A13)*D13)-$B$6*SIN($B$7*A14))/$B$2
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=(-$B$3*(B14+(A15-A14)*C14)-$B$4*(C14+(A15-A14)*D14)-$B$6*SIN($B$7*A15))/$B$2
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=(-$B$3*(B15+(A16-A15)*C15)-$B$4*(C15+(A16-A15)*D15)-$B$6*SIN($B$7*A16))/$B$2
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=(-$B$3*(B16+(A17-A16)*C16)-$B$4*(C16+(A17-A16)*D16)-$B$6*SIN($B$7*A17))/$B$2
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=(-$B$3*(B17+(A18-A17)*C17)-$B$4*(C17+(A18-A17)*D17)-$B$6*SIN($B$7*A18))/$B$2
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=(-$B$3*(B18+(A19-A18)*C18)-$B$4*(C18+(A19-A18)*D18)-$B$6*SIN($B$7*A19))/$B$2
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=(-$B$3*(B19+(A20-A19)*C19)-$B$4*(C19+(A20-A19)*D19)-$B$6*SIN($B$7*A20))/$B$2
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=(-$B$3*(B20+(A21-A20)*C20)-$B$4*(C20+(A21-A20)*D20)-$B$6*SIN($B$7*A21))/$B$2
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=(-$B$3*(B21+(A22-A21)*C21)-$B$4*(C21+(A22-A21)*D21)-$B$6*SIN($B$7*A22))/$B$2
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=(-$B$3*(B22+(A23-A22)*C22)-$B$4*(C22+(A23-A22)*D22)-$B$6*SIN($B$7*A23))/$B$2
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=(-$B$3*(B23+(A24-A23)*C23)-$B$4*(C23+(A24-A23)*D23)-$B$6*SIN($B$7*A24))/$B$2
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=(-$B$3*(B24+(A25-A24)*C24)-$B$4*(C24+(A25-A24)*D24)-$B$6*SIN($B$7*A25))/$B$2
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=(-$B$3*(B25+(A26-A25)*C25)-$B$4*(C25+(A26-A25)*D25)-$B$6*SIN($B$7*A26))/$B$2
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=(-$B$3*(B26+(A27-A26)*C26)-$B$4*(C26+(A27-A26)*D26)-$B$6*SIN($B$7*A27))/$B$2
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=(-$B$3*(B27+(A28-A27)*C27)-$B$4*(C27+(A28-A27)*D27)-$B$6*SIN($B$7*A28))/$B$2
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=(-$B$3*(B28+(A29-A28)*C28)-$B$4*(C28+(A29-A28)*D28)-$B$6*SIN($B$7*A29))/$B$2
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=(-$B$3*(B29+(A30-A29)*C29)-$B$4*(C29+(A30-A29)*D29)-$B$6*SIN($B$7*A30))/$B$2
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The results look like this:
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mu" = -ku-fu' -c sin wx
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m=
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1
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k=
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1
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x0=
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0
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f=
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0.3
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u0=
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1
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d=
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0.02
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u'0=
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0
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c=
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1
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w=
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1.5
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min u in st state
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-0.769651461
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1
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x
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u
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u'
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u"
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0
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1
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0
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-1
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0.02
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0.9997976
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-0.020239955
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-1.0239955
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0.04
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0.999185688
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-0.040951318
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-1.047140848
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0.06
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0.99815498
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-0.062119497
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-1.06967697
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0.08
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0.996696465
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-0.083731975
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-1.091570886
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0.1
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0.994801389
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-0.105775593
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-1.11279094
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0.12
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0.992461268
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-0.128236563
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-1.133306027
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0.14
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0.989667897
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-0.15110048
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-1.153085631
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0.16
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0.986413369
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-0.174352335
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-1.172099856
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0.18
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0.98269008
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-0.197976528
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-1.190319459
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0.2
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0.978490746
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-0.221956881
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-1.207715881
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0.22
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0.973808411
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-0.246276653
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-1.224261277
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0.24
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0.968636459
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-0.270918551
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-1.239928548
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0.26
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0.962968626
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-0.29586475
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-1.254691366
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Using excel we can exhibit graphs by choosing x-y scatter plots of the first
three columns and of the second and third columns.
Realize that you can watch what happens as it happens when you vary parameters
if you do this.
Exercise 33.1 With k = m = 1 and c non-zero, locate the frequency w at which
the periodic steady state u amplitude is greatest for f = 0.1. (home in on it
by divide and conquer means.)
Do the same for k = 2, m = 1.
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