]>
Home | 18.013A | Chapter 33 |
||
|
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 and .
Then I would devote a column to each of and , 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 , and then all subsequent ones to twice the previous value minus the value two before ( ) which means that the intervals in all have the same size.
The following chart shows how the setup might look on a spreadsheet
Column A |
Column B |
Column C |
M= |
1 |
|
k= |
1 |
x0= |
f= |
0.3 |
u0= |
d= |
0.01 |
u'0= |
c= |
1 |
|
w= |
1.5 |
|
x |
U |
u' |
=D3 |
=D4 |
=D5 |
=A10+B5 |
=B10+(A11-A10)*(C10+C11)/2 |
=C10+(A11-A10)*(D10+D11)/2 |
=2*A11-A10 |
=B11+(A12-A11)*(C11+C12)/2 |
=C11+(A12-A11)*(D11+D12)/2 |
=2*A12-A11 |
=B12+(A13-A12)*(C12+C13)/2 |
=C12+(A13-A12)*(D12+D13)/2 |
=2*A13-A12 |
=B13+(A14-A13)*(C13+C14)/2 |
=C13+(A14-A13)*(D13+D14)/2 |
=2*A14-A13 |
=B14+(A15-A14)*(C14+C15)/2 |
=C14+(A15-A14)*(D14+D15)/2 |
=2*A15-A14 |
=B15+(A16-A15)*(C15+C16)/2 |
=C15+(A16-A15)*(D15+D16)/2 |
=2*A16-A15 |
=B16+(A17-A16)*(C16+C17)/2 |
=C16+(A17-A16)*(D16+D17)/2 |
=2*A17-A16 |
=B17+(A18-A17)*(C17+C18)/2 |
=C17+(A18-A17)*(D17+D18)/2 |
=2*A18-A17 |
=B18+(A19-A18)*(C18+C19)/2 |
=C18+(A19-A18)*(D18+D19)/2 |
=2*A19-A18 |
=B19+(A20-A19)*(C19+C20)/2 |
=C19+(A20-A19)*(D19+D20)/2 |
=2*A20-A19 |
=B20+(A21-A20)*(C20+C21)/2 |
=C20+(A21-A20)*(D20+D21)/2 |
=2*A21-A20 |
=B21+(A22-A21)*(C21+C22)/2 |
=C21+(A22-A21)*(D21+D22)/2 |
=2*A22-A21 |
=B22+(A23-A22)*(C22+C23)/2 |
=C22+(A23-A22)*(D22+D23)/2 |
=2*A23-A22 |
=B23+(A24-A23)*(C23+C24)/2 |
=C23+(A24-A23)*(D23+D24)/2 |
=2*A24-A23 |
=B24+(A25-A24)*(C24+C25)/2 |
=C24+(A25-A24)*(D24+D25)/2 |
=2*A25-A24 |
=B25+(A26-A25)*(C25+C26)/2 |
=C25+(A26-A25)*(D25+D26)/2 |
=2*A26-A25 |
=B26+(A27-A26)*(C26+C27)/2 |
=C26+(A27-A26)*(D26+D27)/2 |
Column D is here
Column D |
0 |
1 |
0 |
=MIN(B1000:B2000) |
u" |
=(-$B$3*B10-$B$4*C10+$B$6*SIN($B$7*A10))/$B$2 |
=(-$B$3*(B10+(A11-A10)*C10)-$B$4*(C10+(A11-A10)*D10)-$B$6*SIN($B$7*A11))/$B$2 |
=(-$B$3*(B11+(A12-A11)*C11)-$B$4*(C11+(A12-A11)*D11)-$B$6*SIN($B$7*A12))/$B$2 |
=(-$B$3*(B12+(A13-A12)*C12)-$B$4*(C12+(A13-A12)*D12)-$B$6*SIN($B$7*A13))/$B$2 |
=(-$B$3*(B13+(A14-A13)*C13)-$B$4*(C13+(A14-A13)*D13)-$B$6*SIN($B$7*A14))/$B$2 |
=(-$B$3*(B14+(A15-A14)*C14)-$B$4*(C14+(A15-A14)*D14)-$B$6*SIN($B$7*A15))/$B$2 |
=(-$B$3*(B15+(A16-A15)*C15)-$B$4*(C15+(A16-A15)*D15)-$B$6*SIN($B$7*A16))/$B$2 |
=(-$B$3*(B16+(A17-A16)*C16)-$B$4*(C16+(A17-A16)*D16)-$B$6*SIN($B$7*A17))/$B$2 |
=(-$B$3*(B17+(A18-A17)*C17)-$B$4*(C17+(A18-A17)*D17)-$B$6*SIN($B$7*A18))/$B$2 |
=(-$B$3*(B18+(A19-A18)*C18)-$B$4*(C18+(A19-A18)*D18)-$B$6*SIN($B$7*A19))/$B$2 |
=(-$B$3*(B19+(A20-A19)*C19)-$B$4*(C19+(A20-A19)*D19)-$B$6*SIN($B$7*A20))/$B$2 |
=(-$B$3*(B20+(A21-A20)*C20)-$B$4*(C20+(A21-A20)*D20)-$B$6*SIN($B$7*A21))/$B$2 |
=(-$B$3*(B21+(A22-A21)*C21)-$B$4*(C21+(A22-A21)*D21)-$B$6*SIN($B$7*A22))/$B$2 |
=(-$B$3*(B22+(A23-A22)*C22)-$B$4*(C22+(A23-A22)*D22)-$B$6*SIN($B$7*A23))/$B$2 |
=(-$B$3*(B23+(A24-A23)*C23)-$B$4*(C23+(A24-A23)*D23)-$B$6*SIN($B$7*A24))/$B$2 |
=(-$B$3*(B24+(A25-A24)*C24)-$B$4*(C24+(A25-A24)*D24)-$B$6*SIN($B$7*A25))/$B$2 |
=(-$B$3*(B25+(A26-A25)*C25)-$B$4*(C25+(A26-A25)*D25)-$B$6*SIN($B$7*A26))/$B$2 |
=(-$B$3*(B26+(A27-A26)*C26)-$B$4*(C26+(A27-A26)*D26)-$B$6*SIN($B$7*A27))/$B$2 |
=(-$B$3*(B27+(A28-A27)*C27)-$B$4*(C27+(A28-A27)*D27)-$B$6*SIN($B$7*A28))/$B$2 |
=(-$B$3*(B28+(A29-A28)*C28)-$B$4*(C28+(A29-A28)*D28)-$B$6*SIN($B$7*A29))/$B$2 |
=(-$B$3*(B29+(A30-A29)*C29)-$B$4*(C29+(A30-A29)*D29)-$B$6*SIN($B$7*A30))/$B$2 |
The results look like this
mu" = -ku-fu' -c sin wx |
|||
m= |
1 |
||
k= |
1 |
x0= |
0 |
f= |
0.3 |
u0= |
1 |
d= |
0.02 |
u'0= |
0 |
c= |
1 |
||
w= |
1.5 |
min u in st state |
-0.769651461 |
1 |
|
||
x |
u |
u' |
u" |
0 |
1 |
0 |
-1 |
0.02 |
0.9997976 |
-0.020239955 |
-1.0239955 |
0.04 |
0.999185688 |
-0.040951318 |
-1.047140848 |
0.06 |
0.99815498 |
-0.062119497 |
-1.06967697 |
0.08 |
0.996696465 |
-0.083731975 |
-1.091570886 |
0.1 |
0.994801389 |
-0.105775593 |
-1.11279094 |
0.12 |
0.992461268 |
-0.128236563 |
-1.133306027 |
0.14 |
0.989667897 |
-0.15110048 |
-1.153085631 |
0.16 |
0.986413369 |
-0.174352335 |
-1.172099856 |
0.18 |
0.98269008 |
-0.197976528 |
-1.190319459 |
0.2 |
0.978490746 |
-0.221956881 |
-1.207715881 |
0.22 |
0.973808411 |
-0.246276653 |
-1.224261277 |
0.24 |
0.968636459 |
-0.270918551 |
-1.239928548 |
0.26 |
0.962968626 |
-0.29586475 |
-1.254691366 |
Using Excel we can exhibit graphs by choosing
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
and
non-zero, locate the frequency
at which the periodic steady state
amplitude is greatest for
. (Home in on it by divide and conquer means.)
Do the same for
.
|