]> 33.3 The Forced and Damped Oscillator

## 33.3 The Forced and Damped Oscillator

This system obeys the equation

$d 2 x d t 2 = − k m x − f m d x d t + c m cos ⁡ ( w t )$

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 $t 0 , x ( t 0 ) , x ' ( t 0 ) , m , k , f , c$ 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 $t 0 + d$ , and then all subsequent ones to twice the previous value minus the value two before ( $t j + 1 = t j + ( t j − t j − 1 )$ ) which means that the intervals in $t$ 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 $x y$ 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
$k = 2 , m = 1$ .