Äèíàìè÷åñêàÿ çàäà÷à - ÷àñòü 3

 êà÷åñòâå ðåçóëüòàòîâ ìîäàëüíîãî àíàëèçà SCAD âûäàåò ñîáñòâåííûå ÷èñëà li è ñîáñòâåííûå âåêòîðû YI çàäà÷è (19.17). Ñ íèìè ñâÿçàíû êðóãîâàÿ ÷àñòîòà w = 1/l (ðàä/ñåê), öèêëè÷åñêàÿ ÷àñòîòà q = w/2p (ãåðö) è ïåðèîä Ò = 1/q.

 ñèëó îðòîãîíàëüíîñòè ôîðì ñîáñòâåííûõ êîëåáàíèé ðåøåíèå ëþáîé äèíàìè÷åñêîé çàäà÷è â âèäå ðàçëîæåíèÿ

                                                                         Z(t) =

(t)Yi                                                                                                                                         (19.18)

âåäåò ê ðàñïàäó ñèñòåìû äèôôåðåíöèàëüíûõ óðàâíåíèé (19.12) íà íåçàâèñèìûå îòíîñèòåëüíî îáîáùåííûõ êîîðäèíàò yi(t). Ýòè óðàâíåíèÿ ñ ó÷åòîì ïðîïîðöèîíàëüíîãî ñêîðîñòè äîïîëíèòåëüíîãî ÷ëåíà, ñ ïîìîùüþ êîòîðîãî ó÷èòûâàåòñÿ ñîïðîòèâëåíèå äâèæåíèþ, èìåþò âèä

                                                       d2 yi /dt2 + 2xi dyi /dt +wi2 yi  = Pi(t)/Mi.                                                                         (19.19)

Îáîáùåííûå ñèëû

                                                                          Pi(t) = w2 YTi f(t),                                                                                            (19.20)

ìàññû

                                                                            Mi =

MYi                                                                                                                                              (19.21)

è ïàðàìåòðû çàòóõàíèÿ xi , ñîâìåñòíî ñ íà÷àëüíûìè óñëîâèÿìè  yoi è y1i , ïîëó÷àåìûìè èç Zo = Z(0) è

Z1 = dZ(0)/dt ïî ôîðìóëàì

                                                              yoi =

MZo ,   y1i = MZ1                                                                                (19.22)

ïîëíîñòüþ îïðåäåëÿþò ðåøåíèå çàäà÷è. Ýòî ðåøåíèå äàåòñÿ âûðàæåíèåì

                                                yi = exp[-xi wi t] {[(yoi xi wi + y1i)/ wDi] sinwDit  +  yoi} +  

                                            + (1/wDiMi)

exp[-xi wi (t - t)] sinwDi(t - t)dt ,                                                              (19.23)

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