2.1.5 Diferentsiaalvõrrandi erilahendid

Mittehomogeense diferentsiaalvõrrandi (2.21) vabaliikmele lisame lauskoormusega ekvivalentse üldistatud koormuse [YSM00].

$${\theta}^{IV} - \hspace*{1pt}{{\kappa}}^{2}{\theta}^{\prime\prime... ...t}\frac{\mathit{M}_{x}{\delta}\left(t - x_{0}\right)}{E\hspace*{1pt}I_{\omega}}$$ (2.37)

Siin on

${\mathit{M}_{x}\hspace*{1pt}{\delta}\left(t - x_{0}\right)}/{E\hspace*{1pt}I_{\omega}}$ lauskoormusega ${m_{x}} = q_{z}\hspace*{-2pt}\cdot\hspace*{-2pt}e$ (jn 1.19 a) ekvivalentne koguväändemoment (2.1)), kus $\mathit{M}_{x} = F_{z}\hspace*{-2pt}\cdot\hspace*{-2pt}e$ ning e on jõu $F_{z}$ kaugus lõikekeskmeid ühendavast teljest (jn 1.19 b);

${\delta}\left(t - a\right)$ - Dirac'i ^2.4 deltafunktsioon.

Mittehomogeense diferentsiaalvõrrandi (2.37) erilahendit ${\theta}_{e}\left(x\right)$ (2.22) otsime Cauchy^2.5 valemi [Sad63, lk 40], [Ste59] abil:

$${\theta}_{e}\left(x\right) = \int_{x_{0}}^{x}K\left(x,t\right) f_{n}\left(t\right)dt$$ (2.38)

kus $K\left(x,t\right)$ on vastava homogeense diferentsiaalvõrrandi normeeritud lahend.

Täpsemalt,

$${\theta}_{e}\left(x\right) = \int_{x_{0}}^{x}K_{4}\left(x,t\right... ...4}\left(t\right)dt + \int_{x_{0}}^{x}K_{3}\left(x,t\right)f_{3}\left(t\right)dt$$ (2.39)

Siin kasutame normeeritud lahendite fundamentaalsüsteemi (2.28):

$$K_{4}\left(x,t\right) = {\theta}_{4}\left(x - t\right) = {\frac{1}{\kappa^{3}}\left( \mat... ...space*{1pt}} -{{\kappa \hspace*{1pt}\left(x - t\right) }\hspace*{1pt}}\right) }$$ (2.40)

$$K_{3}\left(x,t\right) = {\theta}_{3}\left(x - t\right) = {\frac{1}{\kappa^{2}}\left( \mat... ...pace*{1pt} {{\kappa \hspace*{1pt}\left(x - t\right) }\hspace*{1pt}} -1 \right)}$$ (2.41)

ja koormusfunktsioone $f_{n}\left(t\right)$:

$$f_{4}\left(t\right) = \hspace*{1pt}\frac{m_{x}\left(t\right)}{E\h... ...ight)}{E\hspace*{1pt}I_{\omega}} f_{2}\left(t\right) = \mathcal{M}_{x}/{EI_{y}}$$ (2.42)

Avaldame seosest (2.9) kooldejäikuse $E\hspace*{1pt}I_{\omega}$:

$$\frac{1}{E\hspace*{1pt}I_{\omega}} = \frac{1}{GI_{t}}{\kappa}^{2}, \qquad {E\hspace*{1pt}I_{\omega}} = {GI_{t}}\frac{1}{\kappa^{2}}$$ (2.43)

Nüüd saame koormusfunktsioonid $f_{n}\left(t\right)$ esitada kujul

$$f_{4}\left(t\right) = \hspace*{1pt}\frac{m_{x}\left(t\right)}{GI_... ...\frac{\mathit{M}_{x}\left(t\right)}{GI_{t}}{\kappa}^{2} <tex2html_comment_mark>$$ (2.44)

Vaatleme juhtu, kui ${m_{x}\left(t\right)} = const$. Erilahendi (2.39) saamiseks tuleb integreerida avaldis

$${\theta}_{4\hspace*{1pt}e}\left(x\right) = \int_{x_{0}}^{x}K_{4}\... ...pt}} - {{\kappa \hspace*{1pt}\left(x - t\right) }\hspace*{1pt}}\right) dt \quad$$ (2.45)

või

$${\theta}_{4\hspace*{1pt}e}\left(x\right) = \int_{x_{0}}^{x}K_{4}\... ...pt}} - {{\kappa \hspace*{1pt}\left(x - t\right) }\hspace*{1pt}}\right) dt \quad$$ (2.46)

Esmalt integreerime integraalide (2.45), (2.46) esimese liikme:

$$\int^{x}_{a} \mathrm{sh}\hspace*{1pt} {{\kappa \hspace*{1pt}\left... ...athrm{ch}\hspace*{1pt} {{\kappa\left(x - a\right)_{+}}\hspace*{1pt}} -1 \right)$$ (2.47)

kus $\left(x -a \right)_{+}$ on Heaviside'i^2.6 funktsioon

$$\left(x -a \right)_{+} = \left\{ \begin{array}{ccc} 0, & kui ... ..., & kui & {\left( x -a \right) \geq 0} \end{array}\right. \quad$$ (2.48)

Integraalide (2.45), (2.46) teise liikme integreerimisel saame

$$\int^{x}_{a} {{\kappa \hspace*{1pt}\left(x - t\right) }\hspace*{1... ...^{x}_{a} = \hspace*{1pt}\kappa\hspace*{1pt}\frac{\left(x - a\right)^{2}_{+}}{2}$$ (2.49)

Asetame leitud integraalid erilahenditesse (2.45), (2.46):

$${\theta}_{4\hspace*{1pt}e}\left(x\right) = \hspace*{1pt}\frac{m_{x}}{E\hspace*{1pt}I_{\omega}}\frac{1}{\kapp... ...space*{1pt}\kappa^{2}\hspace*{1pt} \frac{\left(x - a\right)^{2}_{+}}{2} \right]$$ (2.50)

$${\theta}_{4\hspace*{1pt}e}\left(x\right) = \hspace*{1pt}\frac{m_{x}}{GI_{t}}\hspace*{1pt} \frac{1}{\kappa^{2... ...space*{1pt}\kappa^{2}\hspace*{1pt} \frac{\left(x - a\right)^{2}_{+}}{2} \right]$$ (2.51)

Mittehomogeense diferentsiaalvõrrandi (2.37) teisele vabaliikmele ( ${\mathit{M}_{x}}/{E\hspace*{1pt}I_{\omega}} = const$) vastava erilahendi saame avaldist (2.39) integreerides:

$${\theta}_{3\hspace*{1pt}e}\left(x\right) = \int_{x_{0}}^{x}K_{3}\... ...space*{1pt}\left(x - t\right) }\hspace*{1pt}} -1 \right)}\hspace*{1pt} dt \quad$$ (2.52)

või

$${\theta}_{3\hspace*{1pt}e}\left(x\right) = \int_{x_{0}}^{x}K_{3}\... ...space*{1pt}\left(x - t\right) }\hspace*{1pt}} -1 \right)}\hspace*{1pt} dt \quad$$ (2.53)

Integreerime integraalide (2.52), (2.53) esimese liikme:

$$\int^{x}_{a} \mathrm{ch}\hspace*{1pt} {{\kappa \hspace*{1pt}\left... ... \mathrm{sh}\hspace*{1pt} {{\kappa\left(x - a\right)_{+}}\hspace*{1pt}} \right)$$ (2.54)

Integraalide (2.52), (2.53) teise liikme integreerimisel saame

$$\int^{x}_{a} {1\hspace*{1pt}} dt = \left. - \hspace*{1pt}\left(x - t\right)\right\vert^{x}_{a} = \hspace*{1pt}\left(x - a\right)_{+}$$ (2.55)

Asetame leitud integraalid erilahendisse (2.52), (2.53):

$${\theta}_{3\hspace*{1pt}e}\left(x\right) = \hspace*{1pt}\frac{\mathit{M}_{x}}{E\hspace*{1pt}I_{\omega}}\frac... ...ace*{1pt}} - \hspace*{1pt}{{\kappa\left(x - a\right)_{+}}\hspace*{1pt}} \right]$$ (2.56)

$${\theta}_{3\hspace*{1pt}e}\left(x\right) = \hspace*{1pt}\frac{\mathit{M}_{x}}{{G\hspace*{1pt}I_{t}}}\frac{1}... ...ace*{1pt}} - \hspace*{1pt}{{\kappa\left(x - a\right)_{+}}\hspace*{1pt}} \right]$$ (2.57)

Lisame bimomendi $B_{k} = M\hspace*{-2pt}\cdot\hspace*{-2pt}e$ erilahendi:

$${\theta}_{2\hspace*{1pt}e}\left(x\right) = \hspace*{1pt}\frac{\mathit{B}_{k}}{E\hspace*{1pt}I_{\omega}}\frac... ...ce*{1pt} {{\kappa\left(x - a\right)_{+}}\hspace*{1pt}} - \hspace*{1pt}1 \right]$$ (2.58)

$${\theta}_{2\hspace*{1pt}e}\left(x\right) = \hspace*{1pt}\frac{\mathit{B}_{k}}{G\hspace*{1pt}I_{t}} \left[ \h... ...ce*{1pt} {{\kappa\left(x - a\right)_{+}}\hspace*{1pt}} - \hspace*{1pt}1 \right]$$ (2.59)

Lisame erilahendid $n\hspace*{-2pt}\cdot\hspace*{-2pt}\omega$ jaoks:

$${\theta}^{\left(n\right)}_{3\hspace*{1pt}e}\left(x\right) = \hspace*{1pt}\frac{-n\hspace*{-2pt}\cdot\hspace*{-2pt}\omega}{{G\... ...ace*{1pt}} - \hspace*{1pt}{{\kappa\left(x - a\right)_{+}}\hspace*{1pt}} \right]$$ (2.60)

$${\theta}^{\left(N\right)}_{2\hspace*{1pt}e}\left(x\right) = \hspace*{1pt}\frac{-N\hspace*{-2pt}\cdot\hspace*{-2pt}\omega}{G\h... ...ce*{1pt} {{\kappa\left(x - a\right)_{+}}\hspace*{1pt}} - \hspace*{1pt}1 \right]$$ (2.61)

$${\theta}^{\left(M\right)}_{2\hspace*{1pt}e}\left(x\right) = \hspace*{1pt}\frac{M\hspace*{-2pt}\cdot\hspace*{-2pt}\omega^{\pri... ...ce*{1pt} {{\kappa\left(x - a\right)_{+}}\hspace*{1pt}} - \hspace*{1pt}1 \right]$$ (2.62)

Tabelis 1.1 esitatud koormuste jaoks on koostatud diferentsiaalvõrrandite erilahendite tabel 2.1.

Tabel 2.1. Erilahendid õhukeseseinalise varda väändel

Koormuse skeem	Erilahendid
	${\theta}^{\left(q\right)}_{4{\,}e}\left(x\right) = {\,}\mathlarger{\frac{m}{GI... ...{\,}\kappa^{2}{\,} \mathlarger{\frac{\left(x - a\right)^{2}_{+}}{2}} -1 \right]$
	${\theta}^{\left(M\right)}_{3{\,}e}\left(x\right) = {\,}\mathlarger{\frac{\math... ...x - a\right)_{+}}{\,}} - {\,}{\,}{{\kappa\left(x - a\right)_{+}}{\,}} \right]$
	${\theta}^{\left(B\right)}_{2{\,}e}\left(x\right) = {\,}\mathlarger{\frac{\math... ...left[ {\,} \mathrm{ch}{\,} {{\kappa\left(x - a\right)_{+}}{\,}} - {\,}1 \right]$
	${\theta}^{\left(n\right)}_{3{\,}e}\left(x\right) = {\,}\mathlarger{\frac{n\hsp... ... - a\right)_{+}}{\,}} - {\,}{\,}{{\kappa\left(x - a\right)_{+}}{\,}} \right]$
	${\theta}^{\left(N\right)}_{2{\,}e}\left(x\right) = {\,}\mathlarger{\frac{N\hsp... ...left[ {\,} \mathrm{ch}{\,} {{\kappa\left(x - a\right)_{+}}{\,}} - {\,}1 \right]$
	${\theta}^{\left(M\right)}_{2{\,}e}\left(x\right) = {\,}\mathlarger{\frac{M\hsp... ...left[ {\,} \mathrm{ch}{\,} {{\kappa\left(x - a\right)_{+}}{\,}} - {\,}1 \right]$

Leiame erilahendi ${\theta}_{4\hspace*{1pt}e}\left(x\right)$ tuletised:

$${\theta}_{4\hspace*{1pt}e}\left(x\right) = \hspace*{1pt}\frac{m_{x}}{GI_{t}}\hspace*{1pt} \frac{1}{\kappa^{2... ...ce*{1pt}\kappa^{2}\hspace*{1pt} \frac{\left(x - a\right)^{2}_{+}}{2} -1 \right]$$ (2.63)

$${\theta}^{\prime}_{4\hspace*{1pt}e}\left(x\right) = \hspace*{1pt}\frac{m_{x}}{GI_{t}}\hspace*{1pt} \frac{1}{\kappa} \... ...ace*{1pt}} - \hspace*{1pt}{{\kappa\left(x - a\right)_{+}}\hspace*{1pt}} \right]$$ (2.64)

$${\theta}^{\prime\prime}_{4\hspace*{1pt}e}\left(x\right) = \hspace*{1pt}\frac{m_{x}}{GI_{t}}\hspace*{1pt} \left[ \hspace*{1p... ...ce*{1pt} {{\kappa\left(x - a\right)_{+}}\hspace*{1pt}} - \hspace*{1pt}1 \right]$$ (2.65)

$${\theta}^{\prime\prime\prime}_{4\hspace*{1pt}e}\left(x\right) = \hspace*{1pt}\frac{m_{x}}{GI_{t}}\hspace*{1pt} {\kappa} \left[ \h... ... \mathrm{sh}\hspace*{1pt} {{\kappa\left(x - a\right)_{+}}\hspace*{1pt}} \right]$$ (2.66)