# Příčný pohyb dvojkolí v přímé

Odvození rovnice pohybu dvojkolí – Klingelův pohyb

Autor , 02. srpna, Teorie

Při jízdě železničního vozidla v přímé koleji vykonává konické (kuželovité) dvojkolí nápravy příčný pohyb způsobený rozdílnými poloměry kol v kontaktních bodech s kolejnicí. To vede k periodickým pohybům dvojkolí, které teoreticky popsal Klingel v roce 1883, a proto je tato teorie známa jako Klingelův pohyb. Teorie popisuje dvojkolí jako dva komolé kužely souměrné podle jejich podstavy pohybující se po přímé koleji, viz obrázek 1.

Obrázek 1 zavádí následující parametry:

• $\gamma$ = konicita kola,
• $r$ = poloměr kola v centrické poloze dvojkolí,
• $R$ = poloměr trajektorie nápravy,
• $s$ = vzdálenost styčných kružnic,
• $y$ = příčná poloha dvojkolí,
• $x$ = ujetá vzdálenost.

Dvojkolí se pohybuje po trajektorii definované výchylkou $y$ vztaženou k ose $x$. Ke zjištění výchylky $y$ v daném bodě $x$ je nutné určit funkci výchylky.

obrázku 1 je možné sestavit 2 trojúhelníky: První trojúhelník vychází z trajektorie pohybu dvojkolí (Obrázek 2a), druhý z poloměrů kol v kontaktních bodech s kolejnicí (Obrázek 2b).

Z podobnosti trojúhelníků na obrázku 2a lze sestavit rovnici:

$\frac{r_p}{R + \frac{s}{2}} = \frac{r_l}{R - \frac{s}{2}} \; \; \; \; \; \; (1)$

Z trojúhelníku na obrázku 2b lze sestavit následující rovnici:

$\frac{r}{x} = \tg{\gamma} \Rightarrow x = \frac{r}{\tg{\gamma}} \; \; \; \; \; \; (2)$

Pro malé úhly lze zavést zjednodušení v podobě $\gamma = tg \gamma$. Rovnici 2 lze tedy zapsat jako

$x = \frac{r}{\gamma} \; \; \; \; \; \; (3)$

Dosazením rovnice 3 do rovnice 2 získáme

$\frac{r}{\frac{r}{\gamma}} = \frac{r_p}{y + \frac{r}{\gamma}} \Rightarrow \gamma = \frac{r_p}{y + \frac{r}{\gamma}} \Rightarrow r_p = r + \gamma \cdot y \; \; \; \; \; \; (4)$

Analogicky vyjádříme $r_l$:

$r_l = r - \gamma \cdot y \; \; \; \; \; \; (5)$

Dosazením rovnic 4 a 5 do rovnice 1 získáme rovnici

$\frac{r - \gamma \cdot y}{R - \frac{s}{2}} = \frac{r + \gamma \cdot y}{R + \frac{s}{2}} \; \; \; \; \; \; (6)$

Po několika úpravách rovnice 6 můžeme vyjádřit $R$ jako:

$R = \frac{r \cdot s}{2 \cdot \gamma \cdot y} \; \; \; \; \; \; (7)$

Křivost je převrácenou hodnotou poloměru, proto platí:

$K = \frac{1}{R} = \frac{2 \cdot \gamma \cdot y}{r \cdot s} \; \; \; \; \; \; (8)$

Křivost také určuje míru vychýlení křivky od tečny v daném bodě, a proto ji lze zapsat jako druhou derivaci výchylky $y$:

$K = -\frac{d^2 y}{dx^2} \; \; \; \; \; \; (9)$

Kombinací rovnice 89 získáme diferenciální rovnici

$\frac{d^2 y}{dx^2}+2\frac{\gamma \cdot y}{r \cdot s} = 0 \; \; \; \; \; \; (10)$

Rovnici 10 řešíme obdobně jako v článcích o modelování kolejového roštu (například Timošenkův model koleje). Substitucí $y = e^{\lambda x}$, její dvojitou derivací a dosazením do rovnice 10 získáme charakteristickou rovnici:

$\lambda^2 \cdot e^{\lambda x} + 2 \frac{e^{\lambda x} \cdot \gamma}{r \cdot s} = 0 \; \; \; \; \; \; (11)$

Nyní se snažíme určit kořeny $\lambda$. Rovnici 11 vydělíme $e^{\lambda x}$ a získáme:

$\lambda^2 = - 2 \frac{\gamma}{r \cdot s} \; \; \; \; \; \; (12)$

Z rovnice 12 získáme kořeny

$\lambda_1 = i \sqrt{2 \frac{\gamma}{r \cdot s}} \; \; \; \; \; \; (13)$
$\lambda_2 = -i \sqrt{2 \frac{\gamma}{r \cdot s}} \; \; \; \; \; \; (14)$

Pro další výpočty využijeme kladný kořen (rovnice 13) a zavedeme substituci

$a = \sqrt{2 \frac{\gamma}{r \cdot s}} \; \; \; \; \; \; (15)$

Výše uvedenou rovnici $y = e^{\lambda x}$ můžeme též zapsat jako $y = e^{iax}$. Z Eulerovy rovnice $e^{i \phi}= cos \phi + isin\phi$ můžeme vyjádřit obecné řešení diferenciální rovnice:

$y(x) = A \cdot \cos{(a \cdot x)} + C \cdot \sin{(a \cdot x)}, \; \; \; \; \; \; (16)$
kde
$C = B \cdot i\; \; \; \; \; \; (17)$

Příčný pohyb dvojkolí lze vyjádřit jako harmonickou netlumenou funkci. Můžeme si jej tedy představit jako funkci sinus, viz obrázek 3.

Z průběhu funkce sinus můžeme určit okrajové podmínky pro body $0$, $L$$L/4$:

$y(0) = 0, \; \; \; \; \; \; (18)$
$y(L) = 0, \; \; \; \; \; \; (19)$
$y(L/4) = y_0, \; \; \; \; \; \; (20)$

kde

• $y_0$ = amplituda $[mm]$,
• $L$ = délka vlny $[m]$.

Dosazením první okrajové podmínky (rovnice 18) do obecného řešení (rovnice 16) získáme:

$y(0) = A \cdot \cos{(a \cdot 0)} +C \cdot \sin{(a \cdot 0)} = 0 \; \; \; \; \; \; (21)$

Víme, že $\sin{(0)}=0$ a $\cos{(0)}=1$. Jediný způsob jak dostát podmínce $y(0)=0$ je postavit konstantu $A=0.$ Pokračujeme dosazením druhé okrajové podmínky (rovnice 19). Zároveň za konstantu $A$ dosadíme $0$. Získáme:

$y(L) = C \cdot \sin{(a \cdot L)} = 0 \; \; \; \; \; \; (22)$

K tomu, aby rovnice 22 mohla být rovna $0$, musí být $\sin{(a \cdot L)}=0$ nebo $C=0$. Jelikož $C=0$ by nevedlo k řešení, budeme se zabývat případem $\sin{(a \cdot L)}=0$. Víme, že funkce sinus je rovna $0$ v pravidelných periodách. Tyto periody lze charakterizovat jako celočíselné násobky čísla $\pi$, tedy $k \cdot \pi$, kde $k \in \N$. Jelikož je naše okrajová podmínka definována v $L$, budeme brát v úvahu nejnižší sudý násobek $\pi$, tedy $2 \cdot \pi$. Aby $a \cdot L = 2 \cdot \pi$, musí platit:

$a = \frac{2 \cdot \pi}{L} \; \; \; \; \; \; (23)$

Tímto jsme aplikovali druhou okrajovou podmínku. Poslední okrajovou podmínku (rovnice 20) aplikujeme následovně:

$y(L/4) = C \cdot \sin{(\frac{2 \cdot \pi}{L} \cdot \frac{L}{4})} = y_0 \; \; \; \; \; \; (24)$

Jelikož $\sin{(\pi / 2)} = 1$, můžeme zapsat:

$C = y_0 \; \; \; \; \; \; (25)$

Výsledkem zavedení okrajových podmínek do rovnice 16 je rovnice pro výchylku $y$ v bodě $x$:

$y(x) = y_0 \cdot \sin{(2 \cdot \pi \frac{x}{L})}, \; \; \; \; \; \; (26)$

Kombinací rovnic 15 a 23 můžeme vyjádřit rovnici pro délku vlny $L_k$:

$\frac{2 \cdot \pi}{L_k} = \sqrt{\frac{2 \cdot \gamma}{r \cdot s}} \Rightarrow L_k = 2 \cdot \pi \cdot \sqrt{\frac{r \cdot s}{2 \cdot \gamma}} \; \; \; \; \; \; (27)$

Tímto jsme teoreticky vyřešili pohyb dvojkolí pro přímé koleji. Nutno říci, že do trajektorie pohybu dvojkolí v reálných podmínkách vstupuje řada externalit, které nejsou v této teorii zohledněny. Příkladem může být změna ojetí po délce kolejnic.

O tom, že k oscilačnímu pohybu dvojkolí skutečně dochází se můžete přesvědčit v následujícím videu:

Zdroje:

• [1] ESVELD, Coenraad. Modern railway track. 2nd ed. Zaltbommel: MRT-Productions, c2001. ISBN 90-800324-3-3.
• [2] Civil Engineering RWTH Aachen University. Guidance By Railway Tracks. 2016. Available at: https://www.youtube.com/watch?v=vkzgcJGdUnA&t=53s

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