Periodic boundary value problem of a fourth order differential inclusion

summary:The paper deals with the periodic boundary value problem (1) $L_4 x(t) + a(t)x(t) \in F(t,x(t))$, $t\in J= [a,b]$, (2) $L_i x(a)= L_i x(b)$, $i=0,1,2,3$, where $L_0x(t)= a_0x(t)$, $L_ix(t)=a_i(t)L_{i-1}x(t)$, $i=1,2,3,4$, $a_0(t)= a_4(t)=1$, $a_i(t)$, $i=1,2,3$ and $a(t)$ are continuous on $J$, $a(t)\geq 0$, $a_i(t)>0$, $i=1,2$, $a_1(t)= a_3(t)\cdot F(t,x): J \times R \to$\{nonempty convex compact subsets of $R$\}, $R= (-\infty , \infty )$. The existence of such periodic solution is proven via Ky Fan's fixed point theorem

Brašno - Kruh '15

Proceedings contains 28 original research articles presented at 8th International Congress Flour – Bread ’15 and 10th Croatian Congress of Cereal Technologists Brašno – Kruh ’1

Periodic boundary value problem of a fourth order differential inclusion

summary:The paper deals with the periodic boundary value problem (1) $L_4 x(t) + a(t)x(t) \in F(t,x(t))$, $t\in J= [a,b]$, (2) $L_i x(a)= L_i x(b)$, $i=0,1,2,3$, where $L_0x(t)= a_0x(t)$, $L_ix(t)=a_i(t)L_{i-1}x(t)$, $i=1,2,3,4$, $a_0(t)= a_4(t)=1$, $a_i(t)$, $i=1,2,3$ and $a(t)$ are continuous on $J$, $a(t)\geq 0$, $a_i(t)>0$, $i=1,2$, $a_1(t)= a_3(t)\cdot F(t,x): J \times R \to$\{nonempty convex compact subsets of $R$\}, $R= (-\infty , \infty )$. The existence of such periodic solution is proven via Ky Fan's fixed point theorem

Vehicle speed measurement

Dan je povijesni pregled sustava mjerenja brzine s početkom na kraju 19. stoljeća. Opisani su mehanički brzinomjer i dva najčešća načina elektroničkog mjerenja brzine: Reedov prekidač i Hallove sonde. Uvod u satelitsku navigaciju i GPS. Mjerenje brzine pomoću GPS-a i Dopplerov efekt GPS signala. Radi dodatnog pojašnjenja relativističkog Dopplerovog efekta objašnjeni su osnovni principi specijalne i opće teorije relativnosti. GPS sustav nadopunjen je INS sustavom navigacije i opisan je fizikalni princip funkcioniranja akcelerometra i žiroskopa. Integracija satelitske navigacije i inercijalnih osjetila izvedena je pomoću Kalmanovog filtra. Navedeni su matematički modeli za estimaciju nagiba i brzine vozila. U zadnjem poglavlju prikazan je postav mjerenja i snimljeni valni oblici: uporedba mjerenja brzine GPS signalima frekvencije 1Hz, 10Hz i 20Hz, mjerenje brzine akcelerometrom i estimacija brzine Kalmanovim filtrom. Komentirane su razlike, ograničenja i mogućnosti nadogradnje korištenih estimacijskih modela.Historical review of speed measurement systems starting in late 19th century is given. Mechanical speedometer and two most common electronic speed measurement techniques are described: Reed switch and Hall effect sensors. Introduction to satellite navigation and GPS. Speed measurement using GPS and Doppler effect of GPS signal. To present better explanation of relativistic Doppler effect, basic principles of special and general theory of relativity are clarified. GPS is extended with INS and physical principles of accelerometer and gyroscope are described. Satellite navigation and inertial sensors integration is done with Kalman filter. Mathematical models for incline and speed estimation are suggested. In last chapter there is shown sensor placement and recorded waveforms: comparation of speed measurements conducted with 1Hz, 10Hz and 20Hz GPS signals, accelerometer speed measurement and Kalman filter speed estimation. Differences between waveforms are discussed as well as limitations and improvement possibilities of used estimation models