Article citation info:Chróst, P., Margielewicz, J. Modeling the vertical dynamics of a car on an mechatronic basis. Scientific Journal of Silesian University of Technology. Series Transport. 2015, 88, 19-29. ISSN: 0209-3324. DOI: 10.20858/sjsutst.2015.88.2.Przemysław CHRÓST, Jerzy MARGIELEWICZModeling the VERTICAL dynamics of A car on an mechatronic BASISSummary. This paper presents the results obtained on the basis of the mechatronic model of a passenger car. In order to minimize the induced mechanical vibrations during drive event on a coarse road surface, a modification of the structure of the vehicle suspension has been applied. The proposed modification introduces additional active damping element, which task is to reduce the vertical displacement of the vehicle body. In the real world such a target can be achieved by use of appropriate control units, which drive the electronic and hydraulic components of the suspension. Additionally, the phenomenological models of the passengers have been included into the computer simulation model.Keywords: modelling, mechatronics, vibroisolation, mechanical vibration.MODELOWANIE DYNAMIKI PIONOWEJ SAMOCHODU OSOBOWEGO W UJĘCIU MECHATRONICZNYMStreszczenie Niniejszy artykuł przedstawia wyniki badań modelowych uzyskane na podstawie mechatronicznego modelu samochodu osobowego. W celu minimalizacji drgań mechanicznych wzbudzanych podczas jazdy po nierównym podłożu zastosowano modyfikację konstrukcji zawieszenia samochodu. Proponowana modyfikacja polega na wprowadzeniu do zawieszenia dodatkowego aktywnego elementu, którego zadaniem jest ograniczanie przemieszczenia pionowego nadwozia. Tak sformułowane zadanie badawcze osiąga się przez odpowiednie sterowanie aktywnymi elektrycznymi lub hydraulicznymi urządzeniami wykonawczymi odpowiedzialnymi za współpracę nadwozia z zawieszeniem. Dodatkowo podczas symulacji komputerowych uwzględniono modele fenomenologiczne odwzorowujące dynamikę ciała kierowcy i pasażera.Słowa kluczowe: modelowanie; mechatronika; wibroizolacja; drgania mechaniczne. WPROWADZENIEW ostatnich latach obserwuje się coraz więcej samochodów osobowych i ciężarowych poruszających się po polskich drogach. Potwierdzają to dane publikowane corocznie w rocznikach statystycznych wydawanych przez Główny Urząd Statystyczny. Przykładowo w 2013 roku przybyło około 645 tys. samochodów osobowych i 64 tys. samochodów ciężarowych w stosunku do 2012 roku [32]. Wzrastająca liczebność poruszających się pojazdów po polskich drogach na szczęście nie przekłada się na zwiększenie liczby wypadków z ich udziałem. W tym miejscu warto wspomnieć, że wypadki drogowe w głównej mierze są spowodowane nieprzestrzeganiem przepisów o ruchu drogowym przez kierujących. Zły stan techniczny pojazdów stanowi znikomą część przyczyn wszystkich zarejestrowanych wypadków drogowych. Pomimo faktu, że nowo projektowane samochody są coraz bardziej bezpieczne i niezawodne, to zwiększone prędkości jazdy oraz pracujący silnik wraz z układem przeniesienia napędu są źródłem drgań mechanicznych, na które narażony jest człowiek [2]. Czas oraz wartość oddziaływań dynamicznych w istotny sposób decydują o komforcie podróży. Hałas oraz drgania mechaniczne są ponadto źródłem zagrożenia i chorób zawodowych, które mogą być przyczyną obniżenia wydajności pracy i sprawności psychofizycznej, a w konsekwencji zagrożeniem w bezpieczeństwie ruchu [16]. Długotrwałe narażenie na oddziaływanie drgań mechanicznych jest przyczyną chorób, do których zalicza się chorobę: wibracyjną, lokomocyjną oraz chorobę „białego palca” [12, 30]. Przeprowadzone badania doświadczalne jednoznacznie wskazują, że przy dużych intensywnościach oddziaływań dynamicznych pojawiają się różnego rodzaju nieprzyjemne odczucia, spowodowane wpływem drgań mechanicznych na organizm ludzki, takie jak bóle: w klatce piersiowej (5÷7 Hz), jamie brzusznej (4,5÷10 Hz), grzbietu (8÷12 Hz), trudności w oddychaniu (4÷8 Hz), utrudniona artykulacja dźwięków (13÷20 Hz). Drgania głowy wzbudzone składowymi harmonicznymi zawierającymi się w przedziale od 6 Hz do 8 Hz są odpowiedzialne za rezonans żuchwy. Częstotliwości rezonansowe organów głowy mieszczą się w zakresie od 20 Hz do 30 Hz, a gałek ocznych 60÷90 Hz [5, 9, 22]. Ocena poziomu drgań oddziałujących na organizm człowieka jest jednym z ważniejszych wskaźników, który świadczy o komforcie podróży [4]. Ich minimalizacja jest możliwa do zrealizowania w dwojaki sposób. Jedną z metod jest modyfikacja konstrukcji; o konieczności takich zmian najczęściej decydują badania testowe na prototypach. Dla większości elementów prace projektowe są wykonywane przez niezależne grupy inżynierów, którzy mają odmienne, a nieraz wręcz wykluczające się cele do zrealizowania. Przykładem takiego konfliktu jest zestrojenie zawieszenia pod kątem właściwości jezdnych, w przypadku którego technicy odpowiedzialni za hałas starają się maksymalnie usztywnić elementy oraz połączyć je miękkimi, podatnymi podzespołami w celu izolacji. Natomiast zespół odpowiedzialny za dynamikę jazdy zakłada możliwie maksymalne usztywnienie zawieszenia w kierunku pionowym oraz poprzecznym do toru jazdy. Dodatkowe założenia dotyczące spełniania wymagań testów zderzeniowych oraz wytrzymałości powodują zamrożenie geometrii konstrukcji zawieszenia we wczesnej fazie projektowania. Drgania na gotowym prototypie można rejestrować dopiero w późnej fazie projektowania, co w konsekwencji zawęża możliwość zmian konstrukcyjnych do zastosowania układów tzw. pasywnej wibroizolacji. Główne zadanie takich układów polega na rozpraszaniu energii oraz okresowym jej magazynowaniu, a następnie oddawaniu. Metody pasywnej wibroizolacji bywają również nazywane metodami tradycyjnymi. Cechują się licznymi zaletami, niemniej jednak przy wzrastających wymaganiach odnośnie do ich skuteczności nie zawsze spełniają swoją funkcję w zakresie niskich częstotliwości [15, 17]. Z tego też względu poszukiwano układów aktywnych, które spełniają stawiane przed nimi wymagania, a których nie są w stanie zrealizować układy pasywne. W odróżnieniu od układów pasywnych układy aktywne [11, 18, 20] konstruuje się w wyniku zastosowania pętli sprzężenia zwrotnego. Prowadzono badania, których celem było poprawienie komfortu jazdy przez aktywną wibroizolację kabin kierowcy [25] czy też samego zawieszenia pojazdu [3, 7, 10]. Warto wspomnieć, że układy aktywnej wibroizolacji mają własne źródło zasilania, które odpowiednio sterowane, może dostarczać lub rozpraszać energię w określony sposób.Badania modelowe prowadzone na modelach symulacyjnych uwzględniających aktywną wibroizolację bywają nazywane modelowaniem mechatronicznym. W ogólnym ujęciu słowo mechatronika powstało z połączenia słów mechanika i elektronika. Pojawiło się pierwszy raz w japońskiej firmie Yaskawa Electronic Corporation ok. 1969 roku i w latach 1971 − 1982, jako nazwa handlowa, było chronione prawnie [8]. Pomimo wielu spotykanych w literaturze definicji wciąż pojawiają się trudności z jednoznacznym i precyzyjnym określeniem, które urządzenia można sklasyfikować jako mechatroniczne. Trudności te wynikają przede wszystkim ze zbyt dużej liczby definicji tej nowej dyscypliny naukowej [6]. W ogólnym ujęciu mechatronika jako nauka ujmuje w jedną całość mechanikę, elektrotechnikę oraz teorię sterowania. Z tego też względu nazwa modelowanie mechatroniczne jest w pełni uzasadniona. Pomimo faktu, że mechatronika jest stosunkowo młodą dyscypliną naukową, dysponuje bardzo obszerną teorią oraz szerokim wachlarzem środków, które umożliwiają automatyzację procesów technologicznych [31] oraz transportowych [14, 19, 27].Przeprowadzanie jakiekolwiek badań modelowych, których celem jest minimalizacja skutków oddziaływań dynamicznych zagrażających zdrowiu człowieka, jest mało wiarygodne bez uwzględnienia modelu dynamicznego odwzorowującego właściwości dynamiczne organizmu człowieka. Opublikowano już wiele prac poświęconych numerycznym modelom ciała człowieka. Odpowiedni wybór modelu dynamicznego organizmu ludzkiego nie należy do zadań prostych. Zazwyczaj osoba modelująca posiłkuje się danymi zgromadzonymi podczas badań doświadczalnych [12, 16] oraz informacjami dotyczącymi budowy anatomicznej [23, 24], istniejącymi dyrektywami, normami, a przede wszystkim sformułowanym celem badań. Dostępne w literaturze modele cechują się różnym stopniem złożoności, od kilku [26, 28, 33] do kilkunastu stopni swobody [1, 16]. Modele rozbudowane zwykle stosuje się, gdy przy jednym źródle wymuszenia zamierza się równocześnie badać zachowanie różnych narządów wewnętrznych ciała. Niemniej jednak w większości zastosowań praktycznych zwykle używa się modeli stosunkowo prostych, których niewątpliwą zaletą jest ich funkcjonalność, w wyniku czego można je łatwo zaadaptować do założonych celów badań. SFORMUŁOWANIE MODELU OBLICZENIOWEGOBudowę modelu obliczeniowego najczęściej poprzedza myślowa idealizacja struktury obiektu badań oraz procesów i zjawisk w nim zachodzących. Podejmuje się wówczas decyzję o stopniu oraz zakresie zastosowanych uproszczeń. Podczas komputerowego symulowania procesów drganiowych wywołanych jazdą po nierównym podłożu nadwozie zamodelowano jako bryłę sztywną. Przyjęto takie założenie, gdyż odkształcenia elementów nadwozia są pomijalnie małe w odniesieniu do elementów wykonawczych zawieszenia oraz siedziska kierowcy i pasażera, a uwzględnienie energii akumulowanej w nadwoziu wymaga opracowania dokładnego modelu numerycznego uwzględniającego drgania własne. W takim wypadku konieczna jest znajomość detali geometrii konstrukcji wraz z umieszczeniem pasywnych elementów tłumiących, co wykracza poza zakres tego artykułu. Podczas formułowania modelu obliczeniowego podzielono badany obiekt (rys. 1) na bryły sztywne reprezentujące zespół nadwozia, koła oraz osoby przemieszczające się pojazdem. W rozpatrywanym modelu pominięto silne nieliniowości niektórych elementów zawieszenia, prowadzące do usztywnienia nadwozia już przy niewielkich odkształceniach. Takie efekty są widoczne szczególnie w przegubach kulowych oraz amortyzatorach [37, 38]. Elementy bezwładnościowe połączono dwójnikami mechanicznymi odwzorowującymi właściwości sprężyste oraz tłumiące: zawieszenia, opony oraz ciała człowieka. W sformułowanym modelu obliczeniowym oddziaływanie dynamiczne jest wywołane nierównością podłoża, po jakim porusza się samochód. Rys. 1. Model fenomenologiczny obiektu badańMając na względzie czytelność modelu fenomenologicznego, uwzględniono na nim jedynie niezbędne wielkości wymagane do wyprowadzenia równań ruchu. Równania ruchu dane są w postaci układu równań różniczkowych drugiego rzędu o stałych współczynnikach. Z uwagi na ograniczenia objętościowe niniejszego artykułu same równania nie zostaną zaprezentowane. Równania te można wyprowadzić zarówno metodami klasycznymi, do których zaliczają się równania Lagrange’a II rodzaju, jak i metodami nieklasycznymi, opierającymi się np. na formalizmie grafów. Ruch sformułowanego układu dynamicznego rozpatrzono ze względu na wymuszenia kinematyczne pochodzące od nierówności podłoża, przy czym wyidealizowana analityczna funkcja wymuszeń kinematycznych jest dana zależnością:,(1)gdzie: v – prędkość przejazdu, A – amplituda nierówności powierzchni, – długość, na jakiej przeprowadza się pomiar podłoża, t – czas przejazdu.Formalną podstawą modelowania mechatronicznego jest ujemna pętla sprzężenia zwrotnego, która jest niezbędna do utrzymania wielkości regulowanej na określonym poziomie. Skorygowanie i utrzymanie wielkości regulowanej na określonym poziomie jest zadaniem regulatora. Należy zdawać sobie sprawę, że rodzaj zastosowanego regulatora ma decydujący wpływ na jakość regulacji. Nie bez znaczenia jest również optymalny dobór parametrów nastaw regulatorów, jak również liczba pętli sprzężenia zwrotnego. Należy zdawać sobie sprawę, że układy automatycznej regulacji złożone z pojedynczą ujemną pętlą sprzężenia zwrotnego nie zawsze dostarczają satysfakcjonujących wyników, dlatego też czasami wymagane jest zastosowanie dodatkowej pętli sprzężenia zwrotnego. Jeżeli liczba pętli sprzężenia zwrotnego jest większa od jedności, wówczas takie układy automatycznej regulacji nazywa się kaskadowymi. Sterowanie przez oddziaływanie na układ dynamiczny uzyskuje się przez porównanie wielkości regulowanej z wielkością sterującą. Rezultatem takiego porównania jest błąd regulacji, którego wartość powinna w teorii być bliska zeru. Na podstawie sygnału błędu następnie obliczany jest sygnał sterujący, który poprzez urządzenie wykonawcze oddziałuje na obiekt regulacji. W przypadku sformułowanego modelu obliczeniowego za wielkość regulowaną przyjęto przemieszczenie pionowe nadwozia q4, natomiast wielkością sterującą są siły FS1 i FS2, których kierunki działania są równoległe do elementów sprężystych i tłumiących zawieszenia. Schemat blokowy układu aktywnej wibroizolacji nadwozia samochodu przedstawiono na rysunku 2.Rys. 2. Schemat blokowy aktywnej wibroizolacji nadwozia samochoduRównania różniczkowe ruchu oraz związki przyczynowo-skutkowe, wynikające ze schematu blokowego aktywnej wibroizolacji (rys. 2), stanowią formalną podstawę do przeprowadzenia symulacji komputerowej. Odnośnie do modelu dynamicznego ciała człowieka warto wspomnieć, że struktura połączeń dwójników mechanicznych uproszczonych modeli człowieka nie zawsze odpowiada rzeczywistym parametrom segmentów ciała człowieka. Ich strukturę dobrano w ten sposób, by charakterystyki amplitudowo-częstotliwościowe modelu w miarę wiarygodnie odwzorowywały charakterystyki zarejestrowane w badaniach doświadczalnych. Z tego też względu w prezentowanej pracy ciało kierowcy oraz pasażera zamodelowano układem rekomendowanym przez normę ISO 5982:2001 (rys. 1). Użytego do symulacji komputerowych modelu dynamicznego ciała człowieka nie można wykorzystać do oceny wpływu drgań mechanicznych na poszczególne narządy, gdyż znajduje on zastosowanie przede wszystkim podczas projektowania układów wibroizolacji [13]. Funkcje wrażliwości względnej, odwzorowujące wpływ elementu bezwładnościowego m3 oraz sztywności siedziska cS na strefy rezonansowe modelu ciała człowieka, zobrazowano na rysunku 3.a)b)Rys. 3. Funkcje wrażliwość wyznaczone ze względu na wartość: a) elementu bezwładnościowego m3, b) elementu sprężystego modelującego sztywność siedziskaRozkład stref rezonansowych charakterystyki własnej (rys. 4) wyznaczono przy założeniu, że wymuszenie zewnętrzne jest przyłożone do elementu bezwładnościowego m3. a)b)Rys. 4. Rozkład stref rezonansowych analizowanego modelu dynamicznego ciała człowieka wyznaczono z: a) pominięciem elementów dyssypacyjnych, b) uwzględnieniem elementów dyssypacyjnychWyznaczone funkcje wrażliwości należy interpretować jako stosunek względnej zmiany charakterystyki dynamicznej do względnej zmiany do elementu m3 i cS. Funkcja wrażliwości względnej jest odwrotnością funkcji wrażliwości bezwzględnej, którą pierwotnie zdefiniował Hendrik Bode. Przyjęty do badań symulacyjnych model człowieka jest złożony z czterech elementów bezwładnościowych połączonych dwójnikami mechanicznymi, odwzorowujących właściwości sprężyste oraz dyssypacyjne ciała człowieka w pozycji siedzącej. Jego zaletą jest łatwe dostosowanie parametrów modelu wagi pasażera i kierowcy w zakresie od 55 kg do 90 kg. Osiąga się to, zmieniając jedynie wartość liczbową elementu bezwładnościowego m3. Przeprowadzając symulacje komputerowe, należy jedynie pamiętać, że suma elementów bezwładnościowych modelu dynamicznego człowieka w pozycji siedzącej ma wynosić 73% masy całego ciała. wyniki badań modelowychBadania modelowe dotyczące aktywnej wibroizolacji nadwozia samochodu osobowego zrealizowano w dziedzinie czasu dla pięciu różnych prędkości przejazdu: 50 km/h, 80 km/h, 110 km/h, 140 km/h oraz 170 km/h. Symulacje komputerowe przeprowadzono dwukrotnie, tzn. w pierwszych eksperymentach założono, że kierowca jest lżejszy od pasażera, następnie przyjęto sytuację odwrotną. Uwzględnienie takich założeń umożliwia określenie wpływu wagi przemieszczających się samochodem osób na wartości liczbowe wskaźnika dyskomfortu. Z uzyskanych z symulacji komputerowych przebiegów czasowych przyspieszenia obliczono wartości skuteczne przyspieszeń drgań w ruchu ustalonym. Tak oszacowany wskaźnik należy następnie odnieść do wytycznych zawartych w nomach ISO-2631-1 [34] oraz BS 6841 [35]. Należy mieć na uwadze, że wskaźniki te stanowią formalną podstawę dyskomfortu, jaki mogą odczuwać podróżujący samochodem. Należy zdawać sobie sprawę, że wskaźnik reprezentowany przez wartość skuteczną przyspieszenia drgań nie zawsze dokładnie odwzorowuje wpływ obciążeń o charakterze udarowym [29]. Dane liczbowe, na podstawie których przeprowadzono niezbędne symulacje komputerowe, zestawiono w tabeli 1.Tab. 1Parametry fizyczne i geometryczne badanego układuElementy bezwładnościoweElementy sprężysteElementy tłumiąceWymiary geometryczne[kg][kg m2][N/m][Ns/m][m]m0K = 2J = 231cSK = 65 700bSK = 1 210l1 = 1,03 m1K = 6-cSP = 65 700bSP = 1 210l2 = 1,55m2K = 2-c1K = 9 990b1K = 387l3 = 0,1m3K = 30-c2K = 3 440b2K = 234l4 = 1m0P = 2-c3K = 3 620b3K = 1 390-m1P = 6-c1P = 9 990b1P = 387-m2P = 2-c2P = 3 440b2P = 234-m3K = 55-c3P = 3 620b3P = 1 390-m4 = 1015-c4 = 31 710b4 = 1 309-m5 = 14,7-c5 = 21 880b5 = 1 309-m6 = 14,7-c6 = 165 258----c7 = 165 258--Wartości liczbowe dotyczące sztywności promieniowej opony dobrano na podstawie wyników badań zamieszczonych w pracy [21]. Wpływ współczynnika tłumienia promieniowego opony pominięto ze względu na jego zmienność w zależności od postaci drgań własnych, częstotliwości, temperatury opony oraz prędkości obrotowej. Uwzględnienie wszystkich tych zależności uniemożliwiłoby bezpośrednie porównanie wyników symulacji przeprowadzonych dla różnych prędkości jazdy. Dodatkowo współczynnik tłumienia przy nieznajomości dokładnej geometrii oraz danych materiałowych opony można wyznaczyć jedynie na podstawie danych doświadczalnych [36]. Podczas przeprowadzania obliczeń numerycznych w zakresie aktywnej wibroizolacji nadwozia samochodu parametry nastaw regulatorów odpowiedzialnych za minimalizację oddziaływań dynamicznych wywieranych na podróżujące osoby zestawiono w tabeli 2.Tab. 2Parametry nastaw układu aktywnej wibroizolacjiNazwa parametruOznaczenieRegulator PIDRegulator PIWzmocnienie sygnału błęduK02020Wzmocnienie członu proporcjonalnegoKi2020Stała czasowa całkowaniaTiI3030Stała czasowa różniczkowaniaTD20-Na podstawie danych liczbowych zestawionych w tabelach 1 i 2 przeprowadzono symulacje komputerowe układu aktywnej wibroizolacji modelu nadwozia samochodu. Wyniki badań modelowych przedstawiono w postaci przebiegów czasowych dla wybranych współrzędnych konfiguracyjnych sformułowanego modelu obliczeniowego. Wykresy przedstawione na rysunkach 5 i 6 obrazują przemieszczenie pionowe siedziska kierowcy i pasażera podczas startu i rozpędzania samochodu do maksymalnej prędkości 170 km/h. Podczas rozpędzania samochodu uwzględniono chwile, w których następuje zmiana przełożenia. Przypominamy, że wyniki te uzyskano przy założeniu, że kierowca jest lżejszy od pasażera.a)b)Rys. 5. Przemieszczenie siedziska nadwozia niewibroizolowanego: a) kierowcy, b) pasażeraa)b)Rys. 6. Przemieszczenie siedziska nadwozia wibroizolowanego: a) kierowcy, b) pasażeraAnalogiczne symulacje komputerowe wykonano w odniesieniu do sytuacji, gdy kierowca jest cięższy od pasażera. Uzyskane wyniki badań modelowych układu aktywnej wibroizolacji nadwozia samochodowego nie wykazały, że ciężar osób zlokalizowanych na siedzisku kierowcy i pasażera istotnie statystycznie wpływał na odczuwane drgania podczas podróży.PODSUMOWANIEW pracy zamieszczono mechatroniczny model aktywnej wibroizolacji nadwozia samochodu osobowego, który możne znaleźć zastosowanie podczas projektowania lub modyfikacji zawieszenia pojazdu. Warto zaznaczyć, że przedstawiona koncepcja aktywnego układu minimalizującego oddziaływania dynamiczne wywierane na człowieka podczas jazdy jest skutecznym narzędziem badawczym. Uwzględnienie w symulacjach komputerowych złożonych modeli ciała człowieka umożliwi prognozowanie wpływu oddziaływań dynamicznych na odczuwane dolegliwości przez podróżujących. Na podstawie przeprowadzonych eksperymentów numerycznych możliwe jest sformułowanie wniosków o charakterze ogólnym:Zaproponowane podejście do aktywnej wibroizolacji pozwala na jakościową oraz ilościową ocenę wpływu oddziaływań dynamicznych przenoszonych na kierowcę oraz pasażerów.Zastosowany układ aktywnej minimalizacji spełnił postawione przed nim zadanie. O skutecznym ograniczaniu oddziaływań dynamicznych w istotny sposób decydują parametry nastaw układów automatycznej regulacji. Zastosowanie dodatkowej pętli sprzężenia zwrotnego zdecydowanie poprawia komfort podróży.Niezależnie od tego, czy nadwozie samochodu jest wibroizolowane, czy nie, największe wartości przemieszczeń pionowych są rejestrowane w momencie startu i w początkowej fazie rozpędzania samochodu. 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Lee. 2007. “A study on radial directional natural frequency and damping ratio in a vehicle tire”. Applied Acoustics 68: 538-556.Heiβing B., M. Ersoy. 2001. Chassis Handbook. Fundamentals, Driving dynamics, Components, Mechatronics, Perspectives. ATZ. Komori K., T. Nagataki. 2015. “Friction Behavior of Diamond-Like Carbon Coated Ball Joint: Approach to Improving Vehicle Handling and Ride-Comfort”. SAE Technical Paper 2015-01-1507.Received 20.01.2015; accepted in revised form 15.05.2015Scientific Journal of Silesian University of Technology. Series Transport is licensed under a Creative Commons Attribution 4.0 International LicensePAGE 20P. Chróst, J. MargielewiczPAGE 29Modeling the vertical dynamics of a car… Ford-Werke GmbH, Henry-Ford-Straße 1, 50735 Köln, Germany, Maildrop. F-RLE-4.1, e-mail: pchrost1@ford.com Faculty of Transport, Silesian University of Technology, Krasińskiego 8 Street, 40-019 Katowice, Poland, e-mail: jerzy.margielewicz@polsl.pl Scientific Journal of Silesian University of Technology. Series TransportZeszyty Naukowe Politechniki Śląskiej. Seria TransportVolume 88 2015p-ISSN: 0209-3324e-ISSN: 2450-1549DOI: 10.20858/sjsutst.2015.88.2Journal homepage: http://sjsutst.polsl.pl/9j/4AAQSkZJRgABAQAAAQABAAD//gA+Q1JFQVRPUjogZ2QtanBlZyB2MS4wICh1c2luZyBJSkcg
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