Simulation of a Single and Doule-Span Guideway under Action of Moving MAGLEV Veicles wit Constant Force and Constant Gap Reinold Meisinger Mecanical Engineering Department Nuremerg University of Applied Sciences Astract In 003 te Sangai MAGLEV ransportation System ransrapid starts operation on an eleveted single and doule-span guideway wit speeds over 00 km/. e general simulation prolem of a ig speed MAGLEV veicle on a flexile guideway includes two closely related prolems ot of wic are in a sense limiting cases. In te first case te magnet force is constant and in te second case te magnet gap is constant. o descrie te dynamics of te distriuted parameter system modal analysis tecnique is used. Simulation results are given comparing te solutions for dynamic displacements of a single and a doule-span guideway wen te moving veicle is modelled wit a constant force or wit a constant gap. ISSN 66-076 Sonderdruck Scriftenreie der Georg-Simon-Om-Facocscule Nürnerg Nr. Nürnerg 00
Scriftenreie Georg-Simon-Om-Facocscule Nürnerg Seite 3. Introduction For economic reasons ig speed MAGLEV veicles will operate on elevated periodically supported single or doule-span guideways wic will e ligt and flexile [3]. For qualitative investigations te system can e approximated y a single mass veicle and an undamped flexile single or doule-span eam wit rigid piers. e general control and simulation prolem of te veicle/guideway dynamics at ig speeds includes two limiting cases cf.[]. In te first case te magnet force is constant (only te weigt of te veicle) and in te second case te magnet gap is constant (te veicle is following te guideway). In te paper te elastic structure of te guideway is approximated y omogenous elastic single and doule-span eams descried y partial BERNOULLI-EULER eam equations [3]. For typical guideway configurations tis approximation is permissile since te lengts of te eams are large compared wit te oter dimensions and tese also large compared wit te deflections. e partial differential equation can e transformed into an infinite numer of ordinary differential equations y means of a modal transformation cf.[5]. e kernels of tis transformation consist on te eigenfunctions (modes) of te force-free undamped eam. An approximation is acieved y considering only te first modes. e matematical description of te linear time variale system wit periodic coefficients is given in state space notation wic can e directly used for te digital simulation. In te simulation it is assumed tat te static deflection of te eam y its own weigt is compensated.. Single and Doule-Span Guideway Model e guideway is descried as a omogenous elastic single and doule-span eam mounted wit pivots on rigid piers. Equations of motions are derived on te asis of BERNOULLI-EULER eam teory cf. [3]. If l is te lengt of one span and l te lengt of te eam te guideway is determined y its first viration mode frequency f and its span mass m l. As sown in Fig. te concentrated magnet force F(t) = m f (g-& z& ) moves along te span wit constant velocity v. Fig. : MAGLEV single mass veicle on an elevated doule-span guideway For te furter investigations te following normalized system parameters are introduced: v / l κ = f µ = m / m f l λ = l / l were κ is te span crossing frequency ratio λ is te eam to span lengt ratio wic indicates a single or doule-span guideway and µ is te veicle to span mass ratio. Wit te nondimensional variales ξ = x/l τ = vt/l (ξ = (xt)/ sm were sm is te maximum static span deflection of a single-span caused from te concentrated weigt m f g of
Scriftenreie Georg-Simon-Om-Facocscule Nürnerg Seite te veicle te nondimensional guideway equation of motion is ( ξ + τ π κ ( ξ = F( δ( ξ ξ () wit te oundary and (λ - ) intermediate conditions of te eam ( ξ = 0 ξ ξ = 0 λ and ( ξ = 0 ξ = 0 () λ () In eq. () δ is te DIRAC delta function and F ( is te nondimensional magnet force wic can e written as 8 (l / v) F( = µ z z = π κ sm were & z& is te veicle acceleration and z & z is te corresponding nondimensional variale. (3) Modal approximation Based on te oundary and intermediate conditions of te eam in eq.() eigenfunctions ϕ (ξ) and eigenvalues λ can e otained y solving te eigenvalue prolem ϕ ( ξ) = λ ϕ ( ξ) = (). π κ ξ () Wit te ortonormality relation λ ϕ ( ξ) ϕk ( ξ) dξ = δ k and β = λ π κ 0 δ k = Kronecker symol te following results are otained: a) Eigenfunctions of a single-span (λ = ) and antimetric eigenfunctions of a doule-span (λ = ) ϕ (ξ) = C sin(β ξ) = 3.. for λ = = 35.. for λ = parameter β : coefficient C : β = (λ + - )π/ λ C = /λ ) Symmetric eigenfunctions of a doule-span = 6.. ϕ (ξ) = C [cos(β ) sin(βη) - cos(β ) sin(β η)] were η = ξ for ξ є [0] η = ( - ξ) for ξ є [] parameter β : coefficient C : β = 3.9660 C = 3.906 0 - β = 7.06858 C =.70875 0-3 β 6 = 0.076 C 8 = 7.358789 0-5 β 8 = 3.35768 C 8 = 3.800 0-6 β 0 = 6.9336 C 0 =.373 0-7.... Wen only te first n eigenfunctions ϕ are considered it can e sown tat an approximation of te solution of
Scriftenreie Georg-Simon-Om-Facocscule Nürnerg Seite 5 eq.() can e written as cf. [5] n ( ξ = ϕ ( ξ) ( ξ) = ϕ ( ξ) ( were ϕ is te n - vector of te eigenfunctions and is te n - vector of te modal coordinates wic is otained as solution from te ordinary differential equation 8 = Λ + ϕ( µ & z π κ wit te n n- diagonal matrix Λ = diag(λ). Note: Eq. (5) is exact in te case n. For a finite numer n te approximation converges wit (λ /n). Investigations ave sown [] tat te result is close enoug if approximately n = λ modes are considered. (5) (6) 3. Veicle wit Moving Constant Force Because te magnet force is constant te veicle acceleration in eq.(6) is zero. en te following linear differential equation wit constant coefficients is otained: & = Λ + ϕ( 8 κ π. As sown in [] eq.(7) can e solved analytically. oug eq.(7) ere is solved numerically as a limiting case of te veicle wit moving constant gap (µ = 0). (7). Veicle wit Moving Constant Gap Because te magnet gap is constant te nondimensional veicle acceleration & z in eq.(6) is te same as te nondimensional guideway acceleration under te moving veicle ( ξ = τ ). Wit eq.(5) te acceleration can e written as: & & z = (ξ = = ϕ ( + ϕ ( + ϕ ( were ϕ( = ϕ(ξ = and ϕ &( ϕ&(τ & ) are te according derivatives wit respect to time. ogeter wit eq.(6) and eq.(8) te following linear time variale differential equation wit periodic coefficients is otained: (8) 8 Μ( = K( + D( + ϕ( π κ wit te symmetric n n-matrices (9) τ Μ ( = [ Ε + µ ϕ( ϕ ( )] τ Κ ( = [ Λ µ ϕ ( ϕ& ( )] τ D ( = [ µ ϕ ( ϕ& ( )]. Wit eq.(5) te nondimensional span deflection under te moving veicle (ξ = can e written as: ( = ϕ (. (0)
Scriftenreie Georg-Simon-Om-Facocscule Nürnerg Seite 6 For te computer simulation eq.(9) and eq.(0) will e written in state space notation: = & M 0 E 0 8 + ( K( M ( D( ( ) ( ) M τ ϕ τ π κ [ ] = [ ϕ ( τ ) 0]. () Eq.() as te general form x = A( x + B( y = C( x () were A( is te n n-system matrix B( te n -disturance matrix C( te n-measurement matrix and x y are vectors wit te according dimensions. is linear time variale system wit periodic coefficients can e integrated numerically wit a t order RUNGE-KUA algoritm over one period. o reduce te elapsed computing time in te numerical integration an analytical expression for M - is used: M µ ( = E ϕ( ϕ + µ ϕ ( ϕ( ( (3) were ϕ ( ϕ( is a scalar quantity cf.[6]. Note: For µ = 0 from eq.() te state equation for te moving veicle wit constant force is otained. 5. Simulation Results e normalized system parameters used in te simulation are: κ = 0; 0.5; 0.5; 0.75; ;.5; µ = 0.5; λ = ;. For te single-span n = 3 and for te doule-span n = 6 modes are considered in te approximation. For te limiting case κ = 0 te steady-state solution for te guideway deflection under te veicle n ϕ ( τ ( = 8 β ) is used wic can e otained from eq.(6) and eq.(0). () Fig. and Fig.3 illustrate te influence of span crossing frequency ratio κ on te nondimensional deflection of a doule-span guideway under a moving veicle wit constant force and a moving veicle wit a constant gap. Fig. and Fig.5 sow te same results for a single-span guideway. e results sow tat for low values of span crossing frequency ratio κ te deflections of te doule-span and for ig values of κ te deflections of te single-span guideway are smaller. Note tat te traectories for te first span of te doule-span are closely similar to tose of te single-span guideway. Wen κ approaces zero te two prolems of constant force and constant gap reduce to a static one and te traectories are all exactly alike. An important oservation is te increase in exitation of iger mode components as te veicle to span mass ratio µ is increasing.
Scriftenreie Georg-Simon-Om-Facocscule Nürnerg Seite 7 Fig. : Histories of doule-span deflection under a moving veicle wit constant force Fig. 3: Histories of doule-span deflection under a moving veicle wit constant gap (µ = 0.5)
Scriftenreie Georg-Simon-Om-Facocscule Nürnerg Seite 8 Fig. : Histories of single-span deflection under a Fig. 5: Histories of single-span deflection under a moving veicle wit constant force moving veicle wit constant gap (µ = 0.5) 6. Conclusion In tis paper modal analysis tecnique as een applied to otain te state equation of a moving single mass veicle on an elastic single and doule-span guideway. Simulation results for te guideway traectories were given for te two prolems wen te MAGLEV veicle is moving wit constant magnet force and moving wit constant magnet gap. Comparing te maximum deflections te results sow tat for low values of span crossing frequency ratio κ (quasistatic region) te doule-span guideway is recommended wile for iger values of κ te single-span guideway must e favoured. More realistic simulations wit an actively controlled magnet force are given in [3]. References [] FEIX J.: Planung und Bau der Scangaier ransrapid-rasse Beratende Ingenieure /00 Springer- VDI- Verlag S. -7. [] MILLER L.: ransrapid Innovation für den Hocgescwindigkeitsverker Bayeriscer Monatsspiegel /998 S. 3-5. [3] MEISINGER R.: Control Systems for Flexile MAGLEV Veicles Riding over Flexile Guideways. Proc. of IUAM Symposium on te Dynamics of Veicles on Roads and Railway racks Delft 975. [] MEISINGER R.: Beiträge zur Regelung einer Magnetscweean auf elastiscem Farweg. Dr.-Ing. Dissertation U Müncen 977. [5] GILLES E.D. ZEIZ M.: Modales Simulationsverfaren für Systeme mit örtlic verteilten Parametern. Regelungstecnik Heft 5 (969) S. 0-. [6] MEISINGER R.: Simulation von Stromanemer und Farleitung ei oen Gescwindigkeiten Zeitscrift für angewandte Matematik und Mecanik ZAMM 6 (98) S. 69-70.