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Memoirs Faculty of Engineering ISSN 0078䋭6659 MEMOIRS OF THE FACULTY OF ENG THE FACULTY MEMOIRS OF MEMOIRS OF THE FACULTY OF ENGINEERING OSAKA CITY UNIVERSITY INEERING OSAKA CITY UNIVERSITY VOL. 52 DECEMBER 2011 VOL. 52. 2011 PUBLISHED BY THE GRADUATE SCHOOL OF ENGINEERING OSAKA CITY UNIVERSITY 1202-0321ޓᄢ㒋Ꮢ┙ᄢቇޓᎿቇㇱ Ꮏቇㇱ⧷ᢥ♿ⷐ81.㧔㧕ޓ ⷗ᧄ ࠬࡒ This Memoirs is annually issued. Selected original works of the members of the Faculty of Engineering are compiled herein. Abstracts of paper presented elsewhere during the current year are also compiled in the latter part of the volume. All communications with respect to Memoirs should be addressed to: Dean of the Graduate School of Engineering Osaka City University 3-3-138, Sugimoto, Sumiyoshi-ku Osaka 558-8585, Japan Editors Kenji KATOH Ayumu፧ SUGITA Daisuke MIYAZAKI Hideki AZUMA Tatsuhiko KIUCHI Minako NABESHIMA 1202-0321ޓᄢ㒋Ꮢ┙ᄢቇޓᎿቇㇱ Ꮏቇㇱ⧷ᢥ♿ⷐ81.㧔㧕ޓ MEMOIRS OF THE FACULTY OF ENGINEERING OSAKA CITY UNIVERSITY VOL. 52 DECEMBER 2011 CONTENTS Regular Articles ····································································································· 1 Mechanical and Physical Engineering Applied Mathematics Energy Estimates for Regularly Hyperbolic Operators with Non-classical Symbols Shigeo TARAMA ··········································································································· 1 Physical Electronics and Informatics Information and Communication Engineering An Improved Pedestrian Detection Method for DSS (Driver Supporting System) Using an Attitude Measurement Unit Yukie HASHIMOTO, THI THI ZIN, Takashi TORIU and Hiromitsu HAMA ·················· 15 Urban Engineering Civil Engineering A Numerical Model to Simulate the Time Dependent Behavior of Rock Kenichi NAKAOKA, Koji HATA, Hiroaki KITOH and Yujing JIANG ························· 21 Evaluation of the Bolt Axial Force in High Strength Bolted Joints by Using Strain Gauges Chao PAN, Takashi YAMAGUCHI and Yasuo SUZUKI ·············································· 37 Finite Element Analysis on the Mechanical Behavior of High Strength Bolted Friction Type Joints Considering Fluctuation of Bolt Axial Forces Xue PENG and Takashi YAMAGUCHI ······································································· 43 An Analysis on Influence of Regional and Individual Conditions to Intended Needs and Actual Demands of Bus Service Naoki OKAMOTO and Yasuo HINO ·········································································· 49 Waterfront Functions Required from a Ship Transport Perspective Yasumasa FUKUSHIMA and Takashi UCHIDA ························································ 55 Abstracts of Papers Published in Other Journals ································································· 63 Mechanical and Physical Engineering Mechanical Engineering ···························································································· 65 Intelligent Materials Engineering ··············································································· 75 Machine Workshop ····································································································· 83 Physical Electronics and Informatics Electrical Engineering ································································································ 85 Applied Physics ·········································································································· 87 Information and Communication Engineering ··························································· 97 Applied Chemistry and Bioengineering Applied Chemistry ······································································································ 99 Bioengineering ··········································································································· 109 Urban Engineering Architecture and Building Engineering ······································································· 119 Civil Engineering ······································································································· 123 Environmental Urban Engineering ············································································· 141 ISSN 0078－6659 MEMOIRS OF THE FACULTY OF ENGINEERING OSAKA CITY UNIVERSITY ―――――――― VOL. 52 ―――――――― DECEMBER 2011 PUBLISHED BY THE GRADUATE SCHOOL OF ENGINEERING OSAKA CITY UNIVERSITY Energy estimates for regularly hyperbolic operators with non-classical symbols Shigeo TARAMA∗ (Received September 30, 2011) Synopsis m Energy estimates for regularly hyperbolic operators with S1,2/3 class symbols are drawn. KEYWORDS: energy estimates, regularly hyperbolic 1. Introduction Consider the wave operator with an oscillating coefficient L u = ∂2u a2(t)∂2u =0 ε t − ε x with aε(t)=a(t/ε) . Here a(t) is a positive and periodic smooth function and ε is a small positive parameter. For the standard energy E(t)= u 2 + u 2 of a solution u, we have t x C t E(t) e ε E(0). ≤ ˆ d2 2 2 In order to improve the above estimate, we consider the decomposition of Lε = dt2 + aε(t)ξ : d i d i ( iaε(t)ξ + a0(t) a1(t))( iaε(t)ξ a0(t) a1(t)) = Lˆε + R (t, ξ) (1) dt ± ± ξ dt ∓ − ∓ ξ ± aε (t) 1 2 with a0(t)=− , a1(t)= (a0 (t)+a0(t) ) and 2aε(t) 2aε(t) 2 i a1(t) R (t, ξ)= (2a1(t)a0(t)+a1 (t)) + . ± ∓ξ ξ2 Then we see that 1/2 d i uˆ = aε− (t)( iaε(t)ξ a0(t) a1(t))ˆu ± dt ∓ − ∓ ξ satisfies d i 1/2 ˆ ( iaε(t)ξ a1(t))ˆu = aε− (t)Lεuˆ + R (t, ξ)ˆu . (2) dt ± ± ξ ± ± ± We define the energy e1(t) by 1 2 2 1 d 2 a1 2 e1(t)= ( uˆ+ + uˆ )= ( uˆ a0uˆ + (aεξ + )ˆu . 2 | | | −| aε |dt − | | ξ | Note that C C 1 1 a0(t) , a1(t) , and R (t, ξ) C( + ). | |≤ ε | |≤ε2 | ± |≤ ξ ε3 ξ 2ε4 | | | | Then, when ξ ε is large, we see C 1e (t) ( uˆ 2 + ξ2 uˆ 2) Ce (t) with some C>0. Hence from | | − 1 ≤ | t| | | ≤ 1 (2) follows that a solutionu ˆ of Lˆ uˆ = 0 satisfies, when ξ ε is large, ε | | d C e (t) e (t), dt 1 ≤ ξ 2ε3 1 | | ∗Professor, Laboratory of Applied Mathematics －1－ from which we obtain Ct e (t) e ξ 2ε3 e (0), (t 0). 1 ≤ | | 1 ≥ 2 2 2 Hence, the high frequency energy Eε(t)= 3/2 ( uˆt + ξ uˆ ) dξ with large C of a solution ξ Cε− | | | | L u = 0, has the estimate | |≥ ε E (t) C eC2tE (0) (t 0) ε ≤ 1 ε ≥ (see the related results 1),2)). In order to obtain a similar estimate for the regularly hyperbolic operator m t x m j α P u = D u + a ( , )D − D u, ε t j,α ε ε t x 1 j m, α j ≤ ≤| |≤ we are led to study the property of the operator P in the case where ε ξ 2/3 is large. ε | | Noting, if ε 1 C ξ 2/3, − ≤ | | k α t x (k+ α ) (k+ α ) 2 ∂ ∂ a ( , ) C ε− | | C ξ | | 3 , | t x j ε ε |≤ 1 ≤ 2| | we consider, in this paper, the regularly hyperbolic operator P whose symbol p(t, x, τ, ξ) is given by m m m j p(t, x, τ, ξ)=τ + aj(t, x, ξ)τ − j=1 with real aj(t, x, ξ) satisfying k α β j β +(k+ α ) 2 ∂ ∂ ∂ a (t, x, ξ) C ξ −| | | | 3 | t x ξ j |≤ j that is, aj in S1,2/3 where ξ = (1 + ξ 2)1/2. | | More precisely, for given T>0, we denote by Sk ([0,T]) the set of symbols a(t, x, ξ) C ([0,T] ρ,δ ∈ ∞ × Rn Rn) satisfying for any non-negative integer l and any multi-index α, β Nn, where N = × ∈ 0, 1, 2,... , { } l α β k ρ β +δ(l+ α ) ∂ ∂ ∂ a(t, x, ξ) C ξ − | | | | | t x ξ |≤ l,α,β on [0,T] Rn Rn. × × Let P be an operator on [0,T] Rn: × m m m j Pu = Dt u + Aj(t, x, Dx)Dt − u j=1 whose symbol p(t, x, τ, ξ)=τ m + m a (t, x, ξ)τ m j with real symbols a (t, x, ξ) Sj ([0,T]) has j=1 j − j ∈ 1,2/3 a factorization : if ξ R with some R > 0, | |≥ 0 0 m p(t, x, τ, ξ)= (τ λ (t, x, ξ)) (3) − j j=1 with real λj(t, x, ξ) satisfying, with some δ>0, n n λj(t, x, ξ) λk(t, x, ξ) δ ξ (t, x, ξ) [0,T] R R with ξ R0 (4) | − |≥ ∈ × × | |≥ for any j, k I with j =k. ∈ －2－ Here I = 1, 2,...,m and D = 1 ∂ . For a symbol a(t, x, ξ) Sj ([0,T]), we define the { } t i t ∈ 1,2/3 operator A(t, x, Dx) by 1 i(x y)ξ A(t, x, D )u = e − a(t, x, ξ)u(y) dydξ x (2π)n l l j and for q(t, x, τ, ξ)= j=0 aj(t, x, ξ)τ − l l j Q(t, x, Dt,Dx)= Aj(t, x, Dx)Dt− . j=0 We show the following energy estimate for P . Theorem 1. We assume that the opretator P satisfies the above mentioned conditions (3) and (4). Then we have the following: m 1 − j α Dt Dx u(t, ) (m 1)/2 · − − ≤ j=0 α m 1 j | |≤− − m 1 t − j α C( Dt Dx u(0, ) (m 1)/2 + (P + Psb)u(s, ) (m 1)/2 ds) (5) · − − · − − j=0 α m 1 j 0 | |≤− − n for any u(t, x) C∞([0,T] R ) that is rapidly decreasing with respect to x, where Psb is the operator ∈ × 1 n whose symbol psb(t, x, τ, ξ) is given by psb(t, x, τ, ξ)= 2i µ=0 ∂xµ ∂ξµ p(t, x, τ, ξ) with x0 = t, ξ0 = τ. Here D = 1 ∂ and Dα = Dα1 Dαn . The norm stands for the standard Sobolev norm xj i xj x x1 ··· xn ·σ given by u( ) 2 = (1 + ξ 2)σ uˆ(ξ) 2 dξ with the Fourier transformu ˆ(ξ) of u(x). · σ | | | | Remark 1. The differential operator whose principal symbol has only real roots satisfying (3) and (4) is called the regularly hyperbolic operator. Remark 2. We remark that the above estimate (5) dose not necessarily imply the wellposedness of the Cauchy problem for P + Psb. The application of the above estimate to the operator with oscillating coefficients will be discussed in a forthcoming paper. If m = 1, Theorem 1 is evident. We assume m 2 from now on. In order to draw the estimate ≥ (5), we use the decomposition of P similar to (1). For the simplicity of expression, we use the Weyl calculus of pseudodifferential operators. For the properties of such calculus, refer to the section 14 in 3) l l j Chapter 7 of Taylor’s book . For symbols a(t, x, ξ) and q = j=1 aj(t, x, ξ)τ − , we define operators Aw and Qw by w 1 i(x y)ξ x + y A (t, x, D )u = e − a(t, ,ξ)u(y) dydξ (6) x (2π)n 2 and l w 1 i((t s)τ+(x y)ξ) t + s x + y l j Q (t, x, D ,D )u = e − − a ( , ,ξ)τ − u(s, y) dsdydτdξ t x (2π)n+1 j 2 2 j=0 n n w with some appropriately extended aj(t, x, ξ) on R R R .
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