2016年量纲分析谈庆明教授答案修改版

kyReokXy!kyReRkkdRXL/TL/TkX.BK2MbBQMJ2``BK@q2#bi2`.BK2MbBQMURVUkVUjVU9VU8VUeVUdVU3VRURVRBbiQ`vQ7lMBibM/.BK2MbBQMHMHvbBb-CQ?Ma+?mHKM-J`+?Rd-kyRyRkyReokXyUkVUjVRe3dF=maU9VU8V6QM+2MBtRdeRUeVR3ddUdVR33yRNyyU3VR3Neob+?vΠRNRR_B#Qm+?@BMbFvRNR91Xm+FBM;?KΠUNVjX/BK2MbBQM*:aRNdRaAdddLMTIΘNJ/BKA=LαMβTγIδΘεNζJηαβγXXX/BKV=LT−1/BKa=LT−2/BK2MbBQMkkyReokXyL/TL/T9Xρpv=f(ρ,p)ρpv[v]=LTƐ1[ρ]=MLƐ3[p]=MLƐ1TƐ2!p/ρvv!p/ρ=f(1,1)=*QMbi=⇒v=C!p/ρC≈1.28XhgvtURVhhgt=f(h,g)=⇒t!h/g=*QMbi.v=f(h,g)=⇒v√gh=*QMbi.UkVtvhtgv=f(t,g)=⇒vgt=*QMbi.h=f(t,g)=⇒hgt2=*QMbi.jkyReokXyUjVvhtvgh=f(v,g)=⇒hv2/g=*QMbi.t=f(v,g)=⇒tv/g=*QMbi.eXÇÇÇTmlgαT=f(m,l,g,α)mlgTlgf−2TlgUfV1/2T!l/g=f(1,1,1,α)=f1(α)ÇT√lÇTg−1/2ÇTmÇT/!l/gαdXmkmkmkωω=f(m,k)=⇒ω!k/m=*QMbi.9kyReokXy3XΠÇÇÇΠNa1a2XXXanaaa=f(a1,a2,...,an)kn−kΠ=g(Π1,Π2,...,Πn−k)ΠÇΠÇΠÇggΠNXΠnn−1kn−kn−kRyXρghvv=f(ρ,ρ0,g,h)ρghv√gh=f(1,ρ0ρ,1,1)ρ0/ρv√gh=*QMbi.8kyReokXyRRXαhρgQQ=f(ρ,g,h,α)[Q]=M/T[q]=M/L3[g]=L/T2[h]=L[b]=LQρg1/2h5/2=f(α)=⇒Q=ρg1/2h5/2·f(α)RkXρµldU∆p/l∆pl=f(d,U,ρ,µ)9jdUρ∆pρU2·dl=f(µρUd)=f(1_2)ττ=∆p·πd2/4πd·l=∆p·d4lτcd=τρU2/2=∆pρU2/2·d4l=g(_2)cd=g(_2)=+QMbi_2_2cdRjXR9l′l,...,α,_2#m=l′l,...,α,_2#pekyReokXyWρv2l2#m=Wρv2l2#p=⇒WmWp=(ρv2l2)m(ρv2l2)pÇl′l,...#m=l′l,...#pαm=αpÇ(_2)m=(_2)p(µ/ρ)m̸=(µ/ρ)p(vl)m(vl)p=(µ/ρ)m(µ/ρ)pWmWp=(ρv2l2)m(ρv2l2)p=(µ2/ρ)m(µ2/ρ)p25◦*(vl)m(vl)p=(µ/ρ)m(µ/ρ)p=(8.9×10−4S·b)/(997F;/Kj)(18.1×10−6S·b)/(1.1691F;/Kj)=5.766×10−2α1=vm/vpα2=lm/lpα1α2=+QMbiα1α2=+QMbivp70K/bδp∼0.15K=⇒vpδp∼10Kk/b(vδ)m=(vδ)p∈[10,15]Kk/b∼2KRfdδm∼0.3Kvm(10/15)/0.3∼2.2K/bR9Xll′XXXvαρµWW=f(l,l′,...,v,α,ρ,µ)lvρWρv2l2=fl′l,...,α,_2#_2=ρvlµ=⇒W=ρv2l2·fl′l,...,α,_2#R8XµrvvfgF=f(µ,v,r)dkyReokXyrvµFµvr=+QMbi=⇒F=µvr·+QMbiρsρvf=f(ρs−ρ,µ,g,r)µrρs−ρvfµ/[(ρs−ρ)d]=f(1,1,1,gµ2/[(ρs−ρ)2d3])=⇒vf=µ(ρs−ρ)d·f(g(ρs−ρ)2d3µ2)Fµvfvf=+QMbi·ρs−ρµd2gaiQF2bvf=118·ρs−ρµd2gReXfTsTs=+QMbi·ρR3f2ρRTsfTs=f(ρ,R,f)⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩[R]=L[Ts]=M·T−2[ρ]=M·L−3[f]=T−1ρRfTsρR3f2=*QMbi.=⇒Ts=+QMbi·ρR3f23kyReokXyRdXEI/4w/x4+q(x)=0⎧⎪⎨⎪⎩w=/w/x=0(w=0)w=/w/x=0(w=l)q(x)qmlqww=f(x,l,EI,qm,lq)lEIwl=fxl,qml3EI,lql#R3Xq=ρAARdw=f(x,l,EI,q)=⇒wl=fxl,ql3EI#x=lwKtwKt=l·fql3EI#wKt∝1/EwKt=+QMbi·ql4EIRNXRdqml3/EIlq/lqml3EI#m=qml3EI#pNkyReokXylql#m=lql#pEII=h4/12I=(h4o−h4i)/12(qml3Eh4)m=(qml3E(h4o−h4i))mkyXÇhρghρgh=O(Σ)Çρghρgh=O(p+`)Çρgh/YkRXwCl@9yyÇ9yy;XiQMÇR8y;Ç9X8KÇRX8KƓRXkKƓRX8KRv2/RhGC@8yy500;@ieK8K8XN3i9NXN3ikkkX3RfKBMU9yy;RRdXyfKBMVkkXρEνlω=f(l,l′,l′′,...,ρ,E,ν)RykyReokXylρEωl!E/ρ=f(l′l,l′′l,...,ν)kjXλEρνc=f(ρ,E,ν,λ)ρEλc!E/ρ=f(ν)=⇒c=*Eρ·f(ν)λλk9XkjRc=f(ρ,E,ν,λ,R)ρERc!E/ρ=fν,λR#=⇒c=*Eρ·fν,λR#R≪λc!E/ρ=1−π2ν2Rλ#2RRR/≪1/(πν)c→!E/ρk8Xk9Rhc=*Eρ·fν,λh#RRkyReokXykeXσydσys=1+k˙εnσydσysσyd/σys˙εk,nkdX⎧⎪⎨⎪⎩σ=Eε(ε!YE)σ=Kεn(εYE)H=Fπa2aRH=f(E,ν,Y,n,R,h)hYhHY=fEY,ν,n,Rh#RhHY=fYE,ν,n,θ#n=0Y/E0.4k3XRjRjRkkyReokXykNXRNjyXRk_2kNJ=f(A/A∗)jRXURVρ=7.8×103F;/Kjc=0.5×103L·K/F;·Kλ=50L/b·EE=2.1×1011F;/bk·K1Kt?.+.≈ρcl2λ=7.8×103×0.5×103×1250b=0.78×105bt2.r.≈l!E/ρ=1!2.1×1011/(7.8×103)b=1.9×10−4b=⇒t2.r.t?.+.=1.9×10−40.78×105≈2×10−9UkVρ=2.7×103F;/Kjc=0.88×103L·K/F;·Kλ=237L/b·EE=7×1010F;/bk·K1Kt?.+.≈ρcl2λ=2.7×103×0.88×103×12237b=1.0×104bt2.r.≈l!E/ρ=1!7×1010/(2.7×103)b=2×10−4b=⇒t2.r.t?.+.=2×10−41.0×104≈2×10−8UjVρ=1.738×103F;/Kjc=1.0×103L·K/F;·Kλ=156L/b·EE=4.5×1010F;/bk·K1Kt?.+.≈ρcl2λ=1.738×103×1.0×103×12156b=1.1×104bt2.r.≈l!E/ρ=1!4.5×1010/(1.738×103)b=2×10−4b=⇒t2.r.t?.+.=2×10−41.1×104≈1.8×10−8RjkyReokXyjkXaBHivbQBHbkρ=1.75×103F;/Kjk=10−4K/bE=2.1×107F;/bk·KJ=11K.`+vv=kJ=−k/H/skJtb.T.≈lv=lkJ=110−4b=104bt2.r.≈l!E/ρ=1!2.1×107/(1.75×103)b=9.13×10−3b=⇒t2.r.tb.T.=9.13×10−3104≈9.13×10−7jjXRe`MQH/6`Q?MLQ`#2`i_Qi?.vMKB+bQ7.`QTH2ibk?iiT,ffrrrXb2}M/BXQ`;f7Q`mKf}H2bfTT+nbQBHnT`QT2`iB2bndR3XT/7R9