工程振动试验分析(教材)

111.19060(FFT)1.21.2.11.1mkmÄx+kx=0(1-1)(1-1)x=Xmsin­t(1-1)(k¡m­2)Xmsin­t=0=)k¡m­2=0=)­=pk=m{1{1.2­=pk=m­km1.11.2mÄx+c_x+kx=0(1-2)1.2(1-2)x=e¸t¸(1-2)m¸2e¸t+c¸e¸t+ke¸t=(m¸2+c¸+k)e¸t=0(1-3)¸2+c¸+k=0=)¸1;2=¡c§pc2¡4mk2m=¡c2m§rkmsµc2pmk¶2¡1(1-4)n=c2m;­=rkm;³=c2pmk=n­=ccccc=2pmk³ccc³1¸1;2=¡n§j­p1¡³2x=e¡nt(X1cosp1¡³2­t+X2sinp1¡³2­t)=Xe¡ntsin(p1¡³2­t+µ)(1-5)³=1¸1;2=¡n¸=¡nx=e¡nt(X1+X2t)(1-6)³1¸1;2=¡n§­p³2¡1x=X1e(¡n+­p³2¡1)t+X2e(¡n¡­p³2¡1)t(1-7)(1-6)(1-7)c=ccmkc­mk³cc{2{1nmc1.2.21.3(m1Äx1+(k1+k2)x1¡k2x2=0m2Äx2¡k2x1+k2x2=0(1-8)1.3m100m2#ÃÄx1Äx2!+k1+k2¡k2¡k2k2#Ãx1x2!=Ã00!(1-9)Ãx1x2!=ÃX1X2!sin­t(1-9)á­2m100m2#+k1+k2¡k2¡k2k2#!ÃX1X2!sin­t=Ã00!=)(k1+k2)¡m1­2¡k2¡k2k2¡m2­2#ÃX1X2!sin­t=Ã00!(1-10)¯¯¯¯¯¡­2m100m2#+k1+k2¡k2¡k2k2#¯¯¯¯¯=¯¯¯¯¯(k1+k2)¡m1­2¡k2¡k2k2¡m2­2¯¯¯¯¯=0(1-11)£(k1+k2)¡m1­2¤¡k2¡m2­2¢¡k22=m1m2­4¡(k2m1+k1m2+k2m2)­2+k1k2=0­21;2=(k2m1+k1m2+k2m2)§p(k2m1+k1m2+k2m2)2¡4m1m2k1k22m1m2=12µk2m2+k1+k2m1¶§12sµk2m2+k1+k2m1¶2¡4k1k2m1m2(1-10)ÃX1X2!1=A10@1k2k2¡­21m21A;ÃX1X2!2=A20@1k2k2¡­22m21A(1-12)A1;A2­21­22{3{1.3Ãx1x2!1=ÃX1X2!1sin­1t;Ãx1x2!2=ÃX1X2!2sin­2t(1-13)­21­22(1-13)1.3(c),(d)1.3xtx=Xmsin!tcos!t=sin(!t+¼=2)(1-14)1.3.1x=Xmsin(!t+®)(1-15)1.4x=Xm\(!t+®)(1-16)1.4®!Xm1.3.2¹x¹x=Xm(cosµ+jsinµ)=Xmejµ(1-17)µ=!t+®¹x=Xm[cos(!t+®)+jsin(!t+®)]=Xmej®ej!t(1-18){4{1x=Xmsin(!t+®)=Im(¹x)(1-19)¹x=Xmej®ej!t1.5¹x1.51.61.3.3x=Xmsin(!t+®)(1-20)Xm!®(1-20)v=_x=!Xmcos(!t+®)=!Xmsin(!t+®+¼=2)=Vmsin(!t+®0)(1-21)a=_v=Äx=¡!2Xmsin(!t+®)=!2Xmsin(!t+®+¼)=Amsin(!t+®00)(1-22)Am=!Vm=!2XmVm=Am!=!Xmmmmm/smm/s2gg1g=9800mm/s2!rad/s!=2¼1T=2¼ff1/sf=1=TTs!t+®rad1.3.48:x=Xmsin(!t+®x)v=Vmsin(!t+®v)a=Amsin(!t+®a)(1-23)!Xm=Vm=Am=!®a=®v+¼=2=®x+¼®a¡¼=2=av=®x+¼=2{5{1.4(1-23)8:x=Xm\(!t+®x)v=Vm\(!t+®v)=!Xm\(!t+®x+¼=2)a=Am\(!t+®a)=!2Xm\(!t+®x+¼)(1-24)8:x=Xmej®xej!tv=Vmej®vej!t=!Xmej(®x+¼=2)ej!ta=Amej®aej!t=!2Xmej(®x+¼)ej!t(1-25)(1-24)(1-25)!¼=2!2¼¼=2¼=21.61.41.4.1XpeakXpXp=XmXm1.71.71.71.4.2x=x(t)=x(t+T)Xrms=s1TZT0x2(t)dt(1-26)Tx=Xmsin(!t+®)Xrms=s1TZT0X2msin2(!t+®)dt=Xmp2(1-27){6{11.4.3x=x(t)=x(t+T)Xav=1TZT0jxjdt(1-28)x=Xmsin(!t+®)Xav=1TZT0jXmsin(!t+®)jdt=2TZT=20Xmsin!tdt=2Xm¼(1-29)1.4.4Xrms=¼2p2Xav=1p2Xp(1-30)1.81.8Xrms=FfXav=1FcXp(1-31)FfFcFf=1:11Fc=1:414Ff=1:156Fc=1:732Ff=1Fc=11.5(Fourier)x=x(t)=x(t+T)(1-32){7{1.6x(t)=a02+nXk=1(akcosk!0t+bksink!0t)(1-33)a0=2TZT0x(t)dt;ak=2TZT0x(t)cosk!0tdt;bk=2TZT0x(t)sink!0tdt;!0=2¼T(1-33)x(t)=X0+X1sin(!0t+®1)+X2sin(!0t+®2)+¢¢¢+Xksin(!0t+®k)+¢¢¢(1-34)X0=a02;Xk=qa2k+b2k;®k=arctanakbk!02!03!0¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢x=asin(!0t¡¼4)+a2sin2!0t1.9(b),(c)1.91.9(a)1.9(a)1.9(b)1.9(c)1.61.6.1123{8{11.6.2()1.6.31.101.1021.6.4mamaxPP¸mamax(1-35)PP=P0+P4(1-36)P0P0=k¸¸kP4=kxxP4PtP0P0mamax(1-37)amax=!2XmP0=m!2Xm(1-38)¸mk!2Xm(1-39){9{1.71.71.7.11.111.12Sv=UmVm(1-40)Vmv=Vmsin!tu=Umsin!t1.111.12¡(1-40)(1-40)S==S1S2S3¢¢¢(1-41)S1S2;S3;¢¢¢1.7.2BK20lgSN·5dBSN1.77{10{11.7.31.131.131.141.7.41.14(a)1.14(b)1.7.51.1545±45±10Hzs800Hz10¡6mms1mm10¡3gs103g1.81.8.11010s201s101s1010s100lg10¡lg1=lg100¡lg10{11{1.81.151.160011lgµ11¶=lg1¡lg1=¡1000VD=V=!A=!VV!V¡!(lgV-lg!)D!V¡45±lgD=lgV¡lg!(1-42)A!V+45±lgA=lg(!V)=lgV+lg!(1-43)1.17D-!V-!A-!V1.17V1.18V-!DAV-!VD!+45±A!¡45±lgD=lgV¡lg!;lgA=lgV+lg!DAD1A1lgV=lgD1+lg!;lgV=lgA1¡lg!{12{1A1D1+45±¡45±1.18V!DAfHzf=!=2¼()+45±¡45±()lgya(Nomograph)()1.161.8.2(dB)(bel)BB=lgPP0(1-44)(dB)dB=10lgPP0(1-45)P0PZP=FX=F(F=Z)=F2=ZdB=10lgF2=ZF20=Z=20lgFF0(1-46)FF0(RMS)dB=20lgXX0(1-47)XX0DVAXDX0D1dBD0=10¡6m10¡5m20dB10¡4m40dB1.19dB10¡6m2dBXX0X00=10X0¢dB=20lgX00X0¡20lgX0X0=20lgX00X0=20lg10=20X1020dBXX0100060dBX002X06dBXX0{13{1.8¢dB=20µlgX0X0¡lgXX0¶=20lgX0X320lgABA0B0=20lgAA0+20lgBB04dB20lgXmp2X0=20µlg1p2+lgXmX0¶=20µ¡12lg2+lgXmX0¶=¡3:01+20lgXmX03dB1.201.19dB1.20P0=2£10¡4¹Pa(1¹Pa=0:1N=m2)dBISO1967885dBdB(V)dB(mV)dB1V1mVdB(mV)dB0dB1mV6dB2mV10mV20dB0¡6dB0.5mV20lg0:51=20lg12=20lg2=¡6dB0.1mV¡20dB0.01mV¡40dB{14{222.12.1(a)2.1y=y(t)x(t)0mzz=x+y2.1(a)2.1(b)m¡(kx+c_x)=maama=Äz=Äx+Äym(Äx+Äy)+kx+c_x=0=)mÄx+kx+c_x=¡mÄy(2-1)(2-1)my(2-1)xyy=Ymsin!t(2-2)(2-1)mÄx+kx+c_x=m!2Ymsin!t=)Äx+cm_x+kmx=!2Ymsin!t(2-3)­2=km;2n=cm=)Äx+2n_x+­2x=!2Ymsin!t(2-4){15{2.2yx=Xmsin(!t¡®)(2-5)Xm=Ym(!=­)2p(1¡!2=­2)2+4n2!2=­4=Ym(!=­)2p(1¡!2=­2)2+(2³!=­)2(2-6)®=arctan2n!­2¡!2=arctan2³(!=­)1¡(!=­)2(2-7)(2-6)(2-7)³³=n­=c2pmk=ccc(2-6)XmYm(2-7)xyXm=Ym!=­(2-6)2.2(2-7)2.32.22.32.22.2.1(2-6)Xm=Ym(!=­)2p[1¡(!=­)2]2+4³2(!=­)2(2-8)!=­À1(2-8)Xm=Ym(!=­)2p(!=­)4+2³2(!=­)2=Ymp1+4³2(!=­)2(2-9)!=­À1­=!¿1³1Xm¼Ym(2-10){16{2(2-10)!=­!­(2-5)XmYmx=Xmsin(!t¡®)=Ymsin(!t¡®)(2-11)!=­À1!À­(2-12)!­­2.2.2³1³(2-4)x=Ce¡ntsin(p­2¡n2t+Á)+Xmsin(!t¡®)(2-13)³(n=­³)³22.2³(!=­)³³=0:6s0:72%³=0:62:7fn(fn=­=2¼)³=0:72:13fn32.3®10±³=0:67fn³=0:78fn2.32.4e=Blv£10¡8(V)(2-14){17{2.32.4B(T)l(cm)v(cm/s)v_x(2-14)e=Bl_x£10¡8(V)(2-15)(2-11)x=Xmsin(!t¡®)=Ymsin(!t¡®)(2-16)y=Ymsin!t=)_y=¡!Ymcos!t(2-17)(2-11)(2-15)e=Bl£10¡8[¡!Ymcos(!t¡®)]=¡Bl£10¡8!Ymcos(!t¡®)=¡Emcos(!t¡®)(2-18)Em=Bl£10¡8!Ym(2-19)(2-17)(2-18)®®(2-7)³=0:710HzHz{1mm2.5'()I0Ä'+kc'+mglsin'=0'sin'¼'I0Ä'+(kc+mgl)'=0=)Ä'+kc+mglI0'=0{18{22.5­=rkc+mglI0kcI0mlgI0Ä'+kc'=0=)Ä'+kcI0'=0­=rkcI02.42.4.1y=Ymsin!tÄy=¡!2Ymsin!t=!2Ymsin(!t+¼)(2-20)(2-6)(2-7)!­¿1(2-6)Xm=Ym³!­´2=1­2(!2Ym)(2-21)­!xXm!