機械原理第六章機械的平衡

機械原理第六章機械的平衡第六章機械的平衡Chapter6.BalancingofMachinery

1.Purposes§6-1Introduction

DynamicpressinkinematicpairFrictionandinnerstressinlinkefficiencyandlife-spanInertiaforce(torque)

Compelledoscillation

Thepurposeofmechanicalbalancingistoclearupordecreasethebadeffectbybalancingthecomponent’sunbalancedinertia.Thebalanceofcomponentrotatingaboutafixedaxis

Rotor(轉(zhuǎn)子):Partsconstrainedtorotateaboutafixedaxis.2.Contents(1)Balancingofrigidrotor剛性轉(zhuǎn)子的平衡

staticbalancing(靜平衡)dynamicbalancing(動平衡)(2)Balancingofflexiblerotor繞性轉(zhuǎn)子的平衡

Balancingofmechanisms

rigidrotorflexiblerotormechanism(avi)1.Phenomenaofstaticimbalance

§

6-2Calculationforstaticbalancingofarigidrotor

Ifmasscenterofrotordoesn’tcoincidewiththeaxisofrotation,theireccentricmasswillleadtocentrifugalinertiaforce(離心力)whenrotating,andcausesanadditionaldynamicpress(附加動壓力)inthelinkage.2.Geometriccondition

B/D≤1/5BDm3m1m2m3m1m23.TheoryofstaticbalancingF1F2F3m1m2m3r2r1r3centrifugalforces(離心力）oftheunbalancedmasses（偏心質(zhì)量）：F1=m1r1w2F2=m2r2w2F3=m3r3w2If,F1+F2

+F3≠0FPmPThen,imbalanceF1F2F3m1m2m3r2r1r3

F1+F2

+F3+FP=

0FPmPSomecounterweight質(zhì)量點(mp)canbeaddedtotherotortobalanceitscentrifugalforce.Tobalance:Fp=mprpw2m1r1w2+m2r2w2+m3r3w2+mPrPw2=0m1r1

+m2r2

+m3r3

+mPrP

=0miri----mass-radiusproduct(質(zhì)徑積)Conclusion：Requirementforstaticbalance:∑m1r1=0

AddRemoveConclusion:Abalancecanbeachievedbyaddingorremovingabalancemassinthesameplane.resolution：A.GraphicalmethodF1F2F3m1m2m3r2r1r3FPmPWPW1W2W3Scale(比例尺)：μW=(kgm/mm)miriWiWi=mirim1r1w2+m2r2w2+m3r3w2+mPrPw2=0B.Analyticalmethod

mx1rx1+mx2rx2

+mx3rx3

+mxPrxP

=0my1ry1+my2ry2

+my3ry3

+myPryP

=0F1F2F3m1m2m3r2r1r3FPmPEXAMPLEGiven:ThesystemshowninFIGhasthefollowingdata:m1=1.2kg

R1=1.135m

θ1=113.4°m2=1.8kg

R2=0.822m

θ2=48.8°Find:Themass-radiusproductionanditsangularlocationneededtostaticallybalancethesystem.Solution:Resolvethepositionvectorsintoxycomponents:

R1=1.135m

θ1=113.4°;R1x=-0.451,R1y=-1.042R2=0.822m

θ2=48.8°;R2x=+0.541,R2y=0.618SolveequationsmbRbx=-m1R1x-m2R2x=-(1.2)(-0.451)-(1.8)(0.541)=-0.433

mbRby=-m1R1y-m2R2ySolveequationsSolution:Themass-radiusproductcanbeobtainedwithavarietyofshapes.WhenRb=0.806matrequiredangleof259.6°,themassforthiscounterweightdesignisthen:

mb=2.042kg.m/0.806m=2.980kg

atthechosenradiusofRb=0.806m§6-3Calculationforthedynamicbalancingofarigidrotor剛性回轉(zhuǎn)體的動平衡geometriccondition

B/D>1/5

Massmaybeunevenlydistributedbothrotationallyaroundtheiraxisandalsolongitudinallyalongtheiraxis.

F1F2F3r1m1m2r2m3r32.Force-balance-condition∑Fi=0centrifugalforces

∑Mi=0centrifugalmomentF

FⅠ

FⅡ3.CorrectionplanesTocorrectdynamicimbalancerequireseitheraddingorremovingtherightamountofmassattheproperangularlocationsintwocorrectionplanesseperatedbysomedistancealongtheshaft.l1l2l3lF1F2F3r1m1m2r2m3r3F'3F"3F'1F"1F'2F"2Ⅱ

Ⅰ平衡基面平衡基面4.TheoryL1L2L3LF1F2F3r1m1m2r2m3r3Ⅱ

ⅠF2ⅠF1ⅠF3ⅠF2F1F3ⅡⅡⅡConverteachcentrifugalforcestothecorrectionplaneⅠandⅡ.ThatisF1,F2andF3canbereplacedbyF1Ⅰ,F2Ⅰ,F3ⅠandF1Ⅱ,F2Ⅱ

,F3Ⅱ

spatialforcesystem2planarforcesystems平衡原理：將集中質(zhì)量點所產(chǎn)生的離心力F向兩個平衡基面上分解，得到兩個分力F1和F2；合力F對系統(tǒng)的影響可以完全有兩分力F1、

F2對系統(tǒng)的影響所代替；F-F'-F"L1

L2

LⅠⅡF2F1

在平衡基面上分別對兩個分力F1、

F2進行平衡，得平衡力F'和F"，從而完成對集中質(zhì)量點的平衡。FF'F"

將力F平行分解到兩個平衡基面上，得F1和F2:L1

L2

LⅠⅡF2

=

FL1L(4)F1=

FL2L(3)F2F1F=F1+F2(1)F1L1=F2L2(2)l1l2l3lF1F2F3r1m1m2r2m3r3F'3F"3F'1F"1F'2F"2Ⅱ

Ⅰ平衡基面平衡基面L1L2L3LF1F2F3r1m1m2r2m3r3F'2F'1F'3F"3F"1F"2Ⅱ

ⅠF'1

=

F1

L-L1LF"1

=

F1L1LF'2

=

F2

L-L2LF"2

=

F2L2LLF'3

=

F3

L-L3LF"3

=

F3L3m'r'F'F'1+F'2+F'3+F'=0F"1+F"2+F"3+F"=0從而求得m'r'和m"r"。m"F"r"L1L2L3LF1F2F3r1m1m2r2m3r3Ⅱ

ⅠF2ⅠF1ⅠF3ⅠF2F1F3ⅡⅡⅡ步驟：(1)分別將各回轉(zhuǎn)平面上集中質(zhì)量點mi所產(chǎn)生的慣性力Fi(或質(zhì)徑積、重徑積)向兩個平衡基面上分解，得到F'i和F"i

。(2)分別在兩個平衡基面上用靜平衡的方法求解平衡質(zhì)量點的質(zhì)徑積miri(或重徑積)。1.StaticBalancingExperiment