Chapter 3 Basic Structural Dynamics for Impact,Shock and Explosion

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1、3BasicStructuralDynamicsforImpact,ShockandExplosion3.1GeneralIntroductionMostloadsactingonstructuresaredynamicinorigin.Theseloadscanbesuddenlyappliedorallowedtoreachfullmagnitudeafteraconsiderabledelay.Ontheotherhand,thestructureswillhavevariousdegreesoffreedomwithunclampedorclampedfreeor

2、forcedvibrations.Theseneedtobediscussedpriortotheintroductionofimpactandexplosionanalysisanddesign.3.2Single-Degree-of-FreedomSystemIfasystemisconstrainedsuchthatitcanvibrateinonlyonemodewithasin-gleco-ordinatesystem(geometriclocationofthemasseswithinthesystem),thenitisasingle-degree-of-f

3、reedomsystem.3.2.1UnclampedFreeVibrationsAmassmissuspendedbyaspringwithastiffnessk(forcenecessarytocauseunitchangeoflength).LetthemassmbedisplacedverticallyasshowninFig.3.1.Then,withgivenrestraints,F−kδST=0,(3.1)whereForW=mg,g=accelerationduetogravityandδST=staticdeflection.Themassisrelease

4、danddisplacedfromtheequilibriumposition.Thecoordinatexthendefinesthepositionofthemassmatanytimeandistakentobepositivewhenmovinginadownwarddirection.Figure3.2showsthenewpositions.4003BasicStructuralDynamicsforImpact,ShockandExplosionkkkδSTunstrainedmpositionδSTequilibriummmpositionfreebodyd

5、iagramForW=mgFig.3.1.Themassanditsequilibriumpositionequilibriumkpositionk(δST+x)kx¨xdynamicpositionmmmdynamicfreebodycompletefreebodyForW=mgFig.3.2.ThemassdisplacedfromtheequilibriumpositionMethod1:UsingNewton’sSecondLawofMotionThislawstatesthatthemagnitudeoftheaccelerationofamassispropo

6、rtionaltotheresultantforceactinguponitandhasthesamedirectionandsenseasthisforce.Thefollowingequationsareobtained:3.2Single-Degree-of-FreedomSystem401d2xm=−k(δST+x)+F,(3.2)dt2mx¨=−kxormx¨+kx=0.(3.3)Method2:EnergyMethodForaconservativesystem,thetotalenergyofthesystem(potentialenergy(PE)plus

7、kineticenergy(KE))isunchangedatalltimes.ThusdKE+PE=constant;or(KE+PE)=0,(3.4)dt12KE=mx˙,(3.5)20012PE=[F−k(δST+x)]dx=−kxdx=kx.(3.6)xx2Using(3.5)d22(mx˙+kx/2)=0.(3.7)dtHence(mx¨+kx)˙x=0ormx¨+kx=0.(3.8)Sincetheenergybalanceholdsallthetime,includingthebeginningstage,i