Class Note for MATH 331 at UMass(1)
Class Note for MATH 331 at UMass(1)
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This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Massachusetts taught by a professor in Fall. Since its upload, it has received 15 views.
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Date Created: 02/06/15
BASIC HOMOGENEOUS Recall our classical spring problem deals with modeling the position dis placement of a spring as it vibratesoscillates We start with a spring hanging from some surface7 and we hang an object of mass m on it7 which causes it to elongate L units This is what we call the equilibriumoriginal state or position Letting yt be the function which represents the displace ment from this equilibrium state at time t7 we will use the convention that upwards movements forces are negative quantities7 while downward ones are positive So yt gt 0 A spring is stretched down yt 0 A spring is at equilibrium no displacement yt lt 0 A spring is compressed up We used Newton s Law7 Ft mat7 to get a DE for yt7 since acceler ation is just y t To do this7 we added up all of the forces acting on the object at time t These consisted of gravity F97 resistancedrag F7757 and spring force That is mm Flttgt F9 M no my t my 7 W05 ML W we was 7 mt since mg kLgt So the our spring problem is modeled by the homogeneous DE my t W05 Wt 07 MO 907 MW y where m mass of the object7 y is the dampingresistance constant7 and k is the spring constant ADDITIONAL WEIGHTS AND NONHOMOGENEOUS PROBLEMS We can modify the original homogeneous problem described on the rst page by introducing additional forces in 2 ways 1 ADDITIONAL WEIGHT MODIFIED HOMOGENEOUS EQUATION Suppose that we have the setup described on page 1 for some spring prob lem But now the spring will be set into motion by throwing some additional weight W onto the object which is already hanging at equilibrium on the end of the spring Let mw g be the mass of this additional weight In keeping with the spirit of the description of the homogeneous DE on the rst page we should expect to be able to model this problem by an equation My t MW KW 07 90 Yoi M0 Yo So let s gure out which pieces change and how they do 0 Mass The new mass on the end of the spring is now M m mw o Damping The new damping constant T is probably not drastically different so there s no need to change it P y 0 Spring The new spring constant K is also unchanged since it s just a proportion of elongation and weight K k c Elongation The new elongation L is going to be different now since the additional weight would certainly increase the amountit would stretch before coming to rest But exploiting the fact that L you can gure out the new elongation amount 0 Note that yt will NOW measure displacement from an elongation of f instead of L This shifts things by i 7 L units So Y0 4f 7 L to re ect that you re starting above the new equilibrium state If you don t want to change your reference point you can use the original setup where yt measures displacement from an elongation of L by using the equivalent equation My t W05 Wt W7 y0 907 MW y Of course this is now non homogeneous but it may be easier than moving around your reference point Note that the M here is M m mw 2 EXTERNAL FORCE NONHOMOGENEOUS EQUATIONS Suppose that instead of the dropping a weight on the object there is an externaladditional force Fet which acts on the object Fet can be a function like cosine which might model some sort of wavy or cyclic force a constant which might be due to a constant wind or water current or an impulse more on this in chapter 6 Modifying the page 1 derivation we d get that this spring problem is modeled by my t W05 Wt F805 MO 907 y 0 y which is now a non homogeneous DE If you haven t already done so you can solve the homogeneous equation and use your solutions with a method from 36 or 37 to nd a particular non homogeneous solution
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