The Popeties of Moel ocket Boy Tube Tansitions Date: Septembe 6, 000 Pepae By: J.. Bohm NA 7808 CA S680 ev: (June, 005)
.0 Intouction When esigning moel ockets, esignes often choose to incopoate iffeent iamete boy tubes into the moels esign. Sometimes this is one to accommoate special featues, such as payloa o camea bays, but often it is one just to make the esign moe inteesting an appealing. Component manufactues usually offe stana tansition pats that pemit stana boy tubes of iffeent iametes to be mate, esolving the poblem simply an conveniently. But esignes an builes ae usually left to thei own evices when the mating poblem involves boy tube iametes that ae not suppote by the manufactues, o pehaps equie a non-stana tansition length. Fotunately, the poblem of the non-stana tansition can be solve with a little geomety, an this pape pesents some solutions.. Backgoun Geometically, boy tube tansitions ae a physical eivative of a cone. If one can imagine a cone that has some potion of its top cut off, then the emaine of the cone woul epesent a tansition boy, gaually eucing in iamete fom its base to the point whee the top of the cone was cut off, as shown in Figue, below: Figue : Tansition Boy This tansition boy can be use to mate two iffeent boy tubes of a moel ocket when the iametes of the base an top of the tansition match the iametes of the two boy tubes. The shapness o steepness of the tansition is popotional to the length of the tansition: the longe the length of the tansition the moe gaual the euction in iamete. Once the iametes of the boy tubes that ae to be mate ae known, an the esie length of the tansition is ecie, it shoul be possible to ceate a soli tansition out of a lightweight mateial such as balsa. Ceating a soli tansition is elatively staightfowa if access to the ight tools is available o if one is willing Tansitions Septembe 6, 000
to spen the money to have one mae by a component manufactue, e.g.: BMS. But most often, a soli pat isn t necessay. The most common pactice is to ceate a tansition shou, which when constucte, has the same shape an seves the same pupose as the soli tansition but is nothing moe than a conically-eive coveing of the junction between the two couple boy tubes. The next section efines an explains the geomety behin tansition shous an then eives the basic equations necessay to calculate the paametes of the pat. Tansitions Septembe 6, 000
.0 Tansition Shous. Tansition Shou Geomety Taking the ecision to make a tansition shou means that the buile has esigne some othe iect means to solily couple the two boy tubes. This is usually achieve with some sot of coupling tube/centeing ing aangement. The main point is that the buile is not elying on the tansition shou to povie stength to the aifame: instea, the tansition shou is being use to fai the joint between the two issimila tubes an will only be stong enough to potect itself. Usually, the shou is mae fom pape o ca stock (o even light gauge plastic sheet o fibeglass) epening on the size of the moel. Sometimes unelying einfocement ibs mae of balsa ae installe to suppot an futhe stengthen the shou. The fist step involve in ceating a tansition shou consists of efining the imensions the shou will equie. These values ae then plugge into a set of elations that efine the layout imensions of the shou. But befoe we o this, we must fist establish the elationship that exists between the shou s theeimensional shape an its plan-fom layout. It is the plan-fom layout that we must etemine in oe to fashion it coectly fom an essentially two-imensional mateial such as pape, caboa o plastic castock. et s evisit the geometic popeties of the tansition: ( ) ( ) Figue : Tansition Sie View In Figue, the tansition is shown in the sie view. In this shape the tansition shou is efine by:, the length of the tansition, Tansitions 3 Septembe 6, 000
, the aius of the font of the tansition,, the aius of the ea of the tansition. These imensions ae known to the esigne, as the boy tube iametes an the esie length of the tansition ae paametes that the esigne specifies fo the ocket esign. It is also appaent that an coespon to the aii of the smalle an lage boy tubes espectively. By using the Pythagoean Theoem, the shou length can be efine as ( ). If one wee to take a pai of scissos an cut along the axis of the shou, the tansition coul be lai flat on a table. In oing so, the shou woul assume the shape of a isc secto, as shown in Figue 3 below: Θ Shou ength Figue 3: The Tansition Shou in Plan Fom. The physical chaacteistics of the secto ae efine by:, the aius of the inne cicle,, the aius of the oute cicle, Θ, the secto length in egees. These ae the imensional popeties that must be compute in oe to coectly fabicate the shou.. Solving fo an Tansitions Septembe 6, 000
et s take a close look at these popeties. The ac lengths of the secto can be efine as: S Θ, with Θ expesse in aians. S Θ Since Θ is common to both ac lengths, S S Because the secto is iectly eive fom the shou, it follows that the plan fom popeties ae iectly elate to the tansition shou popeties. We can begin to efine some of these elationships as follows: S π an S π (By efinition, the ac lengths must be equal to the cicumfeences of the espective ens of the tansition). π π ; ; an By compaing the iagams in Figues an 3, it can be seen that the shou length ( ) ( ). ( ). Substituting the above fo ( ) gives: ( ) Tansitions 5 Septembe 6, 000
( ) ( ) ( ) ( ) ( ) ( ).3 Solving fo Θ ecall that: Θ S π, with Θ expesse in aians. Since 80 π, Then Θ 80 π π 360 Θ 360 ( ) ( ). Shou Dimensions Expesse in Boy Tube Diametes The foegoing esults povie a complete solution fo the imensional popeties of a tansition shou. Howeve, the expessions ae base on boy tube aius an this ceates an exta calculation step, an magin fo eo, when one uses the elations to calculate the imensions of a shou. This is so because manufactues geneally expess the imensions of boy tubes in tems of Tansitions 6 Septembe 6, 000
iamete, not aius. With a bit moe algeba, we can e-expess the equations in tems of boy tube iamete. To begin, we know that an whee is the iamete of the smalle boy tube an is the iamete of the lage boy tube. Substituting these vaiables into the solution fo gives: [ ] Solving fo the othe paametes gives: 360 80 360 Θ Tansitions 7 Septembe 6, 000
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3.0 Summay This pape emonstates how tansition shous can be eive fom the popeties of cones an povies a set of fomulae fo calculating thei imensional popeties. Expesse in tems of boy tube aius, the imensions of a tansition shou ae specifie by: ( ) ( ) ( ) ( ) Θ 360 ( ) ( ) Expesse in tems of boy tube iamete, the imensions of a tansition shou ae specifie by: ( ) ( ) ( ) ( ) Θ 360 ( ) ( ) Tansitions 9 Septembe 6, 000