B l 4 P A 1 DYNAMICS OF RECIPROCATING ENGINES

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Nickolas Lawrence
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1 DYNMIS OF REIROTING ENGINES This chapte studies the dnaics of a side cank echaniss in an anatica wa. This is an eape fo the anatica appoach of soution instead of the gaphica acceeations and foce anases. The gas equations and odes fo cobustion is not a concen of this chapte. essue intake copession powe ehaust ank ange Fig: Indicato diaga showing the pessue esus cank otation. G I, G 1 ω I, Fig:Side cank echanis Loop cosue equation can be: sin sin (1) cos + cos () fo tigonoetic identities sin + cos 1 sin 1 cos () fo fist equation

2 sin sin () Substitute this equation into and then esuting equation into () cos + 1 sin (5) Dnaic anasis of ecipocating engines was done in ate 1800 s and b that tie etensie cacuations had to be aoided. So thee ae an appoiations in the anasis to sipif the aithetic. In equation (5) squae oot te can be epaced b sipest epession. Tao seies epansion of squae oot te, fist two te incuded is as foows: 1 sin 1 sin (6) Squaing is aso an aithetica difficut pocess: 1 cos sin (7) Substituting equation 6 and 7 into 5 1 cos cos cos ωt Equation (8) defines the dispaceent of the side. Veocit and acceeation epessions ae b successie diffeentiation of this equation with espect to tie. If we assue that the angua eocit of the cank is constant then eocit and acceeation of the side becoe: ω & ω sinωt sin ωt ω sinωt sin ωt (9) ω & ω ω cos ωt ω cos ωt (10) In dnaic foce anasis, we put inetia and etena foces on top of eisting echanis and then soe statica. Unde the action of etena and inetia foces, too an foces eist on the echanis hence we use supeposition (8) Gas foce: ssue on gas foce eists on the echanis and cacuate the toque on the cank b the gas foce. Foces eated with gas foce wi be denoted b a singe pie.

3 Link is two foces ebe. Link is thee foces ebe. F' F' + F'1 T' F' F' F' F' F' F'1 τ ω F' Taking the oent about, on foce of F ceates oent τ ' F' (11) We aead know, et us die epession fo F using foce pogon: F' tan F' tan sin F ' (1) cos

4 Intoducing sin sin and cos 1 sin into equation 1, 1 F ' sin (1) 1 sin Tao seies epansion of squae oot te becoes; 1 1 sin 1+ sin Substituting this epession into 1 + F ' sin 1 sin () Substituting equation 8 and into 11 + τ ' + + cos ωt sin 1 sin Negecting the highe ode of atio afte necessa anipuation τ ' sin 1+ cos (15) Equiaent asses: To obtain acceeation of thid ink in agebaic epession is a aboious task. fte finding acceeation of the thid ink and putting the inetia foce on cente of ass of thid ink, doing foce anasis is aso aboious. To futhe sipif the pobe, an equiaent ass appoach is bought. In equiaent ass sste pobe, we geneate a ode which has two point asses athe than one (Fig). One of the asses wi be at point. The othe is at. G I G The ass of the ode and ass of the actua ink shoud be equa.

5 (16) + Mass cente of the ode and ass cente of the actua ink shoud be at the sae pace. (17) Mass oent of inetia of the ode and the actua ink shoud be sae. I + (18) Soing these thee equation fo;,, and + and I + G at that point second ass shoud be ocated. It is aso known as cente of pecussion. ente of pecussion is at point whee thee is no inetia oent. On an inetia foce eists. In a connecting od, whee ass cente is neae to point and distance between ass cente and point is e itte., the cente of pecussion is soewhee in between cente of ass and. So, is nea coinciding with point. So, we do not need point and so and shoud then be paced at point ( ). and ae equied, so we hae equation but on two unknowns. One of the equations wi be edundant which one to assue edundant depends onto decision of the designe. Inetia Foces: Using equiaent ass concept side cank echanis can be coneted into two ass sste which ae ocated at and.

6 These equations satisf the equait of ass and ass cente of the coupe ink. 1 G I, G I, G Second ink ass aount assued to concentated can be found b: G This equation satisfies the equait of ass and ass cente fo the cank. Then tota asses at and ae; + and. + -a 1 ωt -a osition ecto defining point ; R cos ωt i + sinωt j V ω sin ωt i + ω j a [ α sinωt ω ] i + [ α ω sinωt] j ω i ω sinωt j fo a constant cank speed. a

7 Inetia foce on a ω i + ω sinωt j a is iateia fo cank shaft toque point of iew. ecause this inetia foce is diected adia and so does not poduce an toque on the cank. a && ω i fo a constant cank speed. Inetia foce at is a. a ω + cos ωt i X and Y coponent of tota inetia foces fo oing pats can be sued up; F F ( + ) ω + ω cos ωt ω sinωt -a F'' F'' -a F''1 τ ω -a F'' Foces eated with gas foce wi be denoted b a singe pie. Link is two foces ebe. Link is thee foces ebe. Taking the oent about, on foce of F ceates oent τ '' F (**) '' We aead know, et us die epession fo F using foce pogon: F'' F'' tan F'' ( && ) tan ( a ) ( && )

8 Substituting, &, and tan into (**) + τ '' + + cos ωt ω + cos ωt sinωt 1 sin ωt Negecting the highe ode of atio and then using identities sinωt cos ωt sin ωt sin ωt sinωt sin ωt τ ω '' sin ω t sin ω t sin ω t (15) Eape 1.0 ete ong ink has a ass of 10 kg and centoida ass oent of inetia 0.5 kg.. Its ass cente is at its id point. ut this bod into an equiaent two point ass sste, one of which is ocated at. Find the ocation of the cente of pecussion, whee wi the othe point ass be ocated? Find the adius of gation of the ink. G G Oigina bod Mode bod + I G + Soing these thee equation fo;,, and +, I, and + G I G at that point second ass shoud be ocated. It is aso known as 0. 5* 10 1 cente of pecussion. Radius of gation is k I G

9 Eape In the figue, an eiptic tae echanis is shown with appopiate diensions, woking in the hoizonta pane. ut the coupe ink into an equiaent fo as two point asses concentated at points and, on the basis of equiaenc of assand ocation of ass cente. Then cacuate the actuation foce equied on the side at, paae to the sidewa if point is oing ightwad with constant eocit of 1 /sec. 10 c, G G 5 c, 60 o 0.5 kg, 0.8 kg, I 0.01 kg.. V V + V V / s V? b V? to V G V V/ Scae: c. stands fo 1 /sec V 1 / s V / s a a a a a a a? + a n + a t b a n a a n V / s fo to a t? to a / s t a

10 Equiaenc of asses + Equiaenc of ass cente G G soing these two equation fo, and G G 0.05* * kg 0. kg kg kg 9 9 D ebet foce a 0.9* N 90 0 a N G F

11 1.856 N F 1 + G F F F 0; F F1 0 F F F 1 1 F ; F 0 F N M F 8 N 0 0; 1.856*0.1*cos60 + F N 1 *0.1*sin 60 0

12 Eape In the figue, an eiptic tae echanis is shown with appopiate diensions, woking in the hoizonta pane. ut the coupe ink into an equiaent fo as two point asses concentated at points and, on the basis of equiaenc of ass and ocation of ass cente. Then cacuate the actuation foce using agebaic appoach equied on the side at, paae to the sidewa if point is oing ightwad with constant eocit of 1 /sec. 10 c, G G 5 c, 60 o 0.5 kg, 0.8 kg, I 0.01 kg.. O O + (1) 0 + * cos(180 ) * cos ais coponents () 0 + *sin(180 ) *sin ais coponents () & 1 / sec Fo definition of the question side is oing with constant eocit. && 0 & * & sin () & * & cos (5) & * && sin * & cos (6) & *&& cos * & sin (7) O 180 Fo equation fo & and substituting into 5, & & & & * *sin *sin cos soing equation 6 fo & and substituting into 7 & & cos sin 0 * & sin * & cos & & cos sin & cos sin * cos * & * & & sin * && sin sin sin D ebet s foce * a * & 90 sin 0

13 F 1 * * sin & + O F 180 F F 0; F F1 0 F F F 1 F 1 * & 0; sin 0 F * sin & + F * & 0; * * cos + F sin * & cos sin M 1 * *sin 0

Discretizing the 3-D Schrödinger equation for a Central Potential

Discretizing the 3-D Schrödinger equation for a Central Potential Discetizing the 3-D Schödinge equation fo a Centa Potentia By now, you ae faiia with the Discete Schodinge Equation fo one Catesian diension. We wi now conside odifying it to hande poa diensions fo a centa