Chapter Direct Method of Interpolation More Examples Mechanical Engineering
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1 Chpte 5 iect Method o Intepoltion Moe Exmples Mechnicl Engineeing Exmple Fo the pupose o shinking tunnion into hub, the eduction o dimete o tunnion sht by cooling it though tempetue chnge o is given by whee oiginl dimete in coeicient o theml expnsion t vege tempetue in/in/ F he tunnion is cooled om 8 F to 8F, giving the vege tempetue s 4F he tble o the coeicient o theml expnsion vs tempetue dt is given in ble ble heml expnsion coeicient s unction o tempetue F in/in/f empetue, heml Expnsion Coeicient,
2 5 Chpte 5 Figue heml expnsion coeicient vs tempetue I the coeicient o theml expnsion needs to be clculted t the vege tempetue o 4F, detemine the vlue o the coeicient o theml expnsion t 4F using the diect method o intepoltion nd ist ode polynomil Solution Fo ist ode polynomil intepoltion (lso clled line intepoltion), we choose the coeicient o theml expnsion given by
3 iect Method o Intepoltion Moe Exmples: Mechnicl Engineeing 5 y x, y x x, y x Figue Line intepoltion Since we wnt to ind the coeicient o theml expnsion t 4F, nd we e using ist ode polynomil, we need to choose the two dt points tht e closest to 4F tht lso bcket 4F to evlute it he two points e F nd 6F hen 6, 6 6 6, 558 gives Witing the equtions in mtix om, we hve Solving the bove two equtions gives Hence , 6 At 4F, in/in/ F
4 54 Chpte 5 Exmple Fo the pupose o shinking tunnion into hub, the eduction o dimete o tunnion sht by cooling it though tempetue chnge o is given by whee oiginl dimete in coeicient o theml expnsion t vege tempetue in/in/ F he tunnion is cooled om 8 F to 8F, giving the vege tempetue s 4F he tble o the coeicient o theml expnsion vs tempetue dt is given in ble ble heml expnsion coeicient s unction o tempetue F in/in/f empetue, heml Expnsion Coeicient, I the coeicient o theml expnsion needs to be clculted t the vege tempetue o 4F, detemine the vlue o the coeicient o theml expnsion t 4F using the diect method o intepoltion nd ist ode polynomil Solution Fo second ode polynomil intepoltion (lso clled qudtic intepoltion), we choose the coeicient o theml expnsion given by y x, y x, y x x, y x
5 iect Method o Intepoltion Moe Exmples: Mechnicl Engineeing 55 Figue Qudtic intepoltion Since we wnt to ind the coeicient o theml expnsion t 4F, nd we e using second ode polynomil, we need to choose the thee dt points tht e closest to 4F tht lso bcket 4F to evlute it hese thee points e 8 F, F nd 6F hen 6 8, 647 6, 6 6 6, 558 gives Witing the thee equtions in mtix om, we hve Solving the bove thee equtions gives Hence , 6 8 At 4F, in/in/ F he bsolute eltive ppoximte eo obtined between the esults om the ist nd second ode polynomil is % Exmple Fo the pupose o shinking tunnion into hub, the eduction o dimete tunnion sht by cooling it though tempetue chnge o is given by whee o
6 56 Chpte 5 oiginl dimete in coeicient o theml expnsion t vege tempetue in/in/ F he tunnion is cooled om 8 F to 8F, giving the vege tempetue s 4F he tble o the coeicient o theml expnsion vs tempetue dt is given in ble whee Since ble heml expnsion coeicient s unction o tempetue F in/in/f empetue, heml Expnsion Coeicient, ) I the coeicient o theml expnsion needs to be clculted t the vege tempetue o 4F, detemine the vlue o the coeicient o theml expnsion t 4F using the diect method o intepoltion nd ist ode polynomil Find the bsolute eltive ppoximte eo o the thid ode polynomil ppoximtion b) he ctul eduction in dimete is given by d oom tempetue F tempetue o cooling medium F 8F 8F 8 d 8 Find out the pecentge dieence in the eduction in the dimete by the bove integl omul nd the esult using the theml expnsion coeicient om pt () Solution ) Fo thid ode polynomil intepoltion (lso clled cubic intepoltion), we choose the coeicient o theml expnsion given by
7 iect Method o Intepoltion Moe Exmples: Mechnicl Engineeing 57 y x, y x, y x x, y x, y x Figue 4 Cubic intepoltion Since we wnt to ind the coeicient o theml expnsion t 4F, nd we e using thid ode polynomil, we need to choose the ou dt points closest to 4F tht lso bcket 4F to evlute it hen the ou points e 8F, F, 6F nd 6 F gives ,, 6, 6, Witing the ou equtions in mtix om, we hve Solving the bove ou equtions gives
8 58 Chpte Hence in/in/ F he bsolute eltive ppoximte eo nd thid ode polynomil is , % obtined between the esults om the second b) In inding the pecentge dieence in the eduction in dimete, we cn enge the integl omul to d nd since we know om pt () tht ( ) , 6 8 we see tht we cn use the integl omul in the nge om 8F to 8F heeoe, d d So 59 in/in using the ctul eduction in dimete integl omul I we use the vege vlue o the coeicient o theml expnsion om pt (), we get
9 iect Method o Intepoltion Moe Exmples: Mechnicl Engineeing 59 6 nd 6 in/in using the vege vlue o the coeicient o theml expnsion using thid ode polynomil Consideing the integl to be the moe ccute clcultion, the pecentge dieence would be % INERPOLAION opic iect Method o Intepoltion Summy Exmples o diect method o intepoltion Mjo Mechnicl Engineeing Authos Aut Kw te Novembe, 9 Web Site
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