SIMPLE NUMERICAL METHOD FOR KINETICAL INVESTIGATION OF PLANAR MECHANICAL SYSTEMS WITH TWO DEGREES OF FREEDOM

Transcription

1 Interdisciplinar Descriptin f Cple Sstes 4(), 6-69, 06 SIMPLE NUMERICAL METHOD FOR KINETICAL INVESTIGATION OF PLANAR MECHANICAL SYSTEMS WITH TWO DEGREES OF FREEDOM István Bíró* Facult f Engineering Universit f Szeged Szeged, Hungar DOI: /indecs.4..6 Regular article Received: 9 Octber 05. Accepted: 6 Nveber 05. ABSTRACT The ai f this article is t denstrate the applicatin f a siple nuerical ethd which is suitable fr tin analsis f different echanical sstes. Fr echanical engineer students it is iprtant task. Mechanical sstes cnsisting f rigid bdies are linked t each ther b different cnstraints. Kineatical and kinetical analsis f the leads t integratin f secnd rder differential equatins. In this wa the kineatical functins f parts f echanical sstes can be deterined. Degrees f freed f the echanical sste increase as a result f built-in elastic parts. Nuerical ethds can be applied t slve such prbles. The siple nuerical ethd will be denstrated in MS Ecel b authr b the aid f tw eaples. MS Ecel is a quite useful tl fr echanical engineers because eas t use it and details can be seen rever failures can be nticed. Se parts f results btained b using the nuerical ethd were checked b analtical wa. The published ethd can be used in higher educatin fr echanical engineer students. KEY WORDS tin analsis, echanical sstes, nuerical ethd, initial-value prbles, nnlinear vibratin CLASSIFICATION ACM: D JEL: O35 *Crrespnding authr, : bir-i@k.u-szeged.hu; ; *Universit f Szeged Facult f Engineering, Mars tér 7, H-674 Szeged, Hungar

2 I. Bíró INTRODUCTION The studing f tin analsis is iprtant part in the educatin f echanical engineers. Kineatic and dnaic analsis f echanical sstes is a fundaental chapter in tin analsis [-4]. Multi DOF echanical sstes can be described b secnd rder differential equatins. Analtical slutin f the in st cases is quite difficult r ipssible. In such cases the applicatin f nuerical ethds can be advantageus. There are different nuerical algriths which are suitable fr slving differential equatins. The results btained in this wa can be pltted in different kineatical diagras. This ethd can help engineer students better learning f schl-wrk and cnnectins ang different phsical quantities. In this article the results f kinetic analsis f tw echanical sstes will be denstrated. PROPOSED SIMPLE NUMERICAL METHOD In general cases tin equatins f tw degree-f-freed echanical sstes as hgenus differential equatin-sste are Mq Sq 0 () where M is ass atri, S spring stiffness atri and q vectr f generalized crdinates. Let us suppse that the echanical sste can be described b q = q(, φ) generalized crdinates. Phsical quantities,,, describe the initial state f the sste. Tie step is ti ti. Applied algriths in MS Ecel can be seen in Table. Table. Applied algriths fr slving differential-equatin sste (Eaple ). t (,,, ) (,,, ) t,,, ) (,,, ) ( t,,, ) t ) t ),,, ) t ) t t ) ( ( t t,,, ) t ) ( ( t ( t t t ( ( t (,,, ) t ) t ) ( ( t ( t t t EXAMPLE : CRANK DRIVE WITH OSCILLATING MASS ( DEGREES OF FREEDOM) In Figure a sketch f a siple echanical sste (tw degrees f freed) can be seen. There is a ass and a crank drive in between spring and viscus daping. Disc n the left side driven b ent M can rtate arund hrizntal ais dented b A. Ment f inertia is arked b J A. There is a frae between disc and ass is driven b slider secured n the disc can ve alng hrizntal ais. Spring stiffness f linear spring and daping factr f viscus daping are dented b s and k. Unladed length f spring arked b l. The tin f the crank drive echanis can be described b the fllwing secnd rder differential equatin sste: 6

3 Siple nuerical ethd fr kinetical investigatin f -DOF planar echanical sste J A r B k M A φ s Figure. Sketch f crank drive echanis and scillating ass. k( r sin) s( l r cs ) 0, () J M. (3) k ( r sin) s( l r cs) r sin 0 Appling nuerical integratin ethd the kineatical functins f disc and ass can be pltted and studied in case f different phsical prperties f eleents f ving structure. During applicatin f prpsed siple nuerical ethd the initial cnditins f rtating disc and translating ass can be varied ptinall. Further effects fr eaple frictin and eternal lads can be taken int cnsideratin. In first case the structure starts fr rest psitin i.e. 0 /s, 0, 0 rad/s, 0 rad. Data: M = 0 N, J A = 4 kg, = 8 kg, r = AB = 0,, l =,0, s = 000 N/, k = 00 Ns/, (tie step: 0,00 s, tie interval: 0 t 5 s). After tw-tie nuerical integratin f tin equatins the kineatical functins can be seen in Figure. EXAMPLE : TWO MASSES SPRING IN BETWEEN MOVE IN CROSS DIRECTION In Figure 4 a sketch a special echanical sste can be seen. There are tw ass pints arked b and rever linear spring in between. Mass pints can slide verticall and hrizntall. The ass f spring between ass pints is neglected. As it can be seen in Figure 4 psitins f ass pints are deterined b crdinates and. Frictin between ass pints and surfaces is neglected. Unladed length f spring is arked b l, its actual length is the spring frce, l, (4) l l s l R s. (5) Algebraic sign f spring frce f pressed spring is psitive. Mtin equatins f ass pints are: 63

4 I. Bíró 64 Figure. Kineatical diagras f rtating disc and translating ass (eaple, first case). sin R g, (6) cs R. (7) Taking int cnsideratin that sin and cs, the fr f tin equatins ields the fllwing: l s g, (9) l s, (0) fr which after rearrangeent g l s, l s. (rad/s) (/s) (rad/s) (/s) (rad) ()

5 Siple nuerical ethd fr kinetical investigatin f -DOF planar echanical sste, rad s,, rad/s, /s, rad/s, /s Figure 3. Kineatical diagras f rtating disc and translating ass (eaple, secnd case). s O φ Figure 4. Sketch f echanical sste cnsisting f tw ass pints and linear spring in between. 65

6 I. Bíró Data and initial cnditins fr the nuerical slutin: unladed length f spring, l =0,5, spring stiffness, s = 000 N/, gravital acceleratin g = 9,8 /s, = 0 kg, = 6 kg, 0,4, /s 0, 0,3, 0 /s. Tieinterval: 0 s t s, tie step: 0,000 s. Kineatic functins f ass pints and the spring frce in functin f tie can be seen in Figure 5-8. Figure 5. Acceleratin functins f ass pints. Figure 6. Velcit functins f ass pints. Figure 7. Psitin functins f ass pints. The level f echanical pwer f the ving sste in functin f tie is suitable t check the reliabilit and accurac f applied nuerical ethd. Because f the fact that the investigated echanical sste is cnservative the level f echanical pwer has t be cnstant. The difference f the level f echanical pwer fr its initial value is nt significant as it can be seen in Figure 9. In nnlinear case the characteristic f the spring can be described in functin f defratin accrding t net equatin, i. e.: 66

7 Siple nuerical ethd fr kinetical investigatin f -DOF planar echanical sste Figure 8. Spring frce in functin f tie. Figure 9. Mechanical pwer f the echanical sste in functin f tie. R l b l l cl,() where a, b and c are paraeters. B dificatin f these paraeters the real nnlinear characteristic f the spring can be appriated with the deanded accurac. In this case the tin equatins in final fr are a l 3 b l l c l a l b l l c l 3 g, (3) 3. (4) Data and initial cnditins are quite the sae like in linear case: unladed length f spring, l = 0,5, characteristical paraeters a = 000 N/, b = N/, c = N/ 3, gravital acceleratin g = 9,8 /s, = 0 kg, = 6 kg, 0,4, 0 /s ; 0,3, 0 /s. Tie interval: 0 s t s, tie step: 0,000 s. Kineatic functins f ass pints and the echanical pwer in functin f tie can be seen in Figures 0-3. The spring frce in functin f tie and defratin (unit: eter) are denstrated in Figure 4. As it can be nticed the applied spring is strngl nnlinear. 67

8 I. Bíró Figure 0. Acceleratin functins f ass pints. Figure. Velcit functins f ass pints. Figure. Psitin functins f ass pints. Figure 3. Mechanical pwer f the echanical sste in functin f tie. 68

9 Siple nuerical ethd fr kinetical investigatin f -DOF planar echanical sste a) b) Figure 4. a) Spring frce in functin f tie and defratin and b) algebraic sign f spring frce f pressed spring is psitive. CONCLUSIONS The denstrated ethd can be applied easil fr engineer students in the higher educatin. The ethd is suitable fr investigatin f siilar echanical sstes having ne r re degrees f freed. B cnsequent dificatin f data (phsical quantities) f sstes a wide range f pssible structures and their kineatical behavir can be analzed. Fr this reasn the applicatin f this ethd can be advantageus fr engineer students. REFERENCES Bíró, I.: Eaples fr Practice in Mechanical Mtin fr Mechanical Engineers. Labert Acadeic Publishing, Saarbrücken, 05, Haug, E. and Shni, V.N.: Cputer Aided Analsis and Optiizatin f Mechanical Sste Dnaics. NATO ASI Series 9, 984, 3 Sársi, J.; Bíró, I.; Néeth, J. and Cveticanin, L.: Dnaic Mdelling f a Pneuatic Muscle Actuatr with Tw-directin Mtin. Mechanis and Machine Ther 85, 5-34, 05, 4 Mester, G.: Intrductin t Cntrl f Mbile Rbts. In: Prceedings f the YUINFO 006, pp.-4, Kpanik,