ON SOME ANALOGIES * UDC (075.1) Radu P. Voinea

Transcription

1 FACTA UNIVERSITATIS Series: Mehanis, Automati Control and Robotis Vol.3, N o 14, 3, pp Invited Paper ON SOME ANALOGIES * UDC (75.1)+6.1 Radu P. Voinea Romanian Aademy, Department of Tehnial Sienes 15 Vitoriei Av., Buharest/1, Romania Abstrat. The paper onsideres the idea of two physial phenomena, whih are essentially different, but an have the same linearized mathematial model. If one of two phenomena is well known, the analogy allwes for better understanding of the seon one. Analogy an be asertained, but it may lead under some irumstanes to new disoveries, at least from a theoretial point of view and if experimentally onfirmed, they an be onsidered as new verities in siene. The idea of analogies is illustrated by numerous examples. 1. INTRODUCTION 1. Analogy represents the idea of two physial phenomena, whih are essentially different, but an have the same mathematial model. If one of the two phenomena is well known, the analogy allows for a better understanding of the seond one. Suh an example is the famous analogy notied by Kirhhoff in the 19 th entury between the movement of a rigid body with a fixed point (Euler's equations): dk K = + K = M (1) dt t and the relations between the efforts in the ross setion of a bar, having as axis a urve in the three-dimensional spae and the exterior load on the bar unit length: dr R + p = + Ω R + p = ds t () Reeived May 3, * Paper presented at international Conferene Mehanial Vibrations Timisoara, as Invited Plenary Leture. Paper is from program ollaboratons between Faulty of Mehanial Engineering University Politehnika Timisoara, Rpmania and Faulty of Meahanial Engineering University of Niš.

2 786 R. P. VOINEA In the relation (1), K is the kineti moment with respet to the fixed point, M is the moment of the exterior fores with respet to the same point, and is the angular veloity vetor. In the relation (), R is the resultant of the N τ (axial), T υ and T β (shearing) efforts on the transversal setion, p is the exterior load on the bar unit length, and 1 1 Ω = τ + β is Rτ R the Darboux vetor, where R τ and R represent the torsion radius and the urvature radius of the bar axis, respetively, and τ, υ, β are the vetors of Frenet's trihedron axes.. Analogy an be asertained as in the above example, but it may lead under some irumstanes to new disoveries, at least from a theoretial point of view and if experimentally onfirmed, they an be onsidered as new verities in siene. This is the ase of the analogy that we ould name De Broglie's Analogy. He was wondering why only the light had orpusular and undulator features, whereas the other entities in nature were either orpusles or waves. Admitting the existene of an analogy between the orpusular and the undulator features of light on the one hand, and the features of the eletron (onsidered at that time to be a orpusle) on the other hand, he has disovered in a theoretial way the undulator features of the eletron. A few years later, the interferene of the eletrons was disovered in laboratory, namely a learly undulator feature resulting in the validation of De Broglie's presumed analogy. * 3. We will further give three examples of analogies: the first two have been aknowledged; the last example leads to a theoretial result waiting to be onfirmed (or invalidated) by pratial knowledge. * 4. Analogy between the small strains of a prismati onsole ated by a onstant moment ouple and the retilinear translation movement of a rigid body ated by a onstant module fore The differential equations in the two ases are respetively: d w EI = M (3) dx d u m = F (4) dt In ase of the boundary onditions x =, w = and x =, w' = for the onsole and in ase of the initial onditions t =, u = and t =, u ' = for the translation movement, the solutions are respetively: M w = x (5) EI u = F m t (6)

3 On Some Analogies 787 In ase of the onsole, we note with EI the rigidity of the bar, with w the arrow in any setion and with M the ouple moment In ase of the translation movement, we note with m the body mass, with u the spae displaement and with F the module of the fore F. The analogy is obvious. 5. Analogy between the large deformations of a prismati onsole load by a onstant moment ouple and the retilinear translation movement large veloity of a rigid body driven by a onstant fore in the module The differential equations in the two ases are respetively: 1 w EI = EI = M (7) ρ [1 + ( w ) ] u m = F 1 1 ( u ) (The differential equation (8) has been taken over from the Relativity Theory). Apparently, there is no perfet analogy, the minus sign of the denominator of the relation (8) foreshadowing a "dramati" situation in ase of the translation movement. It is only an appearane. Both differential equations (7) and (8) result from the more general equation: y 1 = (9) [1 + a( y ) ] b where a and b are two onstants. The solution of the differential equation (9) for the initial onditions x =, y = and x =, y' =, is: (8) x + ay by = (1) namely a oni, what an easily be verified. The differential equation (7) and its solution omply with a = 1, b = EI / M, and the differential equation (8) and its solution omply with a = 1 /, b = m / F, the solutions in the two ases being respetively: Ei x + w + = (11) M 1 m t u u = (1) F The diagrams are shown in the figures 1 and. w u m F O EI M x O m F t Fig. 1 Fig.

4 788 R. P. VOINEA In ase of large deformations of the onsole, the diagram is a irle and in ase of a high veloity translation movement, the diagram is a hyperbola. In both ases, the diagrams represent onis, that are the analogy is perfet, the irle and respetively the hyperbola onditions, resulting from the values assigned to different oeffiients and not from different natures of the diagrams, whih are onis. 6. The analogy between the small deformations of a prismati onsole onsidering the influene of the shearing fore and the retilinear translation movement of a rigid body. The differential equation of a medium deformed fibre of the onsole is in this ase: d w d M EI = M + (1 + υ) k i s (13) dx dx where υ is the oeffiient of Poisson, k is a oeffiient taking into onsideration the nonuniform distribution of the shearing stress in a ross setion and i s is the inertia radius of the ross setion. The retilinear translation movement of a rigid body (if suh an analogy ontinued to exist) would aordingly imply the existene a differential equation, namely: d u m dt 1 d F = F ± (14) dt where is a pulsation that has to be experimentally determined. The plus or minus sign in the seond side of this relation has to be established experimentally, as well. Remarks: a) In pratial designing, the influene of the shearing fores on the deformations of the prismati onsole are negleted, being very small. Aordingly, through analogy, the pulsation is expeted to be very high, so that the term 1 d F ± beomes negligible. b) The deformations due to the shearing fores are nevertheless high in ase of very short onsoles ated by very high fores. Similarly, through analogy, the term 1 d F dt ± dt an not be negleted in ase of short term movements taking plae under the ation of rapidly hanging fores, suh as ollisions, high frequeny elasti waves et. 7. Appliation in ase of a longitudinal wave propagation in a prismati bar. By isolating an element of dx thikness from the respetive bar and introduing the fores ating on it and using the differential equation ma = F, the equation with partial derivatives known as: 1 u u = t x is obtained. It is integrated using for instane the method of separation of the variables. Considering the semiinfinite bar (x ), under the boundary ondition x =, u = u os t, and retaining only the diret wave, the lassi solution is: (15)

5 On Some Analogies 789 u = u os ( x t) (16) If the differential equation (14) is applied for the bar element with the thikness dx, after having introdued all the fores ating on it, the equation with partial derivatives of the fourth order is obtained further to elementary alulation: 1 u u 1 u = ± t x x t It an be integrated through the same method of the separation of variables. Considering also this time the semiinfinite bar (x ) under the boundary onditions x =, u = u os t, and retaining only the diret wave (whih is propagating in the positive sense of the axis Ox), we obtain the solution: 4 (17) / x = u os [ x 1± ( / ) ] t (18) 1± ( / ) pointing out a wave propagating by the speed: = 1± ( / ) (19) Remarks: a) The propagation speed of the elasti longitudinal waves is not onstant ( = E / ρ), but it depends on the pulsation. b) The sign of the relation (/ ) has to be established through pratial experiments. We assume this sign is "minus", that is the formula (19) would beome: = 1 ( / ) () Fig. 3 Fig. 4 Fig. 5 In the figures 3, 4 and 5, the three variants are presented, namely: the lassi: = = t. (fig.3), irle quarter: = 1 ( / (fig.4) hyperbola ar: = 1+ ( / ) (fig.5) )

6 79 R. P. VOINEA ) As a onsequene of the wave propagating speed depending on the pulsation is the fat that in ase the boundary disturbane (x = ) is the sum of different harmonis pulsations, eah harmoni will be propagated by another speed generating a new phenomenon of elasti wave dispersion. d) The fat that no variation of the wave propagation speed depending on the pulsation was stated is aounted for by the fat that the expansion in series of the relation 1± ( / begins with seond order terms: ) = 1± ( / ) = 1± ( / ) ( / ) ±... (1) 8 the order of magnitude of these terms and of the next ones being the same with that of the possible measurement errors due also to the very high pulsation value. The equation (14) is obviously a pure theoretial result based on the assumption that the analogy between the small deformations of a prismati onsole and the spaes overed by a rigid body in retilinear translation, would ontinue to persist even if onsidering the influene of the shearing fores on the deformations of the onsole. This fat has anyway to be onfirmed by pratial experiene in order to beome reality. If onfirmation does not follow, a simple betraying dream will remain like so many other betraying dreams in my areer. And this is all there is to it. REFERENCES 1. Voinea R. P., An analogy, Proeedings of the Romanian Aademy series A, no.1, January-April,, pp O NEKIM ANALOGIJAMA Radu P. Voinea U radu je rezrađena ideja dva fizička fenomena, esenijalnop različitih, ali koji imaju isti linearizovani matematički model. Ako je jedan od ova dva fenomena veoma dobro poznat, analogije omogućavaju bolje razulevanje drugog fenomena. Analogija može biti potvrđena, ali pod određenim uslovima može dovesti do određenih otkrića, barem sa teorijske tačke gledišta, kao i ako se eksperimentalno potvrdi, može se smatrati novim verifikaijama u naui. Ideja analogije je ilustrovana brojnim primerima.