DETERMINING SHEAR ELASTICITY MODULUS BY TORSIONAL PENDULUM

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1 7 th INTERNATIONAL MULTIDISCIPLINARY CONFERENCE Baia Mare, Romania, May 17-18, 007 ISSN-1-36 DETERMINING SHEAR ELASTICITY MODULUS BY TORSIONAL PENDULUM Dr. Sándor Nagy Dr. Gergely Dezső Attila Százvai Abstract: When claming the uer end of a thin bar or wire and fixing a circular late to the bottom end of it, we get torsional oscillating system. The shear elasticity modulus can be calculated after determining the lowest root of the characteristic equation of the second order differential equation describing the torsion oscillation of the linear bar, and measuring exerimentally the smallest natural frequency of the torsion oscillation. 1. INTRODUCTION As it is known, the material equation of homogenous isotroic materials contains three characteristic data: E the coefficient of elasticity, G shear elasticity modulus and ν the Poisson s ratio. If one know two of them, the third can be calculated by the formula E = (1 + ν)g. 1.1 Poisson s ratio is determined from relative longitudinal and cross-directed elongations measured on a tensile-test iece tensioned. Metrological roblems arises when measuring these elongations of wires under 1mm in diamater alied in stranded wire. Because of this reason determining Poisson s ratio is ossible by equation 1.1, if E and G can be measured exerimentally. The coefficient of elasticity E can be obtained from tensile test, the shear elasticity modulus from torsion oscillations. This is the object of the next chater.. DETERMINING SHEAR ELASTICITY MODULUS.1 Theoretical background We get torsional oscillating system if we clam the uer end of a thin bar or wire, which the diameter is d and length is h of, and fix a circular late to the bottom end of it (Figure 1.)

2 M cs q I G M cs + M c dz Fig. 1. The scheme of a torsional oscillating system Eqwuation of motion of torsion oscillation of a linear bar can be derived from the rincile of torque and angular momentum alied to dz thick a slide of it : Mcs djz q = dz,.1 where q generalized coordinate denotes the angular deflection around the centroidal axis of the cross-section, dj z is moment of inertia of the elemental iece of the bar concerning to the axis z: I is olar moment of inertia of the circular bar djz = ρ R dadz = ρ I dz,. I d π =.3 3 and ρ is the mass density of the material of the bar. Suosing elastic deformation of the wire, and using the relation. between the relative angular deflection and the torsional load we can write equation (.1) in the form q Mcs = I G q q ρi = IG.5 t where G denotes the shear elasticity modulus of the bar. While the cross section is constant we get the equation of motion in the following form:. 510

3 q a t 1 q = 0 which is a second order artial differential equation. We introduce for the ratio of G and ρ : a.6 = G 1..7 ρ If we suose that oscillations of all cross sections of the bar have the same α angular frequency, the the solution of the differential equation can be obtained as a roduct of two function, one of them is function of coordinate z, another of them is of time t (Fouriermethod): q(z,t) = ϕ(z)cos( αt + ε),.8 where φ(z) is interreted as amlitude distribution along the bar. The characteristic equation can be written taking into account boundary conditions (claming and resence of circular late) 1 α α Jαa1 sin h cos h = 0 IG a + 1 a.9 1 where J denotes the moment of inertia of the late concerning to the axis z. Introducing the notation we can write the coefficient Jαa 1 /I G int he formula.9 as α β = h.10 a 1 Jαa1 I G Ja1 β J = = β = λβ I Gh hi ρ.11 where J λ =..1 hi ρ Inserting this into the characteristic equation we get λβ = ctg β..13 This shows that roots of characteristic equation β 1, β, can be obtained from intersections of grah of line y=λβ and curve y=ctgβ (Figure ). 511

4 λβ cot( β) 0 6 β1 β Fig. 1. Solutions of characteristic equation It can be shown, that for large values of λ (when the moment of inertia of the bar concerning to the axis of rotation J r =hi ρ is smaller than the moment of inertia of the circular late J ), we can aroximate the smallest normal frequency of the system by the following formula: α min I G 1 h J + J 3 r..1. Realization of the torsion oscillating system To ensure the same boundary conditions, the ends of the wire was soldered into the claming heads shown in Figure 3. Centralization of the wire was guaranteed by the hole 1 mm in diameter in the head, and another hole with same diameter in the circular late. Fig.. Claming heads The mass, serving the moment of inertia fixed to the bottom end of the wire is consisted if two arts: cylinder with hole (1) and cylindrical late (). The cylindrical late and the head are clamed by headless screw (3). Cylinder is clamed to them by the screw 3. (Figure.). The uer end of the wire was clamed by a drillstock linked to a screw sindle M0. The 51

5 length of the wire was measured by an altimeter. Fig. 3. The shae of the weight Fig.. The assembly of the torsional endulum.3 Parameters of the system.3.1 Parameters of the weight: General data: Centre of mass: Mass density: 7,860E-006 kg/mm 3 X: -9,507E-016 mm Volume: 1,70E+005 mm 3 Y:,09E-016 mm Mass: 1,339 kg Z: 31,30 mm Moment of inertia: J = 58,895 kg mm.3. Parameters of wires investigated Parameters of wires investigated are shown in the table below, S.sz. d A h ρ I σ z J [mm] [mm ] [mm] [kg/mm 3 ] [mm ] [Ma] [kgmm ] 1 0,68 0, ,6 8,576E-06 0, ,1571 6,7351E-05 0,68 0, ,55 8,576E-06 0, ,1571 8,1879E ,38 0, ,9 8,09E-06 0, ,783 5,9139E-06 0,38 0, ,1 8,09E-06 0, ,783 7,E

6 where the olar moment of inertia: I stress int he wire: Q σ z =, the moment of inertia: A d π =, the diameter of the wire: 3 J r 1 = ρπhd. 3 π d A =, the axial normal In our exeriments arameters of macaroni with circular cross-section available in commerce were measured. While the cross-section differed from the circle considerably arameters of 10 iece were averaged.: d 1 d A h ρ I σ z J r [mm] [mm ] [mm] [kg/mm 3 ] [mm ] [Ma] [kgmm ],7665 0,98, ,8E-06 5, ,510553,7938E-03 where the olar cross-sectional moment of inertia: I (d1 d ) π =, the mass moment of inertia: 3 J (r1 r ) 1 = πhρ = m(r1 r ). r + Mass moments of inertia of wires (and macaroni ) are negligible comared to the moment of inertia of the weight fixed to the end of them in formula.1.. Investigation of torsional oscillation In our series of exeriment the same wire was measured with two different lengths. Time of 0 eriod was measured in two series. In first series we erformed our measurements with different starting deflections ( ϕ = 10,0,30, 0 ). This does not affects detectably the exerimental results 0 of time of eriod. Of course, the amlitude decreased in time, esecially int he case of macaroni. Because of this only time of 10 eriods was ossible to measure of macaroni. Averaged exerimental results are summarized in the following table: 51

7 Number Tye of wire of Time of one Angular Time measured length diameter eriods eriod frequency measured n T n T averaged H d T α [db] [s] [s] [mm] [mm] [s] [1/s] steel wire I. series 0 39,05 39, ,6 steel wire II. series 0 39,06 1,958 3,1761 0,68 steel wire III. series 0 3,81 3, ,55 steel wire IV. series 0 3,81,190,8685 zither string I. series 0 13,1 13, ,9 zither string II. series 0 13,16 6,157 1,003 0,38 zither string III. series 0 139,83 139,875 31,1 zither string IV. series 0 139,83 6,991 0,89871 macaroni I. series 10 8,10 d 1 =,7665 8, macaroni II. series 10 7,98 d =0,98 0,8038 7, Evaluation of results The quantity λ exressing the gradient of the line is function of the mass of weight m and its mass moment of inertia (I ), the material roerties of the wire (ρ), the length (h) and diameter (d) of it. Tye of wire h λ β α a 1 G G arox [mm] rad [rad/s] [1/s] [Ma] [Ma] Steel wire 359,6 9,035E+06 3,35670E-0 3,1761 3,79155E+06 1,038083E+05 1,037358E+05 Steel wire 51,55 7,195E+06 3,76680E-0,8685 3,7573E+06 1,036036E+05 1,035311E+05 zither string 351,9 9,891E+07 1,005500E-0 1,003 3,57160E+06 1,06967E+05 1,06963E+05 zither string 31,1 8,07E+07 1,1190E-0 0, ,8119E+06 9,98393E+0 9,9868E+0 mavaroni 31,093E+05,185780E-03 7,8173 1,19570E+06,153685E+03,15369E+03 where λ = J α, β is the smallest solution of equation.13, a 1 = h. hi ρ β The shear elasticity modulus exressed from. 11 we get αh G = ρ, and G arox aroximate β 515

8 value calculated by aroximating formula hj G α. I 3. CONCLUSION Determining the time of eriod of the torsional oscillation can be disturbed by two main factors. One of them is the couling between the torsional oscillation and swing oscillation. The other of them is the normal stress coming from the reaction force of the weight. Values of G arox obtained by the aroximating formula shows good aggreement with G calculated from our exerimental measurements. Tis is esecially true int he cases of zither stroing and the macaroni.. REFERENCES [1] J.P. Den Hartog: Mechanical Vibrations. Forth Edition. New York, Mc Graw- Hill Book Comany, INC., 1956 [] Dr. Sályi István: Lengéstan (Theory of vibrations). Kézrat. Nehétiari Műszaki Egyetem. Tanköyvkiadó, Budaest, 1973 [3] Szolgai Péterné, Sályi Zsófia: Torsional oscillating systems with sring loaded by tensile load (Torziós rezgőrendszerek húzásra is igénybevett torziós rugókkal.) Egyetemi Doktori Értekezés. Budaest RETURN ADDRESS HUNGARY, Assistant lecturer, Attila Százvai College of Nyíregyháza H - 00 Nyíregyháza Kótaji út 9-11 POBox: 166 Hungary szazvai@nyf.hu 516