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Full text of "Using the Frequency of Vibration of a Loaded Beam to Determine the Young's Modulus E, or to Determine g, the Rate of Acceleration"

Using the Frequency of Vibration of a Loaded Beam to Determine the Young's Modulus E, or to Determine g, the Rate of Acceleration By Patrick Bruskiewich Abstract The frequency of a vibrating beam can be used to determine the Young's modulus of the beam, or g, the rate of acceleration . This paper was written in 1981. The manuscript remained lost in the author's papers for years, until recently rediscovered with other manuscripts. 1.0 A Loaded Vibrating Beam Consider a beam of length 21 and negligible mass supported at either end, with a weight W at the centre. The beam is of negligible mass and has a moment of inertia I, and a Young's Modulus E (refer to Fig. 1: Loaded, vibrating beam) [1] Fig. 1 : A Loaded Vibrating Beam With a weight W placed at the centre of the beam, the centre is depressed by an amount y from its unloaded equilibrium level given by f 73 A y = 1 v K EIj W 1.1 By Newton's Third Law, the beam exerts an equal but opposite force on the weight, such that the restoring Force F res is given by r K, 6 V / 3 y 1.2 Set the beam vibrating with a slight push to the weight W. Given that the weight W is subject to the acceleration g, the equation of motion for the vibrating beam is W d 2 y g dt 2 ( \EI\ V y + W J 1.3 which is a second order differential equation with constant coefficients, d 2 y , g(*Ei} + dr wy v ) 6 y-g = 1.4 To solve this equation use a trial function of the form y = A + B cos(cot + a) 1.5 where a> is an angular frequency and a is the initial phase. By inspection we find that f 73 N \ y = K EIj W + Bcos f VL g( 6 m w V V J \ t + a 1.6 J 2.0 Using the Period to determine the Young's Modulus The frequency f of the vibrating beam is given by / = - ^ 2x\W 6 V V 2.1 J Provided there is no deformation of the beam, for a known weight W, and a known moment of inertia I, if we can measure the frequency of oscillation experimentally, ceteris paribus, we can determine the Young's Modulus E at that loading, namely f f ;3 A 4tt ? ' 6gl W f An 2 m 3\ V 6gl = E 2.2 3.0 Determination of the Value for g using a Vibrating Beam If we have a known beam, with an accurately determined W, E and I we can use the frequency of the vibration of the beam to determine g, the rate of acceleration, namely / ( An I 3 N \ V 6/ J (W^ \z J = g 3.1 The inquiring and intrepid mind can try their hand at these types of experiments. This technique to determine g is used extensively in mineral and petroleum exploration. References: ,rd [1] H.B. Phillips, Differential Equations, 3 ra ed. John Wiley and Sons, 1951, p. 132-133 1981 Patrick Bruskiewich