oBscence lopsclence. Iop. org Home Search Collections Journals About Contact us My lOPscience Transverse vibrations of a thin loaded rod: theory and experiment This content has been downloaded from lOPscience Please scroll down to see the full text 2015Eur.J.Phys.36055035 (http://iopscience.ioporg/0143-0807/36/5/055035) View the table of contents for this issue, or go to the journal homepage for more Download details P Address:222.66.115.226 This content was downloaded on 05/08/2015 at 07: 00 Please note that terms and conditions apply
This content has been downloaded from IOPscience. Please scroll down to see the full text. Download details: IP Address: 222.66.115.226 This content was downloaded on 05/08/2015 at 07:00 Please note that terms and conditions apply. Transverse vibrations of a thin loaded rod: theory and experiment View the table of contents for this issue, or go to the journal homepage for more 2015 Eur. J. Phys. 36 055035 (http://iopscience.iop.org/0143-0807/36/5/055035) Home Search Collections Journals About Contact us My IOPscience
loP Publishing EuJ.Phys.36(2015)055035(11pp) Transverse vibrations of a thin loaded rod theory and experiment Jue Xu, Yuan-jie Chen and Yong-li Ma Department of Nuclear Science and Technology, Fudan University, Shanghai 200433 Peoples Republic of China artment of Physics, Fudan University, Shanghai 200433, Peoples Republic of E-mail:yIma@fudan.edu.cn Received 3 March 2015, revised 28 April 2015 Accepted for publication 16 June 2015 Published 4 August 2015 Abstract The general formulation of a determinate solution problem is deduced for the transverse vibrations of a thin loaded rod. The vibration frequencies of a thin homogeneous rod carrying a concentrated mass as a function of the loads position and mass are exactly solved. The dynamic measurement method of Youngs modulus of the rods is presented within this theory. Our measure- ments of Youngs modulus in the dynamic method agree with those in the traditional bending method, therefore the theory is verified by our experiments Keywords: transverse vibrations of a load rod, Youngs modulus, dynamic measurement method (Some figures may appear in colour only in the online journal) 1 Introduction In college-level instruction on vibration and waves, the model is usually simplified as vibrations of a single particle even for a tuning fork. Actually, the tuning fork consists of two elastic rods and we need to analyze the transverse vibrations of these rods. The textbooks on he method of mathematical physics have only the longitudinal vibrations of a rod [1-3] and rarely refer to its transverse vibrations [4-6] with and without a load. Even in engineering mathematics [4] and applied mathematics [5], the equation of motion on the vertical vibra- tions of a thin long rod is only directly given and its derivation is not present In the course of the method of mathematical physics, the transverse vibrations of a rod were first introduced by [6], but its deduction on the solving problem is too simple to follow by students. Even in transverse vibrations of a rod is still simple [7, 8] although its longitudinal vibration is 0143-0807/15/055035411$33.00 G 2015 IOP Publishing Ltd Printed in the UK
Transverse vibrations of a thin loaded rod: theory and experiment Jue Xu1 , Yuan-jie Chen2 and Yong-li Ma2 1 Department of Nuclear Science and Technology, Fudan University, Shanghai 200433, People’s Republic of China 2 Department of Physics, Fudan University, Shanghai 200433, People’s Republic of China E-mail: ylma@fudan.edu.cn Received 3 March 2015, revised 28 April 2015 Accepted for publication 16 June 2015 Published 4 August 2015 Abstract The general formulation of a determinate solution problem is deduced for the transverse vibrations of a thin loaded rod. The vibration frequencies of a thin homogeneous rod carrying a concentrated mass as a function of the load’s position and mass are exactly solved. The dynamic measurement method of Young’s modulus of the rods is presented within this theory. Our measurements of Young’s modulus in the dynamic method agree with those in the traditional bending method, therefore the theory is verified by our experiments. Keywords: transverse vibrations of a load rod, Youngʼs modulus, dynamic measurement method (Some figures may appear in colour only in the online journal) 1. Introduction In college-level instruction on vibration and waves, the model is usually simplified as vibrations of a single particle even for a tuning fork. Actually, the tuning fork consists of two elastic rods and we need to analyze the transverse vibrations of these rods. The textbooks on the method of mathematical physics have only the longitudinal vibrations of a rod [1–3] and rarely refer to its transverse vibrations [4–6] with and without a load. Even in engineering mathematics [4] and applied mathematics [5], the equation of motion on the vertical vibrations of a thin long rod is only directly given and its derivation is not present. In the course of the method of mathematical physics, the transverse vibrations of a rod were first introduced by [6], but its deduction on the solving problem is too simple to follow by students. Even in the mechanics of vibration, a fundamental course for a mechanics major, the instruction on transverse vibrations of a rod is still simple [7, 8] although its longitudinal vibration is European Journal of Physics Eur. J. Phys. 36 (2015) 055035 (11pp) doi:10.1088/0143-0807/36/5/055035 0143-0807/15/055035+11$33.00 © 2015 IOP Publishing Ltd Printed in the UK 1
2015)055035 J Xu et al discussed with a load [7]. Earlier papers have addressed the problem of vibrating rods [9, 10] In special textbooks [11], the exact solutions of the rods transverse vibrations are put forward systematically, but only a few solutions at special conditions are given [12, 13].It is necessary to derive a general formulation of the determinant solution problem for the transverse vibrations of a thin loaded rod, and find the exact solutions in general conditions Much work has been done on the more complicated case of anisotropic vibrations of rods. For example, the greatest use of piezoelectric resonators is the famous 32.768 kHz quartz tuning fork, including a review paper [14], the fabrication [15], and the analysis of frequency [16]. Although this work treats only the isotropic case, which is a limiting work in the mechanics major, we emphasize the physical foundation for undergraduates in the physics major Knowledge of Youngs modulus is fundamentally important to understand the mechanical behaviour of materials, such as metals, ceramic grinding stones, dental compo sites, and polymers [17-20). Youngs moduli are determined traditionally by the static and dynamic methods. In static measurements [21, 22], such as the classical tensile or com- pressive test, a uniaxial stress is exerted on the material, and the elastic modulus is calculated from the transverse and axial deformations as the slope of the stress-strain curve at the origin The static methods include the three-point bending [17-20], four-point bending [23], clamped beam, and compression/tension stress, etc. A dynamic method of measuring Youngs mod- ulus of stalloy was described in an earlier paper [24] using a loaded fixed-free bar vibrating in flexure and developed to measure Youngs modulus of elasticity of a solid [25], carrying a heavy mass of precise finite dimensions at the free end and giving the derivation of the quation of motion. Dynamic methods [26-29] are more precise since they use very small strains, far below the elastic limit and therefore are virtually nondestructive and allow repeated testing of the same sample. These include the ultrasonic pulse-echo [27] ,bar resonance methods [22, 28, 29], travelling or standing wave, bending/transversal or long itudinal wave, transient pulse generation, etc. Recently, a new vibration beam technique [30] for the determination of the dynamic Youngs modulus has been developed, but without the dded loads. The dynamic methods redeem the defects that the static bending method cannot be applied to the measurement of fragile materials. Our method has added a variable with the loads at different positions and different masses. For didactic purposes undergraduates majoring in physics, in this paper we derive a general formulation of the determinant solution problem for the transverse vibrations of a thin loaded rod, obtain an exact solution of the problem, and deduce a general relationship between eigenfrequencies and the loads position and mass. Different resonance frequencies are measured by adding both the same mass to different positions of the rod and different masses to the same position of the rod. We deal with Youngs modulus measurement method based on the vibration of a thin long rod with added point mass. The elastic modulus is calculated from the least square fit of frequency versus the square of the wave number calculated from the characteristic equation. According to this model, a new kind of dynamic measurement method of Youngs modulus is presented. This method is more comprehensive and is advanced when it is not convenient to change the length of samples The paper is organized as follows. The mechanical model and the solution for the of the load's position and 2.A practical implementation and the results of the experiments are described in section 3. showing results in agreement with model predictions. A summary is given in section 6
discussed with a load [7]. Earlier papers have addressed the problem of vibrating rods [9, 10]. In special textbooks [11], the exact solutions of the rod’s transverse vibrations are put forward systematically, but only a few solutions at special conditions are given [12, 13]. It is necessary to derive a general formulation of the determinant solution problem for the transverse vibrations of a thin loaded rod, and find the exact solutions in general conditions. Much work has been done on the more complicated case of anisotropic vibrations of rods. For example, the greatest use of piezoelectric resonators is the famous 32.768 kHz quartz tuning fork, including a review paper [14], the fabrication [15], and the analysis of frequency [16]. Although this work treats only the isotropic case, which is a limiting work in the mechanics major, we emphasize the physical foundation for undergraduates in the physics major. Knowledge of Young’s modulus is fundamentally important to understand the mechanical behaviour of materials, such as metals, ceramic grinding stones, dental composites, and polymers [17–20]. Young’s moduli are determined traditionally by the static and dynamic methods. In static measurements [21, 22], such as the classical tensile or compressive test, a uniaxial stress is exerted on the material, and the elastic modulus is calculated from the transverse and axial deformations as the slope of the stress-strain curve at the origin. The static methods include the three-point bending [17–20], four-point bending [23], clamped beam, and compression/tension stress, etc. A dynamic method of measuring Young’s modulus of stalloy was described in an earlier paper [24] using a loaded fixed-free bar vibrating in flexure and developed to measure Young’s modulus of elasticity of a solid [25], carrying a heavy mass of precise finite dimensions at the free end and giving the derivation of the equation of motion. Dynamic methods [26–29] are more precise since they use very small strains, far below the elastic limit and therefore are virtually nondestructive and allow repeated testing of the same sample. These include the ultrasonic pulse-echo [27], bar resonance methods [22, 28, 29], travelling or standing wave, bending/transversal or longitudinal wave, transient pulse generation, etc. Recently, a new vibration beam technique [30] for the determination of the dynamic Young’s modulus has been developed, but without the added loads. The dynamic methods redeem the defects that the static bending method cannot be applied to the measurement of fragile materials. Our method has added a variable with the loads at different positions and different masses. For didactic purposes undergraduates majoring in physics, in this paper we derive a general formulation of the determinant solution problem for the transverse vibrations of a thin loaded rod, obtain an exact solution of the problem, and deduce a general relationship between eigenfrequencies and the load’s position and mass. Different resonance frequencies are measured by adding both the same mass to different positions of the rod and different masses to the same position of the rod. We deal with Young’s modulus measurement method based on the vibration of a thin long rod with added point mass. The elastic modulus is calculated from the least square fit of frequency versus the square of the wave number calculated from the characteristic equation. According to this model, a new kind of dynamic measurement method of Young’s modulus is presented. This method is more comprehensive and is advanced when it is not convenient to change the length of samples. The paper is organized as follows. The mechanical model and the solution for the eigenfrequencies as a function of the load’s position and mass are illustrated in section 2. A practical implementation and the results of the experiments are described in section 3, showing results in agreement with model predictions. A summary is given in section 6. Eur. J. Phys. 36 (2015) 055035 J Xu et al 2
2015)055035 J Xu et al 2. The mechanical model and the solutions We consider a rod of length I along the x-axis in equilibrium. The mass of the rod is m=/dxp(x)s(r) with dx being the line element, P(x) being the volume density, and s(x)=w(r)h(x) being the cross-section area. Here the width of the rod is y= w(r)and the thickness is z= h(x). The turning radius of this cross-section r(x) satisfies zada h(x)Jo 12 Figure 1 shows the diagram and cross-section of the thin homogenous rod. Let the element be dm= p(x)dr with the volume element dr= s()dx, one obtains the rotate inertia to the ov-axis as h(x) When the rod deforms transversely, its every cross-section should produce shearing rces. Let the shearing force on the left of volume element be o(x, t)(down direction), and the right one be Q=o(x, t)+dQ(, t)dx(up direction) with d being an abbreviation of d/dx. These two shearing forces form a force couple, which bends the rod. Figure 2 shows an element of a thin homogenous rod in bending time point (x, D), is u(x, t). The curvature radius of the bending d"&s of a thin loaded rod We consider the characteristic quantity of the transverse vibration the displacement of the rod away from the equilibrium position along the z-direction at space When the rod bends, the central line length dr remains unchanged. However, the upper part of the central line that suffers the tension of the nearby elements is prolonged; the lower part that suffers the pressure is compressed. Consequently, the force couple consists of tension and pressure, the so-called bending moment. Let the bending moment on the left of the volume element be M(x, t)(clockwise), and the bending moment on the right be M=M(x, t)+ dM(r, t)dx(counter-clockwise). The bending moments act as resistance to the bending of the rod, and lead the system to the dynamic equilibrium states. Figure 3 shows the force acting on an element of a thin homogenous rod. As shown in figure 2, we take a lamina with thickness dz at z position and width w(x). We recall that the length is dx at the center line of the volume element The length of the lamina is (R+z)de= dx zdx/R and the relative extension is Z/R. So that the tensile stress is P=-Yz/R with y being Youngs modulus. The tension element is dG= Pwdz, the bending moment element is dM= zdG=-Y22wdz/R and the bending moment is M=-YJ/R withJ=/22wdz=s(x)r2(r) the inertia moment per mass for the cross-section s(x)to the For the mass element p(r)dr of the bending rod, the inertia force is -p(x)drude, the external force is f(x, t)dr, and the external bending moment is m(x, t)dx. The equilibrium equation of moment to the left center C is
2. The mechanical model and the solutions We consider a rod of length l along the x-axis in equilibrium. The mass of the rod is m x xsx = ∫ d () ( ρ ) l 0 with dx being the line element, ρ( ) x being the volume density, and s() () () x wxhx = being the cross-section area. Here the width of the rod is y wx = ( ) and the thickness is z = h x( ). The turning radius of this cross-section r(x) satisfies r x = = ∫ h x z z h x ( ) 2 ( ) d ( ) 12 . (1) h x 2 0 ( ) 2 1 2 2 Figure 1 shows the diagram and cross-section of the thin homogenous rod. Let the mass element be d ( )d m x = ρ τ with the volume element d ( )d τ = sx x, one obtains the rotational inertia to the oy-axis as I = ρ τ h x d x ( ) 12 yy ( )d . (2) When the rod deforms transversely, its every cross-section should produce shearing forces. Let the shearing force on the left of volume element be Q(, ) x t (down direction), and the right one be Q′= +∂ Qx t Qx t x ( , ) ( , )d x (up direction) with ∂x being an abbreviation of ∂ ∂x. These two shearing forces form a force couple, which bends the rod. Figure 2 shows an element of a thin homogenous rod in bending. We consider the characteristic quantity of the transverse vibrations of a thin loaded rod, the displacement of the rod away from the equilibrium position along the z-direction at spacetime point (x,t), is ux t (, ). The curvature radius of the bending is = +∂ ∂ ⎡ ⎣ ⎤ ⎦ R uu 1 . (3) ( ) x xx 2 3 2 2 When the rod bends, the central line length dx remains unchanged. However, the upper part of the central line that suffers the tension of the nearby elements is prolonged; the lower part that suffers the pressure is compressed. Consequently, the force couple consists of tension and pressure, the so-called bending moment. Let the bending moment on the left of the volume element be Mxt ( , ) (clockwise), and the bending moment on the right be M Mxt Mxt x ′= +∂ ( , ) ( , )d x (counter-clockwise). The bending moments act as resistance to the bending of the rod, and lead the system to the dynamic equilibrium states. Figure 3 shows the force acting on an element of a thin homogenous rod. As shown in figure 2, we take a lamina with thickness dz at z position and width w(x). We recall that the length is dx at the center line of the volume element. The length of the lamina is (R z x zx + =+ )d d d θ R and the relative extension is z R. So that the tensile stress is P Yz = − R with Y being Young’s modulus. The tension element is dG = Pwdz, the bending moment element is dd d M z G Yz w z = =− R 2 , and the bending moment is M YJ R = − , (4) with J zw z sxr x = = ∫ d () ( ) 2 2 the inertia moment per mass for the cross-section s(x) to the center oy‐axis. For the mass element ρ( )d x τ of the bending rod, the inertia force is − ∂ ρ τ () d x utt 2 , the external force is fxt ( , )dτ, and the external bending moment is mxt x ( , )d . The equilibrium equation of moment to the left center C is Eur. J. Phys. 36 (2015) 055035 J Xu et al 3
Eur. J. Phy 2015)055035 J Xu et al gure 2. An element of thin homogenous rod in bending (M +M)-M=Qdx-gdxc + m(x, t)dx +p(r)drudxr-dx Omitting the higher order (2 order) small quantities, the reciprocal of the curvature radius is simplified as R-Ig d u(x, t)and equation(5)becomes M(x,t)=Q(x,1)+m(x,D) Without the gravity, the equilibrium equation of force acting in the transverse directions is p(x)dru(x, t)=-de(x, t)+f(r, t) Substituting M=-Ys(x)r2(x)d2 u(x, t)into equation(6)and combining equations(6)and () one obtains the general equation of motion as
( ) d d ( , )d ( ) d M M M Q x Qx mx t x x ux x + ′− = ′ − + + ∂ ρ 1 2 c d . (5) tt 2 Omitting the higher order (⩾2 order) small quantities, the reciprocal of the curvature radius is simplified as ≃ ∂ − R ux t (, ) xx 1 2 and equation (5) becomes ∂ =+ xMx t Qx t mx t ( , ) ( , ) ( , ). (6) Without the gravity, the equilibrium equation of force acting in the transverse directions is ρ x ux t ∂ =∂ + s x () (, ) Qx t f x t 1 ( ) ( , ) ( , ). (7) tt x 2 Substituting M Ys x r x u x t =− ∂ () () (, ) xx 2 2 into equation (6) and combining equations (6) and (7), one obtains the general equation of motion as Figure 1. Diagram and cross-section of a homogenous rod. Figure 2. An element of thin homogenous rod in bending. Eur. J. Phys. 36 (2015) 055035 J Xu et al 4