Flexural Vibration Test of a Beam Elastically Restrained at One End: A New Approach for Young’s Modulus Determination

Abstract

A new vibration beam technique for the fast determination of the dynamic Young modulus is developed. The method is based on measuring the resonant frequency of flexural vibrations of a partially restrained rectangular beam. The strip-shaped specimen fixed at one end to a force sensor and free at the other forms the Euler Bernoulli cantilever beam with linear and torsion spring on the fixed end. The beam is subjected to free bending vibrations by simply releasing it from a flexural position and its dynamic response detected by the force sensor is processed by FFT analysis. Identified natural frequencies are initially used in the frequency equation to find the corresponding modal numbers and then to calculate the Young modulus. The validity of the procedure was tested on a number of industrial materials by comparing the measured modulus with known values from the literature and good agreement was found.

1. Introduction

The Young modulus is a fundamental material property and its determination is common in science and engineering [1, 2]. It is a key parameter in mechanical engineering design to predict the behavior of the material under deformation forces or more to get an idea of the quality of the material. Young’s moduli are determined from static and dynamic tests. In static measurements [3, 4] 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. Dynamic methods [5–12] are more precise and versatile since they use very small strains, far below the elastic limit and therefore are virtually nondestructive allowing repeated testing of the same sample. These include the ultrasonic pulse-echo [6, 7] or bar resonance methods [4, 8–14]. In the sonic pulse technique, the dynamic Young modulus is determined by measuring the sound velocity in the sample. In the resonance method, the linear elastic, uniform, and isotropic material of density 𝜌 usually in the form of a bar of known dimensions is subjected to transverse or flexural vibrations, the natural frequency of 𝑛th mode of which 𝑓_𝑛 related to Young’s modulus 𝐸 by the relation [15, 16] 𝑓_𝑛=𝜆²_𝑛2𝜋𝑙²𝐸𝐼𝜌𝐴,𝑛=1,2,3,… (1) can be accurately measured. In (1) 𝜆_𝑛 is the modal eigenvalue that depends on boundary conditions, 𝑙 is the vibrating length of the bar, 𝐴 is its cross-sectional area, and 𝐼 is the second moment of the cross-section, equal to 𝜋𝑟⁴/4 for a rod of radius 𝑟 and 𝑑ℎ³/12 for a rectangular beam with width 𝑑 and depth ℎ. Knowing the modal numbers, by simply measuring the resonance frequencies, geometry, and density of the specimen, the Young modulus can be determined from (1) as 𝐸=4𝜋²𝑓²_𝑛𝑙⁴𝜌𝐴𝜆⁴_𝑛𝐼. (2) The test sample is usually arranged in a manner to simulate free-free or clamped-free end conditions [10–12], when 𝜆_𝑛, associated with the 𝑛th flexural mode is a constant.

In the present paper, we develop a new approach, in which a rectangular strip-shaped sample attached to a force sensor forms the Euler Bernoulli beam with partial translational and rotational restraints at the fixed end. This feature expands the capabilities of the resonant beam method making it suitable for materials with high stiffness and low density in which case, it is difficult to ascertain the flexural resonance frequencies with high certainty.

2. Theoretical Background

Consider a rectangular bar of uniform density 𝜌, cross-section dimensions of which width 𝑑 and depth ℎ are much less than length 𝑙. The bar is fixed at one end (𝑥=0) to the force sensor with a linear (𝐾_𝑇) and torsion (𝐾_𝑅) springs constants (see Figure 1) and is otherwise free to move in the transverse z-direction. For small deflections, that is, 𝜕𝑧(𝑥,𝑡)/𝜕𝑥≪1, the effects of rotary inertia and shear deformation can be ignored. In this case, neglecting the deflection due to the weight, the flexural displacement of a bar, 𝑧(𝑥,𝑡) at point 𝑥 is governed by the Euler-Bernoulli equation [10] 𝜕𝜌𝐴²𝑧𝜕𝑡²𝜕+𝐸𝐼⁴𝑧𝜕𝑥⁴=0 (3) with boundary conditions at 𝑥=0: −𝐾_𝑇𝑧()𝜕0,𝑡=𝐸𝐼³𝑧𝜕𝑥³_𝑥=0,𝐾_𝑅𝜕𝑧𝜕𝑥_𝑥=0𝜕=𝐸𝐼²𝑧𝜕𝑥²_𝑥=0 (4) which correspond to the force and moment balance, respectively. At the free end 𝑥=𝑙 there are no moment and shear force acting, that is, 𝜕²𝑧𝜕𝑥²_𝑥=𝑙𝜕=0,³𝑧𝜕𝑥³_𝑥=𝑙=0. (5) Equations (3)–(5) define completely the linear flexural vibration problem, in which the natural frequencies of the beam depend on spring constants 𝐾_𝑇 and 𝐾_𝑅. Applying the separation of variables method, the solution of (3) can be cast in the following form: 𝑧(𝑥,𝑡)=𝑤(𝑥)𝑇(𝑡)=^∞_𝑛=1𝑤_𝑛(𝑥)𝑒^{𝑖𝜔𝑛𝑡}, (6) where 𝑤_𝑛(𝑥) describes the 𝑛th normal mode and 𝜔_𝑛 (=2𝜋𝑓_𝑛,  𝑓: resonant frequency) is the angular frequency of the 𝑛th mode. Substituting (6) into (3) gives an eigenvalue problem in the form of a fourth-order ordinary differential equation 𝜕⁴𝑤_𝑛(𝑥)𝜕𝑥⁴−𝜅⁴_𝑛𝑤_𝑛(𝑥)=0, (7) where 𝜅_𝑛 is related to the angular frequency 𝜔_𝑛 and the modal number 𝜆_𝑛 by the dispersion relationship 𝜅⁴_𝑛=𝜆_𝑛𝑙⁴=𝜔²_𝑛𝜌𝐴.𝐸𝐼 (8) With the boundary condition, (4), the solution of (7) admits the form of 𝑛th flexural eigenmode: 𝑤_𝑛(𝑥)=𝐴_𝑛cosh𝜅_𝑛𝑥+𝑎_𝑛sinh𝜅_𝑛𝑥+𝑏_𝑛sin𝜅_𝑛𝑥+𝐶_𝑛cos𝜅_𝑛𝑥−𝑏_𝑛sinh𝜅_𝑛𝑥−𝑎_𝑛sin𝜅_𝑛𝑥, (9) where related coefficients 𝑏_𝑛 and 𝑎_𝑛 are defined as 𝑎_𝑛=1−𝜆⁴_𝑛𝑅_𝑛𝑇_𝑛2𝜆³_𝑛𝑅_𝑛,𝑏_𝑛=1+𝜆⁴_𝑛𝑅_𝑛𝑇_𝑛2𝜆³_𝑛𝑅_𝑛 (10) with 𝑅_𝑛 and 𝑇_𝑛 expressed through experimentally accessible quantities: 𝑅_𝑛=𝐾_𝑅2𝜋𝑓_𝑛²𝜌𝐴𝑙³,𝑇_𝑛=𝐾_𝑇2𝜋𝑓_𝑛²,𝜌𝐴𝑙 (11) and integration constants 𝐴_𝑛 and 𝐶_𝑛 are related by the modal restriction equation (5) as 𝐴_𝑛𝐶_𝑛=cos𝜆_𝑛+𝑏_𝑛sinh𝜆_𝑛−𝑎_𝑛sin𝜆_𝑛cosh𝜆_𝑛+𝑎_𝑛sinh𝜆_𝑛−𝑏_𝑛sin𝜆_𝑛=−sin𝜆_𝑛−𝑏_𝑛cosh𝜆_𝑛+𝑎_𝑛cos𝜆_𝑛sinh𝜆_𝑛+𝑎_𝑛cosh𝜆_𝑛−𝑏_𝑛cos𝜆_𝑛, (12) where the second equality implies the constraints on possible values of 𝜆_𝑛, known as the frequency equation: 1+𝜆⁴_𝑛𝑅_𝑛𝑇_𝑛−1−𝜆⁴_𝑛𝑅_𝑛𝑇_𝑛cos𝜆_𝑛cosh𝜆_𝑛−𝜆_𝑛𝜆²_𝑛𝑅_𝑛+𝑇_𝑛sin𝜆_𝑛cosh𝜆_𝑛+𝜆²_𝑛𝑅_𝑛−𝑇_𝑛sinh𝜆_𝑛cos𝜆_𝑛=0. (13) For given values of 𝑅_𝑛 and 𝑇_𝑛, transcendental equation (13) has infinite (iterative or graphical) solutions. As 𝑇_𝑛→0 and 𝑅_𝑛→0 (13) becomes 1−cos𝜆_𝑛cosh𝜆_𝑛=0 (14) that is the frequency equation for the free-free beam. For 𝑇_𝑛≫𝑅_𝑛 (13) reduces to the frequency equation derived by Chun [17], which in our case is written in the form 𝑅_𝑛𝜆³_𝑛1+cos𝜆_𝑛cosh𝜆_𝑛−sin𝜆_𝑛cosh𝜆_𝑛−sinh𝜆_𝑛cos𝜆_𝑛=0 (15) which, in turn, at 𝑅_𝑛→∞ becomes the frequency equation for the clamped-free beam 1+cos𝜆_𝑛cosh𝜆_𝑛=0. (16)

Figure 1 

Model of a cantilever beam elastically restrained at 𝑥=0.

3. Principles of Operation

The experimental setup consists of a commercial, Fourier Force Sensor DT 272 with rotation 𝐾_𝑅 (=17.837 N·m) and translation 𝐾_𝑇 (=6400 N·m⁻¹) spring constants, accuracy ±2%, and resolution (12 bit) 0.005N for a scale range ±10N. The force sensor mounted on a support is connected through data acquisition system and data studio software to a personal computer (PC) as is shown in Figure 2.

Figure 2 

Equipment used for measuring Young’s modulus with the force sensor.

The strip-shaped specimen with a roughly mass of 𝑚≈10g is attached horizontally to a force sensor at one end and is free at the other end. By simply displacing the free end in the transverse direction and abruptly releasing it, the sample is subjected to free flexural vibrations, so that the condition 𝜕𝑧/𝜕𝑥≪1 was fulfilled. At a given sample rate, 𝑓_𝑠=1000Hz the force sensor detects the dynamic response and through the data acquisition system displays the damped oscillations of the restoring force versus time (Figure 3(a)). This vibration signal is analyzed and processed by an FFT implemented in acquisition software like MultiLogPRO [18] providing the direct information about natural frequencies up to 𝑓_𝑠/2, which appears as single peaks in the frequency spectrum (Figure 3(b)). At this point, using data on the geometrical dimensions of the sample, its density, and spring constants of the force sensor 𝐾_𝑅 and 𝐾_𝑇, identified resonance frequencies 𝑓_𝑛 are used in (11) to calculate dimensionless parameters 𝑅_𝑛 and 𝑇_𝑛. Knowing 𝑅_𝑛 and 𝑇_𝑛, modal number 𝜆_𝑛 was found from the graphical solution of (13) using the mathematical packages MATHEMATICA [19]. The Young modulus is then determined from 𝐸=48𝜋²𝜌𝑓²_𝑛𝑙⁴𝜆⁴_𝑛ℎ². (17)

(a)

(b)

Figure 3 

The flexural vibration test of partially restrained aluminum beam with 𝑙=0.282m, 𝑑=1.35cm, and ℎ=1.55mm. (a) Dynamic response detected by the force sensor at sample rate 1kHz. (b) Identified resonance frequencies: 𝑓1=14.7Hz; 𝑓2=92.8Hz; 𝑓3=249.5Hz.

The relative error of the method arises from the uncertainties in the measurement of the quantities in (17). The relative error in the density 𝜌 determined by the hydrostatic method is |𝑑(ln𝜌)|≤0.7%. The uncertainty in the thickness and length measurement are, respectively, |𝑑(lnℎ)|≤0.4% and |𝑑(ln𝑙)|≤0.25%. The dispersion in the natural frequency determination is |𝑑(ln𝑓₁)|≤0.25%. By applying the error propagation technique, given by |𝑑(ln𝐸)|≤2𝑑ln𝑓₁+𝑑ln𝜌+4|𝑑(ln𝑙)|+2|𝑑(lnℎ)| (18) we find that the relative error in Young’s modulus does not exceed ±3%. The greatest inaccuracies occurred in the measurement of the spacemen dimensions.

4. Results and Discussion

To test the accuracy and validity of the present method, the effect of the sample length on the resonance frequency and Young’s modulus was studied. A commercial brass strip of width 𝑑=16mm, thickness ℎ=1.5mm, and density 𝜌 = 8400 kg·m⁻³ was cut into samples of various length, so that one of the conditions of the Euler-Bernoulli beam theory 𝑙/ℎ>10 remained unchanged, while the second 𝑙/𝑑 ranged from 8 to 16. Since the ration 𝑇_𝑛/𝑅_𝑛≫1 (see Table 1) we used the approximated equation (15) to find the 𝜆₁. The validity of this approach is illustrated in Figure 4, which shows that the solutions of both (15) and (13) practically coincide.

Table 1 

Identified resonant frequencies 𝑓_𝑛 in Hz, modal numbers determined from (13), and Young’s moduli in GPa evaluated by (17) for the brass specimens of different lengths, 𝑑=16 mm, ℎ=1.5 mm, and 𝜌=8400 kg⋅m⁻³.

Figure 4 

Graphical solutions of (13) and (15) for the first mode of flexural vibrations of partially restrained aluminum beam are, respectively, 𝜆1=1.7752 and 𝜆1=1.7745 against 𝜆1=1.8751 for ideal clamped-free case.

Based on the results of measurements of the natural frequencies and modal numbers presented in Table 1, a double logarithm plot of 𝑓_𝑖/𝜆²_𝑖 against 𝑙 predicted by (1) to be linear with the gradient of −2 shows that the slope of the line that best fits these data in a least-squares sense is −2.02 for the first mode and −1.94 for the second one. Based on the same set of data we show in Figure 5 the plot of the ratio 𝑓_𝑖𝑙²/𝜆²_𝑖 versus 𝑙/𝑑. Most of the uncertainty in the ordinates of this plot arises from uncertainty of 𝑙 rather than 𝑓_𝑖. In the range of 𝑙/𝑑>10, the value of 𝑓_𝑖𝑙²/𝜆²_𝑖 is constant to within the experimental uncertainties, showing that 𝑓_𝑖/𝜆²_𝑖∝𝑙⁻² is in agreement with (1). For 𝑙/𝑑<10, that is smaller for validity of the Euler-Bernoulli beam theory, the ratio 𝑓_𝑖𝑙²/𝜆²_𝑖 decreases (increases) with 𝑙 for the first (second) mode. For 𝑙/𝑑≥10, the mean value of Young’s modulus 𝐸=106±2GPa lies within the range 95÷110GPa, listed in an extensive table of ASTM testing [20].

Figure 5 

Plot of 𝑓𝑖(𝑙/𝜆𝑖)2 versus 𝑙/𝑑 based on the flexural vibration test of a rectangular brass strip (𝑑=16.0mm and ℎ=1.5mm). For 𝑙/𝑑≥10, the mean value 0.245 (broken line) is constant to within experimental uncertainty.

Below, we compare the results of measurements of elastic moduli at ambient temperature for a wide class of industrial materials with that accepted in the literature. A set of test specimens used were cut from the commercial sheet materials into strips of the thickness (ℎ) from 0.5 to 3.3 mm, width (𝑑) from 5 to 16.5 mm, and length (𝑙) from 15 to 30 cm. For each specimen, the length, width, and thickness were altered and the value of the Young modulus calculated by (17) for each set varied within the experimental error. Table 2 summarizes data of the specimen dimension, material density, natural frequencies, modal numbers, and Young’s modulus calculated from the first resonant frequency. It can be seen that test results are in good agreement with the accepted those in the literature data.

Table 2 

Young’s moduli of materials 𝐸 in GPa, determined at room temperature by (17) and specimen parameters: length 𝑙 in mm, thickness ℎ in mm, density 𝜌 in kg⋅m⁻³, fundamental frequency 𝑓₁ in Hz, and modal number 𝜆₁ calculated by (13).

In all cases, identified natural frequencies of the partially restrained cantilever beam are lower than those for the clamped-free case. However, this does not necessary mean that the elastic modulus determined from these frequencies should be smaller than in ideal clamped-free case as the modal number, being in the denominator equation (17) in fourth power, decreases as well. Interestingly, the plot of 𝜆⁴₁ versus 1/𝑅₁ in Figure 5 shows a linear dependence 𝜆⁴₁=𝜆⁴_∞11−3𝑅₁, (19) where 𝜆_∞=1.8751 is the first modal number for the clamped-free flexural vibrations of the beam. Taking into account (11), after substituting (19) in (17), we obtain the working equation for the Young modulus determination through the first resonance frequency of flexural vibrations of the partially restrained cantilever beam (Figure 6) 𝐸=48𝜋²𝑓²₁𝜌𝑙⁴ℎ²𝜆⁴_∞1−4𝜋²𝑓²₁𝜌𝐴𝑙³/3𝐾_𝑅. (20)

Figure 6 

The modal number of the first mode of flexural vibrations of partially restrained cantilever beams in the fourth degree as a function of the inverse of the dimensionless frequency parameter, 1/𝑅1. Symbols are experimental data taken from Tables 1 and 2. Solid line is by (19).

5. Summary

A new technique for the fast determination of the dynamic Young modulus was developed, yielding a substantial modification of the classical cantilever beam method. The procedure uses a rectangular beam, partially restrained at one end, flexural vibrations of which are detected with the aid of the force sensor. The relative experimental uncertainty is found to be less than 3%, which is mainly due to the uncertainty in the samples dimensions. The feasibility and accuracy of a new experimental procedure has been demonstrated by measuring the Young modulus for a number of test materials with different material properties. Comparison of obtained results with those accepted in the literature data is good. The relative deviation of measured values from the cited data is less than 5%. The method has potential advantages over other dynamic methods of being very simple and fast and requiring no additional equipment to excite resonance frequencies. It is particularly suitable for composite materials having a high stiffness and low density, such as carbon fiber reinforced plastic. The accuracy can be significantly improved by more precise determination of specimen dimensions.

Acknowledgment

This work was supported by the Ministry of Absorption and Immigration of Israel through the KAMEA Science Foundation.