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

Authors: Rafael M. Digilov; Haim Abramovich
Journal: Advances in Materials Science and Engineering
 (2013)

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## 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.


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## Body

## 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 fn related to Young’s modulus E by the relation [15, 16]
(1)fn=λn22πl2EIρA,n=1,2,3,…
can be accurately measured. In (1) λn 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
(2)E=4π2fn2l4ρAλn4I.
The test sample is usually arranged in a manner to simulate free-free or clamped-free end conditions [10–12], when λn, 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 densityρ, 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 (KT) and torsion (KR) 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]
(3)ρA∂2z∂t2+EI∂4z∂x4=0
with boundary conditions at x=0:
(4)-KTz(0,t)=EI∂3z∂x3|x=0,KR∂z∂x|x=0=EI∂2z∂x2|x=0
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,
(5)∂2z∂x2|x=l=0,∂3z∂x3|x=l=0.
Equations (3)–(5) define completely the linear flexural vibration problem, in which the natural frequencies of the beam depend on spring constants KT and KR. Applying the separation of variables method, the solution of (3) can be cast in the following form:
(6)z(x,t)=w(x)T(t)=∑n=1∞wn(x)eiωnt,
where wn(x) describes the nth normal mode and ωn (=2πfn,  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
(7)∂4wn(x)∂x4-κn4wn(x)=0,
where κn is related to the angular frequency ωn and the modal number λn by the dispersion relationship
(8)κn4=(λnl)4=ωn2ρAEI.
With the boundary condition, (4), the solution of (7) admits the form of nth flexural eigenmode:
(9)wn(x)=An(coshκnx+ansinhκnx+bnsinκnx)+Cn(cosκnx-bnsinhκnx-ansinκnx),
where related coefficients bn and an are defined as
(10)an=1-λn4RnTn2λn3Rn,bn=1+λn4RnTn2λn3Rn
with Rn and Tn expressed through experimentally accessible quantities:
(11)Rn=KR(2πfn)2ρAl3,Tn=KT(2πfn)2ρAl,
and integration constants An and Cn are related by the modal restriction equation (5) as
(12)AnCn=cosλn+bnsinhλn-ansinλncoshλn+ansinhλn-bnsinλn=-sinλn-bncoshλn+ancosλnsinhλn+ancoshλn-bncosλn,
where the second equality implies the constraints on possible values of λn, known as the frequency equation:
(13)(1+λn4RnTn)-(1-λn4RnTn)cosλncoshλn-λn[(λn2Rn+Tn)sinλncoshλn0+(λn2Rn-Tn)sinhλncosλn]=0.
For given values of Rn and Tn, transcendental equation (13) has infinite (iterative or graphical) solutions. As Tn→0 and Rn→0 (13) becomes
(14)1-cosλncoshλn=0
that is the frequency equation for the free-free beam. For Tn≫Rn (13) reduces to the frequency equation derived by Chun [17], which in our case is written in the form
(15)Rnλn3(1+cosλncoshλn)-sinλncoshλn-sinhλncosλn=0
which, in turn, at Rn→∞ becomes the frequency equation for the clamped-free beam
(16)1+cosλncoshλn=0.Figure 1
Model of a cantilever beam elastically restrained atx=0.


## 3. Principles of Operation

The experimental setup consists of a commercial, Fourier Force Sensor DT 272 with rotationKR (=17.837 N·m) and translation KT (=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 ofm≈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, fs=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 fs/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 KR and KT, identified resonance frequencies fn are used in (11) to calculate dimensionless parameters Rn and Tn. Knowing Rn and Tn, modal number λn was found from the graphical solution of (13) using the mathematical packages MATHEMATICA [19]. The Young modulus is then determined from
(17)E=48π2ρfn2l4λn4h2.The flexural vibration test of partially restrained aluminum beam withl=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.
(a)
(b)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
(18)|d(lnE)|≤2|d(lnf1)|+|d(lnρ)|+4|d(lnl)|+2|d(lnh)|
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 widthd=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 Tn/Rn≫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 frequenciesfn 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.
l
/
d
T
n
/
R
n
f
1
λ
1
E
1
f
2
λ
2
E
2
8.13
60.6
37.82
1.595
111.6
232.79
4.068
100
9.06
75.4
31.14
1.619
110.3
193.83
4.14
100
9.813
88.4
26.84
1.638
107.4
170.23
4.179
102
10.69
104.9
23.07
1.653
107.9
147.82
4.215
104.7
12.25
137.8
17.9
1.679
105.2
115.24
4.276
103.7
14.69
198.2
13.0
1.701
108.9
84.10
4.326
109
15.5
220.7
11.75
1.709
108.2
76.25
4.372
106.5Figure 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 Table1, a double logarithm plot of fi/λi2 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 fil2/λi2 versus l/d. Most of the uncertainty in the ordinates of this plot arises from uncertainty of l rather than fi. In the range of l/d>10, the value of fil2/λi2 is constant to within the experimental uncertainties, showing that fi/λi2∝l-2 is in agreement with (1). For l/d<10, that is smaller for validity of the Euler-Bernoulli beam theory, the ratio fil2/λi2 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 offi(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 materialsE 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).
Material
l
h
ρ
f
1
λ
1
E
E
Literature
Al 6061 sheet
282
1.55
2715
14.42
1.778
70.0
70–72
Zn coated steel
193
0.55
7820
12.0
1.868
201
206
Sheet steel 304
193
0.5
7970
11.0
1.864
210
190–213
Cooper alloy
214
1.0
8790
11.89
1.848
106
110–120
Brass stripe
248
1.5
8400
11.75
1.709
108
96–110
Perspex
190
3.0
1190
23.2
1.798
4.2
2.4–4.6
Wood, oak
252
3.5
674
30.3
1.664
12.6
11–12
Compositea
232
2.5
1600
33.0
1.429
92.0
36–150
aGraphite carbon epoxy.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 λ14 versus 1/R1 in Figure 5 shows a linear dependence
(19)λ14=λ∞4(1-13R1),
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)
(20)E=48π2f12ρl4h2λ∞4(1-(4π2f12ρAl3/3KR)).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.



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