Direct Determination of Dynamic Elastic Modulus and Poisson's

vibration

Article Direct Determination of Dynamic Elastic Modulus and Poisson’s Ratio of Timoshenko Rods

Guadalupe Leon and Hung-Liang (Roger) Chen * Department of Civil and Environmental Engineering, West Virginia University, Morgantown, WV 26506, USA; [email protected] * Correspondence: [email protected]

Received: 4 December 2018; Accepted: 1 March 2019; Published: 7 March 2019

Abstract: In this paper, the exact solution of the Timoshenko circular beam vibration frequency equation under free-free boundary conditions was determined with an accurate shear shape factor. The exact solution was compared with a 3-D finite element calculation using the ABAQUS program, and the difference between the exact solution and the 3-D finite element method (FEM) was within 0.15% for both the transverse and torsional modes. Furthermore, relationships between the resonance frequencies and Poisson’s ratio were proposed that can directly determine the elastic constants. The frequency ratio between the 1st bending mode and the 1st torsional mode, or the frequency ratio between the 1st bending mode and the 2nd bending mode for any rod with a length-to-diameter ratio, L/D ≥ 2 can be directly estimated. The proposed equations were used to verify the elastic constants of a steel rod with less than 0.36% error percentage. The transverse and torsional frequencies of concrete, aluminum, and steel rods were tested. Results show that using the equations proposed in this study, the Young’s modulus and Poisson’s ratio of a rod can be determined from the measured frequency ratio quickly and efficiently.

Keywords: Timoshenko beam; dynamic elastic modulus; Poisson’s ratio; ﬁnite element; non- destructive testing

1. Introduction Non-destructive testing (NDT) methods are often used to estimate the elastic properties of a material such as the dynamic elastic modulus and Poisson’s ratio. There are several types of non-destructive testing (NDT) methods available. These include ultrasonic pulse velocity methods, resonance frequency methods, and other wave propagation techniques. Material properties measured with these methods are called dynamic elastic constants. Researchers have used forced vibration and impulse vibration to measure the dynamic elastic constants of steel plates [1] as well as impact-echo resonance and Rayleigh wave velocity [2]. Kolluru et al. proposed a technique to determine the elastic material constants of a standard test cylinder using the longitudinal resonance frequencies of concrete, steel, and aluminum [3]. The ratio between the 2nd and 1st longitudinal resonance frequencies was used to determine the Poisson’s ratio for a variety of length-to-diameter ratios (L/D) and afterward the elastic modulus was estimated. Similarly, Chen and Leon proposed a method to ﬁnd the two elastic constants using the 1st bending mode and the 1st torsional for rectangular prisms [4]. Nieves et al. also estimated the shear modulus and Poisson’s ratio for short cylinders using the two lowest axisymmetric natural frequencies [5]. Veriﬁcation of their methods was made using various cylindrical specimens with an L/D less than 3. Wang et al. proposed a method to evaluate Poisson’s ratio of a rod with only two experimental impact-echo values (fundamental longitudinal and cross-sectional resonant frequencies). However, their methods require an L/D ≥ 25υ for the dynamic Poisson’s ratio to be valid [6].

Vibration 2019, 2, 157–173; doi:10.3390/vibration2010010 www.mdpi.com/journal/vibration Vibration 2019, 2 158

ASTM C215 (2014) and ASTM E1876 (2015) standards are widely used to determine the dynamic elastic modulus using the fundamental resonance frequency. The equations provided in these standards were formulated by simplifying the free-free vibration solution of a Timoshenko beam with the effects of shear and rotational inertia reduced to a simple correction constant. However, the value of the elastic modulus cannot be directly calculated without knowing the material’s Poisson’s ratio. Therefore, in order to calculate the dynamic modulus and Poisson’s ratio, a lengthy iteration process is needed. In this paper, the exact solution of the Timoshenko beam vibration frequency equation under free-free boundary conditions was determined. The fundamental transverse and torsional frequencies were used to determine the dynamic elastic modulus and Poisson’s ratio of rods with an L/D ≥ 2. Then, relationships between the resonance frequencies and Poisson’s ratio were proposed that can directly determine the elastic modulus and the Poisson’s ratio, simultaneously without the need for iteration, using the frequency ratio between the 1st bending mode and the 1st torsional mode, or the frequency ratio between the 2nd bending mode and the 1st bending mode.

2. Analysis For a homogenous and isotropic material, the Timoshenko beam vibration equation can be expressed as follows:

∂4y ∂2y E ∂4y ρ2 I ∂4y EI + m − ρI 1 + + = 0. (1) ∂x4 ∂t2 k0G ∂x2∂t2 k0G ∂t4

In the equation, y is the transverse displacement; t is time; k0 is a shape factor that depends only on Poisson’s ratio; G is the shear modulus; E is the Young’s modulus; I is the moment of inertia of the cross-section; ρ is the mass density; and m is the mass per unit length. In Timoshenko’s publication [7], 3 a shape factor of 4 was assumed for a circular cross-section. However, Hutchinson derived more accurate shape factors for a variety of geometries which match more closely with experimental results [8]. In this paper, Hutchinson’s shape factor, Equation (2), was used to calculate the shape factor for a circular cross-section: 6(1 + v)2 k0 = . (2) 7 + 12v + 4v2 For a rod with free-free boundary conditions, the following can be applied:

00 00 000 000 y (x)|x=0 = 0; y (x)|x=L = 0; y (x)|x=0 = 0; y (x)|x=L = 0.

Using these boundary conditions for a nontrivial solution, the frequency equation can be written as follows [9,10]:

2 γ2 γ4r2r2 − s2 + 3r2 − s2 2 − 2 cos h(α) cos(β) + p sinh(α) sin(β) = 0, (3) 1 − γ4r2s2

4 2 = I 2 = EI γ4 = ρAL 2 where r AL2 , s k0GAL2 , EI ω , and A is the cross-sectional area; α and β represent the eigenvalues of the frequency equation. The eigenvalues α and β can be expressed as follows: v v 2 u s 2 u s γ u 2 2 2 2 2 4 γ u 2 2 2 2 2 4 αi = √ t−(r + s ) + (r − s ) + and βi = √ t(r + s ) + (r − s ) + . (4) 2 γ4 2 γ4

In this paper, a MATLAB program was used to solve for the eigenvalues of the frequency equation. The natural frequencies (Hz) can be expressed using the eigenvalues [11]: Vibration 2019, 2 159

s γ2 EI n = i . (5) i 2π ρAL4

Let i = 1 to obtain the 1st bending mode and i = 2 for the 2nd bending mode.

2.1. Finite Element Analysis Vibration 2019, 3 FOR PEER REVIEW 3 Dynamic analysis using the 3-D ﬁnite element method (FEM) was carried out and compared to the exactDynamic solution analysis for the using transverse the 3‐D and finite torsional element modesmethod of (FEM) a rod. was The carried density out and was compared assumed to to be 2400thekg exact m−3 .solution Six different for the sets transverse of elastic and modulus torsional and modes Poisson’s of a ratiorod. The were density used in was the assumed FEM simulations to be as shown2400 kg in m Table. Six1. different sets of elastic modulus and Poisson’s ratio were used in the FEM simulations as shown in Table 1. Table 1. Elastic modulus and Poisson’s ratio assumed in the ﬁnite element method (FEM). Table 1. Elastic modulus and Poisson’s ratio assumed in the finite element method (FEM). Test Elastic Modulus (GPa) Poisson’s Ratio Test Elastic Modulus (GPa) Poisson’s Ratio 1 125.0 25.0 0.1 0.1 2 225.0 25.0 0.15 0.15 3 25.0 0.2 3 25.0 0.2 4 10.0 0.2 4 510.0 20.0 0.2 0.2 5 620.0 30.0 0.2 0.2 6 30.0 0.2 A total of 199,950 elements of 2.0 mm (0.08 in) cubes, 8-node linear brick with reduced integration A total of 199,950 elements of 2.0 mm (0.08 in) cubes, 8‐node linear brick with reduced andintegration hourglass controland hourglass elements control (C3D8R) elements from (C3D8R) the ABAQUS from the program ABAQUS were program utilized. were A 75 utilized. mm (diameter) A 75 × 300mm mm (diameter) (length) × rod300 wasmm modeled(length) rod as awas free-free modeled beam. as a Thefree‐ 1stfree and beam. 2nd The bending 1st and modes 2nd bending as well as themodes 1st torsional as well mode as the are1st torsional shown in mode Figure are1. shown in Figure 1.

(a) (b)

(c)

Figure 1. Mesh of 75 mm (D) × 300 mm (L) rod: (a) 1st bending mode: 2101.5 Hz, (b) 2nd bending Figure 1. Mesh of 75 mm (D) × 300 mm (L) rod: (a) 1st bending mode: 2101.5 Hz, (b) 2nd bending mode: 4910.6 Hz, (c) 1st torsional mode: 3622.1 Hz. mode: 4910.6 Hz, (c) 1st torsional mode: 3622.1 Hz.

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A comparison between the FEM and the analytical solutions is shown in Table2. Results show that the percent error for all the cases is less than 0.13%. This indicates that the elastic modulus and Poisson’s ratio obtained from the roots of the Timoshenko beam frequency equation in the analysis section above is accurate and the 1st free-free torsional mode from the Euler–Bernoulli beam solution, q 00 1 G n = 2L ρR , is also close to exact, where R is the ratio between the polar moment of inertia and the equivalent moment of inertia, for a rod R = 1.0. Since the FEM usually requires large amounts of computations using commercial software, it would be more efﬁcient to use the analytical solutions to determine the transverse and torsional vibration frequencies.

Table 2. Comparison of the FEM and the analytical solutions.

Transverse Mode (Hz) Torsional Mode (Hz) Test 1st 2nd Exact FEM Error (%) Exact FEM Error (%) Exact FEM Error (%) 1 2103.083 2101.5 0.08% 4914.196 4910.6 0.07% 3626.66 3622.1 0.13% 2 2100.394 2098.9 0.07% 4896.9 4894.0 0.06% 3546.943 3542.4 0.13% 3 2097.75 2096.3 0.07% 4879.969 4877.4 0.05% 3472.262 3467.8 0.13% 4 1326.551 1325.6 0.07% 3085.938 3084.3 0.05% 2195.749 2193.0 0.13% 5 1876.349 1875.0 0.07% 4364.927 4362.6 0.05% 3105.793 3101.8 0.13% 6 2297.917 2296.3 0.07% 5345.615 5342.8 0.05% 3803.585 3798.8 0.13%

2.2. Frequency Ratio Analysis The ratio between the fundamental torsional frequency and the fundamental transverse frequency, χt, can be expressed as follows [11]:

n00 2 χt = 2 (v + 1), (6) n1

00 where n is the fundamental torsional frequency (Hz) and n1 is the 1st fundamental transverse frequency (Hz). Using the roots of the Timoshenko frequency equation, Equation (3), and the solution for the torsional mode, the frequency ratio, χt, is expressed as Equation (7):

L 2 χ = π2 AL2/RIγ4 = 16π2 /γ4. (7) t 1 D 1

Equation (7) was evaluated for a variety of Poisson’s ratios and rod dimensions as shown in Figure2. From Figure2, a linear relationship between the Poisson’s ratio, v, and the ratio χt can be observed. A clear linear trend can be seen after normalizing the frequency ratio, χt , with χt at v = 0 as shown in Figure2c. It was found that the slope of the line and the y-intercept are independent of the elastic modulus and the mass density. The frequency ratio, χt, only depends on the diameter and the length of the rod as shown in Equation (7). Hence, the ratio χt can be expressed as Equation (8):

n00 2 χt = 2 (v + 1) = A1·v + B1. (8) n1

The Poisson’s ratio can then be calculated using the equation below:

2 n00 2 n − B1 v = 1 . (9) 00 2 A − 2 n 1 n1 Vibration 2019, 2 161 Vibration 2019, 3 FOR PEER REVIEW 5

(a) (b)

(c)

Figure 2. Relationship between 𝜒 and Poisson’s ratio with different rod dimensions (diameter × Figure 2. Relationship between χt and Poisson’s ratio with different rod dimensions (diameter × length): (a) 100 mm × 400 mm, (b) 20 mm × 160 mm, and (c) normalized 𝜒 for different L/D ratios length): (a) 100 mm × 400 mm, (b) 20 mm × 160 mm, and (c) normalized χt for different L/D ratios ((L/DL/D == 2, 2, 4, 4, 8, 8, and and 10). 10).

Table 3. Values of 𝐵 and 𝐴 for different length/diameter ratios. Furthermore, after knowing Poisson’s ratio, the elastic modulus can be determined using the experimental torsional frequency with EquationLength/Diameter (10): (L/D) Coefficient 2 3 4 5 8 10 20 50 100 2 00 2 𝐵 2.59893 4.25642 6.50793 E9.37206= 8ρL n21.7075(1 + v ).33.0728 127.7261 790.2198 3156.257 (10) 𝐴 0.26573 0.30686 0.33084 0.34551 0.36556 0.37104 0.37898 0.38135 0.38617 The values of A1 and B1 for different specimen dimensions were found by calculating χt for different length/diameter ratios as well as varying Poisson’s ratio (shown in AppendixA). The constant B was As shown in Figures 3 and 4, polynomials can be used to express the values of 𝐵 and1 𝐴 . Althoughfound by calculatingonly nine L/Dχt when points Poisson’s are shown ratio in wasTable equal 3, a tototal zero. of The97 L/D calculated data points values were for Bused1 are to shown find in Table3. A similar procedure was carried out to determine A . After plotting χ , the slope of the the best‐fit polynomials. Due to the wide range of 𝐵, shown in1 Figure 3, a singlet polynomial will havebest-fit a 3% line difference was used between to find A the1. Table estimated3 shows values the calculated and the theoretical value of A 1valuesfor different found length/diameterusing Equation (8)(L/D when) ratios. 10. Therefore, to reduce the difference to less than 0.06%, two polynomials (sixth‐order) are needed to express 𝐵 for two separate intervals, Equations (11) and (12). Table 3. Values of B1 and A1 for different length/diameter ratios.

Length/Diameter (L/D) Coefﬁcient 2 3 4 5 8 10 20 50 100

B1 2.59893 4.25642 6.50793 9.37206 21.7075 33.0728 127.7261 790.2198 3156.257 A1 0.26573 0.30686 0.33084 0.34551 0.36556 0.37104 0.37898 0.38135 0.38617

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As shown in Figures3 and4, polynomials can be used to express the values of B1 and A1. Although only nine L/D points are shown in Table3, a total of 97 L/D data points were used to ﬁnd the best-ﬁt polynomials. Due to the wide range of B1, shown in Figure3, a single polynomial will have a 3% difference between the estimated values and the theoretical values found using Equation (8) L when D < 10. Therefore, to reduce the difference to less than 0.06%, two polynomials (sixth-order) are B Vibrationneeded 2019 to express, 3 FOR PEER1 for REVIEW two separate intervals, Equations (11) and (12). 6

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(a) (b) (a) (b) Figure 3. Plot of B1 versus the length/diameter ratio: (a) L10 and (b) L10. Figure 3. Plot of B versus the length/diameter ratio: (a) ≤ 10 and (b) > 10. Figure 3. Plot of1 B1 versus the length/diameter ratio: (a) 10D and (b) 10D .

FigureFigure 4. 4.Plot Plot of ofA A11versus versus diameter/length (D/L (D/L) ratio.) ratio.

Figure 4. Plot of A1 versus diameter/length (D/L) ratio. For 10, L For D ≤ 10, For 10, 𝐵 2.20790 10 9.69895 10 1.78941 10 6 5 4 −6 L −5 L −3 L (11) B = −2.20790 × 10 + 9.69895 × 10 − 1.78941 × 10 1 0.0179955 D 0.208881 0.368369 D 0.908491. D 𝐵 2.20790 10 9.69895 10 1.78941 10 (11) 3 2 L L L (11) For 10, +0.0179955 D + 0.208881 D + 0.368369 D + 0.908491. 0.0179955 0.208881 0.368369 0.908491. L 𝐵 2.39473 10 1.38944 10 3.08859 10 ForFor 10> 10,, D (12) 3.34001 10 0.315287 4.89786 10 1.49164. 6 5 4 𝐵 2.39473 10−13 L 1.38944 10−10 L 3.08859 10−8 L B1 = −2.39473 × 10 D + 1.38944 × 10 D − 3.08859 × 10 D + Unlike B1, only a single polynomial expression is needed for A1. Equation (13) was found from (12) 3 2 (12) the best‐fit curve3.34001 shown 10 in Figure−L 4. Equation 0.315287 (13) canL be 4.89786 used to obtain 10 −𝐴 forL 1.49164any diameter/length. 3.34001 × 10 6 + 0.315287 + 4.89786 × 10 3 + 1.49164. (D/L) ratio: D D D Unlike B1, only a single polynomial expression is needed for A1. Equation (13) was found from 𝐴 9.44922 13.8392 5.60087 1.08665 1.27914 the best‐fit curve shown in Figure 4. Equation (13) can be used to obtain 𝐴 for any diameter/length (13) (D/L) ratio: 5.47273 10 0.381757. 𝐴Although 9.44922 the method presented 13.8392 above can 5.60087 be used to determine 1.08665 the dynamic 1.27914 elastic constants easily and quickly, it has two disadvantages. First, it requires a precise measurement of the torsional (13) 5.47273 10 0.381757. mode which is difficult to excite in a cylindrical specimen [6,12]. Additionally, it requires two separate experimental tests. Therefore, it would be beneficial to develop a method based on only the transverse Although the method presented above can be used to determine the dynamic elastic constants bending modes which can be measured simultaneously and accurately. Similar to 𝜒 , the ratio easily and quickly, it has two disadvantages. First, it requires a precise measurement of the torsional mode which is difficult to excite in a cylindrical specimen [6,12]. Additionally, it requires two separate experimental tests. Therefore, it would be beneficial to develop a method based on only the transverse bending modes which can be measured simultaneously and accurately. Similar to 𝜒 , the ratio

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Unlike B1, only a single polynomial expression is needed for A1. Equation (13) was found from the best-ﬁt curve shown in Figure4. Equation (13) can be used to obtain A1 for any diameter/length (D/L) ratio:

6 5 4 3 2 D D D D D A1 = 9.44922 L − 13.8392 L + 5.60087 L + 1.08665 L − 1.27914 L + (13) −3 D 5.47273 × 10 L + 0.381757.

Although the method presented above can be used to determine the dynamic elastic constants easily and quickly, it has two disadvantages. First, it requires a precise measurement of the torsional mode which is difﬁcult to excite in a cylindrical specimen [6,12]. Additionally, it requires two separate experimental tests. Therefore, it would be beneﬁcial to develop a method based on only the transverse bending modes which can be measured simultaneously and accurately. Similar to χt, the ratio between the 2nd bending mode and the 1st bending mode was used to directly calculate Poisson’s ratio and the dynamic elastic modulus. The ratio between the 2nd and 1st fundamental transverse frequencies, χb, can be expressed with the following relationship: n 2 γ4 χ = 2 = 2 , (14) b n 4 1 γ1 where n2 is the 2nd fundamental transverse frequency (Hz). γ1 and γ2 are the roots from the frequency equation, shown in Equation (3), for the 1st and the 2nd bending modes. For a very slender rod, the relationship shown in Equation (15) can be derived to express χb for a Euler–Bernoulli rod with free-free boundary conditions: (7.853)4 χb = = 7.598. (15) (4.73)4

However, for the vibration of a non-slender rod, χb can only be expressed with the roots of the Timoshenko frequency equation using Equation (3). Hence, Equation (14) was evaluated for a variety of Poisson’s ratios and rod dimensions as shown in Figure5. From Figure5, a slightly parabolic relationship between the Poisson’s ratio, v, and the ratio χb can be observed. A clear trend can be seen after normalizing the frequency ratio, χb, with χb at v = 0 as shown in Figure5c. The y-intercept, E1, corresponds to the value of χb when v = 0. It was found that the frequency ratio, χb, is independent of the elastic modulus and the mass density, and χb only depends on the roots of Equation (3). Hence, the ratio χb can be expressed as Equation (16):

2 n2 2 χb = = C1 v + D1ν + E1. (16) n1

C1, D1, and E1 are the coefﬁcients of the second-order polynomial describing the relationship between χb and Poisson’s ratio. Table4 shows the calculated values of γ1, γ2, and χb due to different Poisson’s ratio and length/diameter (L/D). As expected, as the L/D ratio becomes very large, the values of γ1 and γ2 approach 4.73 and 7.853. They are the same values shown in Equation (15) and correspond to a Euler–Bernoulli beam. For a very slender rod (L/D = 200), χb is no longer dependent on Poisson’s ratio and approaches a constant value. The Poisson’s ratio can then be calculated using the equation below: s 2 −D − D2 − 4C E − n2 1 1 1 1 n1 v = . (17) 2C1 Vibration 2019, 3 FOR PEER REVIEW 7 between the 2nd bending mode and the 1st bending mode was used to directly calculate Poisson’s ratio and the dynamic elastic modulus.

The ratio between the 2nd and 1st fundamental transverse frequencies, 𝜒, can be expressed with the following relationship:

𝜒 , (14) where 𝑛 is the 2nd fundamental transverse frequency (Hz). 𝛾 and 𝛾 are the roots from the frequency equation, shown in Equation (3), for the 1st and the 2nd bending modes. For a very slender

rod, the relationship shown in Equation (15) can be derived to express 𝜒 for a Euler–Bernoulli rod with free‐free boundary conditions:

. 𝜒 7.598. (15) .

However, for the vibration of a non‐slender rod, 𝜒 can only be expressed with the roots of the Timoshenko frequency equation using Equation (3). Hence, Equation (14) was evaluated for a variety of Poisson’s ratios and rod dimensions as shown in Figure 5. From Figure 5, a slightly parabolic

relationship between the Poisson’s ratio, v, and the ratio 𝜒 can be observed. A clear trend can be seen after normalizing the frequency ratio, 𝜒, with 𝜒 at v = 0 as shown in Figure 5c. The y‐intercept, 𝐸 , corresponds to the value of 𝜒 when v = 0. It was found that the frequency ratio, 𝜒 , is independent of the elastic modulus and the mass density, and 𝜒 only depends on the roots of Equation (3). Hence, the ratio 𝜒 can be expressed as Equation (16): 𝜒 𝐶 𝑣 𝐷𝜈𝐸. (16)

𝐶,𝐷,and 𝐸 are the coefficients of the second‐order polynomial describing the relationship between 𝜒 and Poisson’s ratio. Table 4 shows the calculated values of 𝛾 , 𝛾 , and 𝜒 due to different Poisson’s ratio and length/diameter (L/D). As expected, as the L/D ratio becomes very large, the values of 𝛾 and 𝛾 approach 4.73 and 7.853. They are the same values shown in Equation (15) andVibration correspond2019, 2 to a Euler–Bernoulli beam. For a very slender rod (L/D = 200), 𝜒 is no longer164 dependent on Poisson’s ratio and approaches a constant value.

Vibration 2019, 3 FOR PEER REVIEW 8 (a) (b)

(c)

Figure 5. Relationship between 𝜒 and Poisson’s ratio with different rod dimensions (diameter × Figure 5. Relationship between χb and Poisson’s ratio with different rod dimensions (diameter × length): (a) 100 mm × 400 mm, (b) 20 mm × 160 mm, and (c) normalized 𝜒 for different L/D ratios length): (a) 100 mm × 400 mm, (b) 20 mm × 160 mm, and (c) normalized χb for different L/D ratios ((L/DL/D == 2, 2, 4, 4, 8, 8, and and 10). 10).

Table 4. Calculation of 𝛾, 𝛾, and 𝜒 for different length/diameter ratios. Table 4. Calculation of γ1, γ2, and χb for different length/diameter ratios. Poisson’s Ratio, L/D = 4 L/D = 20 L/D = 50 L/D = 100 L/D = 200

Poisson’sv 𝜸𝟏L/D =𝜸 4𝟐 𝝌𝒃 𝜸L/D𝟏 = 20𝜸𝟐 𝝌𝒃 𝜸L/D𝟏 = 50𝜸𝟐 𝝌𝒃 𝜸L/D𝟏 = 100𝜸𝟐 𝝌𝒃 𝜸L/D𝟏 = 200𝜸𝟐 𝝌𝒃 Ratio, v 0 γ4.36091 γ6.8012 χ5.511b γ4.71081 γ7.7892 χ7.441b γ4.72691 γ7.8432 χ7.572b γ4.72931 γ7.8512 χ7.592b γ4.72981 γ7.8532 χ7.597b 0.10 4.36094.3526 6.801 6.776 5.511 5.460 4.7108 4.7103 7.789 7.786 7.441 7.435 4.7269 4.7268 7.843 7.842 7.572 7.571 4.7293 4.7292 7.851 7.850 7.592 7.592 4.7298 4.7298 7.853 7.8537.597 7.597 0.20.1 4.35264.3443 6.776 6.753 5.460 5.412 4.7103 4.7097 7.786 7.784 7.435 7.429 4.7268 4.7267 7.842 7.842 7.571 7.570 4.7292 4.7292 7.850 7.850 7.592 7.591 4.7298 4.7298 7.853 7.8527.597 7.597 0.30.2 4.34434.3362 6.753 6.730 5.412 5.365 4.7097 4.7091 7.784 7.782 7.429 7.423 4.7267 4.7266 7.842 7.842 7.570 7.569 4.7292 4.7292 7.850 7.850 7.591 7.591 4.7298 4.7298 7.852 7.8527.597 7.597 0.40.3 4.33624.3281 6.730 6.708 5.365 5.321 4.7091 4.7085 7.782 7.780 7.423 7.418 4.7266 4.7266 7.842 7.841 7.569 7.568 4.7292 4.7292 7.850 7.850 7.591 7.591 4.7298 4.7298 7.852 7.8527.597 7.597 0.50.4 4.32814.3202 6.708 6.686 5.321 5.278 4.7085 4.708 7.7807.778 7.418 7.412 4.7266 4.7265 7.841 7.841 7.568 7.568 4.7292 4.7291 7.850 7.850 7.591 7.591 4.7298 4.7298 7.852 7.8527.597 7.597 0.5 4.3202 6.686 5.278 4.708 7.778 7.412 4.7265 7.841 7.568 4.7291 7.850 7.591 4.7298 7.852 7.597

The Poisson’s ratio can then be calculated using the equation below: Therefore, Poisson’s ratio can be readily calculated using the experimentally measured transverse vibration frequencies, n2 and n1. Furthermore, after knowing Poisson’s ratio, the elastic modulus (17) can be determined using the ﬁrst bending𝑣 mode as shown in. Equation (18). To ﬁnd γ1, one can use Equations (7) and (8) and the calculated Poisson’s ratio from Equation (17). Hence, the elastic modulus can beTherefore, calculated Poisson’s directly asratio follows: can be readily calculated using the experimentally measured transverse vibration frequencies, 𝑛 and 𝑛. Furthermore, after knowing Poisson’s ratio, the elastic modulus can be determined using the64 firstπ2ρ bendingL4n2 mode as shown in Equation (18). To find 𝛾 , one E = 1 = 4ρL2n2(A v + B ). (18) can use Equations (7) and (8) and the calculated4 2 Poisson’s1 ratio1 from1 Equation (17). Hence, the elastic γ1D modulus can be calculated directly as follows:

𝐸 4𝜌𝐿𝑛𝐴𝑣𝐵. (18) The equation can also be used to determine the first bending mode solely from the dimensions of the rod and assumed elastic constants without solving the Timoshenko frequency equation. The

values of 𝐶 , 𝐷 , and 𝐸 for different specimen dimensions were found by calculating 𝜒 for different length/diameter ratios as well as varying Poisson’s ratio (shown in Appendix A). It is noted

from Equation (14) that the frequency ratio 𝜒 is only dependent on the roots of the Timoshenko frequency equation and is independent of the elastic modulus and mass density. Therefore, any set

of material properties would produce the same values of 𝐶, 𝐷, and 𝐸. The values of 𝜒 and its derived 𝐶, 𝐷, and 𝐸shown in this paper are only applicable to the 1st and 2nd bending modes of a Timoshenko rod with free‐free boundary conditions. The constant 𝐸 was found by calculating 𝜒 when Poisson’s ratio was equal to zero. The calculated values for 𝐸 are shown in Table 5. A similar procedure was carried out to determine the coefficients 𝐶 and 𝐷. The transverse modes were determined using Poisson’s ratio ranging from 0.0 to 0.5 in increments of 0.05. After

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The equation can also be used to determine the ﬁrst bending mode solely from the dimensions of the rod and assumed elastic constants without solving the Timoshenko frequency equation. The values of C1, D1, and E1 for different specimen dimensions were found by calculating χb for different length/diameter ratios as well as varying Poisson’s ratio (shown in AppendixA). It is noted from Equation (14) that the frequency ratio χb is only dependent on the roots of the Timoshenko frequency equation and is independent of the elastic modulus and mass density. Therefore, any set of material properties would produce the same values of C1, D1, and E1. The values of χb and its derived C1, D1, and E1 shown in this paper are only applicable to the 1st and 2nd bending modes of a Timoshenko rod Vibrationwith 2019 free-free, 3 FOR boundaryPEER REVIEW conditions. The constant E1 was found by calculating χb when Poisson’s ratio 9 was equal to zero. The calculated values for E1 are shown in Table5. plotting 𝜒, the best‐fit second‐order polynomial was used to find 𝐶 and 𝐷. The coefficient of determination, 𝑅, wasTable above 5. The 0.999 values for of C all1, D the1, and cases.E1 for Table different 5 shows length/diameter the calculated ratios. values for different specimen dimensions. Unlike 𝐸 , the coefficientsLength/Diameter decrease (L/D ) when the length/diameter ratio is enlarged.Coefﬁcient As the rod 2 gets longer, 3 the 4 values 5of 𝐶 and 8 𝐷 approach 10 zero, 20 indicating 50 that 100 the rod behaves moreE1 like3.69886 a Euler–Bernoulli 4.282677 5.51093 slender 5.99205 rod. For 6.78228 larger 7.03545length/diameter 7.44104 ratios 7.57243 (L/D 7.59195 > 20), 𝜒 D1 −0.83478 −0.60134 −0.51647 −0.44455 −0.27264 −0.19948 −0.06117 −0.01043 −0.00263 approaches a constant value equal to only 𝐸. C1 0.22969 0.12914 0.10186 0.08175 0.04227 0.02862 0.00748 0.00120 0.0002986

Table 5. The values of 𝐶, 𝐷, and 𝐸 for different length/diameter ratios. A similar procedure was carried out to determine the coefficients C1 and D1. The transverse modes were determined using Poisson’s ratio rangingLength/Diameter from 0.0 to 0.5 in ( incrementsL/D) of 0.05. After plotting χb, 2 Coefficientthe best-fit second-order2 3 polynomial 4 was used5 to find C18and D1. The10 coefficient20 of determination,50 100R ,

was𝐸 above3.69886 0.999 for all4.282677 the cases. 5.51093 Table5 shows5.99205 the calculated6.78228 values7.03545 for different7.44104 specimen 7.57243 dimensions. 7.59195 Unlike𝐷 E1,− the0.83478 coefficients −0.60134 decrease −0.51647 when the−0.44455 length/diameter −0.27264 ratio−0.19948 is enlarged. −0.06117 As the−0.01043 rod gets − longer,0.00263 the𝐶 values0.22969 of C1 and 0.12914D1 approach 0.10186 zero, indicating0.08175 that0.04227 the rod0.02862 behaves more0.00748 like a0.00120 Euler–Bernoulli 0.0002986 slender rod. For larger length/diameter ratios (L/D > 20), χb approaches a constant value equal to onlyAs shownE1. in Figures 6–8, the three coefficients can be estimated using polynomial equations, As shown in Figures6–8, the three coefficients can be estimated using polynomial equations, and to and to increase the accuracy of the proposed method, two intervals are specified for 𝐶 and 𝐷. A increase the accuracy of the proposed method, two intervals are specified for C and D . A total of 97 L/D total of 97 L/D data points were used to determine the polynomials. The 1proposed1 equations can be data points were used to determine the polynomials. The proposed equations can be used to find the used to find the coefficients for any 𝐿/𝐷 2 and allows for the calculation of the elastic constants coefficients for any L/D ≥ 2 and allows for the calculation of the elastic constants after knowing the after1st knowing and 2nd the bending 1st and modes 2nd as wellbending as the modes rod’s dimensions. as well as The the sixth-order rod’s dimensions. polynomials The were sixth found‐order polynomialsto have a were maximum found percent to have difference a maximum of 0.12% percent with thedifference theoretical of 0.12% values. with On average, the theoretical the percent values. On average,difference the was percent 0.01% difference indicating that was the 0.01% sixth-order indicating polynomials that the cansixth be‐order used adequatelypolynomials to estimate can be used adequatelythe coefficients. to estimate the coefficients.

FigureFigure 6. 6. PlotPlot of of CC1 1versusversus the the diameter/length ( D/L(D/L) ratio.) ratio.

Figure 7. Plot of D1 versus the diameter/length (D/L) ratio.

Vibration 2019, 3 FOR PEER REVIEW 9 plotting 𝜒, the best‐fit second‐order polynomial was used to find 𝐶 and 𝐷. The coefficient of determination, 𝑅, was above 0.999 for all the cases. Table 5 shows the calculated values for different specimen dimensions. Unlike 𝐸 , the coefficients decrease when the length/diameter ratio is enlarged. As the rod gets longer, the values of 𝐶 and 𝐷 approach zero, indicating that the rod behaves more like a Euler–Bernoulli slender rod. For larger length/diameter ratios (L/D > 20), 𝜒 approaches a constant value equal to only 𝐸.

Table 5. The values of 𝐶, 𝐷, and 𝐸 for different length/diameter ratios.

Length/Diameter (L/D) Coefficient 2 3 4 5 8 10 20 50 100

𝐸 3.69886 4.282677 5.51093 5.99205 6.78228 7.03545 7.44104 7.57243 7.59195 𝐷 −0.83478 −0.60134 −0.51647 −0.44455 −0.27264 −0.19948 −0.06117 −0.01043 −0.00263 𝐶 0.22969 0.12914 0.10186 0.08175 0.04227 0.02862 0.00748 0.00120 0.0002986

As shown in Figures 6–8, the three coefficients can be estimated using polynomial equations, and to increase the accuracy of the proposed method, two intervals are specified for 𝐶 and 𝐷. A total of 97 L/D data points were used to determine the polynomials. The proposed equations can be used to find the coefficients for any 𝐿/𝐷 2 and allows for the calculation of the elastic constants after knowing the 1st and 2nd bending modes as well as the rod’s dimensions. The sixth‐order polynomials were found to have a maximum percent difference of 0.12% with the theoretical values. On average, the percent difference was 0.01% indicating that the sixth‐order polynomials can be used adequately to estimate the coefficients.

Vibration 2019, 2 166 Figure 6. Plot of C1 versus the diameter/length (D/L) ratio.

Figure 7. Plot of D1 versus the diameter/length (D/L) ratio. Vibration 2019, 3 FOR PEER REVIEWFigure 7. Plot of D1 versus the diameter/length (D/L) ratio. 10

FigureFigure 8.8. PlotPlot ofof EE11 versus the diameter/length ( (D/LD/L)) ratio. ratio.

For L 10, For D ≤ 10, 𝐶 112.284 6 223.341 5 185.884 4 77.1737 3 15.7309 2 D D D D D C1 = 112.284 L − 223.341 L + 185.884 L − 77.1737 L + 15.7309 L − (19) 0.952765 2.73114 10, (19) D −2 0.952765 L + 2.73114 × 10 , 𝐷 340.388 717.045 588.110 224.277 6 5 4 3 (20) D D D D D1 = −340.388 35.7879+ 717.045 0.603707− 588.110 0.0468.+ 224.277 − L L L L 2 (20) D D For 10, 35.7879 − 0.603707 + 0.0468. L L L > For D𝐶 10, 2541.87 920.988 72.8415 2.22046 3.02631 (21) 6 3.61314 5 10 1.080414 10, 3 2 D D D D D C1 = 2541.87 L − 920.988 L + 72.8415 L − 2.22046 L + 3.02631 L − (21) 𝐷 2744.60 4196.55 −4 1132.34D 11.5492−6 26.1754 3.61314 × 10 L + 1.08041 × 10 , (22) 1.98589 10 6.00293 10. 6 5 4 3 2 D D D D D AD 1single= 2744.60 sixth‐orderL polynomial− 4196.55 canL be+ used1132.34 to estimateL − 11.5492 𝐸 . The equationL − 26.1754 is shownL below:− (22) − D − 1.98589 × 10 3 + 6.00293 × 10 6. 𝐸 39.7722 269.519 530.656L 350.357 96.4206 (23) 1.08854 7.58933. As an example, the FEM was performed for a steel rod with an assumed elastic modulus, Poisson’s ratio, and density of 200 GPa, 0.3, and 7750 kg m , respectively. The values of 𝐴 and 𝐵 were obtained from Equations (11)–(13), whereas 𝐶,𝐷,and 𝐸 were obtained using Equations (19)–(23). Table 6 shows that the 1st and 2nd bending modes as well as the 1st torsional mode frequency calculated using Equations (9), (10), and (16) compare closely with the FEM results with a maximum difference of 0.36%.

Table 6. Steel rod with different length/diameter ratios.

FEM Theoretical Size Bending (Hz) Torsional Bending (Hz) Torsional (mm × mm) A1 B1 C1 D1 E1 1st 2nd (Hz) 1st 2nd (Hz) 100 × 2000 112.09 305.44 784.8 0.381786 127.7267 0.00748 −0.06116 7.4433 112.32 306.079 787.62 100 × 400 2468.2 5718.0 3932.7 0.330818 6.50796 0.10188 −0.51670 5.5142 2470.39 5723.82 3938.11

3. Experiments

Vibration 2019, 2 167

A single sixth-order polynomial can be used to estimate E1. The equation is shown below:

6 5 4 3 2 D D D D D E1 = 39.7722 L + 269.519 L − 530.656 L + 350.357 L − 96.4206 L + (23) D 1.08854 L + 7.58933.

As an example, the FEM was performed for a steel rod with an assumed elastic modulus, Poisson’s −3 ratio, and density of 200 GPa, 0.3, and 7750 kg m , respectively. The values of A1 and B1 were obtained from Equations (11)–(13), whereas C1, D1, and E1 were obtained using Equations (19)–(23). Table6 shows that the 1st and 2nd bending modes as well as the 1st torsional mode frequency calculated using Equations (9), (10), and (16) compare closely with the FEM results with a maximum difference of 0.36%.

Table 6. Steel rod with different length/diameter ratios.

FEM Theoretical Size (mm × mm) Bending (Hz) Torsional Bending (Hz) Torsional A1 B1 C1 D1 E1 1st 2nd (Hz) 1st 2nd (Hz) 100 × 2000 112.09 305.44 784.8 0.381786 127.7267 0.00748 −0.06116 7.4433 112.32 306.079 787.62 100 × 400 2468.2 5718.0 3932.7 0.330818 6.50796 0.10188 −0.51670 5.5142 2470.39 5723.82 3938.11

3. Experiments Aluminum, steel, and concrete rods with different length/diameter ratios were experimentally tested to validate the proposed methods. The density and specimen dimensions can be found in Table7. It should be noted that the dimensions of the rods must be measured accurately for a correct estimation of the elastic modulus and Poisson’s ratio.

Table 7. Material properties of the rods.

Properties Material Diameter (mm) Length (mm) Density (kg m−3) Aluminum 63.50 736.60 2780.56 Steel 46.83 279.4 8143.13 Concrete 99.63 622.30 2288.75

The impact resonance frequency method was used to measure the fundamental transverse and torsional modes. Figure9 shows the instruments and specimens used in the experiment. The rods were suspended with soft ﬂexible rubber tubing placed at approximately 0.15L to allow free vibration in the fundamental transverse and torsional modes as shown in Figure9. The accelerometer was attached to the specimen centered at 0.55L from the edge of the specimen using a hot glue adhesive. The torsional frequency was also tested for each rod. The supports were kept at the same location as the transverse mode experiment. The accelerometer was placed with hot glue adhesive on a rigid metal tab which was located 0.223L from the edge of the specimen. A tangential impact was made using another metal tab located 0.13L from the other edge. A schematic of the torsional mode testing can be found in ASTM C215. For each test, the acceleration time history was plotted on a LabView waveform analyzer. Using fast Fourier transform (FFT), the time domain was converted to the frequency domain. Vibration 2019, 3 FOR PEER REVIEW 11

Aluminum, steel, and concrete rods with different length/diameter ratios were experimentally tested to validate the proposed methods. The density and specimen dimensions can be found in Table 7. It should be noted that the dimensions of the rods must be measured accurately for a correct estimation of the elastic modulus and Poisson’s ratio.

Table 7. Material properties of the rods.

Properties Material Diameter (mm) Length (mm) Density (kg m−3) Aluminum 63.50 736.60 2780.56 Steel 46.83 279.4 8143.13 Concrete 99.63 622.30 2288.75

The impact resonance frequency method was used to measure the fundamental transverse and torsional modes. Figure 9 shows the instruments and specimens used in the experiment. The rods were suspended with soft flexible rubber tubing placed at approximately 0.15L to allow free vibration in the fundamental transverse and torsional modes as shown in Figure 9. The accelerometer was attached to the specimen centered at 0.55L from the edge of the specimen using a hot glue adhesive. The torsional frequency was also tested for each rod. The supports were kept at the same location as the transverse mode experiment. The accelerometer was placed with hot glue adhesive on a rigid metal tab which was located 0.223L from the edge of the specimen. A tangential impact was made using another metal tab located 0.13L from the other edge. A schematic of the torsional mode testing can be found in ASTM C215. For each test, the acceleration time history was plotted on a LabView Vibrationwaveform2019, 2 analyzer. Using fast Fourier transform (FFT), the time domain was converted to the 168 frequency domain.

Figure 9. Picture of experiment: (a) Impact hammer, accelerometer, and specimens; (b) Experimental Figure 9. Picture of experiment: (a) Impact hammer, accelerometer, and specimens; (b) Experimental setup. setup.

Experimental3.1. Experimental Data Data TypicalTypical acceleration acceleration time time histories histories for for the the impactimpact and and response response before before and and after after performing performing the the FFT areFFT shown are shown in Figures in Figures 10 10and and 11 11.. The The excitation excitation frequency range range for for the the transverse transverse and andtorsional torsional impactimpact is approximately is approximately 5000 5000 Hz. Hz. To To find find the the peakpeak frequency, frequency, the the frequency frequency response response function, function, which which is theVibrationisVibration ratio the ratio 20192019 between ,between, 33 FORFOR PEERPEER the the REVIEWREVIEW output output response and and the the input input excitation excitation force forcein the frequency in the frequency domain, was domain,1212 was calculatedcalculated [13]. [13]. Four Four peaks peaks can can be seen be seen in the in output the output response; response; these peaks these correspond peaks correspond to the first to the four TablefundamentalTable 88 presentspresents bending thethe averageaverage modes. measuredmeasured In this study, frequenciesfrequencies only the forfor 1st thethe and fundamentalfundamental 2nd bending transversetransverse modes were andand analyzed; torsionaltorsional first four fundamental bending modes. In this study, only the 1st and 2nd bending modes were frequencieshowever,frequencies the ofof thethe3rd three threeand rods.rods.4th bending TheThe averageaverage modes valuevalue could waswas be determineddetermined used as verificationfromfrom atat leastleast of fourfour the tests.tests. measured TheThe samplingsampling elastic analyzed;frequencyconstants.frequency however, forfor bothboth the impactimpact 3rd and forceforce 4th andand bending accelerationacceleration modes responseresponse could waswas be used25.625.6 kHz,kHz, as verification andand thethe resolutionresolution of the ofof measured thethe elasticmeasuredmeasured constants.Similarly, frequenciesfrequencies Figures waswas 12 andincreasedincreased 13 present toto 0.250.25 the HzHz typical usingusing aaacceleration signalsignal processingprocessing time histories technique.technique. and the frequency response function for the torsional mode. A clear peak can be seen in all the specimens.

((aa)) ((bb))

FigureFigure 10.10. ((aa)) TypicalTypical accelerationacceleration timetime historyhistory andand ((bb)) TypicalTypical responseresponse powerpower spectrumspectrum (transverse(transverse Figure 10. (a) Typical acceleration time history and (b) Typical response power spectrum (transverse mode). mode).mode).

(a)(a) (b)(b)

FigureFigure 11.11. ((aa)) TypicalTypical impactimpact loadingloading timetime historyhistory andand ((bb)) TypicalTypical impactimpact powerpower spectrumspectrum (transverse(transverse Figure 11. (a) Typical impact loading time history and (b) Typical impact power spectrum (transverse mode). mode).mode).

((aa)) ((bb))

FigureFigure 12.12. ((aa)) TypicalTypical accelerationacceleration timetime historyhistory andand ((bb)) TypicalTypical responseresponse powerpower spectrumspectrum densitydensity (torsional(torsional mode).mode).

Vibration 2019, 3 FOR PEER REVIEW 12

Table 8 presents the average measured frequencies for the fundamental transverse and torsional frequencies of the three rods. The average value was determined from at least four tests. The sampling frequency for both impact force and acceleration response was 25.6 kHz, and the resolution of the measured frequencies was increased to 0.25 Hz using a signal processing technique.

(a) (b)

Figure 10. (a) Typical acceleration time history and (b) Typical response power spectrum (transverse mode).

Vibration 2019, 2 169 (a) (b)

Figure 11. (a) Typical impact loading time history and (b) Typical impact power spectrum (transverse mode).Similarly, Figures 12 and 13 present the typical acceleration time histories and the frequency response function for the torsional mode. A clear peak can be seen in all the specimens.

(a) (b)

Figure 12. (a) Typical acceleration time history and (b) Typical response power spectrum density Figure 12. (a) Typical acceleration time history and (b) Typical response power spectrum density Vibration(torsional 2019, 3 mode).FOR PEER REVIEW 13 (torsional mode).

(a) (b)

Figure 13. (a) Typical impact loading time history and (b) Typical impact power spectrum (torsional Figure 13. (a) Typical impact loading time history and (b) Typical impact power spectrum (torsional mode). mode).

4. DataTable Analysis8 presents the average measured frequencies for the fundamental transverse and torsional frequencies of the three rods. The average value was determined from at least four tests. The sampling frequencyThe values for both for impact 𝐴 , 𝐵 force, 𝐶 , and 𝐷, accelerationand 𝐸 can responsebe obtained was for 25.6 each kHz, rod and specimen the resolution using of their the measuredrespective frequenciesL/D ratio. Using was increasedEquations to (9) 0.25 and Hz (17), using the Poisson’s a signal processing ratio for the technique. three rods was calculated and is shown in Table 8. The calculated values for the dynamic elastic modulus using Equations (10) and (18) are also Tableshown 8. inCalculated Table 8. properties Table 8 shows of the aluminum, a comparison steel, between and concrete Method rods. 1 and Method 2.

Method 1 uses 𝜒 to calculate Poisson’s ratio and elastic modulus. As expected, Method 2, which uses Transverse Frequency (Hz) Torsional v E (GPa) the 1stRod and 2nd bending modes, produces Frequencymore reasonable elastic constants. This is mainly due to 1st 2nd Method 1 Method 2 Method 1 Method 2 the inaccuracy of the experimental torsional mode(Hz) frequency. Note, the largest difference occurs in ± ± ± theAluminum Poisson’s ratio531.05 while0.03 the 1415.85 elastic modulus0.07 2,083.02 for both0.45 methods0.434 are almost 0.364 identical. 75.12 75.08 Steel 2589.83 ± 0.33 6460.95 ± 0.43 5686.97 ± 0.87 0.332 0.300 219.10 218.91 Concrete 847.02 ± 0.20 2130.61 ± 0.43 1972.06 ± 0.77 0.283 0.241 35.38 35.35 Table 8. Calculated properties of the aluminum, steel, and concrete rods.

4. Data AnalysisTransverse Frequency Torsional Frequency 𝒗 E (GPa) Rod (Hz) The values for A1, B1, C1, D1, and E1 can be(Hz) obtained for each rod specimen using their respective 1st 2nd Method 1 Method 2 Method 1 Method 2 L/D ratio. Using Equations (9) and (17), the Poisson’s ratio for the three rods was calculated and is Aluminum 531.05 ± 0.03 1415.85 ± 0.07 2,083.02 ± 0.45 0.434 0.364 75.12 75.08 shown in Table8. The calculated values for the dynamic elastic modulus using Equations (10) and (18) Steel 2589.83 ± 0.33 6460.95 ± 0.43 5686.97 ± 0.87 0.332 0.300 219.10 218.91 areConcrete also shown 847.02 in Table ± 0.208 .2130.61 Table8 ±shows 0.43 a comparison1972.06 ± 0.77 between 0.283 Method 10.241 and Method35.38 2. Method35.35 1 uses χt to calculate Poisson’s ratio and elastic modulus. As expected, Method 2, which uses the 1st and 2nd4.1. Sensitivity bending modes, Analysis produces more reasonable elastic constants. This is mainly due to the inaccuracy of the experimental torsional mode frequency. Note, the largest difference occurs in the Poisson’s ratio whileIn the this elastic study, modulus the diameter for both and methods length of are the almost specimens identical. were measured with a resolution of 0.01 mm, while the mass had a resolution of 0.01 kg. A sensitivity analysis of the proposed methods was conducted using the experimental resolution. The experimental frequencies were measured with a resolution of 0.25 Hz, which was achieved using a signal processing technique. It was found that Poisson’s ratio and the elastic modulus had a maximum deviation of 0.3 ± 0.001 and 218.91 ± 0.2 GPa, respectively, when changing the diameter or length by 0.01 mm. Therefore, it is recommended to at least have a resolution of 0.1 mm in order to achieve an accuracy of Poisson’s ratio with ±0.01 deviation. It was also found that a mass change of 0.01 kg can influence the elastic modulus by ±0.5 GPa. Furthermore, if only a 0.25 Hz frequency resolution is obtainable, the estimated Poisson’s ratio can have a large deviation of ±0.05 when a specimen has a large L/D like the aluminum specimen (L/D = 11.6). This is because, as shown in Figure 5 and Table 4, Poisson’s ratio is not sensitive to the

change of the bending frequency ratio 𝝌𝒃 when L/D becomes large. During the experiment, the concrete and steel rods both showed high accuracy in the measurement of Poisson’s ratio using the bending modes as displayed in Table 9. However, since the aluminum rod has an L/D = 11.6, a frequency resolution of ±0.05 Hz was used to reduce the Poisson’s ratio deviation to ±0.009. For an accurate measurement of the Poisson’s ratio, it is recommended that an L/D ≅ 6 be used when using

𝝌𝒃.

Vibration 2019, 2 170

Sensitivity Analysis In this study, the diameter and length of the specimens were measured with a resolution of 0.01 mm, while the mass had a resolution of 0.01 kg. A sensitivity analysis of the proposed methods was conducted using the experimental resolution. The experimental frequencies were measured with a resolution of 0.25 Hz, which was achieved using a signal processing technique. It was found that Poisson’s ratio and the elastic modulus had a maximum deviation of 0.3 ± 0.001 and 218.91 ± 0.2 GPa, respectively, when changing the diameter or length by 0.01 mm. Therefore, it is recommended to at least have a resolution of 0.1 mm in order to achieve an accuracy of Poisson’s ratio with ±0.01 deviation. It was also found that a mass change of 0.01 kg can inﬂuence the elastic modulus by ±0.5 GPa. Furthermore, if only a 0.25 Hz frequency resolution is obtainable, the estimated Poisson’s ratio can have a large deviation of ±0.05 when a specimen has a large L/D like the aluminum specimen (L/D = 11.6). This is because, as shown in Figure5 and Table4, Poisson’s ratio is not sensitive to the change of the bending frequency ratio χb when L/D becomes large. During the experiment, the concrete and steel rods both showed high accuracy in the measurement of Poisson’s ratio using the bending modes as displayed in Table9. However, since the aluminum rod has an L/D = 11.6, a frequency resolution of ±0.05 Hz was used to reduce the Poisson’s ratio deviation to ±0.009. For an accurate ∼ measurement of the Poisson’s ratio, it is recommended that an L/D = 6 be used when using χb.

Table 9. Sensitivity analysis of the aluminum, steel, and concrete specimens.

Measured Aluminum Steel Concrete Properties Method E (GPa) v E (GPa) v E (GPa) v Mass 1 75.12 ± 0.1 - 219.10 ± 0.5 - 35.38 ± 0.03 - (±0.01 kg) 2 75.08 ± 0.1 - 218.91 ± 0.5 - 35.35 ± 0.03 - 0.434 ± 0.332 ± 0.283 ± 1 75.12 ± 0.003 219.10 ± 0.02 35.38 ± 0.002 L (±0.01 mm) 0.00004 0.00009 0.00004 0.364 ± 0.241 ± 2 75.08 ± 0.003 218.91 ± 0.02 0.300 ± 0.0002 35.35 ± 0.002 0.00008 0.00009 1 75.12 ± 0.05 0.434 ± 0.0004 219.10 ± 0.2 0.332 ± 0.0005 35.38 ± 0.001 0.283 ± 0.0002 D (±0.01 mm) 2 75.08 ± 0.05 0.364 ± 0.0009 218.91 ± 0.2 0.300 ± 0.001 35.35 ± 0.001 0.241 ± 0.0005 1 75.12 ± 0.07 0.434 ± 0.001 219.10 ± 0.04 0.332 ± 0.0003 35.38 ± 0.02 0.283 ± 0.0008 n (±0.25 Hz) 1 2 75.08 ± 0.1 * 0.364 ± 0.05 * 218.91 ± 0.06 0.300 ± 0.004 35.35 ± 0.03 0.241 ± 0.01 1 ------n (±0.25 Hz) 2 2 75.08 ± 0.01 ** 0.364 ± 0.02 ** 218.91 ± 0.008 0.300 ± 0.001 35.35 ± 0.004 0.241 ± 0.004 1 75.12 ± 0.07 0.434 ± 0.001 219.10 ± 0.001 0.332 ± 0.0001 35.38 ± 0.0003 0.283 ± 0.0003 n” (±0.25 Hz) 2 ------

* For a frequency resolution of ±0.05 Hz for n1, E = 75.08 ± 0.02 GPa and v = 0.364 ± 0.009. ** For a frequency resolution of ±0.05 Hz for n2, E = 75.08 ± 0.002 GPa and v = 0.364 ± 0.004.

5. Discussion

The frequency ratio between the first torsional mode and the first bending mode, χt, was found to be linearly proportional to the Poisson’s ratio with a slope, A1, and a y-intercept, B1, whereas the frequency ratio between the first two bending modes, χb, was slightly parabolic with coefficients C1, D1, and E1. It was found that the constants A1, B1, C1, D1, and E1 depend only on the dimensions of the specimen and can be tabulated for different specimen sizes. Values for A1, B1, C1, D1, and E1 at different length/diameter ratios were found, allowing the use for any rod dimensions. As the length/diameter ratio increased, the value of A1 could be ignored and χt approached to a constant. In other words, χt is independent of Poisson’s ratio at very large length/diameter ratios. Likewise, as the length/diameter increases, the values of C1 and D1 approach to zero, and χb approaches a constant value of 7.598. Equations for the elastic modulus, E, and Poisson’s ratio, v, were proposed in this paper using the tabulated values of A1, B1, C1, D1, and E1, allowing for a quick calculation of both E and v. The proposed methods do not require any iterations to solve for the material constants. Results show that, using the equations proposed in this study, the Young’s modulus and Poisson’s ratio of a rod can be determined from the measured frequency ratio quickly and efficiently. Vibration 2019, 2 171

6. Conclusions This paper presents a method to calculate Poisson’s ratio and dynamic elastic modulus using experimentally measured transverse and torsional modes. The method is based on the exact solution to the frequency equation for a Timoshenko rod under free-free boundary conditions. The finite element method using the ABAQUS program was shown to match the exact solution with less than 0.15% error for both the transverse and torsional modes. Equations for the elastic modulus, E, and Poisson’s ratio, v, were proposed in this paper using the frequency ratio values, allowing for a quick calculation of both E and v. The frequency ratio between the first bending mode and the first torsional mode or the frequency ratio between the first bending mode and the second bending mode for any rod with a length-to-diameter ratio of L/D ≥ 2 can be directly estimated using the proposed equations. The proposed method does not require any iterations to solve for the material constants and can be used to calculate the elastic constants of rods with different dimension sizes. The proposed equations were used to verify the elastic constants of a steel rod with less than 0.36% error percentage, and the equations were also used to measure elastic constants of three rods with different material properties (aluminum, steel, and concrete) using simple impact vibration. It was found that the torsional mode is more difficult to excite and not as reliable as the bending modes, whereas a more accurate measurement can always be obtained from using the 1st and 2nd bending modes which were measured simultaneously.

Author Contributions: Conceptualization, G.L. and H.-L.C.; Methodology, G.L. and H.-L.C.; Validation, G.L.; Writing—Original Draft Preparation, G.L.; Writing—Review and Editing, H.-L.C. Funding: This research was partially funded by the WVDOH/FHWA research project number [RP#312]. Acknowledgments: The authors acknowledge the support provided by the West Virginia Transportation Division of Highways (WVDOH) and the Federal Highway Administration (FHWA) for the research project WVDOH RP#312. Special thanks are extended to our project monitors, Mike Mance, Donald Williams, and Ryan Arnold of WVDOH. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Appendix

Values of frequency ratios χt and χb and values of the ﬁrst two roots (γ1 and γ2) of the Timoshenko frequency equation for different length/diameter ratios (L/D) at various Poisson’s ratios.

Table A1. Values of χt and χb for different length/diameter ratios at various Poisson’s ratios.

Length/Diameter Poisson’s Ratio γ1 γ2 χt χb 0 3.948 5.476 2.599 3.699 0.1 3.938 5.431 2.626 3.617 0.2 3.928 5.389 2.653 3.541 2 0.3 3.919 5.348 2.679 3.469 0.4 3.909 5.309 2.705 3.402 0.5 3.900 5.271 2.731 3.338 0 4.275 6.336 4.256 4.827 0.1 4.267 6.305 4.288 4.768 0.2 4.259 6.275 4.319 4.712 3 0.3 4.252 6.246 4.349 4.658 0.4 4.244 6.218 4.379 4.607 0.5 4.237 6.191 4.408 4.558 0 4.439 6.801 6.508 5.511 0.1 4.433 6.776 6.542 5.460 0.2 4.427 6.753 6.576 5.412 4 0.3 4.422 6.730 6.608 5.365 0.4 4.417 6.708 6.640 5.321 0.5 4.411 6.686 6.672 5.278 Vibration 2019, 2 172

Table A1. Cont.

Length/Diameter Poisson’s Ratio γ1 γ2 χt χb 0 4.530 7.088 9.372 5.992 0.1 4.526 7.068 9.408 5.948 0.2 4.522 7.049 9.443 5.906 5 0.3 4.518 7.031 9.477 5.866 0.4 4.514 7.013 9.510 5.827 0.5 4.510 6.996 9.543 5.790 0 4.645 7.496 21.707 6.782 0.1 4.643 7.485 21.745 6.755 0.2 4.641 7.475 21.782 6.729 8 0.3 4.639 7.465 21.818 6.704 0.4 4.637 7.455 21.853 6.680 0.5 4.636 7.446 21.888 6.656 0 4.675 7.613 33.073 7.035 0.1 4.673 7.606 33.111 7.016 0.2 4.672 7.598 33.149 6.997 10 0.3 4.671 7.591 33.185 6.978 0.4 4.669 7.584 33.221 6.960 0.5 4.668 7.577 33.256 6.943 0 4.716 7.789 127.726 7.441 0.1 4.715 7.786 127.766 7.435 0.2 4.715 7.784 127.804 7.429 20 0.3 4.715 7.782 127.841 7.423 0.4 4.714 7.780 127.877 7.418 0.5 4.714 7.778 127.913 7.412 0 4.728 7.843 790.220 7.572 0.1 4.728 7.842 790.259 7.571 0.2 4.728 7.842 790.298 7.570 50 0.3 4.728 7.842 790.335 7.569 0.4 4.728 7.841 790.372 7.568 0.5 4.727 7.841 790.408 7.568 0 4.729 7.851 3156.257 7.592 0.1 4.729 7.850 3156.297 7.592 0.2 4.729 7.850 3156.335 7.591 100 0.3 4.729 7.850 3156.373 7.591 0.4 4.729 7.850 3156.409 7.591 0.5 4.729 7.850 3156.446 7.591

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