Free vibration analysis of helicoidal bars with thin- walled circular tube cross-section via mixed f
* Author to whom correspondence should be addressed.
Sigma Journal of Engineering and Natural Sciences 2015, Vol. 33, Issue 2, pp. 200-218; doi.org/10.62051/ytu.sigma-journal-of-engineering-and-natural-sciences-free-vibration-analysis-of-helicoidal-bars-with-thin-walled-circular-tube-cross-
Abstract
Keywords: Timoshenko beam theory; finite element; non-cylindrical helix; thin-walled circular tube section; free vibration.
1. Introduction
Derivation of the differential equations of helicoidal bars goes back to 19th century [1-3]. Shear influence and rotary inertia effects are investigated by [4] and [5] derived the dynamic equations. *
Corresponding Author/Sorumlu Yazar: e-mail/e-ileti: eratli@itu.edu.tr, tel: (212) 285 65 52
Natural frequencies of helicoidal bars depend on different parameters, and in order to address them various numerical methods were employed such as the finite element method [6-8], the transfer matrix method [9] which are the most popular ones. The transfer matrix method is intensively applied to dynamic analysis of cylindrical/non-cylindrical helical springs besides finite element method with circular and rectangular cross-sections by [10-14]. [15] employed the exact element method for the free vibration analysis of non-cylindrical helicoidal beams with circular and rectangular variable cross-sections. [16,17] applied the pseudospectral method to investigate the free vibration analysis of cylindrical and non-cylindrical helical springs with circular cross-sections. In this study, free vibration analysis of cylindrical, conical, barrel and hyperboloidal helices having thin-walled circular tube cross-section is performed via the mixed finite element method. The influence of some parameters (e.g., the thickness-to-section average radius ratio, the helix height-to-helix maximum radius ratio, various parameters of the noncylindrical helicoidal geometry, boundary conditions, and the density of the material) on the fundamental natural frequency of helicoidal bars are investigated.
Element Formulation
Helix geometry: The geometrical properties of the helices in Figure 1 are x = R ( )cos ,
y = R ( )sin , z = p ( ) , p ( ) = R ( ) tan , where denotes the pitch angle, R ( ) and p ( ) signify the centerline radius and the step for unit angle, respectively, of the helix as a function of the horizontal angle . With c ( ) = R 2 ( ) + p 2 ( ) , the infinitesimal arc length becomes ds = c( )d . In the cylindrical helix, since R = R ( ) = constant and it is clear that c = R 2 + p 2 , = R / c 2 , = p / c 2 , p = R tan are all constant. and are the curvature and torsion of the helix axis, respectively. The Frenet unit vectors are as follows: t is the tangent unit vector, n is the normal unit vector, b = t ´ n is the binormal unit vector. In the case of a conical helix, the radius at any point on the helix geometry is R( ) = R max + ( R min- R max )( 2n ) where n is the number of active turns, Rmax and Rmin are the bottom radius and top radius, respectively, of the conical helix geometry and in the case of a 2 barrel, the radius is R ( ) = Rmax + ( R min- R max ) (1 - n , Rmin and Rmax are the bottom radius and the central radius, respectively or in the case of hyperboloidal helix, the radius is 2 R ( ) = Rmin + ( R max- R min ) (1 - n , where Rmax and Rmin are the bottom radius and the central radius, respectively.
The functional: The field equations for the helicoidal bars, which are based on the Timoshenko beam theory and refer to the Frenet coordinate system, are discussed in [7,8]. Using u = ut t + un n + ub b as the displacement vector, Ω = t t + n n + b b as the rotational vector, T = Tt t + Tn n + Tb b as the force vector, M = M t t + M n n + M b b as the moment vector, as the density of material, A as the area of the cross-section, I as the moment of inertia, q and m as the distributed external force vector and moment vector, respectively, the field equations can be written in the form u0 T, s q A M, s t T m I Ω 0
= 2 / t 2 . Eq. (1) is the equation of u = 2u / t 2 , where the accelerations are denoted by motion and Eq. (2) is the difference between the kinematic strain equation and the constitutive strain equation, namely, u - = 0 . The kinematic strain equation is in form u = D k u , where Dk is a differential operator. The constitutive strain equation is in the form = C , where C is the compliance matrix, namely,
1. EI b úû
where A¢ = A / k ¢ and k ¢ is the shear correction factor; E and G are the elasticity and shear modulus, respectively; I t , I n and I b are the moments of inertia with respect to the t, n, b axes, respectively. Eqs. (1)-(2) can be written in operator form as Q = Ly - f ; if the operator is potential, the equality
d Q(y, y ) are Gâteaux derivatives of the operator in the directions of y and y * , respectively. After proving the operator to be potential and considering the harmonic motion of the helix in the free vibration analysis (and also q = m = 0 ), the functional yields to the following form *
I y u, T, s t Ω, T M , s , Ω 21 C M, M 12 C T, T 12 A 2 u, u
ˆ ˆ , Ω u, ˆ T Ω, 12 2 Ω, Ω T Tˆ , u M M M
where is the natural circular frequency and the square parentheses indicate the inner product. The terms with hats in Eq. (4) are known values on the boundary and the subscripts and represent the geometric and the dynamic boundary conditions, respectively. The curved element: Using the subscripts i , j to represent the node numbers of the bar element, the linear shape functions i = ( j - ) / and j = ( - i ) / are employed in the finite element formulation, where = ( j - i ) . The non-cylindrical helix geometry is interpolated from the cylindrical geometry as stated by [7]. The free vibration analysis: The problem of determining the natural frequencies of a structural system reduces to the solution of a standard eigenvalue problem ([K ] - 2 [M ]){u} = {0} where [ K ] is the system matrix, [ M ] is the mass matrix for the entire domain, u is the eigenvector
and is the natural angular frequency of the system. Hence the explicit form of standard eigenvalue problem in the mixed formulation is æ é[K 11 ] [K 12 ]ù é[0] [0] ù ÷öïïì{F}ïïü ïïì{0}ïüï ú -2 ê ú÷÷í = í çç ê ê ú÷ çèç ëê[K 22 ] [K 22 ]ûú ë[0] [M ]û ø÷ïïî{U}ïï ïïî{0}ïï
where {F} denotes the nodal force and the moment vectors and {U} = {u Ω}T signifies the nodal displacement and rotation vectors. The {F} vector is eliminated in Eq. (5) and the eigenvalue problem in the mixed formulation becomes ([K * ] - 2 [M]){U} = {0} where the condensed system matrix is [ K * ] = [ K 22 ] - [K 12 ]T [K 11 ]-1 [K 12 ] .
3.1. Convergence analysis
A barrel helix bar, having circular cross section and fixed at both ends is solved [see Figure 2]. The material and geometrical properties are: the modulus of elasticity E 210GPa ; Poisson's ratio 0.3 ; the material density 7850 kg/m3 ; the number of active turns n 6.5 ; the pitch angle 4.8 ; the ratio of the minor radius to the major radius of the helix Rmin / Rmax 0.4 (where Rmax 25mm ), radius of the circular cross section r 1mm . Through the analysis, the first two natural frequencies of the barrel helix are calculated using 50, 75 and 100 mixed finite elements. The convergence of the first two frequencies compared with [12,17] and SAP2000, and the results are shown graphically in Figure 3. SAP2000 needs more than 500 elements for fulfillment but the result of 500 seems to be satisfactory. In this example, the shear correction factor k 1.18 is used [19] but [12,17] considered the value of shear correction factor as k 1.1 .
Figure 3. The first two frequencies graph for the barrel helix
3.2. Benchmark Examples
The material and geometrical properties of the cylindrical and non-cylindrical (conical, barrel, hyperboloidal) helicoidal bars, which are solved here, are as follows: the modulus of elasticity E = 210GPa ; Poisson's ratio 0.3 ; the density of the material = 7850 kg/m3 ; the number of active turns n (3.5, 7.5, 11.5); the minimum radius of helix-to-maximum radius of helix ratios Rmin / Rmax (0.4, 0.6, 0.8); the height of helix-to-maximum radius of helix ratio H / Rmax (4, 6, 8); the average radius of cross-section ro = 1mm is kept constant; and the thickness-to-cross-section average radius ratios t/ro = 0.01, 0.10, 0.15 and 0.25. The pitch angle has a unique value that refers to the number of active turns n and the ratios Rmin / Rmax and H / Rmax as shown in Table 1. In these examples, some cited parameters are kept constant for the solution, 200 mixed elements are employed. 3.2.1. Fixed-fixed Boundary Condition H / Rmax = 4 = constant [see Figure 1(a)]: The fundamental natural frequencies are listed in Tables 2(a)-(c). An interpretive discussion of each table is as follows: As the thickness-to-section average radius ratio increases, an increasing trend is observed for the fundamental natural frequency. For the thin-walled circular tube sections (t / ro £ 0.1) , this increasing trend is nearly negligible. If the fundamental natural frequency values of each table are compared with the results that correspond to t / ro = 0.01 for each helix type, the percent increase in the fundamental natural frequency, which corresponds to t / ro = 0.25 for Rmax = 10 mm , 20 mm
and Rmax = 40 mm , ranges from 0.70% ~ 0.87% and 0.48% ~ 1.06% , respectively. If the fundamental natural frequencies in each table are compared with the results that correspond to n = 3.5 for each helix type, the percent reduction in the case of n 7.5 and n = 11.5 range between 46% ~ 52% and 64% ~ 69% , respectively. If the fundamental natural frequencies of the non-cylindrical helices are compared with the fundamental natural frequencies of the cylindrical helix, the latter is smaller. For each number of turns, the comparison of the fundamental natural frequencies in each table with the results that correspond to Rmin / Rmax = 0.4 for each helix type reveals that the percent reduction in the case of conical, barrel and hyperboloidal helices range from 14% ~ 36% , 8% ~ 22% and 18% ~ 40% , respectively. The comparison of the fundamental natural frequencies in Tables 2(b)-(c) are compared with the corresponding results of Table 2(a) for each helix type reveal that the percent reductions in Tables 2(b) and 2(c) are approximately 75% and 94% , respectively. Rmax = 20mm=constant [see Figure 1(b)]: The fundamental natural frequencies for
H / Rmax = 6, 8 are listed in Tables 3(a)-(b). The common evaluations of the fundamental natural frequencies in each Table 2(b) and Tables 3(a)-(b) are as follows: As the thickness-to-section average radius ratio increases, an increasing trend is observed for the fundamental natural frequency. For thin-walled circular tube sections (t / ro £ 0.1) , this increasing trend is nearly negligible. The comparison of the fundamental natural frequencies in each table with the results that correspond to the ratio t / ro = 0.01 for each helix type indicates that the percent increases in the fundamental natural frequency, which correspond to t / ro = 0.25 for H / Rmax = 4, 6, 8 , range from 0.68% 0.95% . The comparison of the fundamental natural frequencies in each table with the results that correspond to n = 3.5 for each helix type indicates that the percent reduction in the case of n = 7.5 and n = 11.5 range from 45% 52% and 63% 69% , respectively. The comparison of the fundamental natural frequencies of the non-cylindrical helices with the cylindrical helix reveals that the latter is always smaller. For each number of turns, the fundamental natural frequencies in Table 2(b) and Tables 3(a)-(b) are compared with the results that correspond to Rmin / Rmax = 0.4 ; in the cases of Rmin / Rmax = 0.6, 0.8 , the percent reductions for the H / Rmax = 4, 6, 8 ratios are listed in Table 4. The fundamental natural frequencies shown in Tables 3(a)-(b) are compared with the corresponding values in Table 2(b) [for H / Rmax = 4 ], and the reduction in the fundamental natural frequencies for the cylindrical, conical, barrel and hyperboloidal helices are listed in Table 5.
Table 1. The pitch angles ( ) of helix types for the number of active turns n and the ratios Rmin / Rmax and H / Rmax
Free Vibration Analysis of Helicoidal Bars with Thin- … Sigma 33, 200-218, 2015
Table 2. The fundamental natural frequencies (Hz) for H / Rmax 4 constant and Rmax variable (a) Rmax 10 mm , H 40mm
Free Vibration Analysis of Helicoidal Bars with Thin- … Sigma 33, 200-218, 2015
Table 3. The fundamental natural frequencies (Hz) for Rmax 20mm constant and H / Rmax variable (a) H / Rmax 6 , H 120mm
Free Vibration Analysis of Helicoidal Bars with Thin- … Sigma 33, 200-218, 2015
Table 4. The percent reductions in the fundamental natural frequencies of non-cylindrical helices in the case of Rmin / Rmax = 0.6, 0.8 with respect to Rmin / Rmax = 0.4 . H / Rmax 4 6 8
Table 5. The percent reductions in the fundamental natural frequencies of cylindrical and noncylindrical helices in the case of H / Rmax = 6,8 with respect to H / Rmax = 4 H / Rmax 6
Helix types Conical Barrel 13% 20% 11% 13% 10% 22% 10% 14% 10% 22% 10% 14% 29% 37% 28% 32% 27% 38% 26% 31% 26% 38% 26% 31%
Hyperboloidal 14% 20% 9% 18% 9% 17% 30% 37% 26% 35% 25% 33%
Rmax = 20mm = constant [see Figure 1(b)]: The fundamental natural frequencies are shown in Tables 6(a)-(c), and the common evaluations are as follows: As the thickness-to-section average radius ratio increases, an increasing trend is observed for the fundamental natural frequency. In the case of t / ro = 0.01 for each helix type for t / ro = 0.25 , the percent increase are shown in Table 7. The comparison of the fundamental natural frequency values in each table with the results that correspond to n = 3.5 for each helix type reveals that the percent reduction for n = 7.5 and n = 11.5 range from 48% 53% and 65% 69% , respectively. The comparison of the fundamental natural frequency values for the non-cylindrical helices with the cylindrical helix reveals that the latter is smaller. For each number of turns, the fundamental natural frequency values in each table are compared with the results that correspond to Rmin / Rmax = 0.4 ; the percent reductions in Rmin / Rmax = 0.6 and 0.8 are listed in Table 8. The fundamental natural frequencies shown in Tables 6(b)-(c) are compared with the results in Table 6(a) [for H / Rmax = 4 ], and the percent reduction in the fundamental frequency are listed in Table 9. Table 7. The percent reductions in the fundamental natural frequencies of non-cylindrical helices in the case of t / ro = 0.25 with respect to t / ro = 0.01 H / Rmax 4 6 8
Helix types Conical Barrel 0.60% 1.06% 0.53% 1.16% 0.56% 1.00% 0.00% 1.01% 0.66% 1.16% 0.00% 1.18%
Free Vibration Analysis of Helicoidal Bars with Thin- … Sigma 33, 200-218, 2015
Table 6. The fundamental natural frequencies (Hz) for Rmax 20mm constant and H / Rmax variable (a) H / Rmax 6 , H 80mm
Free Vibration Analysis of Helicoidal Bars with Thin- … Sigma 33, 200-218, 2015
Table 8. The percent reductions in the fundamental natural frequencies of non-cylindrical helices in the case of Rmin / Rmax = 0.6, 0.8 with respect to Rmin / Rmax = 0.4 H / Rmax 4 6 8
Table 9. The percent reductions in the fundamental natural frequencies of cylindrical and noncylindrical helices in the case of H / Rmax = 6, 8 with respect to H / Rmax = 4 H / Rmax 6 8
Helix types Conical Barrel 30% 33% 29% 32% 46% 51% 46% 50%
3.2.3. Influence of Density on the Fundamental Natural Frequency The thickness-to-section average radius ratio t / ro = 0.10 = constant ; the helix height to helix maximum radius ratio H / Rmax = 4 = constant , where Rmax = 20mm ; and the densities of material = 7850kg/m3 and 8300kg/m3 . The fundamental natural frequency results of the fixed-fixed and fixed-free boundary conditions are provided in Table 10(a) and Table 10(b), respectively. For the both boundary conditions, the percent reduction in the fundamental natural frequency values that correspond to 8300kg/m3 with respect to the corresponding results of
= 7850kg/m3 range from 2.3% 3.2% . Table 10. The fundamental natural frequencies (Hz) for two different density of material (a) B.C. (boundary condition): fixed-fixed ( Rmax 20 mm , H / Rmax 4 ) n
65.6. 63.8
Table 10. continuing… (b) B.C. (boundary condition): fixed-free ( Rmax 20 mm , H / Rmax 4 )
4. Conclusion
The mixed finite element formulation is based on the Timoshenko beam theory, and the documentation of the corresponding functional exists in [7,8]. The non-cylindrical helix geometry is derived using exact curvatures at the nodal points and their interpolations through the element. As a convergence test, a barrel type helicoidal bar is handled, results of the present program is compared by the literature and a commercial program, and even with a coarse element mesh excellent agreement is achieved. In this study, four benchmark examples are solved to investigate the influence of the thickness-to-section average radius ratio, the helix height-to-helix maximum radius ratio, the various parameters of the non-cylindrical helicoidal geometry, the boundary conditions, and the density of the material on the free vibration analysis of helicoidal bars having thin-walled circular tube cross-section. Following remarks can be cited: As the thickness-to-section average radius ratio increases, an increasing trend is observed for the fundamental natural frequency. If the fundamental natural frequencies of the non-cylindrical helices are compared with the fundamental natural frequencies of the cylindrical helix, the latter is smaller. As the number of active turns and the ratio Rmin / Rmax increase, a reduction in the fundamental natural frequencies of the non-cylindrical helicoidal bars is observed. For both the cylindrical and non-cylindrical helicoidal bars, an increase of the density of material caused a reduction of the fundamental natural frequencies. Acknowledgments / Teşekkür This research is supported by The Scientific and Technological Research Council of Turkey under project no 111M308 and by the Research Foundation of ITU under project no 38078. These supports are gratefully acknowledged by the authors.
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ERATLI, N.; ERMİŞ, M.; OMURTAG, M.H. Free vibration analysis of helicoidal bars with thin- walled circular tube cross-section via mixed f. Sigma Journal of Engineering and Natural Sciences 2015, Vol. 33, pp. 200-218. https://doi.org/10.62051/ytu.sigma-journal-of-engineering-and-natural-sciences-free-vibration-analysis-of-helicoidal-bars-with-thin-walled-circular-tube-cross-