The formula for the first natural frequency and the frequency spectrum of a spatial regular truss

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1. Introduction The calculation of natural frequencies of structures in practice is carried out by the finite element method [1-3]. In this way, it is possible to calculate the entire spectrum of natural frequencies of rather complex, including statically indeterminate, structures, taking into account various types of fastening, material inhomo- geneities, errors in mounting and manufacturing of structural elements, etc. For simple statically determinate constructions, analytical solutions for the lower and upper bounds of the first frequency are also possible. Such solutions are of particular value for regular constructions with a periodic structure. This is achieved by using the induction method [4-6]. Solutions to deformation problems for planar regular trusses with an arbitrary number of panels using the Maple computer mathematics system were obtained in [7; 8]. Formulas for deflections of some spatial rod systems are derived in [9; 10]. For the first time, the question of the existence of schemes of bar statically determinate structures was raised by Hutchinson R.G. and Fleck N.A. [11; 12], Zok F.W., Latture R.M., and Begley M.R. [13]. The optimization and classification of regular trusses was carried out by Kaveh A. [14; 15]. Analytical methods for calculating elements of building structures using the Maple system are considered in [16; 17]. The handbook [18] contains diagrams of various planar regular trusses and formulas for calculating their deflections, displacements of movable supports, and forces in characteristic rods. An analytical calculation of the deflection of a planar externally statically indeterminate truss with an arbitrary number of panels was performed using the computer mathematics system in [19]. The formula for the dependence of the deflection of a planar truss on the number of panels was derived by induction in [20]. In [21], a lower bound for the first natural oscillation frequency of a flat truss was obtained using the Dunkerley method. The frequency spectrum of a family of regular trusses is also numerically analyzed here. A simplified Dunkerley method for estimating the first natural frequency of a planar truss was proposed in [22]. When simplifying the desired calculation formula for the first frequency, it is proposed here to calculate the sum of the members of the sequence by the average value of its element. The analytical dependence of the deflection of the spatial console of a triangular profile on the number of panels was derived in [23]. The formula for the deflection of a cantilever truss with a cruciform lattice, depending on the redistribution of the cross-sectional areas of the rods and the number of panels, was obtained by induction in [24]. In [25], some exact solutions were found for the problems of deflection of archtype planar trusses. The number of schemes of rod statically determinate regular constructions that are attractive from the point of view of the possibility of obtaining analytical solutions is very limited. In [11], the problem of finding such constructions was even called “hunting”. There are especially few schemes of regular statically determinate spatial trusses. In this paper, we propose a scheme of a spatial truss and derive a formula for the lower limit of its first natural frequency. 2. Construction The truss consists of n panels of length 2a and height h (Fig. 1) of six rods each. The sides of the panel are isosceles rod triangles connected at the bottom by a horizontal rod of length 2b. The side supports of the structure are racks with a height d on one side and buildings with additional horizontal connections on the other. The construction is asymmetrical. In hinge A, it is fixed on a spherical support. Figure 1. Truss scheme, n = 5 A truss of n panels η= 9 3n + contains rods, of which n rods of length 2a in the upper chord, 4n inclined side rods of length c= a2 + +b2 h2 , 2n support posts of height d, n horizontal external support braces, n horizontal braces of length 2b, and three rods simulating the spherical hinge A in upper belt. The number of internal nodes endowed with masses is equal to K = 3n+1. The inertial properties of the structure are modeled by concentrated masses in the nodes, oscillating along the vertical z axis. 3. Methods To calculate the forces in the elements, the coordinates of the truss hinges are entered into the Maple system program: xi = xi n+ = a(2i -1), yi = -yi n+ = -b z, i = zi n+ = d i, =1,.., .n The coordinates of the hinges of the supports on the base: xi+ +3 1n = xi+ +4 1n = xi+ +5 1n = xi , yi+ +3 1n =-yi+ +4 1n =-b z, i+ +3 1n = zi+ +4 1n = 0, yi+ +5 1n =-s z, i+ +5 1n =d. Here s is the length of the lateral horizontal support links. This value will not be included in the calculation, since for the selected type of mass oscillations in the nodes (vertically), the force in these rods will be zero. The structure of the lattice, determined by the order of connection of the rods into nodes (hinges), is programmed by lists of the numbers of the ends of the rods: Ni =[i i, + 2 ]n , Ni n+ =[i i, + 2n +1], Ni+2n = +[i n i, + 2 ]n , Ni+3n = +[i n i, + 2n +1], Ni+4n = +[i 2n i, + 2n +1],Ni+5n =[i i, + n], Ni+6n =[i i, + 3n +1], Ni+7n = +[i n i, + 4n +1], Ni+8n =[i i, + 5n +1], i =1,.., .n The numbering order of the nodes and the choice of the beginning of the bar and its end does not affect either the force or its sign. Figure 2. Numbering of knots and elements, n = 4 Based on the data on the coordinates and structure of the lattice, a matrix of a system of linear equations is compiled to determine the forces in the rods by cutting out nodes. The system of linear equations for the equilibrium of nodes in projections onto the coordinate axes x, y, z is written in matrix form GS B= . Here G - is the matrix of direction cosines in the projection equations, S - is the force vector in the rods, including the support reactions, B is the load vector. The projections of the conditional vectors of the rods look like lx i, = xNi,1 - xNi,2 , ly i, = yNi,1 - yNi,2 ,lz i, = zNi,1 - zNi,2 . Since the forces are applied to one end of the rod and the other in opposite directions, the direction cosines have different signs: G3Ni,1 -2,i = lx i, / ,l Gi 3Ni,1 -1,i = ly i, / ,li G3Ni,1 ,i = lz i, / ,li G3Ni,2 -2,i =-lx i, / ,l Gi 3Ni,2 -1,i =-ly i, / ,l Gi 3Ni,2 ,i =-lz i, / .li where li = lx i2, + +ly i2, lz i2, - is the length of the rod. Rows of the matrix G with numbers 3j - 2, j = 1,..,K contain the coefficients of the projection equation on the horizontal x axis, rows with numbers 3j - 1 correspond to the projection equation on the y axis, and those with numbers 3j correspond to the projection equation on the vertical z axis. The solution of the system of equations - the forces in the elements, is searched for in symbolic form using the Maple system. Calculation of the natural vibration frequencies of the structure is carried out according to a simplified, but widespread truss model, in which the mass is evenly distributed over all internal nodes. If we assume only vertical motions of masses along the z axis, then the number of degrees of freedom of the considered structure is equal to K. In an analytical form for such a system, one can obtain a lower estimate of the first frequency using the Donkerley method. The Donkerley formula [26; 27] for the lower frequency limit has the form K ω = ω-2 -2 D p . (1) p=1 where ωp - are partial frequencies. Here, in fact, the problem of the eigenvalues of a matrix is replaced by the calculation of its trace. The equation of vertical oscillations of a separate mass m, has the form m&&zp +Dp pz =0, p = 1,..., K, (2) where Dp - is stiffness, the reciprocal of compliance δp =1/ Dp . Compliance (linear displacement along the z-axis) is determined by the Maxwell-Mohr formula. Assuming that the stiffnesses of all rods EF are the same, we have the expression: η 2 δp =1/ Dp = (S( )jp ) lj / (EF)., (3) j=1 where S( )jp - is the force in the rod with number j from the action of a vertical unit force applied to the node p, in which the mass is located. The value of the stiffness coefficient and the partial frequency are affected by the location of the mass. For harmonic oscillations: zp =U p sin(ω +t ϕ) , formula ωp = Dp /m follows from (2). Substitution of this expression in (1) gives a formula for calculating the partial frequency: K ωD-2 = m δp = m nΔ( ). (4) p=1 4. Results and discussion 4.1. The Dunkerley’s method. Calculations of the frequencies of trusses with a different number of panels show that the coefficient Δ( )n in (4) has the form Δ( )n =Ca1 3 +C b2 3 +C c23 3 +C h4 2d . (5) EFh Only the coefficients in the numerator depend on the order of the truss n in this expression. Sequential calculation of trusses with a different number of panels gives the following expressions: Δ =(1) (a3 + 2b3 + +c3 6h2d) / (EFh2), Δ(2) = (6a3 + 6b3 +3c3 +13h2d) / (EFh2), Δ(3) = (12a3 +12b3 + 6c3 + 22h2d) / (EFh2), Δ(4) = (20a3 + 20b3 +10c3 +33h2d) / (EFh2),... To identify patterns in the formation of coefficients in these expressions, the sequence should be extended to eight. In this case, the operators of the Maple system can derive formulas for the common members: C1 = C2 = n n( +1), C3 = n n( +1) / 2, C4 = n2 + 4n +1. Thus, the dependence of the lower limit of the first frequency on the number of panels in the truss has the form: ωD = h m n( (1+ n a)( 3 +b3 + cEF3 / 2) + (n2 + 4n +1)dh2) . (6) The solution can be checked against a numerical one in the Maple system using the Eigenvalues operator from the Linear Algebra package of linear algebra. In this case, the same program is used, according to which formula (6) was derived. For example, the following parameters of a steel structure with masses

About the authors

Mikhail N. Kirsanov

National Research University “MPEI”

Author for correspondence.
Email: c216@ya.ru
ORCID iD: 0000-0002-8588-3871

Doctor of Physical and Mathematical Sciences, Professor of the Department of Robotics, Mechatronics, Dynamics and Strength of Machines

Moscow, Russian Federation

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