Analytical calculation of an externally statically indeterminate truss oscillation frequency

The research object is a p lan a r braced jointed frame with a cross-sh a p e d d o u b le l a t t i ce . T he m a ss o f t h e tr uss i s e v e nl y d i st ribu te d a cr o s s its n od e s. T h e ta s k i s t o o b ta in an ana l y t ica l d e p e nd en ce of the f u n d a m e nta l f req u e n cy of t he fr a m e ' s v ib r a tions o n its d imen sio n s , ma ss, an d nu m b er of panels. Method. The f r am e is st a t i ca lly d e t e r mi na te . T he meth o d of cut t ing n o d e s is u se d t o de te r m i n e th e f or ces in t h e r od s an d t h e r e actio n s a t t h e s u p p o r t s. Th e s t iff ness of t h e s t ruc t ur e is calc u la te d using th e M a xw e l l - Mohr f o r mu l a. T h e sim pl ific a t io n o f th e f req u en c y cal cula t i o n f o r sy stem s wit h ma n y d eg re e s of f r e ed o m i s a c h ie v e d by a p ply ing the Dun k e rl ey m e th o d . Results . Compa c t fo r mu las f or t h e d e fl e ctio n of the fram e a nd i t s n a tur a l fr eque nc y o f v i br a tio n h a v e b e e n ob t ai ne d . A c a se of ki n e m a tic v a r ia b il i t y o f t he structure wi t h a n o d d n u m b er o f pa ne l s i n h a lf th e sp an wa s d iscov e red . A co m pa ri son of th e a n a ly tica l r e sul t wit h t h e n um er ica l o n e sh o w s g o o d accur a cy of th e pr op o se d m e t h o d . So m e a sy m p t o t i c so lu t i o n s h av e b ee n found. © The Author 2025

1    Introductions

In practice, computer programs based on the finite element method are used to calculate the natural frequency of rod oscillations [1], [2]. Independently of this, analytical methods are also being developed that are free from the shortcomings of numerical methods associated with the inevitable effect of accumulation of rounding errors, which manifests itself in large-scale structures containing a large number of elements. One of the methods for overcoming the "curse of dimensionality" is analytical, based on the use of computer mathematics systems [3], [4] and the induction method of generalizing a number of particular solutions to the general case [5]. Computer mathematics methods are also used to obtain analytical solutions to problems in structural mechanics using the theory of initial functions and functional series [6], [7]. Some of the first to raise the question of the possibility of analytical studies of rod systems of regular structure were, apparently, professors of the University of Cambridge Fleck, N.A. and Hutchinson, R.G. [8], [9]. Among the problems for which analytical solutions were found, problems for planar trusses can be singled out separately. The reference books [5], [10] contain diagrams of planar regular statically determinate trusses and formulas for deflections and natural vibration frequencies depending on the number of panels.

For m u la s fo r t h e f ir s t fr equ en cy o f a c a n t il ev e r sp a tia l tr u ss b y t h e i n d u c t io n m e t h od wer e de r i ve d in [11], [12]. T he v i brat i o n s of a r ec ta n g ul a r t ru ss w i t h dia go n a ls fo rm in g a fo u r -sl op e sy m m etr ical sp a t i al structure were studied in analytical for m b y th e i n d u ct i o n me t h o d in [13]. Reg ul ar truss sch em e s in so me c a se s h a v e a h id de n a n d d a n ger o u s f e a t u re. Fo r som e n u m b er s of p a n e l s, r ega rd le ss of th e loa d, suc h Kirsanov, M.

A nal y t i c al calcu l at i o n o f a n e xt e r nall y st ati ca l l y i n de termi n a t e t russ o scil l a tion f r eq u e n cy ;

schemes allow kinematic variability [14]. An approximate formula for calculating the fundamental natural frequency of oscillations of a two-span truss by the induction method in the Maple system was obtained in [15]. In [16], analytical estimates of the deflection and the first natural frequency of free oscillations of a thrust planar arch truss with a cross-shaped lattice were found. The joint spectrum of a family of regular trusses was also analyzed, where frequencies were found that were the same for trusses of different orders (spectral constants). The dependence of the deflection of a planar statically determinate arch truss on the number of panels was derived in [17]. An analytical expression for the first natural frequency of a planar truss using the Maple computer mathematics system was obtained in [18]. The static calculation of the deflection of planar regular trusses in analytical form is performed in [19], [20]. The formula for the first natural frequency of a planar truss is derived in [21], [22]. A two-sided analytical estimate of the fundamental frequency of oscillations of a flat truss was obtained in [23], [24]. Nonlinear vibrations of rod structures were studied in [25], [26]. In [27] the problem of symmetric free vibrations of pyramidal trusses is solved taking into account the general large measure of deformation (quadratic and logarithmic).

In this paper, a formula is derived for the dependence of the first natural frequency of a new scheme of a planar lattice externally statically indeterminate truss with two fixed supports. An algorithm is proposed that simplifies the form of the solution and gives it a higher accuracy, an analytical solution is found for the deflection of the truss and the asymptotics of this solution.

2    Materials and Methods 2.1    Truss design

The structure under consideration is a statically determinate planar frame truss with two fixed supports and a double diagonal lattice (Fig. 1) [28]. The length of the truss with 2 n panels in the middle part of the span is L = 2 a ( n + 1), the height is 3 h . The mass of the truss is modeled by concentrated loads of mass m, uniformly distributed over all K = 4 n + 12 nodes of the structure. The truss consists of П = 8 n + 20 rods. This number also includes four rods modeling the left and right fixed supports.

Fig. 1 - Truss, n = 4

The calculation of forces in the rods is performed in analytical form using the computer mathematics system Maple [29]. The forces in the rods are calculated by cutting out nodes in analytical form. The projections of the forces in the rods onto the coordinate axes are determined by the coordinates of the nodes and the structure of the connection of the rods. The numbering of the rods and nodes of the truss is performed in parametric form depending on the number of panels. The numbering of the nodes for n = 2 is shown in Figure 2.

Fig. 2 - Numbering of truss nodes, n = 2

The coo r d in a t e s of th e n od e s a r e a s follows:

x 1 ⁼ У 1 ⁼ 0, x ₂ ⁼ a, У 2 ⁼ 0,

^X i +2 = ^X i+ 2n+8 = ai , У + 2 = h , У + 2„+8 = ^{3 h}, i = ^{1 ,}”^,2 n ^{+ 1},

x 2 n+4

^X 2 n+6

^X 2 n+ 8

^X 4 n+ 11

— L o — a , ^y 2 „ +4 — 0, ^x 2 n +5 ^{— L} 0 ^, y 2n+ 5 ^{— 0,}

0, y2n+6 h, X2n+7    0, y2n+7    2h, a /2, y2„+8   5h /2, X4„+io   Lo — a / 2, y4„+10   5h / 2,

• ^{— X} 4 n + 1 2 ^{— L} 0 , y 4 n+11 ^{— 2h}, y 4n+12 ^{— h}-

• The structure of the connection of individual rods into a lattice is specified by special vectors qi , i = 1,...,n, containing the numbers of the ends of the corresponding rods by analogy with the assignment of a list of edges of a graph in discrete mathematics. The following vectors correspond to the upper and lower chords of the truss:

qi = [i,i +1], i = 1,...,2n + 4, qi+2n+5 = [i + 2n + 5,i + 2n + 6], i = 1,...,2n + 6.

The l a t t i ce ( p o s t s a n d b r a ce s) are se t a s f o llo w s:

q + 4„ + 12 = [ i +1 ,i + 2n + 5 ], q ₊ 6„ ₊ 1₈ = [ i + 2 ,i + 2n + 10], i = 1 , ... ,2n + 2.

The ro d s th at m o d e l th e f i x ed hi ng e sup p o rts c o rre sp on d t o fo ur vector s:

q _{n - 3} = [1,4 n +13] , q _{n - 2} = [1, 4 n +14], q _n - 1 = [2 n + 5, 4 n + 1 5], q^ = [2 n + 5,4 n +16 ] .

Ba se d o n th e s e d at a, the matr ix G o f t h e direct io n co sin es of t h e e f f or t s i s f o r me d . T h e sy st e m o f e q u atio n s of t h e e qui li br i u m of n o d e s in th e matr ix v ers i on h a s t h e f or m:

GS=B, where B is the vector of node loads, and S is the vector of unknown forces and support reactions. The length of these vectors is determined by the number K of truss nodes. The odd elements B2i-1 of the load vector contain projections of nodal loads onto the horizontal x-axis, and the even elements contain projections of forces applied to the truss onto the vertical y-axis. The elements of the matrix B are calculated using data on the lattice structure and node coordinates:

l — (x — x ) / l, l — (y — y ) / l, i — 1,..., n, x,i           qi1        qi^ i’ y,i       '1у:         qi,27i where l =  l"' +1"'  is the length of the rod with number i. The rows of the matrix B include the i             x ,iУ coefficients of the equations of projections on the x and y axes for one node:

G,   == l . / l, G,   . = l . / l,

"r/i 1-1,i         x,i       i       "r/i pi         y,ii

G„   = — —I . / I,G   .= —I . /1. .

"^i 2—1, i             x ,i      i      "^i2, i            У, ii

The in v er se m atr i x m e tho d i n sy m bo l ic f o r m wi t h th e h e lp of M a p le sy st em o p er a tor s is us ed t o fin d a s o l utio n to sy st e m ( 1 ). T h e fi r s t t ri al ca lcu la ti o n s sho wed th a t fo r an o d d nu mb e r o f p a n els n the determinant of the matrix G b ecomes z er o . T h is i n d ica t e s t h e k i nem a tic de ge ne r a t i o n o f t h e s t ru c t u re. T h i s f a ct is co nf i r m e d b y t h e p ictu r e o f po s si bl e no de v e lo citie s f or n = 1 (F ig . 3) . O n l y thr e e n o d e s r e c e iv e p ossib le v e l ocitie s, t h e r e m a i ni ng n o d es ar e m o t io nl es s. The rel at io nsh i p b e tw e e n t h e v e loci t i es v alues s p e ci f ie s th e p os i t i o n o f t he i n s t an ta n e o u s v el o citie s cen te r: U / a = 2V / c, w he r e c = 7 a ² + h² . The a lg o r i t hm f or sequ e n t ia ll y cal c u la tin g t he v e loc itie s of t h e t russ n o d e s us in g t h e k in em at i c re lation sh ip s of plane motion is used [30].

Fig. 3 - Possible velocities of truss nodes, n = 1

• 2.2    The deflection

Two fi x ed h in ge d su p p or ts cr ea t e e x t erna l s t atic in d e t erm in a cy o f t h e s t r u ctur e . I t is im pos si bl e t o fi nd t he sup p o rt r e ac t i on s se pa r ate ly f ro m t h e f or c e s i n th e r o ds o nl y f r om th e e q ui l ib ri u m eq u at ion s o f t h e entire truss as a whole. The suppo r t re a c ti on s of a t wo- p ar t s t ru c tu r e, ar ti cu lat e d by a hi ng e , on t w o fi x ed h ing es ar e e a sil y f o u nd b y di v i di ng th e s t r u cture i nto tw o pa r t s . H er e, h o we v e r , t h e re i s n o a r t i cula t i n g h ing e. T o de termin e th e su p p or t r e a c t i on s, i t i s n e ce ssa ry t o co m p i l e t he en tir e sy stem of eq u ilibriu m e q u atio n s of no d es , in cl ud i ng t h e sup p or t n o d e s . For a tr u s s l oa d e d w ith a u nif o rm no da l l oa d a lo n g t h e lo w e r cho r d , t he n o n z er o co m po nen t s of t h e v e ct o r of t h e r i g h t si d e of the e qu il i b r iu m e q u a t i o n s of n o d es have the form: B _2i = — P , i = 2 ,. . ,2n + 4 .

Fig. 4 - Load on the lower belt, n = 4

The vertical reactions of the supports calculated for the sequence of trusses are as follows:

Y A = Y B = 7 P /2,11P/2,15P/2,19 P /2,... The general formula for this value is obvious:

Y A = Y B = (4k + 3) P/ 2. The calculation shows that the horizontal reactions in this problem do not depend on the order of the truss: X A = X B = 3aP / (2h). The deflection of the truss under the action of the load on the lower chord (Fig. 4) is estimated by the vertical displacement of the central hinge C using the Maxwell – Mohr's formula:

л П Sisili л k = L i=1 EF where Si is the force in the i-th rod from the external load, si — the force in the same rod from the action of a single vertical force on node C, EF is the rigidity of the rods, li is their length. Calculation of the deflection for trusses of different orders gives the following sequence:

А 1 = P(9a³ + 3c³ + 23h³) / (4h² EF ),

A ₂ = P(51a³ + 3c³ +11h³)/ (4h² EF ), A 3 = P(617a³ +17c³ + 63h³)/(4h² EF ),

A ₁₈ = 3P(1183 85a 3 + 97c³ + 25h³)/(4h² EF ),...

In general:

A _k = P (Ca ³ + C₂c ³ + C₃h³ ) /( EFh ²).

To find the general term of this sequence using Maple, a sequence of length 18 was required. The most difficult task was to find the coefficient C 1 . To determine it, using the operator rgf_findrecur , a recurrent equation of the ninth order was obtained, which is satisfied by this coefficient

С =C + 4C -4C   -6C  +6C  + 4C -4C -C

1, k        1, k - 1 ⁺      1, k - 2        1, k - 3        1, k - 4 ⁺      1, k - 5 ⁺      1, k - 6        1, k - 7    ^v'1, k - 8 ⁺    1, k - 9 '

The solution to this equation gives the desired coefficient:

C 1 = (20 k ⁴ + 8(5 - 4(-1) ^k ) k ³ - 2(24(- 1) ^k + 1) k ² + 2(37(- 1) ^k - 11) k + 3(- 1) ^k +15)/ 24.      (2)

Other coefficients are obtained similarly:

C ₂ = (2k² + (2 - 6(- 1) ^k )k + 5(- 1) ^k +1) / 8,

C ₃ = (4(-2(-1) ^k + 3)k + 3)/4.                                       ⁽³⁾

Solution (1) with coefficients (2) and (3) has a quadratic asymptote with respect to the number of panels. Using Maple methods for the relative value Л' = EFЛ _k /(L ₀ P sum ) , where P sum = 2 P(n +1) is the total truss load, one can find the limit limЛ'/k² = 5a² /(96h²). k ^x

• 2.3    Natural frequency of oscillations

To calculate the first natural frequency of oscillations of a truss, assuming that the masses at the nodes perform only vertical movements, the Dunkerley estimate is used in a simplified form:

K

^ = !/< ME6 = F/F6"“K)■ i=i where 6max is the greatest value of deflection 6, i = 1,.., K from the action of a vertical unit force on node i over all nodes, m is the mass in the node. In this case, the truss has two nodes with the greatest possible deflection. One node is in the middle of the lower chord and has the number n+3, the other with the number 3n+9 is in the middle of the upper chord. The calculation is carried out for kinematically permissible numbers of panels: n = 2k, k = 1,2,.... To clarify the result, the half-sum of these values is taken 6max = (6n+3 + 63n+9) / 2. The analytical value of the deflection is calculated using the Maxwell-Mohr formula as a sum over all the rods of the structure, including the four supporting ones:

V \2

6 , = E (^Sd ^l „ /(EF), a = 1

where S(,), , = n + 3, 3n + 9, are the values of the force in the rod with number under the action a of a single load on the node with number , , la is the length of the corresponding rod. The stiffnesses EF of all the rods of the truss are assumed to be the same.

Calculation of the sums in (1) for a sequence of trusses with an increasing number of panels yields the following formulas:

6 1 ^max = ( 8a ³ + 3c ³ + 6h ³ ) / (8h 2 EF ),

6 2 ^max = ( 40a ³ + 5c ³ + 6h ³ ) / (8h ² EF ),

6 m ^ax = ( 112a ³ + 7c ³ + 6h ³ ) / (8h 2 EF ),

6 ₄ ^max = 3 ( 80a ³ + 3c ³ + 2h ³ ) / (8h ² EF), 6 m ^ax = ( 440a ³ + iic ³ + 6h ³ ) / (8h 2 EF),...

For an arbitrary number of panels, this expression has the form:

6 m ^ax = ( C 1 a ³ + C₂c ³ + C 3 h ³ ) / ( EFh² )

The coefficients in this expression are obtained in the Maple system by induction:

C 1 = k ( k + 1 )( 2k + 1 ) /6, C₂ = ( 2k + 1 ) /8, C₃ = 3/4.                      (5)

Hence, the calculation formula for the first natural frequency is:

h           EF

2 M (2k + 3) ( c 1 a ³ + C₂c ³ + C 3 h ³ )

3    Results and Discussion

The approximate dependence of the first natural oscillation frequency on the number of truss panels found should be compared with the minimum oscillation frequency obtained numerically taking into account all degrees of freedom of the system. Trusses made of steel rods with an elastic modulus of E = 2.1 - 10 5 MPa and with a rod cross-sectional area F = 9 sm ² are considered. All masses concentrated at the nodes are the same M = 100 kg , the panel size is a = 3m. In Figure 5, the dotted lines ^ 1

correspond to the numerical solution obtained as the minimum natural frequency of the entire spectrum of natural frequencies of the truss.

Fig. 5 – Dependence of the first frequency on the number of truss panels

The solution uses the Eigenvalues operator for calculating the eigenvalues of the matrix from the LinearAlgebra package of the Maple system. The curve w * is plotted using the analytical solution (6) with coefficients (5). The numerical and analytical methods yield very close results; the curves, intersecting at n =1, almost merge. With an increase in the number of panels, the frequency of natural oscillations decreases monotonically. For a more accurate estimate of the error in the analytical solution, we can consider the dimensionless relative value s = | ro 1 -w _D | / W j . The curves in Figure 5, plotted for two values of the truss height, show that starting with k=6, the error is within 5% and decreases with an increase in the number of panels.

Fig. 5 – Relative error of method.

The height of the structure h has little effect on the accuracy of the analytical solution.

4    Conclusions

At the very beginning of the work on solving the problem, an unexpected feature of the proposed truss design was revealed. With an odd number of panels in half a span, the determinant of the matrix of the nodes equilibrium equations system turned out to be equal to zero for any dimensions of the truss. A thorough kinematic analysis of the scheme confirmed this by the presence of virtual velocities of nodes. Finding a pattern of velocity distribution turned out to be difficult. The above-mentioned author's kinematic algorithm for finding virtual velocities does not automatically provide a solution. It was necessary to try several options, assigning instantaneous angular velocities to some rods and leaving others motionless. Excluding odd numbers of panels from the calculations made it possible to solve the problem of oscillation frequency in analytical form. The basis for calculating the natural frequency was the simplified Dunkerley method, in which the sum of displacements is replaced by its average value. In this case, instead of one characteristic node, by the partial frequency of which the sought oscillation frequency of the entire structure is calculated, two nodes are used (in the middle of the upper and middle of the lower chord). The half-sum of the coefficients corresponding to these nodes provides a solution to the problem with high accuracy and in a compact form. The obtained solution turned out to be more accurate than the solution using the original Dunkerley method.

The following main conclusions can be drawn:

• 1.    The estimate of the fundamental natural frequency obtained using the modified method is quite compact and at the same time provides good accuracy, increasing with the number of panels.

• 2.    External static uncertainty did not prevent finding an analytical solution for support reactions and deflection values.

• 3.    Asymptotic characteristics of the solution to the deflection problem were found.