UDC DAMAGE DIAGNOSTICS IN A VERTICAL BAR ON THE ELASTIC SUSPENDER WITH CONCENTRATED MASS

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1 1 UDC DAAGE DIAGNOSTICS IN A VERTICAL BAR ON THE ELASTIC SUSPENDER WITH CONCENTRATED ASS A Ilgamov, BZ Sultanov, AN Tazhitdinov, AG Khakimov Institute of ehanis, Ufa Branh RAS, Ufa, Russia Using two natural longitudinal vibration frequenies we an determine both plae and size of a ross inision in a vertial bar hanged on the elasti suspender and strethed by its own weight and the gravity of the load at the upper end of the bar Keywords: bar, vertial bar-type olumn, load, natural frequenies of longitudinal vibration, inision parameter, inision oordinate In the ase of the bars of finite length for deteting defets we an use hanges in the natural frequeny spetrum of flexural vibrations [1] or those in natural longitudinal vibrations [] In [3] the solution is given for determining the variable ross-setional area on the longitudinal oordinate by the dependene of the bar s free end movement on the perturbing fore frequeny Paper [4] deals with the solution of inverse problems on longitudinal travelling waves in bars of finite length Consideration is given to a stress-strain state for a vertial bar hanged on the elasti suspender of rigidity 1 and strethed by its own weight and the gravity of the load of mass (Fig 1) It is assumed that there is a short portion in the bar (small when ompared to its total length) with a lesser ross-setional area This inision does not ause any bar bending and serves as a model for damage in the bar, partiularly for a lateral open rak A stress-strain state in the limits of flexibility for a thin bar is under onsideration only Considering that the rak appears as a result of the growth of a small nuleus (not neessarily in the most stressed plae) we assume that the inision might be loated in any plae along the length of the bar Our

2 goal is to determine the oordinate of this inision and its size in the approximation of plane ross setions x x L F f l Fig 1 Let us denote the length and ross-setional of the bar by L, F, modulus of elastiity, density and internal frition oeffiient by E, ρ, µ, length and rosssetional area of the inision by l, f, the latter s oordinate by x, movement and tensile fore of the bar by ut, The relation between tension σ and deformation ε is taken as follows ε u σ= E ε+µ, ε= t x In aordane with the aforesaid we have 3 u u u u u E +µ, T EF ρ = = +µ x x t t x x t easuring oordinate x from the point of fixation let us write the boundary onditions u T = с1u ( x=, ) T = ( x= L) t

3 Within the bounds of the inision of omparatively short length 3 and in its lose proximity there is a omplex spatial stress-strain state [5] However, for the sake of simpliity we assume uniaxial tension-ompression and also take no aount of inertial fores As shown by experimental data [6], on impat at the lower end, the average value of the longitudinal damping oeffiient for the hanged bar with inision is % greater than the same oeffiient for the bar without inision Let us examine the dynami problem [] 3 u u ρ u u u +µ =, T = EF, +µ x x t E t x x t u T = с1u ( x=, ) T = ( x= L), t u u u1 u1 u1 u u1 +µ = +µ, u u1 = ml µ, x= x x x t x x t x t t l ( ) where lf m = Lf Parameter m and oordinate x are thus involved in the simplest model of inision In parameter m the ratio of the ross setional area to the length of the bar F/L is onsidered to be known In ase of the diret problem the ratio of the length of inision to its area is also known, the inverse problem neessitates the determination of this ratio Variables l and f by themselves are not determined in the model [] The partial solution of problem (1) at µ = has the form ( os sin ) sin (, ) u = A α x+ B αx ωt α=ω a a = E ρ Four onstants in this solution written for domains x x and x x L, are speified by four boundary onditions (1) To make it so that A1, A, B1, B will never be zero together, the following determinant should equals zero where (1) det( a ij ) =, ()

4 11 1 ω a = +, a = EFα, a =, a =, 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( x ) ( ) ( ) a =, a =, a = sin α L, a = os α L, a = sin α x, a = os α x, a = sin α x, a = os α x, a = os αx + mlαsin α x, a = sin αx mlαos α, 41 4 a = os α x, a = sin α x Condition () gives the frequeny equation ( ( ) ( ) ( ) ( )) + α( tg( α ) sin( α ) os( α ) tg( α ) sin ( α ) )( ) mlα sin αx sin αx os αx tg αl + ml L x x L x + D 1 ω tg αl D,, D ( ) + = = = EFα When oeffiient EFα, the frequeny equation takes the form ( ( ) ( ) ( ) ( ) ( ) ) + (3) mlα tg αl sin αx os αx tg αl sin αx 1= (4) In this ase load mass has no effet on natural vibration frequenies of the bar The term involving m in (4) gives a hange in these frequenies formula At At the known oordinate of the inision parameter m is determined by the 1 + D tg( αl), ( 1+ ( + )) ( ) ( ) ( ) ( L) ( ) ( ) ( ) ( x ) m = αld d D d = sin αx sin αx os αx tg α, d = tg αl os αx sin αx os α 1 m = αl tg αl os αx sin αx os αx ( ( ) ( ) ( ) ( ) ) The mass of the load hanged on the bar an be determined by the formula ( ) + α( 1+ ) ω ( 1 mlαd ) EFα tg αl ml d d = The mass of the load hanged on the undamaged bar (m = ) is determined by the formula

5 5 ( ) 1 EFαtg αl = ω For the bar without inision (m = ) and at from the equation os αl = natural frequenies are equal [] to αl = (k 1)π/ (k = 1,, ) or ω k = (k 1)πa/L To determine m and x it is neessary to analyze natural longitudinal vibration frequenies of the bar with inision Suh an investigation was performed for bar s flexural vibrations in [7] The numerial solution to equation (3) is made for the following parameters of the set: E = 1 11 Па, ρ = 78 kg/m 3, L = 1 m, F = 1 m = 5 kg, 1 7 =1 N/m Sound veloity a = 5636 m/s In this ase the first, seond and third natural frequenies of the bar without inision are ω1 = 116 rad/s, ω = 175 rad/s, ω 3 = 339 rad/s, respetively Fig shows the dependenes of parameter m on irular frequenies of the bar s longitudinal vibrations ω 1, ω, ω 3 (rad/s) at different x / L These dependenes for small values of m are linear Analysis of the urves shows that it is quite diffiult to determine parameter m by the first natural frequeny Fig 3 gives the dependenes of ratio x / L on irular frequenies ω 1, ω, ω 3 of longitudinal vibrations at different m One an see the periodial dependene of irular frequenies of longitudinal vibrations ω 1, ω, ω 3 on x / L It is also orret to onlude that determination of the oordinate of inision by the first irular frequeny presents diffiulty beause of auray Big hanges in the oordinate of inision and parameter m orrespond to insignifiant ones in the first irular frequeny

6 6 m m m x/l= x/l= x/l= ω ω ω 3 Fig Dependene of parameter m on irular frequenies of bar s longitudinal vibrations ω 1, ω, ω 3 (rad/s) at different x / L ω m= ω m= 1 15 ω m= x/l x/l x/l Fig 3 Dependene of x / L ratios on irular frequenies of bar s longitudinal vibrations ω 1, ω, ω 3 (rad/s) at different m Fig 4 shows the dependenes of load mass on irular frequenies of the bar s longitudinal vibrations ω 1, ω, ω 3 at m = 1 for different x / L With an inrease in irular frequenies of the bar s longitudinal vibrations load mass is enhaned as well The load mass an be determined by the first frequeny

7 7 51 x/l= x/l= x/l= ω ω ω 3 Fig 4 Dependene of load mass in kg on irular frequenies of bar s longitudinal vibrations ω 1, ω, ω 3 (rad/s) at m = 1 for different x / L x m 4 1 ω = ω = ω ω Fig 5 Dependenes of inision oordinate x in m and parameter m on irular frequenies of bar s longitudinal vibrations ω, ω 3 (rad/s) If we write the frequeny equation for two frequenies of free longitudinal vibrations, using the obtained set of equations we an determine the oordinate of inision x and parameter m Fig 5 gives the dependenes of inision oordinate x in

8 8 m and parameter m on irular frequenies of the bar s longitudinal vibrations ω, ω 3 (rad/s) The performed researh shows the possibility of determining the oordinate of inision x and parameter m by two natural frequenies of free longitudinal vibrations This work has been done under RFFR grant r_povolzhie_a Referenes 1 Vankov YuV, Kazakov RB, Yakovleva ER Natural frequenies of hardware produt as informative mark of damage Eletroni Journal Tehnial Aoustis, 5, 5 Ilgamov A Diagnostis of defets in vertial bar Proeedings of Institute of ehanis of Ufa Branh, RAS Vol 5 Ufa: Gilem 7 P Vatulian AO Inverse problems in deformable body mehanis osow: Fizmatlit 7 4 p 4 Vatulian AO, Soluyanov NO On determining plae and size of avities in elasti bar // Defetosopy 5 No 9 P Parton VZ, orozov E Elasti-plasti frature mehanis // osow: Nauka p 6 Razyantsev AO Vibroaousti diagnostis of pumping rods in servie Cand Si Thesis Ufa: USPTU, p 7 Okrushko EI, Urakseev A Defetosopy of well pump bars // osow: Nedra p