JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 2, pp. 279-298, Warsaw 2006 FREE VIBRATIONS OF A COLUMN LOADED BY A STRETCHED ELEMENT Lech Tomski Janusz Szmidla Institute of Mechanics and Machine Design Foundations, Technical University of Częstochowa szmidla@imipkm.pcz.pl The problem of free vibrations of a two-member column loaded by a sretched element is considered in the paper. The influence of rigidity asymmetry on the bending between the stretched element and column rods, and the influence of the rigidmounting of the loading element and its length on the course of natural frequency in relation to the external load are analysed. The regions, for which the tested systems appear to be of divergence or divergence-pseudo-flutter type, are determined for presented physical and geometrical parameters of the column and sys- tems for border values of the coefficient of bending asymmetry. Numeri- cal computations are supported by appropriate results of experimental investigations. Keywords: elastic column, divergence instability, pseudoflutter systems, natural frequency 1. Introduction In the scientific literature dealing with vibration and stability of slender elastic systems (columns, frames), the following types of systems can be di- stinguished depending on the course of a curve in the plane: load P –natural frequency ω (characteristic curves): • divergence systems – loosing their stability due to buckling (conservative systems) (Gajewski andŻyczkowski, 1969a,b; Leipholz, 1974; Timoshen- ko andGere, 1963; Ziegler, 1968), • flutter systems – loosing their stability due to growing amplitudes of oscillatory vibrations (non-conservative systems) (Beck, 1953; Bogacz and Janiszewski, 1986; Bolotin, 1963; Langthjem and Sugiyama, 2000), 280 L.Tomski, J.Szmidla • hybrid systems – loosing their stability by flutter or divergence due to certain geometrical or physical parameters (non-conservative systems) (Dzhanelidze, 1958; Sundararajan, 1973, 1976; Tomski and Przybylski, 1985; Tomski et al., 1990, 2004), • divergence-pseudo-flutter systems (Bogacz et al., 1998; Tomski et al., 1994, 1995, 1996, 1998, 1999, 2004) – loosing their stability due to buc- kling (conservative systems) forwhich the function in theplane: load P – natural frequency ω (Fig.1) has the following course: – for P ∈< 0,Pc) (Pc – the critical load) the angle of the tangent to the characteristic curve can take a positive, zero or a negative value, – for P ≈Pc the slope of the characteristic curve is negative, – change of the natural vibration form (from the first to the second and inversely) takes place along the characteristic curves, (M1,M2 denote the first and second form of vibrations, respectively). Fig. 1. The course of characteristic curves for a divergence-pseudo-flutter system (cf. Tomski et al., 1995, 1996, 1998, 1999, 2004) 2. Formulation of the problem Two systems (Fig.2b,c) are considered in this paper: • a column loaded by a stretching beam B, Fig.2b (Tomski et al., 1997, 2004), • a column loadedbya force througha stringC(the force directed towards the positive pole), Fig.2c (Fieodosjew, 1969; Gajewski, 1970; Gajewski and Życzkowski, 1969). Free vibrations of a column... 281 Fig. 2. Physical models of considered systems: (a) column loaded by a follower force applied to the positive pole A, (b) column loaded by a stretching beamB, (c) column loaded by a force directed towards the positive pole C The method of mounting and loading the considered columns as well as the shape of axes of deflected rods are shown in Fig.2. The systems are com- posed of two rods with the flexural rigidities (EJ)1 and (EJ)2, respectively, and the mass per unit length (ρ0A)1 and (ρ0A)2 (while: (EJ)1 = (EJ)2, (ρ0A)1 =(ρ0A)2, (EJ)1+(EJ)2 =EJ, (ρ0A)1+(ρ0A)2 = ρ0A). The column rods have the same cross-section and they are made of the same material. 282 L.Tomski, J.Szmidla For the column B, see Fig.2b, the above mentioned elements are loaded by a compressive force P through infinitely rigid element (5) and a stretching beam with the flexural rigidity (EJ)3 and mass per unit length (ρ0A)3. The mounting elasticity of the stretched element is determined by the rigidity of a rotational spring C1. Rods (1,2,3) are connected at the free end by cube (4) with the concentratedmass m. This is done in a rigid way, that is to say that the deflection angles and displacement of the free end are equal for every rod. Changeable length l3 of the stretched element: bolt, beam or string (position of point O) is realized bymechanical system (6). The coefficient of asymmetry of the flexural rigidity µ1 is defined for the considered system µ1 = (EJ)3 (EJ)1+(EJ)2 (2.1) For µ1 = 0 (system C) it was assumed that element (3) is not characterized by flexural rigidity (string). In the case of 1/µ1 = 0 (system A), the flexural rigidity of the considered element is multitudinously higher for the stretching rods of the column (rigid bolt). In this case, themodel of the column (Fig.2a) loaded by the follower force directed towards the positive pole is obtained (Tomski et al., 1998, 2004). System A can be of a divergence or divergence- pseudo-flutter type. The influence of thementionedbelowparameters on the type of the system (divergence, divergence-pseudo-flutter) is analysed for columns with: • asymmetry of the flexural rigidity between the stretched element and compressive rods of the column (system B) • mounting elasticity of the stretched element: – rigid mounting (1/C1 =0) – system B – hingedmounting (C1 =0) – systems B, C • length of the stretched element of the column l3 – systems B, C, and with the constant value of the concentrated mass m at free end of the column. 3. Equations of motion and boundary conditions. Solution to the boundary value problem The equations of motion for the considered structures have been determi- ned from relationships (3.1), for column B – the three-rod system, and from relationship (3.1)1 for column C) – the two-rod system Free vibrations of a column... 283 (EJ)i ∂4Wi(x,t) ∂x4 +Si ∂2Wi(x,t) ∂x2 +(ρ0A)i ∂2Wi(x,t) ∂t2 =0 (3.1) (EJ)3 ∂4W3(x1, t) ∂x41 +S3 ∂2W3(x1, t) ∂x21 +(ρ0A)3 ∂2W3(x1, t) ∂t2 =0 where i is the ith stretching rod of the system (i=1,2) and S1 =S2 = P 2 S3 =−P (3.2) The geometrical boundary conditions for the rigidly restrained point (x=0) are W1(x,t) ∣ ∣ ∣ x=0 =W2(x,t) ∣ ∣ ∣ x=0 =W ′1(x,t) ∣ ∣ ∣ x=0 =W ′2(x,t) ∣ ∣ ∣ x=0 =0 (3.3) The remaining conditions, necessary for solving the boundary value problem, are given in the form — column B ∂W1(x,t) ∂x ∣ ∣ ∣ x=l1 = ∂W2(x,t) ∂x ∣ ∣ ∣ x=l1 = ∂W3(x1, t) ∂x1 ∣ ∣ ∣ x1=l3 W1(l1, t)=W2(l1, t)=W3(l3, t) W3(0, t) = 0 (EJ)3 ∂2W3(x1, t) ∂x21 ∣ ∣ ∣ x1=0 =C1 ∂W3(x1, t) ∂x1 ∣ ∣ ∣ x1=0 (3.4) 2 ∑ i=1 (EJ)i ∂2Wi(x,t) ∂x2 ∣ ∣ ∣ x=l1 +(EJ)3 ∂2W3(x1, t) ∂x21 ∣ ∣ ∣ x1=l3 =0 2 ∑ i=1 (EJ)i ∂3Wi(x,t) ∂x3 ∣ ∣ ∣ x=l1 +(EJ)3 ∂3W3(x1, t) ∂x31 ∣ ∣ ∣ x1=l3 −m ∂2W(l1, t) ∂t2 =0 — columnC ∂W1(x,t) ∂x ∣ ∣ ∣ x=l1 = ∂W2(x,t) ∂x ∣ ∣ ∣ x=l1 W1(l1, t)=W2(l1, t) 2 ∑ i=1 (EJ)i ∂2Wi(x,t) ∂x2 ∣ ∣ ∣ x=l1 =0 (3.5) 2 ∑ i=1 (EJ)i ∂3Wi(x,t) ∂x3 ∣ ∣ ∣ x=l1 +P (∂W1(x,t) ∂x ∣ ∣ ∣ x=l1 − W1(l1, t) l3 ) −m ∂2W(l1, t) ∂t2 284 L.Tomski, J.Szmidla The column undergoes small vibrations, therefore Wi(x,t)= yi(x)cos(ωt) W3(x1, t)= y3(x1)cos(ωt) (3.6) The solution to equations of motion (3.1), after previous separation of variables towards time and displacement (3.6), is yi(x)=C1i cosh(αix)+C2i sinh(αix)+C3i cos(βix)+C4i sin(βix) (3.7) y3(x1)=C13 cosh(β3x1)+C23 sinh(β3x1)+C33 cos(α3x1)+C43 sin(α3x1) where Cnj are integration constants n=1,2,3,4, j =1,2,3 and α2j =− 1 2 k2j + √ 1 4 k2j +Ω 2 j β 2 j = 1 2 k2j + √ 1 4 k2j +Ω 2 j while Ω2j = (ρ0A)jω2 (EJ)j k2i = Si (EJ)i k23 = P (EJ)3 Substitution of solutions (3.7) into the boundary conditions (3.3) and (3.4) for column B or (3.3) and (3.5) for column C (after previous separation of variables towards time and displacement) allows one to receive a system of twelve or eight homogeneous equations. The characteristic equation for the natural frequency of the considered column is obtained when the determinant of the characteristic system of equations equals zero. 4. Characteristic curves in the plane: load-natural frequency A characteristic curve in the plane: load-natural frequency determines the type of a system. This is why the results of theoretical research are presented below. Leipholz (1974) introducedacriterion for the loss of stabilitybydivergence. He stated that for conservative columns (divergence systems) describedby the boundary conditions y′′(x)y(x) ∣ ∣ ∣ x=l1 x=0 =0 (4.1) [ y′′′(x)+ P EJ y′(x) ] y(x) ∣ ∣ ∣ x=l1 x=0 =0 Free vibrations of a column... 285 the course of curves of eigenvalues Ω in relation to the external load λ has the negative slope in the whole range of the load (see curve (a) in Fig.3) dΩ dλ = − l1 ∫ 0 [y′(x)]2 dx l1 ∫ 0 [y(x)]2 dx < 0 (4.2) where y(x) is the lateral displacement of the column, Ω=Ω21, λ=P/(EJ). Fig. 3. The course of the basic characteristic curve of the parameter Ω Leipholz’s research, concerning characteristic curves for conservative and non-conservative loads, was generalised inworks byTomski et al. (1996, 1997, 1998, 2004). Adequate relationships, representing the eigenvalues in the plane Ω-λ, i.e. the parameter of the natural frequency Ω vs. the parameter of the external load λ, are: — for a generalised load (conservative systems, cf. Tomski et al., 1996, 2004) dΩ dλ = − l1 ∫ 0 [y′(x)]2 dx+ρ[y′(l1)]2−γ[y(l1)]2+2νy(l1)y′(l1) l1 ∫ 0 [y(x)]2 dx+ m[y(l1)] 2 ρ0A (4.3) where µ, γ, ρ, ν are established coefficients of the generalised load (Tomski et al., 2004), 286 L.Tomski, J.Szmidla — for a follower force directed towards the positive pole (A), see Fig.2a, (conservative systems, cf. Tomski et al., 1998, 200) dΩ dλ = − l1 ∫ 0 [y′(x)]2 dx+y(l1)y′(l1) l1 ∫ 0 [y(x)]2 dx+ m[y(l1)] 2 ρ0A (4.4) — for a column loaded by a stretched beam (B), see Fig.2b (conservative system, cf. Tomski et al., 1997) dΩ dλ = − l1 ∫ 0 [y′1(x)] 2 dx− l2 ∫ 0 [y′2(x)] 2 dx+ l3 ∫ 0 [y′3(x1)] 2 dx1 l1 ∫ 0 [y1(x)]2 dx+ l2 ∫ 0 [y2(x)]2 dx+ (ρ0A)3 (ρ0A)1 l3 ∫ 0 [y3(x1)]2 dx1+ m[y1(l1)]2 (ρ0A)1 (4.5) — for non-potential systems (non-conservative system, cf. Tomski et al., 2004) (i) Beck’s generalised column dΩ dλ = − l1 ∫ 0 [y′(x)]2 dx+ηy(l1)y′(l1)−λη [ y′(l1) ∂y(l1) ∂λ −y(l1) ∂y′(l1) ∂λ ] l ∫ 0 [y(x)]2 dx+ m[y(l1)] 2 ρ0A +λη [ y′(l1) ∂y(l1) ∂Ω −y(l1) ∂y′(l1) ∂Ω ] (4.6) (ii) Reut’s generalised column dΩ dλ = − l1 ∫ 0 [y′(x)]2 dx+ηy(l1)y′(l1)+λη [ y′(l1) ∂y(l1) ∂λ −y(l1) ∂y′(l1) ∂λ ] l ∫ 0 [y(x)]2 dx+ m[y(l1)] 2 ρ0A −λη [ y′(l1) ∂y(l1) ∂Ω −y(l1) ∂y′(l1) ∂Ω ] (4.7) where η is a coefficient of the follower force (Beck’s column) or a coefficient of the follower moment (Reut’s column). In the case of columns A, B, C and generalised loads (conservative sys- tems), the slope of a curve of the natural frequency may be negative (diver- gence system – curve (a) in Fig.3), positive (divergence-pseudo-flutter system – curve (c)) or equal to zero (curve (b)) depending on geometrical parameters of the loading and receiving heads. In the case of the generalised load (cf. Tomski et al., 1996), the parameter of natural frequency (4.8)2 wasdetermined on the basis ofRayleigh’s quotient. Free vibrations of a column... 287 Relationship (4.3), conservative condition of load (4.8)1 and boundary condi- tions at the restrained and free end of the columnwere used ν+µ−1=0 (4.8) Ω= l1 ∫ 0 [y′′(x)]2 dx l1 ∫ 0 [y(x)]2 dx+ m[y(l1)] 2 ρ0A +λ dΩ dλ 5. Results of numerical computations Numerical computations were accomplished on the basis of the solution to the boundary value problem for the considered systems. The influence of length l3 of the stretched element of the columnand the asymmetry of flexural rigidity between the compressed rods and stretched element on the type of the system were determined. The range of µ1, l∗3 parameters, for which the con- sidered columns were of the divergence type (D) or divergence-pseudo-flutter type (PF), was specified taking into account an educed criterion (systems A, B,C– compareTomski et al., 1997, 1998, 2994) describing eigenvalue curves in theplane: load P -natural frequency ω, (4.3) and (4.5). The computationswe- re carried out for two extreme cases of themounting of the stretched element, i.e. hingedmounting (c∗1 =0), see Fig.4a, and rigidmounting (1/c ∗ 1 =0), see Fig.4b, with a constant value of the concentrated mass m∗ at the free end of the systems, where l∗3 = l3 l1 c∗1 = C1l1 EJ m∗ = m l1ρ0A (5.1) In the case of the hingedmounting (Fig.4a), the influence of parameter l∗3 on the type of the system for a column loaded by a force directed towards the pole (C−µ1 = 0) and a column loaded by a follower force directed towards the pole (A−1/µ1 =0) was additionally determined. Numerical computations were carried out for the considered systems in order to determine the course of changes in natural frequencies in relation to the external load for chosen parameters µ1, l∗3 (lines 1-4 inFig.4). The charac- ter of changes of the first two natural frequencies in a dimensionless form Ω∗t (t=1,2) and additional symmetrical natural frequencies Ω∗s2 (Tomski et al., 288 L.Tomski, J.Szmidla Fig. 4. The effect of µ1, l∗3 parameters on the type of systems: divergence (D), divergence-pseudo-flutter (PF) for: (a) hinged mounting – columns A, B, C, (b) rigid mounting – column B 1997) in relation to the dimensionless loading parameter λ∗ were specified (Fig.5-Fig.8). It was assumed that λ∗ =λl21 = Pl21 EJ Ω∗ =Ωl41 = ρ0Aω 2l41 EJ (5.2) Fig. 5. Characteristic curves for Ω∗ parameter in relation to l∗ 3 for c∗ 1 =0 Free vibrations of a column... 289 Two cases of mounting of the stretched element; i.e. hinged mounting (Fig.5, Fig.6) and rigid mounting (Fig.7, Fig.8) with the fixed value of the concentrated mass at the free end of the system were considered similarly as in Fig.4. Fig. 6. Characteristic curves for Ω∗ parameter in relation to µ1 for c∗1 =0 The slope of the basic natural frequency Ω∗ for λ∗ = 0 may be negati- ve, positive or equal to zero. This is depicted in graphs concerning different parameters µ1, l∗3 see lines 1-4 in Fig.4). 290 L.Tomski, J.Szmidla Fig. 7. Characteristic curves for Ω∗ parameter in relation to l∗ 3 for 1/c∗ 1 =0 Changes in the slope of the considered natural frequency curve allows one to rank the considered systems among one of the two types: divergence (D) – (∂Ω∗/∂λ∗)|λ∗=0 < 0 or divergence-pseudo-flutter (PF) – (∂Ω ∗/∂λ∗)|λ∗=0 > 0. The course of eigenvalueswasdistinguishedbybroken lines (Fig.5-Fig.8).The quality (∂Ω∗/∂λ∗)|λ∗=0 =0 is received for eigenvalues at thedeterminedpara- meters m∗, µ1, l∗3, c ∗ 1. The course of the natural frequency Ω ∗s 2 corresponding to the symmetrical form of vibrations is identical for every presented graph due to the constant total rigidity EJ and length l1 of the stretching rods of the system assumed in calculations. The value of the critical load is determi- ned for Ω∗ =0 for the presented curves of changes in the natural frequencies (Fig.5-Fig.8). Free vibrations of a column... 291 Fig. 8. Characteristic curves for Ω∗ parameter in relation to µ1 for 1/c∗1 =0 6. Results of experimental research Numerical computations were carried out for the systems considered in the paper. The course of natural frequencies in relation to the external load for the column loaded by the stretching rod B and for the column loaded by the force directed towards the pole C, was verified on experimental set-ups (Tomski et al., 1996, 1998, 2004). Physical and geometrical parameters are given in Tables 1 and 2. The results of experimental research (points) and numerical computations (lines) arepresented inFig.9 andFig.10 (columnB)and inFig.11 (columnC). SystemsB1, B2, B3, B5, B6, C are characterised by a hingedmounting of the stretched element (c∗1 =0) for x1 =0. The rigidmounting was applied to the remaining cases. Experimental investigation of the column B was limited to the first three basic natural frequencies (M1, M2, M3) and to two additional frequencies (M2e, M3e) corresponding to the symmetrical form of vibrations (Tomski et al., 1997). In the case of columnC, numerical computations and experimental investigation were carried out for the first two natural frequencies (M1, M2) for six chosen positions of the pole (length l3). The results of numerical com- putations and experimental investigations are in good agreement. 292 L.Tomski, J.Szmidla Fig. 9. The course of characteristic curves for column: B1 (a), B2 (b), B3 (c) Free vibrations of a column... 293 Fig. 10. The course of characteristic curves for column: B4 and B5 (a), B6 andB7 (b), B8 and B9 (c) 294 L.Tomski, J.Szmidla Fig. 11. The course of characteristic curves for columnC Table 1. Physical and geometrical parameters of the column loaded by the stretching rod B Column EJ ρ0A (EJ)3 (ρ0A)3 l1 l3 m [Nm2] [kg/m] [Nm2] [kg/m] [m] [m] [kg] B1 716.58 2.401 2132.6 2.918 0.63 0.31 0.39 B2 362.062 2.586 2132.6 2.918 0.63 0.31 0.35 B3 716.58 2.401 76.32 0.315 0.63 0.31 0.43 B4 152.68 0.631 589.04 0.877 0.61 0.305 0.34 B5 152.68 0.631 589.04 0.877 0.61 0.305 0.34 B6 362.062 2.586 76.32 0.315 0.61 0.305 0.35 B7 362.062 2.586 76.32 0.315 0.61 0.305 0.35 B8 716.58 2.401 38.81 0.219 0.9 0.9 0.58 B9 716.58 2.32 831.49 1.041 0.9 0.9 0.58 Table 2. Physical and geometrical parameters of the column loaded by the force directed towards the positive pole C Column EJ ρ0A (EJ)3 (ρ0A)3 l1 m [Nm2] [kg/m] [Nm2] [kg/m] [m] [kg] C 206.17 1.199 – – 0.6 1.03 Theabove presented changes of natural frequencies in relation to the exter- nal load are typical for divergence-pseudo-flutter systems. Free vibrations of a column... 295 7. Summary On the basis of experiments and carried out numerical simulations for the presented two-member column loaded by the stretching rod, it can be stated in this paper that: • Different characteristic cases of column loading, i.e. from Euler’s load, through loads by forces directed towards the positive pole to loads by a follower force directed towards the positive pole, can be obtained in relation to assumed values of parameters determining the elasticity of mounting of the stretching rod. • The considered systems (a column loadedby the stretched element anda column loaded by the force directed towards the positive pole) appear to be ones of the two characteristic types, namely divergence or divergence- pseudo-flutter systems. This can be resolved on the basis of the course of the natural frequency in relation to the external load for the given geometrical and physical parameters. • The obtained results of numerical computations and experimental inve- stigations regarding the course of natural frequencies in relation to the external load showed good agreement. Acknowledgment The authors would like to express their gratitude to drM.Gołębiowska-Rozanow for her partnership in the experimental research andA. Kasprzycki for preparing the loading heads. Thepaperwassupportedby theStateCommittee forScientificResearch,Warsaw, Poland, under grantNo. 7T07C04427 and statutory funds No. BS – 1-101-302/99/P. References 1. BeckM., 1953,Die knicklastdes einseitig eingespannten tangential gedruckten stabes, ZAMP, 4, 225/228, 476-477 2. BogaczR., Imiełowski Sz., Tomski L., 1998,Optymalization and stability of columns on example of conservative and nonconservative systems, Machine Dynamics Problems, 20, 35-47 3. 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Tomski L., Szmidla J., Gołębiowska-Rozanow M., 2004a, Free vibra- tion of a two member column loaded by a stretching rod, XXI Symposium – Vibrations in Physical Systems, Poznań-Kiekrz, 379-382 26. TomskiL., SzmidlaJ.,Gołębiowska-RozanowM., 2004b,Vibrationsand stability of a two-rod column loaded by the sector of a rolling bearing, J. of Theoretical and Applied Mechanics, 42, 4, 905-926 27. ZieglerH., 1968,Principles of Structural Stability,Waltham,Blaisdell Publi- shing Company Drgania swobodne kolumn obciążonych poprzez rozciągany element Streszczenie W pracy rozważa się zagadnienie drgań swobodnych dwuprętowej kolumny ob- ciążonej poprzez rozciągany element. Analizuje się wpływ asymetrii sztywności na zginanie pomiędzy elementem rozciąganymukładu a prętami kolumny,wpływ sztyw- ności zamocowania elementu obciążającego oraz jego długości na przebieg częstości 298 L.Tomski, J.Szmidla drgańwłasnychw funkcji obciążenia zewnętrznego. Dla prezentowanych parametrów fizycznych i geometrycznych kolumny oraz układów dla granicznych wartości współ- czynnika asymetrii na zginanie wyznacza się obszary, w których omawiane układy są typudywergencyjnego lubdywergencyjnegopseudoflaterowego.Obliczenia numerycz- ne poparte są odpowiednimi wynikami badań eksperymentalnych. Manuscript received November 14, 2005; accepted for print January 12, 2006