Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 4, pp. 991-1004, Warsaw 2015 DOI: 10.15632/jtam-pl.53.4.991 ANALYTICAL SOLUTION OF EXCITED TORSIONAL VIBRATIONS OF PRISMATIC THIN-WALLED BEAMS Elżbieta Augustyn, Marek S. Kozień Cracow University of Technology, Institute of Applied Mechanics, Kraków, Poland e-mail: kozien@mech.pk.edu.pl In the paper, the analytical solution of excited torsional vibrations of prismatic thin-walled beams for different types of boundary conditions and different types of external excitation of torsional moment are formulated. The presented solution can be applied, among others, to preliminary analysis of the optimal position of the actuators and the value of the applied voltage to the elements for minimization of the vibrations of the beams. Keywords: thin-walled beams, torsional vibrations, excited vibrations 1. Introduction Knowledge of the analytical solution of externally excited vibrations for a considered model of realistic structures is very useful for preliminary analysis in the design process or choosing parameters and algorithms for vibration control of the systems. The model of an excited pri- smatic beam response for torsional or coupled bending-torsional vibrations can be applied for analysis of vibration of somemachine elements, e.g. turbine-blades (Gryboś, 1996; Łączkowski, 1974; Pust and Pesek, 2014; Rao, 1991). Sometimes, the beam cross-section takes form of an open thin-walled structure. In this paper, thin-walled beams of closed cross-sections as general models of blades are analyzed (Librescu and Na, 1998; Song and Librescu, 1993; Song et al., 2002). Recently, application of piezoelectric elements for vibration reduction of thin structures are commonly discussed in the literature (Elliott and Nelson, 1997; Hansen and Snyder, 1997; Mo- heimani andFleming, 2006; Preumont, 2006). See also the papers byFerdek andKozień (2013), Kozień andKołtowski (2011). These types of vibration cancellation were considered by the au- thors in preliminary analysis of the possibility of reduction of torsional vibrations of prismatic beams with solid cross-sections (Augustyn andKozień, 2014). There are well known analytical solutions for problems of the dynamics of thinwalled beams with open cross-section, startingwithGere (1954), Gerewith Lin (1958), Aggrawal withCranch (1967), and Carr (1969) and later e.g. by Bishop et al. (1989), Dokumaci (1987), Kaliski and Solarz (1992), Tao (1964), Timoshenko et al. (1974) andYaman (1997). Different cases of vibra- tions in such a type of structures are discussed in many of articles. A review of these different approaches, theories and models for static and dynamic cases is given by Sapountzakis (2013). The effect of variable cross-section on natural frequencies is discussed in (Eisenberger, 1997). Non-linear models are discussed by CrespoDa Silva (1988a,b), Rozmarynowski with Szymczak (1984) andDiEgidio et al. (2003a,b). The steady-state forced vibrations are discussedbyCrespo Da Silva (1988b) and Di Egidio et al. (2003b). The problem of optimal design of a thin-walled beam for a given natural frequency is analyzed by Szymczak (1984). Torsional vibrations of composite thin-walled beams are analyzed by Song and Librescu (1993). The models with the effect of rotary inertia are analysed by Arpaci et al. (2003) and with adaptive capabilities by Song et al. (2002). A detailed solution for the general case of excited vibrations, especially in the 992 E. Augustyn,M.S. Kozień transient case, are not easily found in the literature. This formulation was given by one of the authors of this paper for bending vibrations of a beam for different types of external excitations (Kozień, 2013). Themain aim of this paper is to formulate the analytical solution for describing the excited torsional-type vibrations of a thin-walledbeam fordifferent combinations of boundaryconditions (simply supported, free, fixed) and different types of external torsional moment type excitations (harmonic concentrated, harmonic distributed, pulse concentrated, pulse distributed). The pre- sented solution, among others, can be applied to preliminary analysis of the optimal position of the actuators and to calculations of voltage applied to the elements for minimization of vi- brations of beams when the application of piezoelectric elements are considered. In this model, the influence of external piezoelectric elements can be modeled by the external concentrated moment of a suitable value, as it was done by Elliott and Nelson (1997), Hansen and Snyder (1997) for pairs of elements in simulations of bending vibrations of beams, as was proposed by Augustyn and Kozień (2014) for double pairs of elements for torsional vibrations of beams. 2. Formulation of the problem 2.1. Equation of motion Let us consider a prismatic beamwith a thin-walled cross section of an open type (Murray, 1986; Piechnik, 2007; Vlasov, 1959) in which it is also assumed that the material is isotropic. The geometry of the cross-section is shown in Fig. 1, where G is the gravity center (centroid), S – shear center. The origin of the co-ordinate system Sxyz lies in the shear center, and the axes Sy and Sz are parallel to the principal axes of the cross-section (Gη,Gζ). Fig. 1. Geometry of the cross-section The equations of coupled vibrations of the beam take the following form EJζ ∂4v(x,t) ∂x4 +ρA [∂2v(x,t) ∂t2 −zG ∂2ϕ(x,t) ∂t2 ] = qy(x,t) EJη ∂4w(x,t) ∂x4 +ρA [∂2w(x,t) ∂t2 +yG ∂2ϕ(x,t) ∂t2 ] = qz(x,t) EJω ∂4ϕ(x,t) ∂x4 −GJs ∂2ϕ(x,t) ∂x2 +ρJ0 ∂2ϕ(x,t) ∂t2 +ρA [ −zG ∂2v(x,t) ∂t2 +yG ∂2w(x,t) ∂t2 ] = ms(x,t) (2.1) where v(x,t) is the displacement of the beam in the y direction,w(x,t) is the displacement of the beam in the z direction, ϕ(x,t) is the rotation of the axis of beam around the x direction, Jη is the principalmoment of inertia of the cross-sectionwith respect to the η axis, Jζ is the principal Analytical solution of excited torsional vibrations... 993 moment of inertia of the cross-sectionwith respect to the ζ axis,J0 is thepolarmoment of inertia with respect to the shear center, Js is the equivalent moment of inertia of the cross-section due to torsion, E is Young’s modulus, G is the shear modulus, ρ is the material density, yG, zG are the positions of the gravity center G (co-ordinates), qy(x,t) is the distributed force acting in the y direction (external excitation), qz(x,t) is the distributed force acting in the z direction (external excitation), ms(x,t) is the distributed twisting moment (external excitation). When the considered cross-section is such that the position of the shear center and gravity center is the same (e.g. for the cross-sectionwith two axes of symetry), the following relationship is valid yG = zG =0. It means, that the equation of motion for twisting existing in system (2.1) is separated from the equations of bending vibrations, and takes simplified and independent form from bending vibrations form EJω GJs ∂4ϕ(x,t) ∂x4 − ∂ 2ϕ(x,t) ∂x2 + ρJ0 GJs ∂2ϕ(x,t) ∂t2 = 1 GJs ms(x,t) (2.2) Hence, for the considered cross-section, it ispossible toanalyze torsional vibrations separately with the bending ones. If the cross-section is a monolithic type, equation of motion (2.3) takes the following form −GJs ∂2ϕ(x,t) ∂x2 +ρJ0 ∂2ϕ(x,t) ∂t2 = ms(x,t) (2.3) In general, vibrations of the analyzed structure can be described in form of a three- dimensional model of a solid structure. But for some structures, for which one characteristic dimension (the so-called length) is high enough in comparison to the characteristic dimensions of the cross-section, then so-called one-dimensional models of structures are built for simplici- ty of the analysis. The results of their application must give good enough results of analysis in comparison with three-dimensional models. The same case takes place during detailed con- sideration of the cross-section shape. If one characteristic dimension of the cross-section (the so-called thickness) is small enough in comparison with the second one (the so-called width), the whole structure is called a thin-walled one. A detailed rule states that for a thin-walled cross-section, the thickness is more than eight times smaller than the highest way measured along the middle-line of the cross-section between its two end-points. Moreover, the length of this way should bemore than eight times smaller than the length of the beam (Piechnik, 2007). For such a beam the assumptions of the Bernoulli-Euler beam theory or the de Saint-Venant rule of loadings, for example, are invalid. Theirmotionmust be analyzed as a three-dimensional body or by the application of an especially formulated theory for thin-walled beams. Due to the formof themiddle line of the cross-section, three types of thin-walled beams are defined:with an open cross-section,with a closed cross-section andwith amixed one (Piechnik, 2007). Depending on the type of cross-section, the suitable theory of thin-walled beams should be applied. For a thin-walled beamwith an open cross-section, as considered in this article, theVlasov theory can be applied (Piechnik, 2007; Vlasov, 1959). 2.2. Boundary conditions The equation of motion of the thin-walled beam is the fourth order due to the spatial variable x. Therefore, a set of the four boundary conditions must be formulated for each beam element. Their formulation is connected with the Vlasov theory of thin-walled beams andmust take into account the warping effect. The following types of boundary conditions can be formulated in a natural way (Gere, 1954; Piechnik, 2007): 994 E. Augustyn,M.S. Kozień • Simply supported: — no rotation of the cross-section around the axis of the beam ϕ =0 (2.4) — zero normal stress (free of warping of the cross-section) ∂2ϕ ∂x2 =0 (2.5) • Fixed: — no rotation of the cross-section around the axis of the beam ϕ =0 (2.6) — plain cross-section (blocked warping of the cross-section) ∂ϕ ∂x =0 (2.7) • Free: — zeroes for the total torsional moment (free for rotation) GJs ∂ϕ ∂x −EJω ∂ϕ3 ∂x3 =0 (2.8) — zero normal stress (free for warping of the cross-section) ∂2ϕ ∂x2 =0 (2.9) 3. Eigen-problem, natural vibrations Now, the eigen-problem analysis of the thin-walled prismatic beam with length l is considered. The equation of motion has the form EJω GJs ∂4ϕ(x,t) ∂x4 − ∂2ϕ(x,t) ∂x2 + ρJ0 GJs ∂2ϕ(x,t) ∂t2 =0 (3.1) The solution to the problem is proposed in form (3.2)1 – the Fourier method of solution. After suitable manipulations, it leads to a solution to the problem of separated variables (3.2)2. It can be written as two independent ordinary differential equations (3.2)3,4. The first of the fourth order for the independent spatial variable x, and the second of the second order for the independent variable t (time). The form of unknown functions in the spatial domain can be written in form (3.2)5). For the above given homogeneous boundary conditions, these series of functions satisfy orthogonality conditions (3.4) – see Appendix A for detailed analysis. Finally, the solution can bewritten in form (3.7), where the series of constants an and bn are determined based on initial conditions (3.6) ϕ(x,t)= +∞∑ n=1 Xn(x)Tn(t) EJω GJs d4Xn(x) dx4 Xn(x) − d2Xn(x) dx2 Xn(x) =− ρJ0 GJs d2Tn(t) dt2 Tn(t) = λ4n EJω GJs d4Xn(x) dx4 − d 2Xn(x) dx2 −λ4nXn(x)= 0 d2Tn(t) dt2 +λ4n GJs ρJ0︸ ︷︷ ︸ ω2n Tn(t)= 0 Xn(x)= An sin(αnx)+Bncos(αnx)+Cn sinh(βnx)+Dncosh(βnx) (3.2) Analytical solution of excited torsional vibrations... 995 where αn = √ −GJs + √ (GJs)2+4EJωGJsλ4n 2EJω βn = √ GJs + √ (GJs)2+4EJωGJsλ4n 2EJω ωn = λ 2 n √ GJs ρJ0 (3.3) and l∫ 0 Xn(x)Xm(x) dx = { γ2n n = m 0 n 6= m (3.4) Finding the solution to the system of differential equation (3.2)2 or (3.2)3,4 for a given boundary conditions is the well-known eigen-mode problem which gives a set of eigen-values (powered natural frequencies of a system ω2n – solution of equation (3.2)4 and a set of eigen- functions (eigen-modes, waveforms of eigen-functions Xn(x) – solution of equation (3.2)3). The equations, which make possible the finding of natural frequencies ωn (3.3)1,2,3 or exact formula for ωn for a different combination of typical types of boundary conditions (fixed, simply sup- ported, free) are given in Table 1. Moreover, the analytical form of eigen-functions for the same combination of boundary conditions (simply supported, free, fixed) are also given in Table 1. The other name of the problem is the modal problem and the detailed solution of this problem was done by Gere (1954). If the solution to the equation of motion fulfills initial conditions (3.6), where ϕ0(x) is the initial angle of torsion of the beam for t = 0, and Ω0(x) is the initial angular velocity of the beam for t =0, the problem of natural vibrations is completely defined. These initial conditions should be applied for the detailed solution of the second equation of system (3.2)3,4 ϕ(x,0)= ϕ0(x) ∂ ∂t ϕ(x,0)= Ω0(x) (3.5) Solution (3.6) represents the so-called natural vibrations, which are the response of the system to initial conditions written in the time and spatial domains ϕ(x,t) ϕ(x,t) = +∞∑ n=1 Xn(x)[an sin(ωnt)+ bn sin(ωnt)] an = 1 ωnγ2n l∫ 0 Ω0(x)Xn(x) dx bn = 1 γ2n l∫ 0 ϕ0(x)Xn(x) dx (3.6) 4. Excited vibrations When analyzing the excited vibrations of a realistic beam (with internal damping), the general solution of the homogeneous differential equation is a function relatively fast tending to zero with respect to time due to the internal and external damping. The complete solution to the problem of excited vibrations, understood as the particular solution to the non-homogeneous equation (2.2), for the excitation function of harmonic type, has the form of a sum of two components: connected with a set of natural frequencies ωn and connected with the external loading frequency ν. Note, that the set of constants An, Bn, Cn and Dn existing in particular solution (2.2)5, must be found taking into account the complete solution. 996 E . A u gu sty n , M .S . K ozień T a b le 1 . F orm u las for eigen freq u en cies an d eigen m o d es (G ere, 1954) Boundary Equation/formula for natural conditions frequency ωn or Eigenfunction Xn(x) parameters αn and βn Simply supported- ωn = nπ l2 √ n2π2EJω+l2GJs ρJ0 sin ( nπ l x ) -simply supported Fixed-fixed αnβn = ωn √ ρJ0 EJω βn[cosh(βnl)−cos(αnl)] αn sinh(βnl)−βn sin(αnl) sin(αnx)− cos(αnx) − αn[cosh(βnl)−cos(αnl)] αn sinh(βnl)−βn sin(αnl) sinh(βnx)+cosh(βnx) Simply supported- tanh(βnl) βn = tan(αnl) αn sin(αnx)− sin(αnl)sinh(βnl) sinh(βnx)-fixed Fixed-free α4n+β 4 n α2nβ 4 n cos(αnl)cosh(βnl) βn αn α2n sinh(βnl)−αnβn sin(αnl) α2ncosh(βnl)+β 2 ncos(αnl) sin(αnx)− cos(αnx) −α 2 n−β2n αnβn sin(αnl)sinh(βnl)+2=0 −α 2 n sinh(βnl)−αnβn sin(αnl) α2n cosh(βnl)+β 2 ncos(αnl) sinh(βnx)+cosh(βnx) Simply supported- β2n tanh(βnl)= α 2 n tan(αnl) sin(αnx)+ βn αn cos(αnl) cosh(βnl) sinh(βnx)-free Free-free (β4n −α4n)sin(αnl)sinh(βnl) −αnβn β3n[cosh(βnl)−cos(αnl)] β3n sinh(βnl)−α3n sin(αnl) sin(αnx)+ βn αn cos(αnx) +2α2nβ 2 n[cos(αnl)cosh(βnl)−1]= 0 − β3n[cosh(βnl)−cos(αnl)] β3n sinh(βnl)−α3n sin(αnl) sinh(βnx)+cosh(βnx) D u e to th e sam e resu lt of action of th e in tern al an d ex tern al d am p in g (as for free v ib ra- tion s), th e com p on en ts con n ected w ith th e set of n atu ral freq u en cies ω n are fu n ction s relatively fast ten d in g to zero w ith resp ect to tim e. T h erefore, th e solu tion of ex ited v ib ration s is u su ally con n ected w ith th e on ly ex tern al load in g freq u en cy ν (Ł ączkow sk i, 1974). S u ch a case of v ib ra- tion s is called th e stead y -state case. T h e com p lete solu tion of th e p rob lem w h ich in clu d es free v ib ration s an d ex cited v ib ration s in gen eral form u lation is k n ow n as tran sien t v ib ration s. A s d iscu ssed in th e In tro d u ction , let u s con sid er th e gen eral form of th e solu tion of ex cited v ib ration s. In su ch a case, th e solu tion can b e p rop osed in form (4.1) in term s of th e fu n c- tion s X n (x ) an d a series of u n k n ow n fu n ction s of tim e H n (t) Analytical solution of excited torsional vibrations... 997 ϕ(x,t) = +∞∑ n=1 Xn(x)Hn(t) (4.1) Putting proposed solution (4.1) into equation (2.2) and taking into account the first of equation of system (3.2)3,4 one can obtain +∞∑ n=1 [(EJω GJs d4Xn(x) dx4 − d2Xn(x) dx2︸ ︷︷ ︸ λ4nXn(x) ) Hn(t)+ ρJ0 GJs d2Hn(t) dt2 Xn(x) ] = 1 GJs ms(x,t) (4.2) and then after suitable manipulations +∞∑ n=1 [d2Hn(t) dt2 + GJs ρJ0 λ4n ︸ ︷︷ ︸ ω2n Hn(t) ] Xn(x)= GJs ρJ0 1 GJs ms(x,t)= 1 ρJ0 ms(x,t) (4.3) The external load function existing in equation (2.2) can be represented in a series form (4.4), in terms of functions Xn(x) and a series of the known functions of time Qn(t) 1 ρJ0 ms(x,t)= 1 ρJ0 +∞∑ n=1 Xn(x)Qn(t) Qn(t)= 1 γ2n l∫ 0 ms(x,t)Xn(x) dx Q ∗ n(t)= 1 ρJ0 Qn(t) (4.4) Substituting formulas (4.1) and (4.4) into equation (2.2), the differential equation for the determination of the unknown functions Hn(t) takes the form d2Hn(t) dt2 +ω2nHn(t)= Q ∗ n(t) (4.5) The solution to this equation has the form Hn(t)= 1 ωn t∫ 0 Q∗n(τ)sin[ωn(t− τ)] dτ (4.6) known as the Duhamel integral. The given formulasmake possible the formulation of the analytical solution of excited vibra- tions of a beamwith defined boundary conditions. Let us formulate the general form of theDuhamel integral for the following types of external excitations and different boundary conditions given by the eigen-functions Xn(x): —Harmonically distributedmoment with a constant amplitude m0 ms(x,t)= m0 sin(νt) [m0] = N Hn(t)= m0 1 ρJ0 1 γ2n 1 ω2n −ν2 [ sin(νt)− ν ωn sin(ωnt) ] l∫ 0 Xn(x) dx ϕ(x,t) = M0 1 ρJ0 ∞∑ n=1 1 γ2n 1 ω2n −ν2 [ sin(νt)− ν ωn sin(ωnt) ] l∫ 0 Xn(x)Xn(x) dx (4.7) 998 E. Augustyn,M.S. Kozień —Pulsed distributed moment with a constant amplitude Qm acting at the time t = t0 ms(x,t) =Qmδ(t− t0) [Qm] = Ns Hn(t)=    0 t < t0 QM 1 ρJ0 1 γ2n 1 ωn sin[ωn(t− t0)] l∫ 0 Xn(x) dx t ­ t0 ϕ(x,t)=    0 t < t0 QM 1 ρJ0 ∞∑ n=1 1 γ2n 1 ωn sin[ωn(t− t0)] l∫ 0 Xn(x)Xn(x) dx t ­ t0 (4.8) —Harmonic concentrated moment with a constant amplitude M0 applied to the point x = x0 ms(x,t) =M0δ(x− c)sin(νt) [M0] = Nm Hn(t)= M0 1 ρJ0 1 γ2n 1 ω2n −ν2 [ sin(νt)− ν ωn sin(ωnt) ] Xn(x0) ϕ(x,t)= M0 1 ρJ0 ∞∑ n=1 1 γ2n 1 ω2n −ν2 [ sin(νt)− ν ωn sin(ωnt) ] Xn(x0)Xn(x) (4.9) —Pulsed concentrated moment with a constant amplitude Qm acting at the time t = t0 at the point x = x0 ms(x,t) =QMδ(x−x0)δ(t− t0) [QM] = Nms Hn(t)=    0 t < t0 QM 1 ρJ0 1 γ2n 1 ωn sin[ωn(t− t0)]Xn(x0) t ­ t0 ϕ(x,t)=    0 t < t0 QM 1 ρJ0 ∞∑ n=1 1 γ2n 1 ωn sin[ωn(t− t0)]Xn(x0)Xn(x) t ­ t0 (4.10) 5. Example – excited torsional vibrations of a simply supported thin-walled beam 5.1. Formulation of the problem As an example, let us consider excited vibrations of a thin-walled beam simply supported on both ends. The beam is made of steel (E =2.1 ·1011Pa, G =8.1 ·1010Pa, ρ =7800kg/m3) and has length l =6m. The shape of cross-section with its detailed dimensions is shown in Fig. 2. Fig. 2. Cross-section of the analyzed beam (dimensions given in mm) Analytical solution of excited torsional vibrations... 999 The boundary conditions take form ϕ(0, t) = 0 ∂2 ∂x2 ϕ(0, t) = 0 ϕ(l,t) = 0 ∂2 ∂x2 ϕ(l,t) = 0 (5.1) 5.2. Natural vibrations – a comparison of the results for different models The following models of beam are applied for comparison of the lowest natural frequencies for torsional vibrations: • three-dimensional solid model solved by application of the finite element method, • thin-walled Vlasov analytical model, • model with a monolithic cross-section. The finite element solution has been obtained by application of the Ansys finite element package. The model is built of 76800 three dimensional 8-nodes solid elements of the type solid45. The generated mesh is shown in Fig. 3. Due to the degrees of freedom of the applied elements, the boundary conditions on the two ends of the beam are modeled as free sliding along the axis of the beam and as blocked displacements for all nodes in the plane of its ends. Therefore, deplanation of the ends is possible. Fig. 3. Part of the FEMmesh of the beam The second group of values of the natural frequencies is obtained based on the thin-walled model described in the text, especially based on the formula given in the first row in Table 1. The values of Js, Jω and J0 are calculated based on the formulas given by Gere (1954). The third group of results is obtained for the beam modeled with the assumption of a monolithic type cross-section (shaft) with free-free (ϕ(0, t) = ϕ(l,t) = 0) or fixed-fixed (∂ϕ(0, t)/∂x = ∂ϕ(l,t)/∂x = 0) boundary conditions. The values of natural frequencies are the same for these cases of boundary conditions (Woroszył, 1984). InTable 2, a comparison of the lowest natural frequencies for the analyzed beam for the three considered models and the relative percentage error is given to one model obtained from three dimensional solid models, as the reference model. It should be noted that for the considered beam, the values of natural frequencies are correctly estimated theoretically with a monolithic cross-section only for the first few torsional modes. For higher modes, only the theory for thin- walled beam gives good enough results. 1000 E. Augustyn,M.S. Kozień Table 2. Comparison of the lowest natural frequencies for the analyzed beam [Hz] Mode 3-D Solid Thin-walled Monolytic No. value [Hz] value [Hz] error [%] value [Hz] error [%] 1 25.15 25.54 1.6 25.40 1.0 2 50.97 51.89 1.8 50.81 -0.3 3 78.11 79.84 2.2 76.21 -2.4 4 107.16 110.05 2.7 101.62 -5.2 5 138.64 143.16 3.3 127.02 -8.4 6 172.98 179.66 3.9 152.42 -11.9 7 210.56 219.94 4.5 177.83 -15.5 8 251.65 264.36 5.1 203.23 -19.2 9 296.50 313.14 5.6 228.64 -22.9 10 345.26 366.49 6.1 254.04 -26.4 A comparison of the natural frequencies for the two beammodels – the thin-walled one and the monolithic one are shown in Fig. 4 in form of the relative percentage error in comparison to the values obtained for the thin-walledmodel as the referencemodel. On the horizontal axis, there are identification numbers of the first hundred torsional modes (n =1,2, . . . ,100). It can be shown that for the considered beam, the error of the estimated values of natural frequencies found by application of the theory with monolithic cross-section grow rapidly with an increase in the number of torsional modes. Fig. 4. Relative error of the natural frequencies for torsional modes for the beammodels 5.3. Excited vibrations The vibrations are excited by a concentrated moment with an amplitude M0 =5Nmacting at the point distanced 0.4l (2.4m) from the end of beam. The initial conditions are zeroes (i.e ϕ0(x)= 0 and Ω0 =0 in formulas (3.6)) ms(x,t)= M0δ(x−0.4l)sin(νt) (5.2) The angular frequency of the excitation is ν =200rad/s.Material damping is neglected. The analytical solution has the form ϕ(x,t)= 2 l 1 ρJ0 M0 ∞∑ n=1 1 ω2n −ν2 [ sin(νt)− ν ωn sin(ωnt) ] sin(λn0.4l)sin(λnx) (5.3) In the analysis, the first five modes are considered. Analytical solution of excited torsional vibrations... 1001 The displacement of the middle of the beam just after the beginning of vibrations and for the steady-state are shown in Fig. 5. The control point is the place of action of the concentrated excitation moment. It is distanced by 0.4l (2.4m) from the end of the beam. Fig. 5. Displacement at the control point – complete analytical solution, transient and steady-state The plots show the necessity of taking into account the full solution instead of the steady- state case if the results are important just after the start of action of external loadings. Due to internal material damping (not taken into account in the considered model), the transient component existing in solution is a function that relatively fastly tends to zero.Therefore, enough time after the start of action of theharmonic excitation, the solution takes the steady-state form. 6. Conclusions • The solution of the problem for amonolithic cross-section is not an asymptotic case of the solution of the thin-walled case if the value of Jω tends to zero. • The given formulas can be applied to the analysis of reduction of torsional vibrations by coupled sets of piezoelectric elements. The action of piezoelectric elements can be appro- ximately modeled by a concentrated moment put in the suitable cross-section, see Elliott and Nelson (1997), Hansen and Snyder (1997) for beams. • The given formulasmake it possible to find an analytical solution for the transient case of the response (just after the excitation is applied) and for the steady-state case. • The solution can be generalized by assuming the Voigt-Kelvin model of the viscous type of damping for description of the internal material damping. Appendix A. Orthogonality of the eigen-functions Formula (3.2)2 can be written in the following form GJsλ 4 nXn(x)= EJω d4Xn(x) dx4 −GJs d2Xn(x) dx2 GJsλ 4 mXm(x)= EJω d4Xm(x) dx4 −GJs d2Xm(x) dx2 (A.1) Let us multiply the first equation of system (A.1) by the eigen-function Xm(x) and the second equation suitably by Xn(x), and then subtract the equations by sides and integrate. 1002 E. Augustyn,M.S. Kozień Finally, formula (A.1) is GJs(λ 4 n −λ4m) l∫ 0 Xn(x)Xm(x) dx = EJω l∫ 0 d4Xn(x) dx4 Xm(x) dx−EJω l∫ 0 d4Xm(x) dx4 Xn(x) dx −GJs l∫ 0 d2Xn(x) dx2 Xm(x) dx+GJs l∫ 0 d2Xm(x) dx2 Xn(x) dx (A.2) By integrating the right side of equation (4.2) by parts, one obtains the following formulas for each component EJω l∫ 0 d4Xn(x) dx4 Xm(x) dx = EJω [ d3Xn(x) dx3 Xm(x) ∣∣∣ l 0 − d 2Xn(x) dx2 dXm(x) dx ∣∣∣ l 0 + l∫ 0 d2Xn(x) dx2 d2Xm(x) dx2 dx ] EJω l∫ 0 d4Xm(x) dx4 Xn(x) dx = EJω [ d3Xm(x) dx3 Xn(x) ∣∣∣ l 0 − d 2Xm(x) dx2 dXn(x) dx ∣∣∣ l 0 + l∫ 0 d2Xm(x) dx2 d2Xn(x) dx2 dx ] GJs l∫ 0 d2Xn(x) dx2 Xm(x) dx = GJs dXn(x) dx Xm(x) ∣∣∣ l 0 − l∫ 0 dXn(x) dx dXm(x) dx dx GJs l∫ 0 d2Xm(x) dx2 Xn(x) dx = GJs dXm(x) dx Xn(x) ∣∣∣ l 0 − l∫ 0 dXm(x) dx dXn(x) dx dx (A.3) Finally, the relationship (A.2) can be written in the form GJs(λ 4 n −λ4m) l∫ 0 Xn(x)Xm(x) dx = ( EJω d3Xn(l) dx3 −GJs d2Xn(l) dx2 )Xm(l) − ( EJω d3Xn(0) dx3 −GJs d2Xn(0) dx2 ) Xm(0)− ( EJω d3Xm(l) dx3 −GJs d2Xm(l) dx2 ) Xn(l) + ( EJω d3Xm(0) dx3 −GJs d2Xm(0) dx2 ) Xn(0)−EJω d2Xn(l) dx2 dXm(l) dx +EJω d2Xn(0) dx2 dXm(0) dx EJω d2Xm(l) dx2 dXn(l) dx −EJω d2Xm(0) dx2 dXn(0) dx (A.4) For anyphysicallypossible combination of thediscussedboundaryconditions, all components existing on the right-hand side are equal to zero. 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