Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 47, 1, pp. 193-210, Warsaw 2009 FREE AND FORCED VIBRATIONS OF TIMOSHENKO BEAMS DESCRIBED BY SINGLE DIFFERENCE EQUATION Leszek Majkut AGH University of Science and Technology, Faculty of Mechanical Engineering and Robotics, Cracow, Poland e-mail: majkut@agh.edu.pl In the paper, a new approach to description of the Timoshenko beam free and forced vibrations by a single equation is proposed.The solution to such an equation is a function of vibration amplitudes. The boundary conditions corresponding to such a description of the beamvibration are also given. It was proved that the form of solution to the differential equation de- pends on the vibration frequency.The changeof the solution formoccurs when the frequency crosses a specific value ω= √ GkA/(ρI). Thecorrectnessof proposeddescriptionwascheckedthroughtheanalysis of free vibration frequencies and amplitudes of forced vibrations with different boundary conditions as well as comparison with the results of finite element analysis. Key words: Timoshenko beam, Green function, boundary conditions, forced vibrations 1. Introduction Beam vibrations described by the Timoshenko model have been studied over the years bymany authors (Stephen and Puchegger, 1982; Kaliski, 1992; Zhu et al., 2006; Si et al., 2007). This has been always a description through a systemof second order differential equations, inwhich the vibration amplitude and the angle due to pure bending were the searched functions. Boundary conditions related to the initial-boundary value problem under consideration were described by a proper differential equation of both or only one of those functions. In the bookKaliski (1992), the free vibration equation is written in the form of a single equation depending only on a single function – vibration 194 L. Majkut amplitude. However, there is a remark that such a description concerns only a simply supported beam. This remark arises from the fact that in the case of the simply supported beam the boundary conditions are described by the same differential equations as in the case of the Euler-Bernoulli beam. In this paper, amethod to derive a single differential equation of the fourth order describing free and forced vibrations of a Timoshenko beam is given. In addition, the equations formed during this derivation serve to define the boundary condition equations. It was also shown that the solution form of the vibration differential equ- ation dependson the examinedvibration frequency.The change of the solution form occurs when the frequency crosses a specific value. This value is known from literature as the cut-off frequency (Stephen and Puchegger, 2006) or critical one (Chan et al., 2002). The correctness of such a description for the Timoshenko beam vibrations was checked through the analysis of free vibration frequencies with different boundary conditions and amplitudes of vibrations excited by a harmonic force and through comparison with the results of finite element analysis. Construction of the dynamic Green functions was proposed to solve the problem of forced vibration amplitudes of the beam, excited by an arbitrary function of time t and applied to a beam in an arbitrary way, as a function of the spatial coordinate x. This is the function of beamvibration amplitudes forced by the harmonic unit force. Themethods of determining this function, described in literature (Kukla, 1997; Lueschen et al., 1996), consist in defining the Green matrix. This is necessary for description of the beam vibration by a system of two equations. The single-equation description of vibrations substantially facilitates deve- loping the inverse models, which (for the Euler-Bernoulli beam) are used by the present author for diagnostic (Majkut, 2004, 2005a,b, 2006) and structural modification (Majkut, 2008; Majkut andMichalczyk, 2002). 2. Equations of vibration of the Timoshenko beam TheTimoshenkomodel is an extension of theEuler-Bernoullimodel by taking into account two additional effects: shearing force effect and rotary motion effect. In any beam except one subject to pure bending only, a deflection due to the shear stress occurs. The exact solution to the beam vibration problem requires this deflection to be considered. So, the angle ∂y(x,t)/∂x between Free and forced vibrations of Timoshenko beams... 195 the beam axis and x axis is a sum of the angle Θ(x,t) due to pure bending and the shear angle γ(x,t) i.e.: ∂y(x,t)/∂x=Θ(x,t)+γ(x,t). Another factor that affects the lateral vibration of the beam, neglected in Euler-Bernoulli’s model, is the fact that each section of the beam rotates slightly in addition to its lateralmotionwhen the beamdeflects. The influence of the beam section rotation is taken into account through the moment of inertia, which modifies the equation of moments acting on an infinitesimal beam element: dMB(x,t) = −Iρ∂2Θ(x,t)/∂t2dx, where: ρ is density of the beammaterial, I – the secondmoment of area. By applying d’Alembert’s principle, the system of coupled differential equ- ations for transverse vibration of the uniform Timoshenko beam with a con- stant cross section is given by − ∂Q(x,t) ∂x +ρA ∂2y(x,t) ∂t2 = q(x,t) (2.1) − ∂M(x,t) ∂x +Q(x,t)−Iρ ∂2Θ(x,t) ∂t2 =0 where: Q(x,t) = kGAγ(x,t) is the shear force, M(x,t) = EI∂Θ(x,t)/∂x – bending moment, k – Timoshenko shear coefficient depending on the cross- section of the beam (Stephen and Puchegger, 2006; Rubin, 2003), G – shear modulus, A – cross sectional area, EI – bending stiffness, y(x,t) – vibration amplitude, q(x,t) – external force. The system of differential Eqs. (2.1) describes the Timoshenko beam vi- bration, where the searched functions are the vibration amplitude y(x,t) and the angle due to pure bending Θ(x,t). 3. Free vibrations described by the single equation TheFourier method of variable separation is employed to find functions satis- fying systemof Eqs. (2.1). It is assumed that each function y(x,t) and Θ(x,t) can be presented in the form of a product of a function dependent on the spa- tial coordinate x and a function dependent on time t (with the same time function). y(x,t)=X(x)T(t) Θ(x,t)=Y (x)T(t) (3.1) 196 L. Majkut With such an assumption, after several simple transformations of system (2.1), it can be rewritten as X′′(x)+aX(x)−Y ′(x)=0 Y ′′(x)+ bY (x)+ cX′(x)= 0 (3.2) T̈(t)+ω2T(t)= 0 where a= ω2ρ kG b= ρω2 E − c c= GkA EI and ω – vibration frequency. By eliminating the function Y (x) from the first two equations of system (3.2) Y (x)=− 1 b [X′′′(x)+(a+ c)X′(x)] (3.3) one can get an equation for the transverse displacement X(x). So, the single Timoshenko beam equation of free vibration is obtained in the form X(4)(x)+dX′′(x)+eX(x)= 0 (3.4) where d= a+b+ c= ω2ρI ( 1+ E Gk ) EI e= ab= ω2 ( ω2ρ2 I Gk −ρA ) EI In the next section of the paper, the solution to Eq. (3.4) will be searched. The function Y (x) (Eq. (3.3)) depends on derivatives of the vibration amplitude function X(x). This equation will be used to define boundary con- ditions dependent only on the vibration amplitude function X(x) and its derivatives. 4. Solution to the Timoshenko beam differential equation The characteristic equation of Eq. (3.4) has the form r4+dr2+e=0 (4.1) Replacing r2 = z̃, Eq. (4.1) can be rewritten in the form z̃2+dz̃+e=0 Free and forced vibrations of Timoshenko beams... 197 Its roots are z̃1 = 1 2 (−d+ √ ∆) z̃2 = 1 2 (−d− √ ∆) where ∆= d2−4e=ω4ρ2I2 ( 1− E kG )2 +4EIω2ρA It is easy to observe that ∆> 0 ∀ω (for all ω). Now one should carry out a discussion about signs of the roots z̃1 and z̃2 as the frequency function z̃2 < 0 ∀ω z̃1 > 0 ⇔ √ d2−4e>d ⇒ e< 0 for ω2 < GkA ρI z̃1 < 0 for ω> √ GkA ρI Two possible solutions to Eq. (3.4) come from the above discussion: • For ω< √ GkA/(ρI) The roots of Eq. (4.1) are r1 = √ z̃1 r2 =− √ z̃1 r3 = i √ −z̃2 r4 =−i √ −z̃2 This gives a solution in the form X(x) =C1e √ z̃1x+C2e − √ z̃1x+C3e i √ z̃2x+C4e −i √ z̃2x (4.2) With the use of Euler’s formulae, solution (4.2) may be also expressed in the form of trigonometric and hyperbolic functions X(x)=P1coshλ1x+P2 sinhλ1x+P3cosλ2x+P4 sinλ2x (4.3) where λ21 = |z̃1|= −d+ √ ∆ 2 λ22 = |z̃2|= d+ √ ∆ 2 • For ω> √ GkA/(ρI) In this case, the roots of Eq. (4.1) are r1 = i √ −z̃1 r2 =−i √ −z̃1 r3 = i √ −z̃2 r4 =−i √ −z̃2 198 L. Majkut and the solution for free vibration of a Timoshenko beam (Eq. (3.4)) for such frequencies has the form X(x) =Q1cosλ1x+Q2 sinλ1x+Q3cosλ2x+Q4 sinλ2x (4.4) where λ21 = |z̃1|= d− √ ∆ 2 λ22 = |z̃2|= d+ √ ∆ 2 The integration constants Pi and Qi (i = 1, . . . ,4) depend on the bo- undary conditions associated with the initial-boundary value problem under consideration. 5. Boundary conditions for the Timoshenko beam model The boundary conditions are described in such a way that the letters (ã) and (b̃) denote their physical models, and the letters (ai) and (bi) denote their mathematical models after separation of variables. 1. Hinged end (xi =0 or xi = l) or internal support at xi (ã) y(xi, t)= 0 (b̃) M(xi, t)=EI ∂Θ(xi, t) ∂x =0 The conditions (a) and (b) after separation of variables and taking into account Eq. (3.3) have the forms (a1) X(xi)= 0 (b1) Y ′(xi)= 0⇔X′′(xi)+aX(xi)= 0 2. Fixed clamped end (xi =0 or xi = l) (ã) y(xi, t)= 0 (b̃) Θ(xi, t)= 0 after separation of variable (a2) X(xi)= 0 (b2) Y =0⇔X′′′(xi)+(a+ c)X′(xi)= 0 3. Free end (xi =0 or xi = l) (ã) M(xi, t)=EI ∂Θ(xi, t) ∂x =0 (b̃) Q(xi, t)=GkA (∂y(xi, t) ∂x −Θ(xi, t) ) =0 Free and forced vibrations of Timoshenko beams... 199 after separation of variables (a3) X ′′(xi)+aX(xi)= 0 (b3) dX ′(xi)+X ′′′(xi)= 0 4. Generally supported beam —at x=0 – Fig.1a (ã) Q(0, t)=−kTy(0, t) (b̃) M(0, t) = kR ∂y(0, t) ∂x after separation of variable (a4) GkA[(a+ b+ c)X ′(0)−X′′′(0)]+kTX(0)= 0 (b4) X ′′(0)+aX(0)− kR EI X′(0)= 0 — at x= l – Fig.1b (ã) Q(l,t)= kTy(l,t) (b̃) M(l,t) =−kR ∂y(l,t) ∂x after separation of variable (a4) GkA[(a+ b+ c)X ′(l)−X′′′(l)]−kTX(l) = 0 (b4) X ′′(l)+aX(l)+ kR EI X′(l)= 0 Fig. 1. General boundary conditions of the beam (a) at x=0 (b) at x= l The thus described boundary conditions will be used for determining na- tural frequencies of the beamwith different boundary conditions. 200 L. Majkut 6. Determination of the natural frequencies The choice of the beammodel (Euler-Bernoulli or Timoshenko), which should be used in the analysis of transverse vibrations of a given beam, depends on the ratio of its height h to half of the wavelength corresponding to the vibration frequency (the distance between two adjacent nodes l). Because the difference between the free vibration frequencies for h/l≈ 8% is equal to 5% and increaseswith increasing h/l, it is accepted thatTimoshenkomodel should be employed for beams and frequencies forwhich this ratio is larger than 10%. Such beams are called stocky beams, while the beams for which the Euler- Bernoulli model is sufficient are called the slender ones. But the Timoshenko beam model must be taken in the analysis of high-frequency vibration of all beams. To check correctness of the proposed description of beam vibrations, the calculations were carried out both for a slender beam (small differences in the free vibration frequencies obtained from both models of the beam) and for a stocky beam, for which these differences should be larger. The calculations were carried out for the beam with: E = 2.1 · 1011Pa, G = 8.1 · 1010Pa, ρ = 7860kg/m3, length l = 1m and cross-section b×h = 0.02× 0.08m2 (stocky beam) and b×h = 0.02× 0.03m2 (slender beam). In order to check the correctness of free vibration frequencies calcu- lations, FEM analysis was carried out by the author with the use of proper one-dimensional finite elements (Cheung and Leung, 1991). 6.1. Simply supported beam The boundary conditions for a simply supported beam are: at x = 0, X(0)= 0 and X′′(0)+aX(0)= 0; at x= l,X(l) =0 and X′′(l)+aX(l)= 0. The form of the solution to the vibration equation depends on the interval to which the searched natural frequency belongs: • For ω< √ GkA/(ρI), the solution to Eq. (3.4) has form (4.3). From the boundary conditions at x = 0, the following equations are obtained P1+P3 =0 λ 2 1P1−λ22P3 =0 This system of equations is satisfied when P1 = P3 = 0, or in the case when λ21 +λ 2 2 = 0, i.e. when √ d2−4e = 0, and this is possible only when ω=0, which describes motion of the beam as a rigid body, what is the impossible because of the boundary conditions. Free and forced vibrations of Timoshenko beams... 201 So, the boundary conditions at x=0 require that P1 =P3 =0, which corresponds to the similar solution for the Euler-Bernoulli beam. The boundary conditions at x= l are expressed by thematrix equation [ sinhλ1l sinλ2l λ21 sinhλ1l −λ22 sinλ2l ][ P2 P4 ] = [ 0 0 ] (6.1) The nontrivial solution to Eq. (6.1) is obtained from the condition that the main matrix determinant is equal to zero. • Solution to Eq. (3.4) for ω> √ GkA/(ρI) has form (4.4). From the boundary conditions at x = 0, the following equations are obtained: Q1+Q3 =0 λ 2 1Q1+λ 2 2Q3 =0 This system of equations is satisfied either for Q1 = Q3 = 0 or for λ21 − λ22 = 0. Fulfilling the second condition is possible only when ω = 0, which describes motion of the beam as a rigid body, what is the impossible because of the boundary conditions. Therefore, the boundary conditions at x=0 require that Q1 =Q3 =0. The boundary conditions at x= l are expressed in the matrix form [ sinλ1l sinλ2l λ21 sinλ1l −λ22 sinλ2l ][ Q2 Q4 ] = [ 0 0 ] (6.2) The roots of the main matrix determinant are the eigenvalues of the initial-boundary value problem under consideration. Table 1 contains the first five natural frequencies of the stocky beam to- gether with the corresponding frequencies calculated for the Euler-Bernoulli model. In order to check the correctness of the proposed approach, the frequ- encies were compared with the results of FEM analysis carried out with the use of proper 1-D finite elements (Cheung and Leung, 1991). Table 2 contains the natural frequencies for the slender beam. The results presented in Tables 1 and 2 prove correctness of the proposed description of the Timoshenko beam vibration. The relative difference in the values of the free vibration frequencies determined with the use of different models for the slender beam is equal to 0.2% for the first free vibration fre- quency and increases to 5.4% for the fifth frequency. The difference increases because the wavelength decreases (the ratio h/l increases) with the growth of 202 L. Majkut Table 1.Natural frequencies of the stocky simply supported beam Timoshenko beam E-B beam Timoshenko beam E-B beam (analytical method) (analytical method) (FEM) (FEM) 1 1159.4 1178.1 1160.5 1178.1 2 4436.8 4712.6 4447.4 4712.6 3 9357.6 10603.3 9386.1 10603.0 4 15410.0 18850.0 15439.0 18850.0 5 22163.0 29453.5 22183.0 29454.0 Table 2.Natural frequency of the slender simply supported beam Timoshenko beam E-B beam Timoshenko beam E-B beam (analytical method) (analytical method) (FEM) (FEM) 1 440.8 441.8 440.8 441.8 2 1751.3 1767.2 1752.2 1767.2 3 3897.4 3976.2 3901.2 3978.2 4 6825.5 7068.8 6736.9 7068.9 5 10473.2 11045.0 10494.0 11045.0 the free vibration frequency. In the case of the stocky beam, the differences in the determined frequencies are from 1.6% up to 33%. In both cases. there are no significant differences in the frequency values obtained from the proposed description and from the FEM analysis. 6.2. Cantilever beam Boundary conditions for the cantilever beam are: at x=0, X(0)= 0 and X′′′(0)+(a+c)X′(0)= 0; at x= l,dX′(l)+X′′′(l)= 0and X′′(l)+aX(l)= 0. The form of the solution for the free vibration problem depends on the interval to which the searched natural frequency belongs: • For ω < √ GkA/(ρI), the solution to Eq. (3.4) has form (4.3), the bo- undary conditions are expressed by amatrix equation AC=0 where Free and forced vibrations of Timoshenko beams... 203 A=   1 0 1 0 0 λ1(λ 2 1+a+ c) 0 a24 (λ21−d)λ1 sinhλ1l (λ21−d)λ1coshλ1l (λ22+d)λ2 sinλ2l a34 (λ21+a)coshλ1l (λ 2 1+a)sinhλ1l (−λ22+a)cosλ2l a44   C ⊤ = [P1,P2,P3,P4] and a24 =λ2(−λ22+a+ c) a34 =(−λ22−d)λ2cosλ2l a44 =(−λ22+a)sinλ1l The coefficients a, b, c, d and e are defined in Section 3. Natural frequencies of the beam are determined from the equation: detA=0. • For frequencies ω > √ GkA/(ρI) (the so-called second spectrum of the beam (Stephen and Puchegger, 1982)), the solution has form (4.4), and the main matrix is A=   1 0 1 0 0 λ1(−λ21+a+ c) 0 a24 (λ21+d)λ1 sinλ1l (−λ21−d)λ1cosλ1l (λ22+d)λ2 sinλ2l a34 (−λ21+a)cosλ1l (−λ21+a)sinλ1l (−λ22+a)cosλ2l a44   Table 3 contains the first five natural frequencies of the cantiilever beam together with the corresponding frequencies calculated for theEuler-Bernoulli model of the beam. Table 3.Natural frequency of the stocky cantilever beam Timoshenko beam E-B beam Timoshenko beam E-B beam (analytical method) (analytical method) (FEM) (FEM) 1 424.1 419.6 424.1 419.3 2 2653.7 2666.8 2656.8 2630.6 3 7145.3 7387.3 7148.1 7367.6 4 13016.0 14284.0 13024.0 14443.0 5 19645.0 23383.0 19653.0 23886.0 Table 4 contains results obtained for the slender beam. The results presented in Tables 3 and 4 prove correctness of the proposed description of the Timoshenko beam vibration and the boundary condition equations. 204 L. Majkut Table 4.Natural frequency of the slender cantilever beam Timoshenko beam B-E beam Timoshenko beam E-B beam (analytical method) (analytical method) (FEM) (FEM) 1 157.6 153.6 159.2 154.7 2 987.7 962.6 993.5 986.5 3 2752.5 2695.3 2761.4 2762.8 4 5344.2 5281.6 5350.0 5416.0 5 8716.1 8730.9 8741.1 8957.4 7. Forced vibrations The forced vibrations for the Timoshenko beammodel, written in the form of two differential equations, are as follows ρA ∂2y(x,t) ∂t2 −GkA (∂2y(x,t) ∂x2 − ∂Θ(x,t) ∂x ) = q(x,t) EI ∂2Θ(x,t) ∂x2 +GkA (∂y(x,t) ∂x −Θ(x,t) ) − Iρ ∂2Θ(x,t) ∂t2 =0 After several transformations, similar to transformation of the homogene- ous equation, it is possible to obtain an equation for the Timoshenko beam forced vibrations in a formdependent only on the function of the displacement y(x,t) EI ∂4y(x,t) ∂x4 − (EIρ Gk + Iρ )∂4y(x,t) ∂x2∂t2 + Iρ2 Gk ∂4y(x,t) ∂t4 +ρA ∂2y(x,t) ∂t2 = (7.1) = q(x,t)− EI GkA ∂2q(x,t) ∂x2 + Iρ GkA ∂2q(x,t) ∂t2 This is the equation of transverse vibrations of theTimoshenko beamwith an arbitrary exciting function q(x,t) (the excitation is an arbitrary function of time t and spatial coordinate x). It is impossible to find its solution in the general case. In the paper, a different approach to the problem of solution for the beam forcedvibrations isproposed. In theproposedmethod, thebeamcanbeexcited by an arbitrary function of time t applied to the beam in an arbitraryway, as a function of the spatial coordinate x. The problem of time variability of the exciting function is proposed to be solved in the following way: to transform the excitation function from the Free and forced vibrations of Timoshenko beams... 205 time domain into frequency domain, then to find the system response in the frequency domain and employ the inverse Fourier transform, finding in this way the vibration amplitudes function y(x,t) in the time domain. The spatial distribution of the forcing function can be taken into account by using an integral equation, i.e. by using the superposition rule. The kernel of the integral equation is the Green function. It comes, form the above discussion, that finding the dynamicGreen func- tion is enough to find the forced vibrations function y(x,t) of the beamwith an arbitrary excitation function. ThedynamicGreen function is a functionof thebeamvibrationamplitudes excited by the unit harmonic force. 7.1. Dynamic Green function To determine the dynamic Green function, one should find the solution to inhomogeneous differential equation (7.1), in which the excitation function has the form q(x,t)= 1 exp(iωwt)δ(x−xf), where δ(x−xf) is theDirac delta function and 1 – unit excitation. In such a case, the functionbeing its solution in the steady state, can be expressed in the form y(x,t)=G(x,xf)exp(iωwt). The function G(x,xf) is the searched dynamic Green function, in which xf is the coordinate of the point where the force is applied to the beam. Comment: In the paper, both internal and external damping of the beam was neglected. This is not a problem in the case of determining the eigenfre- quencies and eigenvectors. In the case of forced vibration, especially of high frequency, the negligence of damping leads to large errors in the determined vibration amplitudes. Because of that, the solution to the forced vibration equation was searched only for frequencies below the cut-off frequency, i.e. for ω< √ GkA/(ρI). The function G(x,xf) will be determined as a sum of the general solution to the related homogeneous equation and the particular solution G(x,xf)=G0(x)+G1(x,xf)H(x−xf) (7.2) The function G0(x) is the solution to the homogeneous equation in form (4.3), for x ∈ (0, l), where G1(x,xf) is the solution to the inhomogeneous equation determined for x>xf. H(x−xf) is the step function. The solution to the inhomogeneous equation G1(x,xf) can be expressed in the form G1(x,xf)=R1cosh[λ1(x−xf)]+R2 sinh[λ1(x−xf)]+ +R3cos[λ2(x−xf)]+R4 sin[λ2(x−xf)] 206 L. Majkut The constants Ri were determined from the continuity conditions and the jump condition at x=xf: G(x+ f ,xf)−G(x−f ,xf)= 0 Θ(x + f )−Θ(x− f )= 0 M(x+ f )−M(x− f )= 0 Q(x+ f )−Q(x− f )= 1 The relationships between the individual physical quantities and theGreen functionaregiven in thedescriptionof theboundaryconditions (seeSection5). The constants of integration, determined from the continuity conditions, are: R1 =R3 =0 R2 = 1 b GkA λ22−a− c λ1(λ 2 1+λ 2 1)(a+c+d) R4 = 1 b GkA λ21+a+ c λ2(λ 2 1+λ 2 1)(a+c+d) Finally, the dynamic Green function for the Timoshenko beam has the form G(x,xf)=P1coshλ1x+P2 sinhλ1x+P3cosλ2x+P4 sinλ2x+ (7.3) +R2 sinh[λ1(x−xf)]H(x−xf)+R4 sin[λ2(x−xf)]H(x−xf) In Fig.2, one can see the functions of amplitudes for the forced vibra- tions determined for the simply supported beam with: E = 2.1 · 1011Pa, G = 8.1 · 1010Pa, ρ = 7860kg/m3, length l = 1m and cross-section b×h=0.02×0.08 (stocky beam) – Fig.2a and b×h=0.02×0.03 (slender beam) – Fig.2b. In both figures, the solid line (—) denotes the amplitude for the Timoshenko beammodel, while the dashed line (- -) denotes the amplitu- de for the Euler-Bernoulli model of beam. There are no significant differences in values of vibration amplitudes obtained from the proposed description and FEM analysis. The functions of amplitudes for the forced vibrations determined for the cantilever beam with the same data are shown in Fig.3, those determined for the stocky beam are shown in Fig.3a and those for the slender beam – in Fig.3b. In both figures, the solid line (—) denotes the amplitude for the Timoshenko beam model, while the dashed line (- -) denotes the amplitude for the Euler-Bernoulli model of beam. There are no significant differences in values of vibration amplitudes obtained from the proposed description and FEM analysis. The graphs of the harmonically excited vibration amplitudes, shown in Fig.2 and Fig.3 prove correctness of the proposed description of the Timo- shenko beam vibration and the boundary condition equations. Free and forced vibrations of Timoshenko beams... 207 Fig. 2. Vibration amplitudes of the simply supported beam (a) stocky beam (b) slender beam Fig. 3. Vibration amplitudes of the cantilever beam (a) stocky beam (b) slender beam 8. Summary In the paper, the method to derive a single equation for free and forced vi- brations of the Timoshenko beam model was proposed. The solution to such an equation is a function of vibration amplitudes. Searching for the function describing the angle due to pure bending, necessary in descriptions of the Ti- moshenko beam vibration known so far in the literature, is unnecessary when using such an approach. The boundary conditions corresponding to such a description of the beam vibration were also given. Itwas proved that the formof solution to the differential equation depends on the examined vibration frequencies. The change of the solution formoccurs 208 L. Majkut when the frequency crosses a specific value, determined in thepaper,whichde- pendson thematerial properties and thebeamcross-section ω= √ GkA/(ρI). The correctness of such a description of the Timoshenko beam vibrations was checked by analysis of free vibrations of the beamwith different boundary conditions. The results presented in Tables 1-4 indicate correctness of the proposedmethod.Themaximaldifferencesbetween thevalues of freevibration frequencies determinedwith the use of theTimoshenko beammodel and those determined with the use of the Euler-Bernoulli beam model are 33% for the stocky beam and 5.4% for the slender one. There are no significant differences in the frequency values obtained from the proposed description and FEM analysis. Construction of the dynamic Green functions was proposed to solve the problem of vibration amplitudes excited by an arbitrary function of time t and applied to the beam in an arbitrary way, as a function of the spatial coordinate x. This is a function of beam vibration amplitudes forced by the harmonic unit force. Knowing the dynamic Green function, one can determine the frequency response function of the system (i.e. system transmittance) and next, by em- ploying the inverse Fourier transform, the impulse response function. The dynamic response of the beam to the point force with an arbitrary change in time can be found by using convolution of this time function and the impulse response of the beam. Another possibility is to transform the time function of the excitation from the time domain into the frequency domain and next to find the system response in the frequency domain for the excitation by each of the frequency component. For this purpose, the dynamic Green function can be used (the frequencies and the corresponding amplitudes of the excitation are known). Next, the inverse Fourier transform can be employed to obtain in this way the vibrations in the time domain. The spatial distribution of the excitation can be taken into account by using the superposition rule, i.e. the beam integral equation. The kernel of the integral equation is the Green function. Form the above discussion, it can be conducted that finding the dynamic Green function is enough to find the forced vibrations function of time t and spatial coordinate x i.e. the function y(x,t) due to an arbitrary excitation function. Acknowledgement This investigationwas supportedby theMinistry ofScience andHigherEducation of Poland under research grant No. N50404232/3443. Free and forced vibrations of Timoshenko beams... 209 References 1. 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Zmiana formy rozwiązania zmienia się, gdy częstość osiąga określoną w pracy wartość ω= √ GkA/(ρI). Poprawność zaproponowanego opisu sprawdzono przez analizę częstości drgań własnych i amplitudy drgań wymuszonych belek z różnymi warunkami brzegowymi i porównaniem z wynikami otrzymanymi z analizyMES belki. Manuscript received September 19, 2008; accepted for print October 22, 2008