Free vibration analysis of elastically restrained cantilever Timoshenko beam with attachments

Abstract


Introduction
Many authors have studied the free and forced vibration of Euler-Bernoulli beams under various boundary conditions.Low [1] studied the vibration of a beam carrying several masses on the beam at different locations, but he did not include spring attachment.However, for the cases where the rotary and shear effects must be considered, Timoshenko beam theory must be utilized.Several authors have investigated the free and forced vibration of Timoshenko beam with attachments under various boundary conditions.Majkut [2] proposed a method to obtain a single equation for both free and forced vibration of the Timoshenko beams.Several papers are also available on the free vibration of cantilever beams carrying a concentrated mass.Laura et al. [3], studied the free vibration of a clampedfree beam which carries a finite mass at the free end and obtained the natural frequencies and modal shapes.Chang [4] investigated the vibration characteristics of a simply supported beam with a heavy concentrated mass at its centre.Banerjee [5] investigated the free vibration of a beam carrying a spring-mass system using the dynamic stiffness method.He obtained the natural frequencies and the first five mode shapes.Rossit and Laura investigated the lateral vibration of a beam with a mass attached to the end with a linear spring.Relatively simpler Bernoulli beam theory has been utilized in the analysis [6].However, for a thick beam carrying a mass load such as an electric motor or engine, Timoshenko beam theory must be used [7]- [10].
In addition, when the mass load is too heavy, the assumption of semi-rigid root must also be made due to the elastic nature of the end.Some researchers have focused on a cantilever Timoshenko beam.Rossit and Laura [11] studied a cantilever Timoshenko beam with a spring-mass system attached to the free end.A cantilever Timoshenko beam with a tip mass at the free end and having rotational and translational springs has been studied by Abramovich and Hamburger [12].Salarieh and Ghorashi [13] analysed the free vibration of a cantilever Timoshenko beam with rigid mass and compared with other beam theories.In the work by Jafari-Talookolaei and Abedi [14], a new method was presented to obtain the exact solution for the free vibration of a Timoshenko beam with different boundary conditions.The vibration analysis of a cantilever beam with an eccentric three dimensional object has been investigated by Kati and Gökdag [15].There are also several research works on the tapered Timoshenko beams.Lateral vibration analysis of a Timoshenko beam of variable crosssection carrying several masses is carried out in [16].In that study, differential quadrature element method (DQEM) is used and the changing of the frequencies of the beam is studied in terms of parameters of the mass.Cekus [17] studied the free vibration of a cantilever tapered Timoshenko beam by using Lagrange multiplier formalism.The governing equations for the Timoshenko beams with geometrical non-uniformity and material inhomogeneity along the beam axis have been simplified by a new method [18].
In the present study, the vibration analysis of a cantilever beam carrying a tip mass using Timoshenko theory is investigated.The free end carries a mass attached to the beam by means of a linear spring while the left hand side is semi-rigid with a rotational spring.Natural frequencies and related mode shapes are determined in terms of non-dimensional parameters.

Frequency Analysis
Let us consider a Timoshenko beam with a semi-rigid root (Figure 1).The mass M is attached to the free end of the beam by means of a spring of coefficient  0 . is the length of the beam,   is the rotational rigidity.It is well known that the transversal motion of the beam is governed by the equations [19] as follows: Here,  is the moment of inertia,  is the modulus of elasticity,  is the shear modulus of elasticity,  is the cross-sectional area,  is the density of the beam,  is the shape factor,  is the vertical displacement,  is the bending angle.Where,  is the time,  is the angular frequency.Here,  is substituted for brevity.Substituting Eqs.(2) into Eq.( 1), we have Where; By eliminating () and its derivatives in Eqs.
(3), it can be combined into a single equation as follows: Where; In order to solve Eq. ( 5), we assume () = e  .The characteristic equation and its roots are obtained as Here, The solution of Eq. ( 5) can be written as We now utilize the first of Eqs.( 3) to obtain Ψ().Inserting () into the first of Eqs.(3) yields Where; Integrating Eq.( 9) gives Where; After Eq. ( 9) is integrated, a constant value would surely appear in Eq. ( 11).However, by substituting the solution forms obtained into Eqs.(1), it is quite simple to show that it is indeed zero.The coefficients  1 ,  2 ,  3 ,  4 must be determined by using the boundary conditions at both ends of the beam.These boundary conditions can be written as follows: At  = 0: Here,  is the force exerted on the beam by the spring at  = 1.
In order to find the force , we write the equation of motion for the mass : Here,  1 is the displacement of the mass, and  2 is the deflection of the end.Let us assume  2 −  1 = .Inserting this form into Eq.( 14) gives Let us now assume the following solution forms for  2 and : Substituting these forms into Eq.( 15), we obtain The force  acted upon by the spring now reads By combining Eq.(13d) and Eq. ( 18), the last form of the last condition in Eqs.( 13) can be rewritten as Where; By utilizing the boundary conditions, the four equations are obtained in terms of the unknown coefficients as follows: Where; Eqs. (21) can also be written in matrix form as Where; The frequency equation is obtained by taking det[] = 0.The explicit form of the frequency equation is as follows: In the case of a rigid wall, by taking  1 = ∞, Eq.( 25) can be simplified into the form

Eigen-Function analysis
Eigen-functions of the problem can be determined by writing the coefficients  2 ,  3 ,  4 in terms of  1 : Thus, () is obtained in the form of Where; To find the value of  1 , the condition of orthogonality can be utilized: ∫( 2   ()  () +   ()  ()) Here,   is Kronecker delta.Inserting Eqs.(27) into Eq.( 30) and evaluating the integral, the constant  1 can be obtained.

Results and discussion
In order to validate the present solution, a cantilever Timoshenko beam with carrying mass-spring system at the free end is considered by taking  1 = 10 12 , whose equation is given in Eqs. ( 22).The reason why  1 is taken high is that the rotational spring's effect becomes inactive and therefore behaves as a fixed support.The results are compared with the study of Rossit and Laura [11] in Table 1, and it is seen that they are in good agreement.After validation study, numerical studies have been carried out for different combinations of dimensionless variables.They are tabulated in Tables 2-7.The beam properties used in the analysis are  = 210 GPa,  = 80.76 GPa,  = 5/6,  = 7800 kg/m 3 ,  = 1 m, ℎ = 0.1 m,  = 0.05m.
For ease of interpretation, dimensionless parameters  4 and  5 are defined as follows: 4 = / and  5 =  0  3 / Here,  4 is the ratio of the beam mass to the added mass and  5 is the ratio of the linear spring coefficient to the bending stiffness.
The variation of natural frequencies with  1 are shown in Tables 2, 3.In Table 3, the mass is assumed to be zero.It can be seen from these tables that, for increasing values of  1 , which is associated with the rotational spring coefficient (  ), all natural frequencies increase.The frequency values obtained for zero mass are higher than those found for the case of non-zero mass (Table 2).Thus, the mass attached decreases the values of frequencies.In addition, for the beam with a non-zero end mass, the rates of increase in each natural frequency are slightly higher than those for the case with no mass, except for the first mode.The variation of frequencies with  4 involving mass  is shown in Table 5.As expected, the frequencies increase with increasing  4 .However, unlike the frequencies with low modes, the frequencies with high modes are not influenced by the change of  4 .
Table 5.The variation of frequencies with  4 for  5 = 0.In Table 6, in the case of nearly rigid wall (  ≫ 0), frequencies are obtained for varying values of  5 .The increase in  0 (or  5 ) values has a higher effect on the frequencies with low modes than the higher ones.
Table 6.The variation of frequencies with  5 for  4 = 0.5 ,  1 = 10 10 (  ≅ ∞).Euler-Bernoulli beam approaches.In the case of rigid wall, the results for Euler-Bernoulli beam are taken from the literature [6].The difference between these results becomes smaller at lower modes.In Table 7(b), the constant  1 has been taken as  1 = 10 12 , while it is  1 = 10 10 in the Table 7(a).It is shown that  1 = 10 10 is the value that can be considered as a rigid wall since the changes of frequencies become negligible.From the comparison of both tables, it can be concluded that the difference between the results of both models is not due to the rotational spring's coefficient (  ), but due to the shear and rotary effects of the beam.It can be seen from Figures 6, 7 that if the end mass is assumed to be zero, the linearity of the initial part of the mode shapes is changed as in the case where the end mass exists.On the whole, it can be mentioned that mode shapes are slightly affected by the change of  1 in this case.Moreover, Figures 2, 6 and Figures 5,7 show that the first mode shape is not observed in the case where the end mass does not exist.Namely, the th mode shape in the case where there is no mass shows resemblance with the ( + 1)th mode shape in the case where the end mass exists.

Conclusion
In this study, mode shapes and natural frequencies were analysed in terms of some parameters such as  4 ,  5 ,  1 for the elastically restrained cantilever Timoshenko beam carrying a spring mass system at its free end.Here,  5 ( 5 =  0  3  ⁄ ) and  1 ( 1 =   /) contain the ratios of linear spring and rotational spring coefficients to bending stiffness, respectively. 4 ( 4 = /) is also the ratio of the beam mass to the added mass.The results have been tabulated in order to see the effects non-dimensional parameters on the frequencies.Mode shapes have also been obtained in terms of various values of parameters.In the general case, it has been seen that the increase in the end mass decreases the natural frequencies.Large values of  5 (or  0 ) increase the values of natural frequencies, except for the frequencies with high modes.Also, the natural frequency increases with increasing  1 value.The frequency equation for extremely large values of  1 has also been obtained.In order to see the effects of the parameters on the mode shapes, Figures 2-11 have been plotted.Except for the shape of the first mode, the other mode shapes are affected by the increment of  5 .Changing the value of  1 also affects the mode shapes.Location of peak points and the shape of the initial parts are slightly influenced by  1 .

Author contribution statements
In the scope of this study, Yasar PALA in the formation of the idea, the formulation, the design and the literature review, the assessment of obtained results and writing; Caglar KAHYA in the formulation, numerical computations, writing, obtaining and examining the results.

Ethics committee approval and conflict of interest statement
There is no need to obtain permission from the ethics committee for the article prepared.
There is no conflict of interest with any person / institution in the article.

Tables 7 (
a), 7(b) show the difference between Timoshenko and . The first four ones are associated with the changing values of  1 in the case that end mass exists.On the contrary of these mode shapes, end mass has been considered not existing by taking  4 = 10 10 in Figures 6, 7. Figures 8-11 show how  5 affects the mode shapes.The first four Figures show that the increasing value of  1 changes the linearity of the beginning part of the mode shapes, and peak points slightly move to the free end.

Figures 8 -
show that although mode shapes are more affected as the value of  5 increases in terms of changing amplitudes and shapes, they are less affected in the first mode after the value of  5 = 10.

Table 4 ,
the values of  5 , which equals to  0  3 /, is increased and natural frequencies are tabulated.It is observed that natural frequencies increase with increasing values of  5 .This is valid for all frequencies.However, at frequencies with high modes, the values are not appreciably affected by the varying values of  5 .It can also be concluded from Table4that the frequencies do not change too much for large values of  5 .This means that the spring becomes ineffective and behaves like a massless rigid body.

Table 7
Mode shapes are demonstrated in Figures2-11