Simulation of the Vibrations of a Non-uniform Beam Loaded with Both A Transversely and Axially Eccentric Tip Mass

Important components of engineering structures can be idealized as cantilever beams with a concentrated mass at its free end. Examples of this are antenna structures, large aspect aircraft wings fitted with external tanks or stores, flexible robot arms and atomic force microscopes. The presence of the tip mass plays an important role in the dynamic characteristics of the beam and exerts an inertial force, which is a function of the system motion [1]. The problem of a uniform cantilever beam with an eccentric tip mass has received much attention, for example, Prescott [2], Goel [3], and, Rama Bhat and Wagner [4]. Exact solutions have been found, for example, To [5] derived an exact expression for natural frequencies and mode shapes of a uniform beam with base excitation and an axially eccentric tip mass and Auciello [6] has suggested an exact analysis of free vibration of a linearly tapered cantilever beam with tip mass with rotary inertia and axial eccentricity. Work has been carried out for non-uniform beams which can achieve a better distribution of strength and weight. The non-uniformity, which usually arises from a variable beam cross section or inhomogeneous material properties, leads to a fourth-order partial differential equation with variable coefficients. In general these fourth-order equations cannot be solved analytically. However, for some specific non-uniform beams, exact solutions of the eigenvalue problem have been reported in terms of power series [7], orthogonal polynomials [8] and Bessel functions [9]. More recent work has been carried on beams with tip mass and with uniform [10] and non-uniform [1] cross sections. Matt [10] simulated transverse vibrations of a cantilever beam, with an eccentric tip mass, in the axial direction using integral transforms which included an implicit filter scheme. Malaeke and Moeenfard [1] modelled large amplitude flexural-extensional free vibrations of non-uniform beams carrying a transversely and axially eccentric tip mass with the effects of the variable axial force taken into account.

The method of solution here is the Adomian modified decomposition method, which is a wide ranging method of solution of problems involving algebraic, differential [11], integro-differential [12] and partial differential equations [13]. Specific to this work, the Adomain decomposition and Adomian modified decomposition method have been used by several groups [14, 15] for uniform and non-uniform beams, starting with either the Euler-Bernoulli or Timoshenko formulations. Mao [14] applied the AMDM to rotating uniform beams and included a centrifugal stiffening term while Adair and Jaeger [15] applied the AMDM to rotating non-uniform beams which also included a centrifugal stiffening term. Yaman [16] has used the Adomian decomposition method to investigate the influence of the orientation effect on the natural frequency of a cantilever beam carrying a tip mass.

In this study we use the computational approach of AMDM to analyze free transverse vibrations of non-uniform cantilever beams carrying a transversely and axially eccentric tip mass. As the beam is assumed to be under the effect of gravity, the spatial variation of axial force is taken into account. The natural frequencies and mode shapes of the beam for three test cases were found and compared with results reported in the literature.

The schematic view of a non-uniform beam with an eccentric tip mass is shown in Fig. 1. The distances a and b are the transverse (z) and axial ( x ) eccentric values, respectively, CoG is the centre of gravity of the tip mass M and l is the length of the beam. The polar mass moment of inertia of the tip mass around CoG is J and, in this analysis, the axial stretching of the beam due to both gravitational and inertial axial forces is considered.

In order the find the governing equations of motion of the beam, Hamilton’s principle is employed. Every system behaves in such a way to satisfy.

where δ is the variation operator, t₁ and t₂ are arbitrary times, K and U are the total kinetic and potential energies of the system respectively and We is the work done by external forces.

For this system, the Euler-Bernoulli assumption that the rotation of cross-sections of the beam is small in comparison to the bending deformation is made. Also it is assumed that the angular distortion due to shear is considered small compared to the bending deformation and can be ignored. Therefore, the beam in question is long and slender with the length much greater than the thickness. With the assumption that the material of the beam is linear elastic, then the overall strain energy can be found from

where A(x) and I(x) are the area and the second moment of area of the beam’s cross-section around the neutral axis and E is Young’s modulus of elasticity of the beam’s material. The total kinetic energy of the system is given as K = K_column + K_mass, where

Here, ρ is the material density of the beam, r_G is the radius of gyration of the tip mass, and θ and V_CoG are the rotational and translational velocities of the tip mass and its centre of gravity, respectively. Here V_CoG is obtained using the relative velocity between the tip mass’s centre of gravity and the point of attachment to the beam.

The virtual work done on the system with the assumption that no viscous damping is present is

where δW_g1 and δW_g2 are the virtual gravitational work done on the tip mass and the beam, respectively. The virtual gravitational work done on the system can be written as

Hamilton’s principle is now applied to obtain the governing equations of vibration of the system

The variable axial force of the beam, N(x,t) is defined as

For a long and slender beam, with a small slenderness ratio, the longitudinal inertia is small compared to the restoring force and so

Substitution for the terms N(x,t) and ∂N(x,t)/∂x, as defined in eqns (9) and (10) into the expanded form of eqn (7) gives

where the superscript ' mean differentiation w.r.t. x . According to modal analysis for harmonic free vibration, w (x,t) and u (x,t), can be separable in space and so eqn (11) can be reduced to

The dimensions of the beam in the y and z direction may be given as

where αd = d d 1 0 / , a ′ = ′ ′ d d d 1 0 / ; d d 0 0 , ′ are the cross-sectional widths at x = 0 and d d 1 1 , ′ are the cross-sectional widths at x l = . The area and moment of inertia of the beam vary according to

where A d d 0 0 = 0 ′ and I d d 0 0 0 3 = ′ /12 are the cross-sectional area and moment of inertia at x = 0 , respectively, and β α d d = −1 , b a ′ ′ d d = −1 .

Without loss of generality, the following dimensionless quantities are now introduced.

Equation (12) can now be written in the form

The governing equation, eqn (17), cannot be solved as a closed-form solution and so some approximate approach must be adopted. Here the Adomian modified decomposition method (AMDM) is chosen.

The Adomian decomposition method (ADM) is an iterative method, which has proved successful in treating non-linear equations. It is based on the search for a solution in the form of a series in which the non-linear terms are calculated recursively using the Adomian polynomials. The ADM gives the solutions φ(η) of eqn (17) in a series form of the infinite sum

Substitution of eqn (18) into the linear and nonlinear terms of eqn (17) yields

where

Here L is an invertible operator, which is taken as the highest-order derivative and R is the remainder of the linear operator.

N₁ [ψ, φ] in eqn (19) is a product nonlinearity and the respective Cauchy product can be computed using Adomian polynomials (A_n) as

N₂[φ] in eqn (19) is also a product non-linearity, and accordingly the respective Cauchy product can be computed using Adomian polynomials (B_n) as

According to the Adomian modified decomposition method (AMDM) the terms φ ηj ( ) in eqn (18) can be expressed as Cj j η so giving the infinite series

where the unknown coefficients Cj are determined concurrently. Using a similar approach ψ η( ) can also be equated to the series j j j C = +∞ ∑ ′ 0 h .

Using the linear operator L of Eq. (20) then the inverse operator of L is a 4-fold integral operator defined as

Equation (19) can be written as

where R is now

The Adomian’s polynomials in eqn (25) are found using the standard method [17] for Bj j φ φ 0 ( ) , , … and the generalized method for several variables as reported by Adomian and Rach [18] is used for Aj j ψ ψ j φ φ 0 0 ( ) , , … … ; , , .

The term Φ(η) found in eqn (25) is the initial term and defined as

On substituting eqns (23), (26) and (27) into eqn (25), the following is obtained

The recurrence relation for the coefficients Cj can now be stated and the solution for φ η( ) can be calculated from eqn (28). The series solution is φ η( ) = η = ∞ ∑ j j j C 0 , although all of the coefficients Cj cannot yet be determined, and thus, the solutions must be approximated by the truncated series j n j j C = − ∑0 1 η , with the successive approximations being φ η η n j n j j C [ ] = − ( ) = ∑0 1 as n increases and the boundary conditions are met.

Thus φ η φ η φ η η φ φ η η 1 0 2 1 1 3 2 2 [ ] [ ] [ ] [ ] [ ] 2 ( ) = C C , , ( ) = ( ) + = ( ) + C serve as approximate solutions with increasing accuracy as n → ∞. The three coefficients C j j ( ) = 0 1, ,2 3, depend on the boundary equations. In this case, the two coefficients C0 and C1 can be chosen as arbitrary constants, and the other two coefficients C2 and C3 are stated in terms of the problem parameters or as the functions of the other coefficients.

The initial term Φ( ) η in eqn (25) is a function of C0 and C1 and from the recurrence relation obtained from eqn (28) the coefficients C j j ( ) ≥ 4 are functions of C0 and C1 and λ4 . By substituting φ η [ ] n ( ) into the boundary conditions we then have

For non-trivial solutions, C₀ and C₁, the frequency equation is given a

The ith estimated eigenvalue λ2 i n ( ) [ ] corresponding to n is obtained from eqn (30), i.e. the ith estimated dimensionless natural frequency Ωn i n i n ( ) [ ] ( ) [ ] = λ2 is also obtained and n is determined by the following equation

where Ωn i n ( ) [ ] −1 is the ith estimated dimensionless natural frequency corresponding to n − 1, and ε is a preset sufficiently small value. If eqn (31) is satisfied, then Ωn i n ( ) [ ] is the ith dimensionless natural frequency Ωn i( ) . By substituting Ωn i n ( ) [ ] into eqn (29)

and all of the other coefficients Cj can be obtained from recurrence relations. Furthermore, the ith mode shape φi [ ] n corresponding to the ith eigenvalue Ωn i n ( ) [ ] is obtained by

where Cj [ ]i ( ) η is Cj ( ) η in which λ4 is substituted by λ4( )i and φi [ ] n is the ith eigenfunction corresponding to the ith eigenvalue λ4( )i . By normalizing eqn (33) the ith normalized eigenfunction is defined as

where φ η i [ ] n ( ) is the ith mode shape function of the beam corresponding to the ith natural frequency

Results are given for the calculations of natural frequencies and mode shapes for a non-uniform beam, a uniform beam with constant axial load, i.e. the non-linear terms of eqn (17) were made constant, and an eccentric mass, and finally, a non-uniform beam with an eccentric mass. The calculated results are compared with results found in the literature.

The test case present here is a clamped-free constant width tapered beam (wedge) and the boundary conditions used were

For the wedge beam α α d d ’ = = 1 0 , ; α βd d ′ = = ,β β the area and moment of the section can now be written as

where β is known as the taper ratio.

In this case the coefficients C₀ and C₁ were set to zero and the coefficients C₂ and C₃ are arbitrary constant. By substituting φ η η n j n j j C [ ] = − ( ) = ∑0 1 into the last two boundary conditions of eqn (35) the following algebraic equations involving C₂ and C₃ are obtained

and for non-trivial solutions of C₂ and C₃ the frequency equation can be written as

By solving eqn (37) for n and taking the real root for λ4 2 = Ωn, it was found that for n = 30

with

The results for the wedge beam for different taper ratios are compared with those reported by Banerjee et al. [19], as shown in Table 1. The calculated results are in reasonable agreement with those reported by Banerjee et al. [19]. Values for β = 0 are not reported by Banerjee et al. [19] as their method suffered numerical overflow when β = 0.

Reasonable agreement was found between the two methods, although the current calculations generally gave slightly larger values. The maximum difference between the two methods was of the order of 1% and occurred for Ω_n(3).

Following [1] calculations were made for a uniform beam experiencing a constant axial loading with an eccentric tip mass. The material and geometric properties are listed in Table 2. The boundary conditions used for these calculations and those for Case 3 are An exact solution for this eigenvalue problem can be found [1] and is used here for comparison.

It can be seen form Table 2 that generally the current calculations give a closer result to the exact solution.

Here a linearly varying cross section cantilever beam with a both transversely and axially eccentric tip mass with rotary inertia is considered. The beam here is tubular and the physical properties are given in Table 4.

Using the present method, the normalized natural frequencies and mode shapes of the beam are obtained as on Figure 2.

A method has been developed for calculating characteristics of a vibrating non-uniform cantilever beam with an eccentric tip mass. The developed approach is nontrivial and complicated due to the nature of the problem. No exact solution of this general problem is available. Several test cases were performed with quite reasonable agreement between the results of the current approach and those reported in the literature found.