# Revisited on the Free Vibration of a Cantilever Beam with an Asymmetrically Attached Tip Mass

Authors: Xiangsheng Lei; Yanfeng Wang; Xinghua Wang; Gang Lin; Shihong Shi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Cantilever with an asymmetrically attached tip mass arises in many engineering applications. Both the traditional method of separation of variables and the method of Laplace transform are employed in the present paper to solve the eigenvalue problem of the free vibration of such structures, and the effect of the eccentric distance along the vertical direction and the length direction of the tip mass is considered here. For the traditional method of separation of variables, tip mass only affects to the boundary conditions, and the eigenvalue problem of the free vibration is solved based on the nonhomogeneous boundary conditions. For the method of Laplace transform, the effect of the tip mass is introduced in the governing equation with the Dirac function, and the eigenvalue problem then can be solved through Laplace transform with homogeneous boundary conditions. The computed results with these two methods are compared well with the numerical solution obtained by finite element method and approximate analytical solutions, and the effect of tip mass dimensions on the natural frequencies and corresponding mode shapes is also given.


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## 1. Introduction

In engineering practices, the problem of a beam carrying a concentrated mass at its end or middle may arise. For instance, offshore wind turbines [1], mast antenna structures, wind tunnel stings carrying an airplane or a missile model, large aspect ratio wings carrying heavy tip tanks, or launch vehicles with payload at the tip [2], and all these structures can be modeled as a beam carrying a concentrated mass at its end or middle. In these applications, the concept of an ideal concentrated mass or moment of initial is often not applicable, as the attachment point does not coincide with the center of gravity of the mass. Researchers had paid more attention to the free vibrations of this subject. Generally, two approaches are adopted to solve this free vibration problem: the traditional method of separation of variables (MSV) and the method of Laplace transform (MLT).The most widely used approach solves the homogeneous partial differential equation that describes the free vibration with separation of variables to yield a pair of ordinary differential equations, and then, with the requirement of the nontrivial solution of the linear equations based on the introduction of nonhomogeneous boundary conditions, frequency equation and mode shape can be obtained consequently. This approach is described as the traditional method of separation of variables (MSV), as it mainly depends on the nonhomogeneous boundary conditions. Bhat et al. gave the natural frequencies of a uniform cantilever with a tip mass slender in the axial direction based on the perturbation procedure, in which rotary inertial is also considered [2, 3]. Recently, Mousavi Lajimi and Heppler corrected some errors in Bhat and Wagner’s paper [4], and Bhat responded back with a closure [5]. Wang et al. also proposed an improved analytical method of calculations for natural frequencies and mode shapes of a uniform cantilever beam carrying a tip mass under base excitation [6]. It is noted that they did not consider the effect of the eccentric distance along the vertical direction of the tip mass [6]. Anderson pointed out that due to the importance in airplane and missile design, it is of interest to consider the problem that the centroid of the tip mass does not coincide with its point of attachment to the beam which offsets an arbitrary distance perpendicular to the extended neutral axis of the beam [7]. Anderson also showed that the longitudinal and transverse deflections in the beam become coupled through the boundary conditions because of the presence of the asymmetrically attached tip mass. Natural frequencies and mode shapes of a cantilever beam with a base excitation and asymmetric tip mass were given by To [8]. Many other researchers also used this approach to investigate the vibrations of a cantilever with concentrate mass under different boundary conditions [9–16].The other approach introduces the tip mass by means of a Dirac function to make the constant density beam a variable density one [17], and the governing partial differential equations can be solved based on the method of separation of variables and the Laplace transform under homogeneous boundary conditions. This approach is described as the method of Laplace transform (MLT), as it mainly depends on the Laplace transform to solve the differential equation. The advantage of this method is that if there are many concentrated mass located along the length direction of the beam, it is unnecessary to solve the problem by considering many individual spans separated by these concentrated masses. Chen adopted this method to solve the free vibration and forced vibration problem of a simply supported beam with a middle concentrated mass [18], and the concentrated mass is just considered as a point mass. When the dimensions of the concentrated mass are too big to be ignored, the problem of how to consider the tip mass effect arises. Goel investigated vibrations of a beam carrying a concentrated mass that one end of the beam is free and the other end is hinged by a rotational spring of constant stiffness with this method [19, 20]. Liu and Huang [21] and Chang [22] investigated the free vibration and forced vibration of a beam carrying a concentrated mass at the beam tip and another concentrated mass at an intermediate point, respectively. Park et al. investigated a Bernoulli–Euler beam fixed on a moving cart and carrying a concentrated mass attached to an arbitrary position along the beam length [23].It should be pointed out that, most of the investigations either ignore the rotational inertia of the tip mass or ignore the eccentric effect of the tip mass, and in some situations, these effects cannot be neglected. Therefore, the purpose of this paper is to conduct the free vibration analysis of the cantilever with an asymmetrically attached tip mass with both the conventional method of separation of variables and method of Laplace transform and to clarify the introduction of concentrate force and moment in the method of Laplace transform.


## 2. Mathematical Model

A schematic sketch of the cantilever is shown in Figure1, and Lb and Lm denote the length of the beam and the tip mass, respectively, eh and eL=Lm/2 are the distance between point of attachment A and the center of gravity of the tip mass C along the vertical and length direction, respectively, and hb and hm denote the heights of the beam and the tip mass, respectively. The tip mass and the beam are assumed to have the same width b. The total mass of the tip mass is m.Figure 1
Schematic of a cantilever with an asymmetrical tip mass.


### 2.1. Method of Separation of Variables

In this section, the homogeneous partial differential equation which describes the free vibration of the cantilever beam is solved by the MSV. For the method of separation of variables, the effect of tip mass to the free vibration is introduced by the nonhomogeneous boundary conditions [24].Hamilton’s principle is applied here to obtain the governing equations of the system:(1)∫t0t1δT−Vdt=0,where T and V denote the kinetic and potential energies, respectively, δ is the variation operator, and t0 and t1 are arbitrary times. The potential energy is given by(2)V=12∫0LbEIw″dx.The total kinetic energy consists of the contributions from the beamTb and the tip mass Tm:(3)T=Tb+Tm,(4)Tb=12ρbA∫0Lbw˙2dx,(5)Tm=∫LbLb+Lm∫eh−hm/2eh+hm/212ρmbmvm2dxdy,where the velocity of the representative elemental volume vm is given by(6)vm=w˙L′y2+w˙L+w˙L′x−Lb2.In the above formulations, primes and dots denote differentiation with regard to coordinatex and time t, respectively, E denotes Young’s modulus for the beam material, A and I are constant cross-section area and moment of inertia, respectively, and ρb and ρm denote density of the beam and tip mass material, respectively. The flexural deflection of the beam is denoted by w, and wL is the deflection of the beam at x=Lb.Substituting equations (2)–(6) into equation (1) and carrying out the necessary variations, the following equation for undamped free vibration under small deflection is obtained:(7)EIw‴′+ρbAw¨=0.The corresponding boundary conditions at the fixed endx = 0 are(8)w0,t=0,(9)w′0,t=0,and at the tip x=Lb are(10)EIw‴Lb,t−mw¨Lb,t−meLw¨′Lb,t=0,(11)EIw″Lb,t+Jm+meL2+eh2w¨′Lb,t+meLw¨Lb,t=0,where I=bhb3/12 and Jm=mLm2+hm2/12 are the moment of inertia of the beam cross-section and the tip mass, respectively. These vibration equation and corresponding boundary conditions are the same as in [7].For undamped free vibration of natural frequencyω, one may assume that(12)wx,t=ϕxηt.Substituting equation (12) into equation (7), we have(13)EIϕ″″xηt+ρbAϕxη¨t=0,or(14)ϕ‴′−ρbAω2EIϕ=0,(15)η¨+ω2η=0.The general solution for equation (14) is given by(16)ϕx=Asinλx+Bcosλx+Csinhλx+Dcoshλx,where A, B, C, and D are constants to be determined by boundary conditions (8)–(11) and λ4=ρbAω2/EI.Substituting equation (16) into boundary conditions (8)–(11), we have(17)ϕ0=0,(18)ϕ′0=0,(19)ϕ‴Lb+meLω2EIϕ′Lb+mω2EIϕLb=0,(20)ϕ″Lb−Jm+meL2+eh2ω2EIϕ′Lb−meLω2EIϕLb=0.Considering equation (17) and (18), we have(21)ϕx=Asinλx−sinhλx+Bcosλx−coshλx.Substituting equation (21) into boundary conditions (19) and (20), then they can be written in matrix form:(22)Ω11Ω12Ω21Ω22AB=00,where Ω11=−λ3cosλLb+coshλLb+c2λcosλLb−coshλLb+c1sinλLb−sinhλLb, Ω12=λ3sinλLb−sinhλLb−c2λsinλLb+sinhλLb+c1cosλLb−coshλLb, Ω21=−λ2sinλLb+sinhλLb−c3λcosλLb−coshλLb−c2sinλLb−sinhλLb, and Ω22=−λ2cosλLb+coshλLb+c3λsinλLb+sinhλLb−c2cosλLb−coshλLb and c1=mω2/EI,c2=meLω2/EI,andc3=Jm+meL2+eh2ω2/EI.Based on the untrivial solution condition of equation (22), the frequency equation can be obtained, in principle, by setting the coefficient determinant to zero:(23)Ω11Ω22−Ω12Ω21=0.The roots of equation (23) gives the natural frequencies ωii=1,2,3,…; then, by substituting ωi into equations (22) and (21), one can obtain the constant A and B and the corresponding mode shape ϕix. It is also noted that the natural frequencies were determined by a trial and error method based on interpolation and the bisection approach with MATLAB. The iterative computations were terminated when the value of ωi reached the relative error of 10−5.


### 2.2. Method of Laplace Transform

In order to describe the effect of the asymmetrically attached tip mass, the Diracδ-function is used in the differential equation which is defined by the following equation:(24)∫−∞+∞δx−Lbdx=1.The effect of the tip mass is introduced by the concentrated equivalent forceFm and moment Mm at the attachment point A of the beam, and then, the undamped free vibration equation under small deflection is(25)EIw′‴+ρbAw¨+Fmδx−Lb+Mmδ′x−Lb=0,where(26)Fm=mw¨Lb,t+meLw¨′Lb,t,Mm=−meLw¨Lb,t−Jm+meL2+eh2w¨′Lb,t.With this treatment, the boundary conditions become homogeneous, which is given by(27)w0,t=0,(28)w′0,t=0,(29)w‴Lb,t=0,(30)w″Lb,t=0.Based on separation of variables, for undamped free vibration of natural frequencyω, one may assume that(31)wx,t=ϕxsinωt.Substituting equation (31) into equations (25)–(30), we have(32)ϕ″″−λ4ϕ−δx−Lbc1ϕLb+c2ϕ′Lb+δ′x−Lbc2ϕLb+c3ϕ′Lb=0,(33)ϕ0=0,(34)ϕ′0=0,(35)ϕ‴Lb=0,(36)ϕ″Lb=0,where λ4=ρbAω2/EI and c1=mω2/EI,c2=meLω2/EI,andc3=Jm+meL2+eh2ω2/EI.Equations (32)–(36) define an eigenvalue problem, and the solution can be obtained by the Laplace transform method. Let the transformed function of ϕ be denoted by ϕ¯, and the definition of the Laplace transform is given by(37)ϕ¯s=Lϕx=∫−∞+∞ϕxe−sxdx.Then, we have(38)Lϕ″″=s4ϕ¯−ϕ‴0−sϕ″0−s2ϕ′0−s3ϕ0,(39)L−λ4ϕ=−λ4ϕ¯.Considering the boundary conditions (33) and (34), we have(40)Lϕ″″=s4ϕ¯−ϕ‴0−sϕ″0.The shifting property of the Diracδ-function is given by(41)∫−∞+∞ϕxδx−x0dx=ϕx0,and then, we have(42)L−c1δx−LbϕLb=−c1ϕLb∫−∞+∞δx−Lbe−sxdx=−c1ϕLbe−sx|x=Lb=−c1ϕLbe−sLb,(43)L−c2δx−Lbϕ′Lb=−c2ϕ′Lb∫−∞+∞δx−Lbe−sxdx=−c2ϕ′Lbe−sx|x=Lb=−c2ϕ′Lbe−sLb.The definition of the derivatives of the Diracδ-function is given by(44)∫−∞+∞ϕxδ′x−x0dx=−ϕ′x0,and then, we have(45)Lc2δ′x−LbϕLb=c2ϕLb∫−∞+∞δ′x−Lbe−sxdx=−c2ϕLbe−sx′|x=Lb=−c2ϕLbse−sLb,(46)Lc3δ′x−Lbϕ′Lb=c3ϕ′Lb∫−∞+∞δ′x−Lbe−sxdx=−c3ϕ′Lbe−sx′|x=Lb=−c3ϕ′Lbse−sLb.Substituting equations (39)–(46) into equation (32), then we have(47)ϕ¯=ss4−λ4ϕ″0+1s4−λ4ϕ‴0+c1ϕLb+c2ϕ′Lbe−sLbs4−λ4−c2ϕLb+c3ϕ′Lbse−sLbs4−λ4.Then, the inverse Laplace transform is defined by(48)ϕx=L−1ϕ¯s=12πj∫−j∞+j∞ϕ¯se−sxdx.Through the inverse Laplace transform, we have(49)L−1ss4−λ4=12λ2coshλx−cosλx,(50)L−11s4−λ4=12λ3sinhλx−sinλx,(51)L−1e−sLbs4−λ4=12λ3sinhλx−Lb−sinλx−LbHx−Lb,(52)L−1se−sLbs4−λ4=12λ2coshλx−Lb−cosλx−LbHx−Lb,where Hx−Lb is the unit step function at x=Lb.Substituting equations (49)–(52) into equation (47), then we have the general solution of equation (32) as follows:(53)ϕx=ϕ″02λ2coshλx−cosλx+ϕ‴02λ3sinhλx−sinλx+c1ϕLb+c2ϕ′Lb2λ3sinhλx−Lb−sinλx−LbHx−Lb−c2ϕLb+c3ϕ′Lb2λ2coshλx−Lb−cosλx−LbHx−Lb,(54)ϕ′x=ϕ″02λsinhλx+sinλx+ϕ‴02λ2coshλx−cosλx+c1ϕLb+c2ϕ′Lb2λ2coshλx−Lb−cosλx−LbHx−Lb−c2ϕLb+c3ϕ′Lb2λsinhλx−Lb+sinλx−LbHx−Lb,(55)ϕ″x=ϕ″02coshλx+cosλx+ϕ‴02λsinhλx+sinλx+c1ϕLb+c2ϕ′Lb2λsinhλx−Lb+sinλx−LbHx−Lb−c2ϕLb+c3ϕ′Lb2coshλx−Lb+cosλx−LbHx−Lb,(56)ϕ‴x=λϕ″02sinhλx−sinλx+ϕ‴02coshλx+cosλx+c1ϕLb+c2ϕ′Lb2coshλx−Lb+cosλx−LbHx−Lb−λc2ϕLb+c3ϕ′Lb2sinhλx−Lb−sinλx−LbHx−Lb.It is noted that items withδx−Lb, δ′x−Lb, and δ″x−Lb are ignored in equations (54)–(56), as they will not affect the final results.Substituting equations (55) and (56) into boundary conditions’ equations (35) and (36), then the constants ϕ″0 and ϕ‴0 can be obtained by(57)ϕ″0=λb1coshλLb+cosλLb+b2sinhλLb+sinλLbλ1+cosλLbcoshλLb,ϕ‴0=λb1sinλLb−sinhλLb−b2coshλLb+cosλLb1+cosλLbcoshλLb,where b1=c2ϕLb+c3ϕ′Lbandb2=c1ϕLb+c2ϕ′Lb.Substituting equation (57) into equations (53) and (54), let x=Lb, and the following two equations are obtained:(58)Λ11Λ12Λ21Λ22ϕLbϕ′Lb=00,where Λ11=λ31+cosλLbcoshλLb+c1cosλLbsinhλLb−coshλLbsinλLb−c2λsinλLbsinhλLb, Λ12=c2sinhλLbcosλLb−coshλLbsinλLb−c3λsinhλLbsinλLb, Λ21=c2λcosλLbsinhλLb+coshλLbsinλLb+c1sinλLbsinhλLb, and Λ22=c3λcosλLbsinhλLb+coshλLbsinλLb+c2sinλLbsinhλLb−λ21+cosλLbcoshλLb.Based on the untrivial solution condition of equation (58), the frequency equation can be obtained, in principle, by setting the coefficient determinant to zero:(59)Λ11Λ22−Λ12Λ21=0.The roots of equation (59) gives the natural frequencies ωii=1,2,3,…; then, by substituting ωi into equations (53) and (58), one can obtain the constant A and B and the corresponding mode shape ϕix.


## 2.1. Method of Separation of Variables

In this section, the homogeneous partial differential equation which describes the free vibration of the cantilever beam is solved by the MSV. For the method of separation of variables, the effect of tip mass to the free vibration is introduced by the nonhomogeneous boundary conditions [24].Hamilton’s principle is applied here to obtain the governing equations of the system:(1)∫t0t1δT−Vdt=0,where T and V denote the kinetic and potential energies, respectively, δ is the variation operator, and t0 and t1 are arbitrary times. The potential energy is given by(2)V=12∫0LbEIw″dx.The total kinetic energy consists of the contributions from the beamTb and the tip mass Tm:(3)T=Tb+Tm,(4)Tb=12ρbA∫0Lbw˙2dx,(5)Tm=∫LbLb+Lm∫eh−hm/2eh+hm/212ρmbmvm2dxdy,where the velocity of the representative elemental volume vm is given by(6)vm=w˙L′y2+w˙L+w˙L′x−Lb2.In the above formulations, primes and dots denote differentiation with regard to coordinatex and time t, respectively, E denotes Young’s modulus for the beam material, A and I are constant cross-section area and moment of inertia, respectively, and ρb and ρm denote density of the beam and tip mass material, respectively. The flexural deflection of the beam is denoted by w, and wL is the deflection of the beam at x=Lb.Substituting equations (2)–(6) into equation (1) and carrying out the necessary variations, the following equation for undamped free vibration under small deflection is obtained:(7)EIw‴′+ρbAw¨=0.The corresponding boundary conditions at the fixed endx = 0 are(8)w0,t=0,(9)w′0,t=0,and at the tip x=Lb are(10)EIw‴Lb,t−mw¨Lb,t−meLw¨′Lb,t=0,(11)EIw″Lb,t+Jm+meL2+eh2w¨′Lb,t+meLw¨Lb,t=0,where I=bhb3/12 and Jm=mLm2+hm2/12 are the moment of inertia of the beam cross-section and the tip mass, respectively. These vibration equation and corresponding boundary conditions are the same as in [7].For undamped free vibration of natural frequencyω, one may assume that(12)wx,t=ϕxηt.Substituting equation (12) into equation (7), we have(13)EIϕ″″xηt+ρbAϕxη¨t=0,or(14)ϕ‴′−ρbAω2EIϕ=0,(15)η¨+ω2η=0.The general solution for equation (14) is given by(16)ϕx=Asinλx+Bcosλx+Csinhλx+Dcoshλx,where A, B, C, and D are constants to be determined by boundary conditions (8)–(11) and λ4=ρbAω2/EI.Substituting equation (16) into boundary conditions (8)–(11), we have(17)ϕ0=0,(18)ϕ′0=0,(19)ϕ‴Lb+meLω2EIϕ′Lb+mω2EIϕLb=0,(20)ϕ″Lb−Jm+meL2+eh2ω2EIϕ′Lb−meLω2EIϕLb=0.Considering equation (17) and (18), we have(21)ϕx=Asinλx−sinhλx+Bcosλx−coshλx.Substituting equation (21) into boundary conditions (19) and (20), then they can be written in matrix form:(22)Ω11Ω12Ω21Ω22AB=00,where Ω11=−λ3cosλLb+coshλLb+c2λcosλLb−coshλLb+c1sinλLb−sinhλLb, Ω12=λ3sinλLb−sinhλLb−c2λsinλLb+sinhλLb+c1cosλLb−coshλLb, Ω21=−λ2sinλLb+sinhλLb−c3λcosλLb−coshλLb−c2sinλLb−sinhλLb, and Ω22=−λ2cosλLb+coshλLb+c3λsinλLb+sinhλLb−c2cosλLb−coshλLb and c1=mω2/EI,c2=meLω2/EI,andc3=Jm+meL2+eh2ω2/EI.Based on the untrivial solution condition of equation (22), the frequency equation can be obtained, in principle, by setting the coefficient determinant to zero:(23)Ω11Ω22−Ω12Ω21=0.The roots of equation (23) gives the natural frequencies ωii=1,2,3,…; then, by substituting ωi into equations (22) and (21), one can obtain the constant A and B and the corresponding mode shape ϕix. It is also noted that the natural frequencies were determined by a trial and error method based on interpolation and the bisection approach with MATLAB. The iterative computations were terminated when the value of ωi reached the relative error of 10−5.


## 2.2. Method of Laplace Transform

In order to describe the effect of the asymmetrically attached tip mass, the Diracδ-function is used in the differential equation which is defined by the following equation:(24)∫−∞+∞δx−Lbdx=1.The effect of the tip mass is introduced by the concentrated equivalent forceFm and moment Mm at the attachment point A of the beam, and then, the undamped free vibration equation under small deflection is(25)EIw′‴+ρbAw¨+Fmδx−Lb+Mmδ′x−Lb=0,where(26)Fm=mw¨Lb,t+meLw¨′Lb,t,Mm=−meLw¨Lb,t−Jm+meL2+eh2w¨′Lb,t.With this treatment, the boundary conditions become homogeneous, which is given by(27)w0,t=0,(28)w′0,t=0,(29)w‴Lb,t=0,(30)w″Lb,t=0.Based on separation of variables, for undamped free vibration of natural frequencyω, one may assume that(31)wx,t=ϕxsinωt.Substituting equation (31) into equations (25)–(30), we have(32)ϕ″″−λ4ϕ−δx−Lbc1ϕLb+c2ϕ′Lb+δ′x−Lbc2ϕLb+c3ϕ′Lb=0,(33)ϕ0=0,(34)ϕ′0=0,(35)ϕ‴Lb=0,(36)ϕ″Lb=0,where λ4=ρbAω2/EI and c1=mω2/EI,c2=meLω2/EI,andc3=Jm+meL2+eh2ω2/EI.Equations (32)–(36) define an eigenvalue problem, and the solution can be obtained by the Laplace transform method. Let the transformed function of ϕ be denoted by ϕ¯, and the definition of the Laplace transform is given by(37)ϕ¯s=Lϕx=∫−∞+∞ϕxe−sxdx.Then, we have(38)Lϕ″″=s4ϕ¯−ϕ‴0−sϕ″0−s2ϕ′0−s3ϕ0,(39)L−λ4ϕ=−λ4ϕ¯.Considering the boundary conditions (33) and (34), we have(40)Lϕ″″=s4ϕ¯−ϕ‴0−sϕ″0.The shifting property of the Diracδ-function is given by(41)∫−∞+∞ϕxδx−x0dx=ϕx0,and then, we have(42)L−c1δx−LbϕLb=−c1ϕLb∫−∞+∞δx−Lbe−sxdx=−c1ϕLbe−sx|x=Lb=−c1ϕLbe−sLb,(43)L−c2δx−Lbϕ′Lb=−c2ϕ′Lb∫−∞+∞δx−Lbe−sxdx=−c2ϕ′Lbe−sx|x=Lb=−c2ϕ′Lbe−sLb.The definition of the derivatives of the Diracδ-function is given by(44)∫−∞+∞ϕxδ′x−x0dx=−ϕ′x0,and then, we have(45)Lc2δ′x−LbϕLb=c2ϕLb∫−∞+∞δ′x−Lbe−sxdx=−c2ϕLbe−sx′|x=Lb=−c2ϕLbse−sLb,(46)Lc3δ′x−Lbϕ′Lb=c3ϕ′Lb∫−∞+∞δ′x−Lbe−sxdx=−c3ϕ′Lbe−sx′|x=Lb=−c3ϕ′Lbse−sLb.Substituting equations (39)–(46) into equation (32), then we have(47)ϕ¯=ss4−λ4ϕ″0+1s4−λ4ϕ‴0+c1ϕLb+c2ϕ′Lbe−sLbs4−λ4−c2ϕLb+c3ϕ′Lbse−sLbs4−λ4.Then, the inverse Laplace transform is defined by(48)ϕx=L−1ϕ¯s=12πj∫−j∞+j∞ϕ¯se−sxdx.Through the inverse Laplace transform, we have(49)L−1ss4−λ4=12λ2coshλx−cosλx,(50)L−11s4−λ4=12λ3sinhλx−sinλx,(51)L−1e−sLbs4−λ4=12λ3sinhλx−Lb−sinλx−LbHx−Lb,(52)L−1se−sLbs4−λ4=12λ2coshλx−Lb−cosλx−LbHx−Lb,where Hx−Lb is the unit step function at x=Lb.Substituting equations (49)–(52) into equation (47), then we have the general solution of equation (32) as follows:(53)ϕx=ϕ″02λ2coshλx−cosλx+ϕ‴02λ3sinhλx−sinλx+c1ϕLb+c2ϕ′Lb2λ3sinhλx−Lb−sinλx−LbHx−Lb−c2ϕLb+c3ϕ′Lb2λ2coshλx−Lb−cosλx−LbHx−Lb,(54)ϕ′x=ϕ″02λsinhλx+sinλx+ϕ‴02λ2coshλx−cosλx+c1ϕLb+c2ϕ′Lb2λ2coshλx−Lb−cosλx−LbHx−Lb−c2ϕLb+c3ϕ′Lb2λsinhλx−Lb+sinλx−LbHx−Lb,(55)ϕ″x=ϕ″02coshλx+cosλx+ϕ‴02λsinhλx+sinλx+c1ϕLb+c2ϕ′Lb2λsinhλx−Lb+sinλx−LbHx−Lb−c2ϕLb+c3ϕ′Lb2coshλx−Lb+cosλx−LbHx−Lb,(56)ϕ‴x=λϕ″02sinhλx−sinλx+ϕ‴02coshλx+cosλx+c1ϕLb+c2ϕ′Lb2coshλx−Lb+cosλx−LbHx−Lb−λc2ϕLb+c3ϕ′Lb2sinhλx−Lb−sinλx−LbHx−Lb.It is noted that items withδx−Lb, δ′x−Lb, and δ″x−Lb are ignored in equations (54)–(56), as they will not affect the final results.Substituting equations (55) and (56) into boundary conditions’ equations (35) and (36), then the constants ϕ″0 and ϕ‴0 can be obtained by(57)ϕ″0=λb1coshλLb+cosλLb+b2sinhλLb+sinλLbλ1+cosλLbcoshλLb,ϕ‴0=λb1sinλLb−sinhλLb−b2coshλLb+cosλLb1+cosλLbcoshλLb,where b1=c2ϕLb+c3ϕ′Lbandb2=c1ϕLb+c2ϕ′Lb.Substituting equation (57) into equations (53) and (54), let x=Lb, and the following two equations are obtained:(58)Λ11Λ12Λ21Λ22ϕLbϕ′Lb=00,where Λ11=λ31+cosλLbcoshλLb+c1cosλLbsinhλLb−coshλLbsinλLb−c2λsinλLbsinhλLb, Λ12=c2sinhλLbcosλLb−coshλLbsinλLb−c3λsinhλLbsinλLb, Λ21=c2λcosλLbsinhλLb+coshλLbsinλLb+c1sinλLbsinhλLb, and Λ22=c3λcosλLbsinhλLb+coshλLbsinλLb+c2sinλLbsinhλLb−λ21+cosλLbcoshλLb.Based on the untrivial solution condition of equation (58), the frequency equation can be obtained, in principle, by setting the coefficient determinant to zero:(59)Λ11Λ22−Λ12Λ21=0.The roots of equation (59) gives the natural frequencies ωii=1,2,3,…; then, by substituting ωi into equations (53) and (58), one can obtain the constant A and B and the corresponding mode shape ϕix.


## 3. Numerical Results

In this section, numerical results obtained by ANSYS simulation and analytical solution with the present method of separation of variables and the method of Laplace transform are given and compared for different cantilevers. The effect of several key parameters on the natural frequencies and mode shapes is also presented.


### 3.1. Verification of the Present Methods

ANSYS is used here to determine the natural frequency and mode shape, and the obtained numerical results are considered as the benchmark solution. The cantilever is modeled as a three dimensional structure and meshed with SOLID186 element. The convergence test is performed in advance to make sure that the numerical solution can be treated as the benchmark solution.The approximate analytical solution of the first-order natural frequency of a cantilever with the tip mass is given by(60)f1apr=12π3EI/Lb333/140ρbALb+m.The geometry and physical parameters used in the computation are given in Table1, and three beam structures with different length Lb are investigated. The first three natural frequencies of different beam structures with different methods are given in Table 2.Table 1
Geometry and physical parameters for the beam structure.
ParameterValueLb100, 150, and 200 mmLm15 mmhb1 mmhm30 mmb10 mmeh10 mmeL7.5 mmρm8900 kg/m3ρb7800 kg/m3E210 GPaTable 2
Comparisons of the first three natural frequencies of different beam structures with different methods.
Lb = 100 mmLb = 150 mmLb = 200 mmf1 (Hz)f2 (Hz)f3 (Hz)f1 (Hz)f2 (Hz)f3 (Hz)f1 (Hz)f2 (Hz)f3 (Hz)Results from equation (59)15.960——8.903——5.825——ANSYS15.849153.42576.048.9144103.98286.845.849072.251186.85Present MSV15.780153.98579.128.8868103.99288.025.835072.148187.36Present MLT15.780153.98579.128.8868103.99288.025.835072.148187.36Results without eccentric effect15.925200.75616.908.9241123.41326.905.834479.434217.52Results without inertial effect15.960322.031035.08.9021150.85483.085.824687.373278.82The approximate analytical solution of the first-order natural frequency of the three cantilevers is 15.960 Hz, 8.903 Hz, and 5.825 Hz, respectively. It can be seen that, for all the other five methods, the first-order natural frequency results compare well with the approximate analytical solution, as the inertial effect does not play an important role in the first-order free vibration. For the second- and third-order natural frequencies, results with the present MSV and MLT are the same and in good agreement with the ANSYS solution. At the same time, the inertial effect plays a very important role in higher-order natural frequencies for the cantilever with an asymmetrically attached tip mass, and the inertial effect makes the structure more flexible. If inertial effect is ignored, the higher-order natural frequencies can be unconceivable large. It also can be seen that, although inertial effect is taken into consideration, the eccentric effect is also nonignorable. For the 100 mm length cantilever, the second- and third-order natural frequencies are 154 Hz and 579 Hz, respectively. If the inertial effect is neglected, the natural frequencies increase to 322 Hz and 1035 Hz, respectively, and even if we consider the inertial effect, but neglect the eccentricity of the tip mass, the natural frequencies are still as large as 201 Hz and 617 Hz, respectively, which are much larger than those of 154 Hz and 579 Hz, respectively. The comparisons of the first three mode shapes with different methods for different beams are given in Figures2–4 , and the mode shapes are normalized by the tip displacement.Figure 2
Comparisons of the mode shapes with the beam length of 100 mm. (a) The first-order mode shape. (b) The second-order mode shape. (c) The third-order mode shape.
(a)(b)(c)Figure 3
Comparisons of the mode shapes with the beam length of 150 mm. (a) The first-order mode shape. (b) The second-order mode shape. (c) The third-order mode shape.
(a)(b)(c)Figure 4
Comparisons of the mode shapes with the beam length of 200 mm. (a) The first-order mode shape. (b) The second-order mode shape. (c) The third-order mode shape.
(a)(b)(c)It can be seen from Figures2–4 that the present results with both the method of separation of variables (MSV) and method of Laplace transform (MLT) compare well with ANSYS results, while both results without eccentric effect and without inertia effect have an obvious different with the ANSYS results. It is also noted that both natural frequencies and mode shapes obtained by MSV and MLT are exactly the same with each other. It is clear that the eccentric effect and inertial effect play an important role in the free vibration of the cantilever beam with an asymmetrically attached tip mass, and the quantitative investigations of some key parameters on the natural frequencies and mode shapes are given in the following section.


### 3.2. Parametric Investigations

The geometry and physical parameters of the basic structure used in the parametric investigations are given in Table1, and the variations of the first three natural frequencies and corresponding mode shapes with relative length of the tip mass to the beam are given in Figures 5 and 6 , respectively. The mode shapes are normalized by the tip displacement, and the natural frequencies are normalized by ωii=1,2,and3 given below:(61)ωi=βiLb2EIρbA,where β1=1.875, β2=4.694, and β3=7.855.Figure 5
Variations of the nondimensionless natural frequencies with relative length of the tip mass to the beam. (a) The first-order natural frequency. (b) The second-order natural frequency. (c) The third-order natural frequency.
(a)(b)(c)Figure 6
Variations of the nondimensionless mode shapes with relative length of the tip mass to the beam (Lb = 150 mm). (a) The first-order natural frequency. (b) The second-order natural frequency. (c) The third-order natural frequency.
(a)(b)(c)It can be seen from Figure5 that, with the increment of the relative length of the tip mass, all the first three orders of natural frequencies decrease due to the large tip mass effect. It is also noted that, for all the beams, the first- and third-order frequencies have the same tendency with respect to the nondimensionless tip mass length ratio, while for the second-order free vibration, the nondimensional natural frequency decreases from 1 to 0.21 with the increment of nondimensional tip mass length from 0 to 0.2 for the 100 mm length beam, while for the 150 mm and 200 mm beam, the nondimensional natural frequency decreases from 1 to 0.28 and 0.33, respectively. In other words, longer beam corresponds to smaller tip mass effect, and the frequency reduction phenomena with the increment of the tip mass length is more sensitive for the second-order natural frequency than the first-and third-order frequencies. Figure 6 shows the variations of the first three mode shapes of the 150 mm length beam with different nondimensionless tip mass length ratios, and it is very clear that, for the first-order free vibration, big tip mass can only reduce the natural frequency, while the first-order mode shape almost does not change. For higher-order mode shapes, the tip mass has more obvious effect, and the accurate mode shapes as well as the strain nodes are very important for higher-order vibration-based piezoelectric energy harvesting applications [25].


## 3.1. Verification of the Present Methods

ANSYS is used here to determine the natural frequency and mode shape, and the obtained numerical results are considered as the benchmark solution. The cantilever is modeled as a three dimensional structure and meshed with SOLID186 element. The convergence test is performed in advance to make sure that the numerical solution can be treated as the benchmark solution.The approximate analytical solution of the first-order natural frequency of a cantilever with the tip mass is given by(60)f1apr=12π3EI/Lb333/140ρbALb+m.The geometry and physical parameters used in the computation are given in Table1, and three beam structures with different length Lb are investigated. The first three natural frequencies of different beam structures with different methods are given in Table 2.Table 1
Geometry and physical parameters for the beam structure.
ParameterValueLb100, 150, and 200 mmLm15 mmhb1 mmhm30 mmb10 mmeh10 mmeL7.5 mmρm8900 kg/m3ρb7800 kg/m3E210 GPaTable 2
Comparisons of the first three natural frequencies of different beam structures with different methods.
Lb = 100 mmLb = 150 mmLb = 200 mmf1 (Hz)f2 (Hz)f3 (Hz)f1 (Hz)f2 (Hz)f3 (Hz)f1 (Hz)f2 (Hz)f3 (Hz)Results from equation (59)15.960——8.903——5.825——ANSYS15.849153.42576.048.9144103.98286.845.849072.251186.85Present MSV15.780153.98579.128.8868103.99288.025.835072.148187.36Present MLT15.780153.98579.128.8868103.99288.025.835072.148187.36Results without eccentric effect15.925200.75616.908.9241123.41326.905.834479.434217.52Results without inertial effect15.960322.031035.08.9021150.85483.085.824687.373278.82The approximate analytical solution of the first-order natural frequency of the three cantilevers is 15.960 Hz, 8.903 Hz, and 5.825 Hz, respectively. It can be seen that, for all the other five methods, the first-order natural frequency results compare well with the approximate analytical solution, as the inertial effect does not play an important role in the first-order free vibration. For the second- and third-order natural frequencies, results with the present MSV and MLT are the same and in good agreement with the ANSYS solution. At the same time, the inertial effect plays a very important role in higher-order natural frequencies for the cantilever with an asymmetrically attached tip mass, and the inertial effect makes the structure more flexible. If inertial effect is ignored, the higher-order natural frequencies can be unconceivable large. It also can be seen that, although inertial effect is taken into consideration, the eccentric effect is also nonignorable. For the 100 mm length cantilever, the second- and third-order natural frequencies are 154 Hz and 579 Hz, respectively. If the inertial effect is neglected, the natural frequencies increase to 322 Hz and 1035 Hz, respectively, and even if we consider the inertial effect, but neglect the eccentricity of the tip mass, the natural frequencies are still as large as 201 Hz and 617 Hz, respectively, which are much larger than those of 154 Hz and 579 Hz, respectively. The comparisons of the first three mode shapes with different methods for different beams are given in Figures2–4 , and the mode shapes are normalized by the tip displacement.Figure 2
Comparisons of the mode shapes with the beam length of 100 mm. (a) The first-order mode shape. (b) The second-order mode shape. (c) The third-order mode shape.
(a)(b)(c)Figure 3
Comparisons of the mode shapes with the beam length of 150 mm. (a) The first-order mode shape. (b) The second-order mode shape. (c) The third-order mode shape.
(a)(b)(c)Figure 4
Comparisons of the mode shapes with the beam length of 200 mm. (a) The first-order mode shape. (b) The second-order mode shape. (c) The third-order mode shape.
(a)(b)(c)It can be seen from Figures2–4 that the present results with both the method of separation of variables (MSV) and method of Laplace transform (MLT) compare well with ANSYS results, while both results without eccentric effect and without inertia effect have an obvious different with the ANSYS results. It is also noted that both natural frequencies and mode shapes obtained by MSV and MLT are exactly the same with each other. It is clear that the eccentric effect and inertial effect play an important role in the free vibration of the cantilever beam with an asymmetrically attached tip mass, and the quantitative investigations of some key parameters on the natural frequencies and mode shapes are given in the following section.


## 3.2. Parametric Investigations

The geometry and physical parameters of the basic structure used in the parametric investigations are given in Table1, and the variations of the first three natural frequencies and corresponding mode shapes with relative length of the tip mass to the beam are given in Figures 5 and 6 , respectively. The mode shapes are normalized by the tip displacement, and the natural frequencies are normalized by ωii=1,2,and3 given below:(61)ωi=βiLb2EIρbA,where β1=1.875, β2=4.694, and β3=7.855.Figure 5
Variations of the nondimensionless natural frequencies with relative length of the tip mass to the beam. (a) The first-order natural frequency. (b) The second-order natural frequency. (c) The third-order natural frequency.
(a)(b)(c)Figure 6
Variations of the nondimensionless mode shapes with relative length of the tip mass to the beam (Lb = 150 mm). (a) The first-order natural frequency. (b) The second-order natural frequency. (c) The third-order natural frequency.
(a)(b)(c)It can be seen from Figure5 that, with the increment of the relative length of the tip mass, all the first three orders of natural frequencies decrease due to the large tip mass effect. It is also noted that, for all the beams, the first- and third-order frequencies have the same tendency with respect to the nondimensionless tip mass length ratio, while for the second-order free vibration, the nondimensional natural frequency decreases from 1 to 0.21 with the increment of nondimensional tip mass length from 0 to 0.2 for the 100 mm length beam, while for the 150 mm and 200 mm beam, the nondimensional natural frequency decreases from 1 to 0.28 and 0.33, respectively. In other words, longer beam corresponds to smaller tip mass effect, and the frequency reduction phenomena with the increment of the tip mass length is more sensitive for the second-order natural frequency than the first-and third-order frequencies. Figure 6 shows the variations of the first three mode shapes of the 150 mm length beam with different nondimensionless tip mass length ratios, and it is very clear that, for the first-order free vibration, big tip mass can only reduce the natural frequency, while the first-order mode shape almost does not change. For higher-order mode shapes, the tip mass has more obvious effect, and the accurate mode shapes as well as the strain nodes are very important for higher-order vibration-based piezoelectric energy harvesting applications [25].


## 4. Conclusions

Free vibration analysis of a cantilever beam with an asymmetrically attached tip mass is performed. Both the traditional method of separation of variables (MSV) and the method of Laplace transform (MLT) are employed in the present paper, and the equivalent concentrate force and moment of the tip mass in the MLT are clarified. Numerical results show the accuracy of the present MSV and MLT, and the effect of the tip mass to the natural frequencies and corresponding mode shapes is also numerically investigated. Results show that tip mass has an obvious effect on the natural frequencies and mode shapes of the cantilever, other than the first-order mode shape. The present approach can also be used to solve beam structures with other boundary conditions and many concentrated mass locates at any position of the beam. It is also noted that it is easy for the Laplace transform method to obtain the orthogonality relations defined with respect to the variable density beam in the forced vibration analysis.



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*Source: 8024879-2021-09-13.xml*