# Frequency Equation of Flexural Vibrating Cantilever Beam Considering the Rotary Inertial Moment of an Attached Mass.

1. Introduction

The cantilever beam is a simple structure, and it is an important simplified model for many engineering problem in the fields of mechanical engineering, civil engineering, and so forth. However, there is a vast number of papers concerned with the determination of the eigen frequencies of the cantilever beam subject to various boundary conditions, which can be found in the classic book . And more and more issues of cantilever beam with complicated boundary conditions  or external load conditions [3, 4] have been studied by theoretical deduction  or numerical method , or it is application to determine Young's modulus .

The cantilever beam model is also used in geotechnical earthquake engineering as a simplified model  for the ground responses due to earthquake. It is also a basic principle of measuring the soil's dynamic shear modulus which is an indispensable parameter for analyzing the earthquake response of site caused by far-field ground. The apparatus used to obtain the dynamic shear modulus is well known as resonant column apparatus. The soil column, installed in this apparatus, is driven by electromagnetic force at free end  producing its torsional vibration. If flexural vibration of the soil column occurs, the dynamic Young modulus can be obtained, which is also an important parameter for dynamic analysis of site suffered by near-field ground motion. Essentially, the frequency equation of flexural vibrating cantilever beam with an additional mass is needed. However, the vibrating frequency and shape mode of soil column are effected by not only the shear force but also the moment force, generated by the motion of the additional mass attached at the free end of the soil column. So the influence of these two forces on the vibrating of soil column must be discussed. Cascante et al.  derived a frequency equation of this vibrational problem using Rayleigh's method assuming that the mode shape is a third-order polynomial. Laura  derived the frequency equation of a cantilever beam attaching an additional mass, which is considered as shear force acted on the free end of beam but did not consider the moment force generated by the mass. And recently, Giirgoze [12-14] studies the eigen frequencies of a cantilever beam carrying a tip mass or spring-mass.

The paper tried to derive the frequency equation of the cantilever beam in order to obtain more accurate solution for the soil column and to analyze the frequency characteristics of this system effected by the bending moment generated by the additional mass.

2. Flexural Motion Equation of Cantilever Beam with an Additional Mass

The flexural cantilever beam to be considered is the straight, uniform beam with an additional rigid mass attached by fixed connection. The significant physical properties of this beam are assumed to be the flexural stiffness El and the mass per unit length [bar.m], both of which are constant along the span L. The transverse displacement response y(x,t) is a function of position and time. The free vibration equation of motion for the system as shown in Figure 1 is easily formulated by directly expressing the equilibrium of all forces acting on the differential segment of beam. And it becomes 

EI [[partial derivative].sup.4]y (x,t)/[partial derivative][x.sup.4] + [bar.m] [[partial derivative].sup.2]y (x,t)/[partial derivative][x.sup.2] = 0. (1)

Since El and [bar.m] are constant, one form of solution of this

equation can be obtained by separation of variables using

y (x,t) = [phi] (x) Y (t) (2)

which indicates that the free vibration motion is of a specific shape [phi] (x) having a time-dependent amplitude Y(t). Substituting this equation into (1) can yield two ordinary differential equations.

[mathematical expression not reproducible], (3)

in which [[beta].sup.4] = [[omega].sup.2][bar.m]/EI, and [omega] is the circular frequency. These equations have the solution separately as follows:

Y (t) = [B.sub.1] sin ([omega]t) + [B.sub.2] cos ([omega]t), (4)

[phi] (x) = [A.sub.1] cos [beta]x + [A.sub.2] sin [beta]x + [A.sub.3]ch [beta]x + [A.sub.4]sh[beta]x (5)

in which constants [B.sub.1] and [B.sub.2] depend upon the initial displacement and velocity conditions; and real constants A; must be evaluated so as to satisfy the known boundary conditions at the ends of the beam.

The cantilever beam considered has a fixed end, so its two known boundary conditions are

[phi] (0) = 0' [phi]' (0) = 0. (6)

Making use of (5) and its first partial derivative with respect to x, from (6) one obtains [A.sub.3] = -[A.sub.1] and [A.sub.4] = -[A.sub.2], So (5) becomes

[phi](x) = [A.sub.1] (cos [beta]x - ch [beta]x) + [A.sub.2] (sin [beta]x - sh[beta]x). (7)

An additional rigid mass [m.sub.1] having a rotary mass moment of inertia [J.sub.1] is attached by fixed connection to its free end as also shown in Figure 2. These internal force components are along with the translational and rotary inertial force components [m.sub.1][w.sup.2] [phi](L) and [J.sub.1] [w.sup.2] [phi]'(L), respectively. So the force and moment equilibrium of the additional mass requires the boundary conditions

M (L) = EI [phi]" (L) = - [w.sup.2][phi]'(L)[J.sub.1], J4, V (L) = EI [phi]'" (L) = - [w.sup.2][phi]'(L)[J.sub.1], [m.sub.1]. (8)

Making use of (5) and its first, second, and third partial derivative with respect to x and substituting them into (8) yield

[mathematical expression not reproducible], (9)

For coefficients [A.sub.1] and [A.sub.2] to be nonzero, the determinant of the square matrix in this equation must equal zero, thus giving the frequency equation

[mathematical expression not reproducible], (10)

which can reduce to the following form making use of [w.sup.2]/El = [[beta].sup.4]/[bar.m]

[mathematical expression not reproducible]. (11)

Setting

z = [beta]L = 4[square root of [w.sup.2] [bar.m]/EI] (12)

is a power function of the frequency of the flexural vibrating cantilever beam. And setting [[alpha].sub.j] = [J.sub.1]/[bar.m][L.sup.3] = [J.sub.1]/[m.sub.T][L.sup.2] is a ratio of the rotary mass moment of inertia for the additional mass and for a rigid mass of [m.sub.T] having a rotary arm length of L; and [[alpha].sub.m] = [m.sub.1] /[bar.m]L = [m.sup.1]/[m.sub.T] is a ratio of the mass between the additional mass and the cantilever beam. So the frequency equation becomes

[mathematical expression not reproducible]. (13)

This frequency equation is a transcendental equation that contained two parameters with unambiguous physical meaning. The solution of this equation, which is the frequency of the system of the flexural vibrating cantilever beam, can be only obtained by numerical method for now.

3. Solution of the Frequency Equation

In order to solve the frequency equation, the term of the right side of (13) can be moved to the left side, and then suppose the left side equals f(z, [[alpha].sub.j] [[alpha].sub.m]). And f =0 is equivalent to (13). Suppose the parameters [[alpha].sub.j] and [[alpha].sub.m] are constant, and then the relationship between f(z) and z can be calculated and drawn as shown in Figure 3, in which the parameters [[alpha].sub.j] = 10 and [[alpha].sub.m] = 1. However, the value of f(z) rises abruptly. So the y-axis alters to log f(z) in order to show the curve is up and down across the x-axis obviously. The intersection points of the curve and x-axis as marked in Figure 3 with circle points are the roots of the equation f(z) = 0, where its approximate roots can be obtained using numerical methods for solving system of nonlinear equations, such as method of bisection and Newton's method.

For a specific system, where the [[alpha].sub.j] and [[alpha].sub.m] are determined, the solution of the frequency equation (13) can be obtained. And the roots of the parameter z are varied with the parameters [[alpha].sub.j] and [[alpha].sub.m]. Figures 4 and 5 show their effects. With the increment of the parameter [[alpha].sub.j], the root z is increased identically with a sigmoid function as shown in Figure 4. It means that the frequency of the system will increase if the rotary mass moment of inertia for the additional mass is increased. And the rate of the increment is larger around [[alpha].sub.j] = [[alpha].sub.m] than others. For example, the increment of z at the range of [[alpha].sub.j] = 0.1~10 is fastest while [[alpha].sub.m] = 1. With the increment of the parameter [[alpha].sub.m], the root z is decreasing, as shown in Figure 5, which means that the frequency of the system will decrease with the additional mass increasing.

4. Natural Frequencies and Shape Modes Characteristics

The natural frequency [w.sub.i] of the flexural vibration of cantilever beam considering the bending moment generated by the attached mass can be calculated using the transformation of (12). Table 1 lists the results of the first five natural frequencies of this system with the varied parameters values [[alpha].sub.j] and [[alpha].sub.j]. The natural frequency increases with the parameter [[alpha].sub.j] decreasing, which means the quality of the additional mass reduction compared to that of the cantilever beam. This is consistent with the result of others [7,16]. And the frequency also increases with the increment of the rotary mass moment of inertia for the additional mass compared to the beam, which is defined as the parameter [[alpha].sub.j]. In order to make comparisons, the results of frequencies when the parameters [[alpha].sub.j] = 0 and [[alpha].sub.j] =1 are also listed, which means the free end of beam has no moment force generated by the additional mass. As expected, these natural frequency values are the same as that obtained from , in which only the shear force is considered generated by the attached mass. And the larger value of [[alpha].sub.j] causes the higher frequency of the cantilever beam as shown in Table 1.

However, the interval between the two frequencies for [[alpha].sub.j] equal to zero and for [[alpha].sub.j] unequal to zero is considerable, especially that between the two higher frequency numbers. So the ignored rotary mass moment of inertia may be too ideal for engineering problem and cause obviously error.

The vibration shape mode can also be obtained by substituting the ratio of [A.sub.1] and [A.sub.2] into the shape equation (7). And Figure 6(a) shows the first four mode shapes of the cantilever beam considering the moment force with the parameters [[alpha].sub.m] = 1 and [[alpha].sub.j] = 10. The first four mode shapes of the cantilever beam are shown without considering the moment force in . It is clearly shown that the vibration shape is effected by the moment force from the additional mass significantly. However, the existence of the additional mass restrains the development of displacement at the free end of beam.

In order to analyze the influence of af on the mode shape, the 3rd to 5th mode shapes of considering the moment force with [[alpha].sub.m] = 1 and [[alpha].sub.j] = 0.001 and without considering situation are given, as shown in Figure 6(b). Even if the parameter [[alpha].sub.j] is of very small value, the 5th mode shape of these two situations has obviously divergence, though the smaller the serial number of mode shape is, the less obvious the divergence becomes.

5. Conclusion

The frequency equation of cantilever beam with an additional mass exciting flexural vibration was derived considering the rotary inertial moment of inertia of an attached mass, including the shear force. It is a transcendental equation, and it contains two parameters with unambiguous physical meaning, which can be defined as the ratio of rotary mass moment of inertia and the ratio of the mass, respectively.

These two parameters effect both the natural frequency and the shape mode of the beam. As the ratio of rotary mass moment of inertia increases, the natural frequency climbs. Even a little increment of the ratio may cause higher variance between considering and not considering the rotary mass moment of inertia, especially for the high natural frequency. And the ratio of the rotary mass moment of inertia also effects the mode shape of this system. The higher the serial number of mode shape investigated is, the more obvious the divergence becomes.

http://dx.doi.org/10.1155/2017/1568019

Competing Interests

There are no competing interests regarding this paper.

Acknowledgments

This paper is supported by National Science Foundation of China (NSFC 51309121) and Natural Science Foundation of Jiangsu Province (BK20130463).

References

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Binghui Wang, (1) Zhihua Wang, (2) and Xi Zuo (3)

(1) School of Architecture and Civil Engineering, Jiangsu University of Science and Technology, 2 Mengxi Road, Zhenjiang, Jiangsu, China

(2) Institute of Geotechnical Engineering, Nanjing Tech University, 200 North Zhongshan Road, Nanjing, China

(3) Institute of Architectural Engineering, Jinling Institute of Technology, Nanjing, China

Correspondence should be addressed to Zhihua Wang; wzhnjut@163.com

Received 13 December 2016; Accepted 5 February 2017; Published 22 February 2017

Caption: Figure 1: Diagram of the cantilever beam with an additional mass.

Caption: Figure 2: Diagram of force equilibrium analysis at the free end.

Caption: Figure 3: Relationship between f(z) and z.

Caption: Figure 4: Varied z with [[alpha].sub.j].

Caption: Figure 5: Varied z with [[alpha].sub.m].

Caption: Figure 6: The first five mode shapes of considering the moment force with [[alpha].sub.m] = 1 and [[alpha].sub.j] = 10 and without considering situation (a) and the 3rd to 5th mode shapes of considering the moment force with [[alpha].sub.m] = 1 and [[alpha].sub.j] = 0.001 and without considering situation (b).
```Table 1: The first five natural frequencies of the cantilever
beam considering the bending moment.

Nondimensional natural frequency [w'.sub.i] =
[square root of EI/[bar.m][L.sup.4][z.sub.i]

Frequency     [[alpha].sub.m] = 100    [[alpha].sub.m] = 10
number (i)     [[alpha].sub.j] = 1     [[alpha].sub.j] = 1

1                   0.1749603               0.6054297
2                    22.30366                22.47413
3                    61.66046                61.83822
4                   120.90684                121.0850
5                   199.86943                200.0480

Nondimensional natural frequency [w'.sub.i] =
[square root of EI/[bar.m][L.sup.4][z.sub.i]

Frequency      [[alpha].sub.m] = 1     [[alpha].sub.m] = 1
number (i)    [[alpha].sub.j] = 100    [[alpha].sub.j] = 10

1                    2.950242                2.91158
2                    23.93841                23.93129
3                    63.43232                63.42955
4                    122.7265                122.7251
5                    201.7208               201.71991

Nondimensional natural frequency [w'.sub.i] =
[square root of EI/[bar.m][L.sup.4][z.sub.i]

Frequency     [[alpha].sub.m] = 1     [[alpha].sub.m] = 1
number (i)    [[alpha].sub.j] = 1    [[alpha].sub.j] = 0.1

1                   2.55044                 1.70226
2                   23.86031                23.17405
3                   63.40190                63.1286
4                  122.71062                122.5668
5                  201.71107                201.6229

Nondimensional natural frequency [w'.sub.i] =
[square root of EI/[bar.m][L.sup.4][z.sub.i]

Frequency      [[alpha].sub.m] = 1       [[alpha].sub.m] = 1
number (i)    [[alpha].sub.j] = 0.01   [[alpha].sub.j] = 0.001

1                    1.571098                  1.55867
2                    19.13831                 16.57872
3                    60.72695                 53.50208
4                    121.2014                 113.2826
5                    200.7639                 194.2422

Nondimensional natural
frequency [w'.sub.i] =
[square root of
EI/[bar.m][L.sup.4]
[z.sub.i]                  [[omega]'.sub.i] *

Frequency     [[alpha].sub.m] = 1        [[alpha].sub.m] = 1
number (i)    [[alpha].sub.j] = 0             (ref )

1                   1.557298                   1.557298
2                   16.25009                   16.25009
3                   50.89584                   50.89584
4                   105.1983                   105.1983
5                   179.2320                   179.2320

* Notes. The values of this column were calculated by authors using the
frequency equation from . And the results were carefully checked
by comparing the values of [z.sub.i] (using the character [y.sub.i]
in this reference).
```