#### Abstract

The novel method of an analogous beam is studied, which the flexural rigidity and mass per unit length correspond was described as the reciprocal of the mass per unit and the reciprocal of the flexural rigidity of the beam. It is shown that both beams possess the same natural frequencies of flexural vibration. In order to approximate calculation of these frequencies, the continuously distributed mass of the original beam is substituted for a number of concentrated masses. The analogous beam then becomes a chain of rigid links connected by pins and equipped with springs restraining the relative rotation of adjacent links. The equations of motion for the analogous beam can be solved by a procedure which consists of assuming a value for the natural frequency and calculating the deflections successively from one end of the beam to the other. Under normal circumstances, there will be a certain error, and one boundary condition will not be satisfied. The procedure is repeated with different values of the frequency until the error is removed. The method is illustrated by an example of a Crossbeam for which the fundamental frequency is found.

#### 1. Introduction

The heavy-type numerical control milling planer mainly consists of some critical functional and structural components such as crossbeam, column, slip board, slippery pillow, milling head, and worktable. It is an economic machine tool which has the characteristics of large span and high efficiency in modern large-sized workpiece machining equipments. It can realize profile milling surface processing and obtain a high machining accuracy. The crossbeam, which is a significant support component, is divided into fixed girder and dynamic beam. The crossbeam we studied has characteristics of large span and heavy load. In addition, it connects with columns and other components, bearing complex loads in working conditions. Therefore, one urgent problem that arises is how to evaluate the static and dynamic performances of crossbeam that have a great influence on machine performance and machining quality.

B. P. Zhang and N. S. Zhang adopted a self-evolutionary compensation approach to reduce the deformation induced by the gravity of an 8.8 m long crossbeam [1]. Xu et al. conducted a simulation research about a 6.3 m long crossbeam and analyzed the influence of the junction plane parameter changes on static and dynamic performance of the crossbeam sliding box system [2]. Xie et al. discussed the influence of internal stiffened plate layout on the dynamic performance of crossbeam, took vibration modal relative displacement as the reference basis for design improvement, and put forward several suggestions on crossbeam improvement [3]. Zatarain et al. utilized finite element method for the machine model by the modal analysis. Finally, he selected the reasonable structure through the comparison of several schemes [4]. Guo et al. [5] analyzed a large span and heavy load crossbeam by simulation analysis and experiment research.

The analogy of Christian Otto Mohr (1835–1918) allowed the computation of displacements and sloped in a linear elastic Euler-Bernoulli beam as bending moments and shear forces in a beam loaded by auxiliary forces and with modified support conditions. Since displacements and slopes can be obtained from static considerations, the analogy had found widespread attention in the engineering community. Williams’ book [6] showed that the column analogy method provided the most useful means for the determination of fixed-end moments, stiffness, and carryover factors. Ellakany et al. [7] provided the analysis of composite beams which is carried out using a combination of the transfer matrix and the analog beam methods. An extension of Mohr’s analogy to bending of shear-deformable beams with eigenstrain-type actuation by Irschik and Naderb [8]. Using Mohr’s analogy, it was shown that the auxiliary loading of the adjoint beam must form a self-equilibrated system of loading in order to achieve the latter goal. Gamer [9] studied the application of the Mohr method to bending of beams with elastic joints. Irschik [10] presented a review on static and dynamic shapes control of structures by piezoelectric actuation. A simplified grillage beam analogy was performed to investigate the behaviour of railway turnout sleeper system with a low value of elastic modulus on different support moduli by Manalo et al. [11]. This study aimed at determining an optimum modulus of elasticity for an emerging technology in railway turnout application-fibre composites sleeper. Refined theories into Mohr’s analogy had been motivated by the work of Aldraihem and Khdeir [12], who pointed out that situations may occur for which the beam behavior in the region of the patches must be accurately described by higher-order beam theories. Sato et al. [13] presented the mathematical hypothesis that a beam on equidistant elastic support (BOES) can be considered as a beam on an elastic foundation (BOEF) in static and free vibration problems. A unifying numerical method was provided by Rubin in [14]. For a general discussion of the influence of shear on the deflection of beams including thermal loading, see the book by Mang and Hofstetter [15]. This class under consideration was identified by Irschik [16], who showed that various shear deformable beam theories and one-dimensional versions of plate theories can be put into a common mathematical form. Al-Sarraf and Ali [17] researched vibration analysis of plates using beam-column analogy the percentage of error depends on mesh size. El-Mously [18] established a Timoshenko beam on Pasternak-foundation model that was developed for the analysis of thin elastic cylindrical shells. Szyszkowski and Grewal [19] solved optimal control problems for linear dynamic systems with quadratic performance index using the beam analogy. Dong and Meng [20] presented the thermal analogy method to predict the dynamic behavior of complex structures with piezoelectric actuators. A practical efficiency factor of circular and spiral shear reinforcements for solid and hollow core circular shear truss analogy for concrete members to be used for design purposes is also presented by Turmo et al. [21]. Based on a force analogy method, Zhang et al. [22] proposed a damage index for special moment resisting steel frames considering both maximum deformations and cumulative effects.

In this paper, a method of an analogous beam is studied, which the flexural rigidity and mass per unit length correspond was described as the reciprocal of the mass per unit length and the reciprocal of the flexural rigidity of the beam. It is shown that both beams possess the same natural frequencies of flexural vibration. In order to approximate calculation of these frequencies, the continuously distributed mass of the original beam is substituted for a number of concentrated masses. The analogous beam then becomes a chain of rigid links connected by pins and equipped with springs restraining the relative rotation of adjacent links. The equations of motion for the analogous beam can be solved by a procedure which consists of assuming a value for the natural frequency and calculating the deflections successively from one end of the beam to the other. The method is illustrated by an example of a large span and heavy load crossbeam for the research object. In this example, a three-dimensional model of crossbeam was built by using the UG system, and then, according to the actual working load conditions of the crossbeam, process dynamic simulation for the whole machine was made by using the mechanical system multibody dynamic simulation software ADAMS. Secondly, the load curve, natural frequency, and modal shape are obtained, respectively. The presented novel analogous method is verified by comparing with the analogy analysis results, experimental data, and simulation results.

#### 2. Analogous Method

##### 2.1. Differential Equation of Beam

The differential equation describing the free, flexural vibration of an elastic beam is

The variable may be eliminated by introducing the new variable defined by

Then, dividing both sides of (3) by , differentiating twice with respect to , and changing the order of differentiating in the second term (it is assumed that and are continuous),

And, since , are not functions of , so

A comparison of (1) and (5) shows that may be interpreted as the deflection of a beam, which hereafter will be referred to as the analogous beam, having a rigidity and a mass per unit length given by

If satisfies (1), then will satisfy (5). One implication is that the two beams will have the same natural frequencies of vibration.

##### 2.2. Boundary Conditions

The boundary conditions imposed on the original beam can be transformed readily into conditions on the analogous beam. For example, if for all values of , at one end, then (3) indicates that there; also, if for all values of , , then, from (3),

That is, the shear force on the analogous beam is zero; finally, if for all values of , , then according to (2); that is, the deflection of the analogous beam is zero. Thus, a simple support at one end of the original beam corresponds to a simple support at the corresponding location in the analogous beam, a built-in support to a free end, and a free end to a built-in support. Also, a simple support in the original beam at a point other than an end results in a hinge at the corresponding location in the analogous beam.

An end condition of frequent occurrence is that of a rigid object attached to the free end of a cantilever. If this object has a mass and a mass moment of inertia (about an axis perpendicular to the plane of vibration of the beam and through the mass center of the object), then the shear force and bending moments at the end of the beam are, respectively,

This indicates that, in the analogous beam, a bending moment proportional to the slope and a shear force proportional to the deflection must be provided by the support; that is, the support consists of a torsional spring of modulus and a linear spring of modulus .

##### 2.3. Natural Frequency

The natural frequencies of vibration can be found by solving either (1) or (5). If, as an approximation, the mass of the original beam is assumed to be concentrated at few points along its length with mass less elastic portions connecting these points, then the analogous beam correspondingly will consist of rigid links connected by elastic joints which may be pictured as pinned connections with spiral springs resisting deformation of the chain from the straight configuration. The contrast between the original and the analogous beams in the character or their configurations, as shown in Figure 1, suggests that the computations necessary to obtain the natural frequencies may naturally follow different courses, depending on whether the original form of the analogous beam is considered. The original beam as depicted in Figure 1 serves as a basis for well-known method. The method to be developed here, on the other hand, results from the formulation of equations of motion for the analogous beam as depicted in Figure 1.

Figure 2 shows three consecutive rigid portions, or links, of the analogous beam. There are links and () joints; both are numbered successively from left to right starting with number 1. The displacement of the th joint is . For the sake of simplicity, it will be assumed that all links have the same length, , and that each one is a uniform bar. As the beam vibrates, the links execute plane motions.

The angle that the th link with the axis is , where the usual small deflection approximation is used. The displacement of the mass center of the th link is . Thus, the kinetic energy in the system is where and are, respectively, the mass and mass moment of inertia about the mass center of the th link. The strain energy is concentrated in the springs acting at the joints. The angular deflection of the spring at the th joint, that is, the angle of rotation of the th link relative to the ()st, is for small deflections, , where . It will be assumed that the angular deflections of the springs at the 1st and ()st joints are, respectively, and ; these expressions apply at least in the important cases of simple supports, free ends, and built-in ends. If represents the torque developed by the th spring per radian of deflection at the th joint, the total strain energy in the system is Interpreting the as generalized coordinates in a Lagrangian formulation, the () equations of motion are obtained by substituting for and according to (9) into the following equations: where = upward force on beam at left end, = upward force on beam at right end, , and .

It has been assumed that external forces act only at the ends of the beam. The formulation can, however, be easily generalized to include other cases.

The equations of motion which result after the indicated substitutions made in (10) are where

For a normal mode, the solution of (11) may be written as , , , where , are the amplitudes of ,, respectively, and is the circular frequency of the vibration. Inserting this solution in (11) and solving in the th () equation and in each of the last two,

The simultaneous solution of the homogeneous equations (13)–(16) for the ’s and ’s may be carried out in the following manner.(1)value of is assumed.(2)If the boundary conditions at the left end of the analogous beam are such that , then is set equal to unity. If, on the other hand, , then is set equal to unity. In either case, (13) gives in terms of . Only boundary conditions of these types will be considered, but the method can easily be adapted to others.(3)The expression for in terms of is substituted in the right side of (14). Thus, is expressed in terms of alone. If and are now eliminated from the first of (15) by using the expressions for them in terms of , an expression for in terms of will result. are each found in terms of alone. In every case, a linear function of will result.(4)The expression forms , , , in terms of are now inserted in (13) and (16), which can then be regarded as simultaneous linear equations for and . Their solution will, in general, be incompatible with the boundary conditions at the right end of the analogous beam; for instance, the condition .(5)The procedure is repeated starting with step 1 and using a new value of until the boundary conditions at the right end are satisfied.

##### 2.4. Calculation of Analogous Beam

Before the calculations described earlier can be carried out, it is necessary to know the numerical values of all the , , and . Consider the original beam with its mass continuously distributed. Since the rigidity of the analogous beam is , the curvature that it will acquire due to the bending moment will be

Thus, where is the mass of the original beam in the length . If the mass is now assumed to be concentrated at a number of discrete points and if the interval between and contains one of these points, then the right side of (18) becomes , being the concentrated mass. But is the stiffness of the spring in the analogous beam corresponding to the concentrated mass in the original beam. Thus,

The mass of the th link of the analogous beam is

And its mass moment of inertia is

#### 3. Example

Three-dimensional models of crossbeam and the whole machine have been built by the UG system, as shown in Figure 3. Meanwhile, in order to avoid lots of finite element units created by little features and small structures which would increase the computer calculation time, this paper makes some necessary amendments and predigestions to the crossbeam model.

Then, the model is imported into ANSYS in IGES format, element with eight nodes and six faces with appropriate mesh size, and the parameters as following: the material is gray cast iron, density kg/m^{3}, and the young’s module GPa.

According to the position and function in the machine tool, the crossbeam is simplified to both ends fixed supported form, as shown in Figure 4.

The crossbeam is subjected to complex space load in actual operation. Crossbeam gravity is uniformly distributed load and causes static deformation. Gravity of slippery pillow, main milling head, and slip board is concentrated load and causes bending deformation when the three components move along the crossbeam guide. Cutting force is a fluctuating and external load. The crossbeam contact surface with the force is as shown in Figure 5, is the forces existing on the contact surface between slippery pillow and the upper guiding face, and , are the force existing between slippery pillow and the bottom guiding face.

In order to truly reflect the actual static characteristics of crossbeam, process dynamic simulation for the whole machine has been made by using the mechanical system multibody dynamic simulation software ADAMS, considering the influence of cutting force. The contact surface between slippery pillow and crossbeam can be simplified to spring damping system, realized with the method of the constraint pair combination. The contact surface between crossbeam and columns can be simplified to the fixed constraint pair. Finally, the load condition can be obtained.

##### 3.1. Analogous Method

The fundamental frequency of a uniform Euler beam has been found by the method described in this paper. The mass was assumed to be concentrated at five points equally spaced along the beam as depicted in Figure 6, where the analogous beam is also shown.

From (20) and (21), and , , and , the following expressions for the analogous beam parameters result: from which the constants , are evaluated by using (12). These constants, and , can be put in dimensionless forms by multiplying them, respectively, by and . If in (13)–(16), and are written in terms of their dimensionless counterparts, and if the numerator and denominator of each coefficient of are multiplied by , one is led to consider the dimensionless quantity , rather than .

The boundary conditions for the analogous beam can be expressed as , . By assumed different values of and solving (13)–(16) successively as described before, corresponding values of the dimensionless force at the right end of the analogous beam, , were obtained as in Table 1. The interpolated value of for which is 0.0168, which corresponds to a frequency . The exact value for the numerical coefficient in the expression for is 1.875.

##### 3.2. FEA Method

Figure 7 presents the finite element crossbeam model. This study defines the constraint condition as fixed constraint on both ends, and we can obtain the calculated loads N, and N, N through the force model and simulation method previously presented.

Then, the calculated results are loaded on finite element crossbeam model, and the intermediate position where crossbeam is in the most dangerous state for solving is selected. In the deformation cloud diagram of stress, as shown in Figure 8, static comprehensive maximum deformation is 0.2626 mm, and the maximum deformations of direction and direction are −0.0681 mm and −0.2484 mm, respectively. However, direction deformation created by crossbeam and components gravity is much larger. It shows that antibending ability of crossbeam structure is slightly feeble.

In order to reduce or eliminate the deformation when crossbeam is at work and achieve the purpose of holding crossbeam guide face in horizon, it is necessary to render and design the load curve. The static deformation of a 14.350 m long crossbeam has been obtained by using the finite element analysis software ANSYS.

The forecast load curves as shown in Figure 9 can be constructed with data-fitting method, and the fitted curve equations have been obtained through numerical computation in the meantime.

**(a)**

**(b)**

**(c)**

Modal analysis and model experiment research Static stiffness of crossbeam only reflect the capability of crossbeam resisting deform caused by cutting force and crossbeam components gravity which are regarded as static force, but in fact, since crossbeam is an elastic body, it is exciting force created under the conditions such as the cutting force variable that usually produces vibration.

Through modal analysis, designers can distinguish what kinds of vibration appears according to deformations under the modes of different ranks motivated by exciting force and avoid sympathetic vibration by taking control of the exciting force frequencies away from the resonance zone and reducing the amplitude of vibration. Meanwhile, the highest spindle speed is 8000 rpm, and only the preliminary limited rank modal frequencies and the exciting force frequencies may overlap; thus, this study takes the preliminary limited rank modal frequencies and modes for research objects. The foremost modals obtained by ANSYS software are as shown in Figure 9. The first-, second-, and third-order frequencies are 39.556 Hz, 75.807 Hz, and 95.312 Hz.

##### 3.3. Experiment Method

Experimental objective is to obtain the frequency response curves of multimeasurement points, which reflect the dynamic characteristics of the combined three-section crossbeam, through dynamic test on the combined three-section beam. Test equipment selection is as follows. Since the mass of the beam is great, so it is necessary to use a huge excitation hammer if the natural frequencies are excited by hammer excitation method; however, it will create great impact on the excitation point and engender local destroy on the structure. Therefore, exciter method is adopted, and the frequency sweep method is used to excite the natural frequency of the structure. Other instruments include multichannel data acquisition, front-end, power amplifier, force sensor, and some acceleration sensors. Arrangement of measuring points is as follows. Measuring site should follow two principles: (1) the pick-up point of connection can sketch the outline of the combined three-section crossbeam; (2) layout more measuring points on the parts of some concerns. Since the combined three-section crossbeam has the characteristic of symmetry, we distributed all measuring points evenly to avoid node locations effectively. Experimental area and arrangement of measuring points are shown in Figures 10 and 11.

It can be seen that there are four frequency places appearing among all frequency response functions which have been blazoned with graph, as shown in Figure 12. Finally, the model experiment results in which natural frequency within 100 Hz can be obtained. Table 2 shows the comparison of analogy data, simulation data, and experimental data.

From Table 2, it can be seen that analogy data, simulation data, and experimental data are inosculate. Therefore, analogy data is reliable and can be used as a reference in the structure optimization process.

#### 4. Concluding Remarks

(1)Based on analogous beam method, this paper proposes a mathematical model of beam. In this method, a continuous beam is decomposed into unit mass. Then, the flexural rigidity and mass per unit length of an originally given beam are corresponded to the reciprocal of the mass per unit length and the reciprocal of the flexural rigidity of an analogous beam. Through the deduction of the mathematical formula, we obtained that both beams possess the same natural frequencies of flexural vibration. (2)The analogous method is illustrated by an example of a large span and heavy load crossbeam for the research object. In the example, three-dimensional model of crossbeam has been built by UG system. According to the actual working load conditions of the crossbeam, using this method, FEA simulation method obtained natural frequency, respectively. The presented novel analogous method is verified by comparing with the analogy data and simulation data.(3)Under the real conditions of the crossbeam, we conducted related experiments and obtained experimental data. It is shown that analogy data and experimental data are inosculate, which further confirmed the presented analogous method. Therefore, the analogous method is reliable and can be used as a reference in the structure optimization process.

#### Nomenclature

: | Coefficients in equations of motion for analogous beam |

: | Cross-sectional area of original beam |

: | Cross-sectional area of analogous beam |

: | Modulus of elasticity of original beam |

: | Modulus of elasticity of analogous beam |

: | Force at joint of analogous beam |

: | Amplitude of |

: | Length of each section into which beam is divided |

: | Mass moment of inertia of portion of analogous beam between two joints |

: | Moment of inertia of cross-section of original beam about the neutral axis |

: | Moment of inertia of cross-section of analogous beam about the neutral axis |

: | Mass moment of inertia of rigid body attached to end of original beam |

: | Torque developed at joint of analogous beam per unit rotation of adjacent portions of beam |

: | Length of beam |

: | Concentrated mass in original beam, or mass of rigid body attached to end of original beam |

: | Bending moment acting on analogous beam |

: | Number of links in analogous beam |

: | Frequency of normal vibration |

: | Deflection of analogous beam |

: | Amplitude of |

: | Time |

: | Kinetic energy of analogous beam |

: | Strain energy of analogous beam |

: | Coordinate along beam axis |

: | Deflection of original beam |

: | Angle of slope of analogous beam |

: | |

: | Mass of a link of analogous beam |

: | Density of original beam |

: | Density of analogous beam |

: | Indices |

: | . |

#### Acknowledgments

This work was supported by the National Science and Technology Major Project of China (Grant no. 2012ZX04010-011), and the National Natural Science Foundation of China (Grant no. 51075006). The authors are grateful to other participants of the projects for their cooperation.