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.
The heavytype 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 largesized 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 selfevolutionary compensation approach to reduce the deformation induced by the gravity of an 8.8 m long crossbeam [
The analogy of Christian Otto Mohr (1835–1918) allowed the computation of displacements and sloped in a linear elastic EulerBernoulli 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 [
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 threedimensional 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.
The differential equation describing the free, flexural vibration of an elastic beam is
The variable
From (
Then, dividing both sides of (
And, since
A comparison of (
If
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
That is, the shear force on the analogous beam is zero; finally, if for all values of
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
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
The natural frequencies of vibration can be found by solving either (
Original and analogous beams.
Figure
Three consecutive links of the analogous beam.
The angle that the
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 (
For a normal mode, the solution of (
The simultaneous solution of the homogeneous equations (
If the boundary conditions at the left end of the analogous beam are such that
The expression for
The expression forms
The procedure is repeated starting with step 1 and using a new value of
Before the calculations described earlier can be carried out, it is necessary to know the numerical values of all the
Thus,
The mass of the
And its mass moment of inertia is
Threedimensional models of crossbeam and the whole machine have been built by the UG system, as shown in Figure
Crossbeam and Heavyduty CNC machine tools.
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
According to the position and function in the machine tool, the crossbeam is simplified to both ends fixed supported form, as shown in Figure
Crossbeam force model.
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
Crossbeam contact surface with the force.
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.
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
Original and analogous beams of original example.
From (
The boundary conditions for the analogous beam can be expressed as
Values of the natural frequencies for the beam.




0.0200  −0.8655  44.263 
0.0170  −0.1506  76.354 
0.0143  −0.0346  97.953 
Figure
Finite element model of crossbeam.
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
The deformation cloud diagram of stress.
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
The first, second, and third mode shapes of beam.
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
Experimental objective is to obtain the frequency response curves of multimeasurement points, which reflect the dynamic characteristics of the combined threesection crossbeam, through dynamic test on the combined threesection 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, frontend, power amplifier, force sensor, and some acceleration sensors. Arrangement of measuring points is as follows. Measuring site should follow two principles: (1) the pickup point of connection can sketch the outline of the combined threesection crossbeam; (2) layout more measuring points on the parts of some concerns. Since the combined threesection 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
Exciting position and suspension method of exciter for threesection crossbeam.
The diagram of sensors arrangement.
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
The comparison of analogy data, simulation data, and experimental data.
Order  Analogy data 
Simulation data 
Experimental data 

1  44.263  39.556  41.796 
2  76.354  75.807  76.021 
3  97.953  95.312  96.675 
Frequency response functions of all test points.
From Table
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.
The analogous method is illustrated by an example of a large span and heavy load crossbeam for the research object. In the example, threedimensional 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.
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.
Coefficients in equations of motion for analogous beam
Crosssectional area of original beam
Crosssectional 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 crosssection of original beam about the neutral axis
Moment of inertia of crosssection 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
This work was supported by the National Science and Technology Major Project of China (Grant no. 2012ZX04010011), and the National Natural Science Foundation of China (Grant no. 51075006). The authors are grateful to other participants of the projects for their cooperation.