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采用绝对节点坐标法对同步带传动系统进行弹性动力学分析外文文献翻译、中英文翻译、外文翻译.doc
采用绝对节点坐标法对同步带传动系统进行弹性动力学分析外文文献翻译、中英文翻译、外文翻译
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采用绝对节点坐标法对同步带传动系统进行弹性动力学分析外文文献翻译、中英文翻译、外文翻译,采用,绝对,节点,坐标,同步带,传动系统,进行,弹性,动力学,分析,外文,文献,翻译,中英文
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英文原文Nonlinear DynDOI:10.1007/s11071-015-2076-3ORIGINAL PAPERElastic dynamic analysis of synchronous belt drive system using absolute nodal coordinate formulationAbstractWhen synchronous belts are engaging with sprockets, they have complicated dynamic behaviors because of their complex composition. The inertial and damping forces of synchronous belt cannot be neglected when the belt is in a state of high acceleration or high-speed transmission, and these factors considerably influence the selection of belt type and the overall design of the system. The pretensioning force also significantly influences the stress state and fatigue life of synchronous belts, so a pretensioning analysis should be carried out before the elastic dynamic analysis. Our goalinthisworkistostudythepreciseinfluenceofiner- tial and damping forces on the dynamic performance of synchronous belts under high acceleration or high speed. Toward this end, the pretensioning and elastic dynamic analysis of a synchronous belt drive system using two-dimensional beam elements without shear deformation based on the absolute nodal coordinate are studied in this paper. To verify the results of the elastic dynamic analysis, a rigidflexible coupling dynamic simulation of synchronous belt drive system with the dynamic software RecurDyn is also carried out. The results are presented in the form of stress curves for the synchronous belt.Keywords： Synchronous belt drive system ； Beam element ； Elastic dynamic analysis ； Absolute nodal coordinate ； Rigidflexible coupling dynamic simulation1Introduction同步带链轮Fig. 1 Typical synchronous belt driveFigure 1 shows a typical synchronous belt drive for power-transmission equipment involving rotating mechanical elements. A synchronous belt commonly consists of tensile cords, a rubber compound that forms the teeth bulk and the backing of the belt, and facing fabric covering the belt face and teeth face. The tensile cords bear the external loads, and the facing fabric reinforces the wear resistance of the belt and teeth. Synchronous belt drives have higher transmission precision and carrying ability than flat rubber belt drives and produce less noise than chain drives. Synchronous belt drives are broadly used in automotive applications, production lines, industrial machines, and other industrial applications. The form of the teeth determines largely the carrying capacity of synchronous belts. Typically, the carrying capacity of arc teeth is much greater than that of trapezoidal teeth. With the development of the tensile-cord materials, the speed of synchronous belts has increased, with the instantaneous tangential velocity of the belt now reaching up to 50m/s. When syn chronous belts operate at high speeds, high acceleration, or under heavy loads, inertial and damping forces must be taken into account in designing belts and to select the proper type of belt for a given application. The sliding friction between a synchronous belt and the sprockets affects the fatigue life of the belt. Current research on synchronous belt drives mainly focuses on static analyses, experiments, and computer-aided engineering via finite-element analysis.A method is presented to calculate the loads on belt and sprocket teeth given the belt strand forces as a function of time by Nestor and Peter in Ref. 1, and the belt-tension members are modeled by a number of identical extension springs. The load and friction forces on the teeth are measured by using a pair of semiconductor strain gauges installed on a sprocket tooth while the sprocket rotates at the low speed of 16 rpm. The experiment of Dalgarno et al. 2 shows that the extensional stiffness of a synchronous belt gradually decreases with running time before becoming constant. The stiffness loss of synchronous belts is considered to arise from the development of cracks in the cords that reinforce the belt. Ergin et al. 3 have presented a technique based on structured neural networks to estimate the position for synchronous belt drives used in precision machinery. The position of a carriage is calculated via a structured neural network topology that accepts input from a position sensor on the actuator side of the synchronous belt. Factors affecting transmission errors in helical synchronous belts are studied by Kagotani and Ueda 4, who have determined that, under bidirectional operation, the error occurs on the belt-side face. A study on noise in synchronous belt drives is presented in Ref. 5 by Weiming and Tomio, who have experimentally and theoretically analyzed impact sound. Multibody modeling and simulation of synchronous belt drives can be done with commercial software packages, as demonstrated by Callegari et al. 6. Zhanguo et al. 7 have analyzed the dynamics of an automotive trapezoidal synchronous belt drive system by using the software RecurDyn.The elastic dynamic analysis of flat rubber belt drives based on absolute nodal coordinate formulation hasdevelopedquickly,primarilyduetothecontribution of Berzeri and Shabana 8, who have proposed a two-dimensional beam element without shear deformation and based on absolute coordinate. A flat rubber belt drive is modeled using the two-dimensional beam elements without shear deformation by Cepon 9,10. In addition, a two-dimensional shear-deformable beam element based on absolute coordinate proposed by Omar and Shabana 11 is modified by Kerkkanen et al. 12,13 to obtain a belt-like element for a flat rubber belt drive formulation, and the modification of the shear-deformable beam element reduces the bending stiffness. The thickness of synchronous belts is small, similar to that of flat rubber belts, and belt deformations are mainly distributed through the thickness and pitch. The bending stiffness of synchronous belts is also much lower than the axial stiffness, as is the case for flat rubber belts. Thus, the mechanical model of synchronous belt drive system can be simplified to a two-dimensional problem, as is the case for flat rubber belt drive system. The technique of using two-dimensional beam elements without shear deformation, as proposed by Berzeri and Shabana 8, can be used to model synchronous belt drive system, and the teeth of synchronous belt can be simplified by considering them as cantilever beam elements.However, the curvature of the beam element proposed by Berzeri and Shabana 8 is simplified for the case of small axial deformation, and this simplification is not accurate for large axial deformation. To solve this problem, we use the curvature expression, which couples bending and axial strains, proposed by Gerstmayr and Irschik 14. This curvature is accurate for the case of large axial deformation.The formulation of synchronous belt drive system can be carried out by using the beam elements proposed by Berzeri and Shabana 8 with the correct curvature representation proposed by Gerstmayr and Irschik 14, and therefore the accurate bending stiffness of the synchronous belt can be guaranteed. And the lower bending stiffness of the synchronous belt can be obtained as detailed in the literature 12.s=lsTight spanYs=0 s=2ls+2Rp drivendriver s=ls+RpOXSlack spans=2ls+Rpls2 Modeling of the arc-teeth synchronous belt drive systemFig. 3 Simplified schematic diagram of synchronous belt driveAs shown in Fig. 3, the driver sprocket of a synchronous belt drive system is assumed to rotate in the counterclockwise direction. The origin of the coordinate system is midway between the rotation centers of two sprockets. The X axis of the coordinates is horizontal, andtheYaxisofthecoordinatesisvertical.Theposition s = 0 is the engaging-out position of the synchronous belt on the driven sprocket, s = ls is the engaging-in position of the synchronous belt on the driver sprocket, s = ls + Rp is the engaging-out position of the synchronous belt on the driver sprocket, and s = 2ls + Rp is the engaging-in position of the synchronous belt on the driven sprocket, where ls is the central distance of the two sprockets and Rp is the radius of the pitch circle of the synchronous belt2.1 Element dividing of synchronous beltFor the sake of convenience for modeling the arc- teeth synchronous belt drive formulation, the tensile cords of the belt are simplified as a layer of tensile- cord material according to the equivalent-volume rule. Based on this assumption, the rubber part is separated into two components: the bulk of teeth and the backing of the belt. The layer of tensile material is defined as the tensile-cord layer, and the backing layer of the belt is defined as the backing-rubber layer. The facing fabric is thin and lightweight, so the facing fabric is considered as the medium layer that transmits forces. The tensile-cord and backing-rubber layers transmit interaction forces to each other through the contact surface. Because the belt teeth and tensile-cord layer are the main components of synchronous belt that transport loads, the arc teeth are best fixed on the tensile-cord layer. A cantilever beam is a beam that is fixed on an object of high strength; the tensile-cord layer usually has high strength, so the arc teeth of synchronous belt can be considered as cantilever beams. One arc tooth can be simplified as one beam element. The rubber layer separated by the belt teeth is also considered to transmit forces, and the inertial force of this layer is considered. To ensure computational accuracy, one tensile-cord layer element corresponds to one backing-rubber layer element and one tooth element.In an arc-teeth synchronous belt drive formulation, the backing-rubber layer, the tensile-cord layer, and the belt teeth (which engage with the sprockets) are the main objects to study. The loads from the driven sprocket and the driving force from the driver sprocket are transmitted to the tensile-cord layer and the backing-rubber layer through the belt teeth that engage with the sprockets. The inertial forces of synchronous belt drive system components, such as belt layers, belt teeth, and the driven sprocket, can be transformed to apply to the tensile-cord layer. The damping forces of the backing-rubber layer can also be transformed to apply to the tensile-cord layer. If the belt teeth and tensile-cord layer are strong enough to transmit the driving force from the driver sprocket to the driven sprocket, the influence of the belt teeth and the belt layers on the transmission accuracy of synchronous belt drive system gives the response delay of the driven sprocket with respect to the driver sprocket. And if the belt teeth and tensile-cord layer are not strong enough, the teeth of the synchronous belt drive system may slip or the tensile cords may break. Compared with a flat rubber belt drive, a synchronous belt drive has a higher transmission accuracy because flat belts continually slip on the pulleys, and this phenomenon cannot be eliminated.2.2 Statics analysis of synchronous belt drive systemThe pretensioning force is very important in determining the carrying ability and the fatigue life of a synchronous belt. If the pretensioning force is less than required, a synchronous belt cannot carry the desired loads and tooth slippage will easily occur. If the pretensioning force is greater than required, the synchronous belt may wear out rapidly or prematurely crack because offatiguefactors.Usually,asynchronousbeltdrivesys- tem with a same-size transmission ratio is pretensioned by increasing the central distance between the sprockets. The elements of the backing-rubber layer and the tensile-cord layer are elongated in the pretensioning process. The elongation of each element is assumed to be uniform, and the elements in the free span are assumed to carry the forces from the same layer. The sprockets transmit the pretensioning forces to the belt by contacting with the teeth and grooves of the belt. The sprocket teeth are higher than the synchronous belt teeth, so the top surfaces of the sprocket teeth can always maintain contact with the bottom surfaces of the synchronous belt grooves.The backing-rubber layer bears the external forces from the tensile-cord layer through the contact surface between the two layers. Based on the characteristics of element node forces, some two-dimensional surface force distribution coefficients are defined to solve the boundary forces. The boundary forces are calculated from the tight span to the slack span on the driver sprocket or the driven sprocket. When one element node is engaging with the sprocket and the other node is at free span of the belt, the boundary forces are first cal- culatedfor thenodeengagingwiththesprocket because the node at the free span is in equilibrium. The same work is then carried out one node after another.Fig. 3Surface-force distribution coefficients between two layersFigure 3 shows a portion of a synchronous belt engaging with a sprocket. The backing-rubber layer element j is assumed to increase in the counterclockwise direction around the sprocket axis. Four two-dimensional surface force distribution coefficients are defined to calculate the boundary forces for node i. The boundary forces of the backing-rubber layer are the contact forces from the tensile-cord layer. The boundary forces are calculated as follows: Fb1 j+ Fb2 j+ I1GbSb + I2GbSb + I1Feb = 0 jThe driving forces from the driver sprocket and the loads from the driven sprocket are applied to the belt in the tangential direction with respect to the belt, and the initial driving forces or loads from the sprockets on each tooth are assumed to be uniform. The pretensioned belt has a tight span and a slack span due to the effect of the driving forces and loads from the sprockets, and the tight span changes gradually to the slack span through the engaging span on the sprocket. Accordingly, the tensile force of the tight span changes gradually to that oftheslackspan;thetensile-cordlayerandthebacking- rubber layer both obey this law. For the beam elements of each layer, the element tensile forces of the tight span and the slack span are uniform, but the tensile forces of the elements of each layer engaging with the sprocket change gradually as per tooth-load size. So one tensile-cord layer element corresponds to one tooth element and one backing-rubber layer element, and this makes the formulation convenient and describes the tensile forces of each layer clearly. After the driving forces from the driver sprocket and the loads from the driven sprocket are applied to the synchronous belt, the tensile-force distribution on the belt is clear.2.3 Solving elastic dynamic equation for synchronousbelt drive systemSolving the elastic dynamics equation mainly involves focusing on the dynamics of the backing-rubber layer, the tensile-cord layer, and the belt teeth engaging with the sprockets. The belt teeth at the free span can be treated as rigid bodiesFor each temporal integration step, the elastic dynamic equation of the backing-rubber layer is solved firstlytoobtaintheexternalforcesthatprovidedynamic increments such as inertial and damping forces for the backing-rubberlayer,andtheseexternalforcesarefrom the tensile-cord layer. The dynamic increments of the backing-rubber layer are applied to the tensile-cord layer, and the elastic dynamic equation of the tensile- cord layer is then solved. Then, dynamic increments of the tensile-cord layer are applied to the belt teeth engaging with the sprockets, and the elastic dynamic equation of the belt teeth engaging with the sprocket is solved. The entire elastic dynamic equation is(34)where M is the entire mass matrix, C is the entire damping matrix, K is the entire stiffness matrix, et is the entire node-displacement matrix, and Qt is the entire generalized external force vector matrix. Direct inte gration is used to solve the entire elastic dynamic equation (34) as follows:(35)where Mv is the equivalent-mass matrix, which can be obtained from(36)(37)(38)(39)When the node acceleration et at time t is obtained, the node velocity and node displacement at time t can be calculated as follows:(40)(41)2.4Rigidflexible coupling dynamic simulation of synchronous belt drive system in RecurDynIn the rigidflexible coupling dynamic simulation of synchronous belt drive system, the arc-teeth synchronous belt is considered as a flexible body and the sprockets are considered as rigid bodies. The multi-body dynamic software RecurDyn is not convenient for working with the element mesh of a composite- material body such as a synchronous belt, so the synchronous belt is meshed in the other finite-element software. The synchronous belt is meshed in the finite-element software Ansys using solid185 elements, and different layers have common nodes at the contact surface.The flexible synchronous belt is imported into the dynamic software RecurDyn via a constant database file (see Fig. 4). The sprocket teeth are connected with the belt teeth by the rigidflexible coupling contact pairs. The contact stiffness of the contact pairs is 9 106 N/m, the friction coefficient is 0.5, and the damping coefficient is 0.57.Fig. 4 Rigidflexible coupling model used in dynamic software RecurDynFig.5 Stress distribution in synchronous beltAs shown in Fig. 4, the tensile-cord layer has two layers of elements, which is also true of the backing- rubber layer. The stress of element node is the average of the stresses of the adjacent elements, so the nodal stress on the axis of each layer approximates the element stress. In RecurDyn, only the element nodal stress is simulated, so the nodal stress is the main thing to describe.To conveniently tension the synchronous belt, a slider block is used to link the ground and driven sprocket (see Fig. 5b). The slider block can move slowly to increase the distance between the driver sprocket and the driven sprocket. After pretensioning the synchronous belt, constant angular acceleration is applied to the driver sprocket and constant torque is applied to the driven sprocket. As shown in Fig. 5, the larger stresses in the belt occur in the tensile-cord layer and in the belt teeth. To clearly display the stress distributions of the belt teeth, the maximum stress of scale plate is kept at a reasonable value3 Numerical resultsIn this section, the application of the absolute nodal coordinate formulation for modeling the synchronous belt drive is done using a simple two-sprocket system with sprockets of equal size (for an equal transmission ratio). The results of the absolute nodal coordinate formulation are compared with the results of the software simulation.Table 1 Parameters of thestudied synchronous beltSynchronous belt drive parameterSymbolAssigned valuedrive systemAddendum circle radius of the driver sprocketR150.245 103 (m)Addendum circle radius of the driven sprocketR250.245 103 (m)Span lengthls0.96 (m)Length of the synchronous belt pitch lineLp2.24 (m)Total number of belt teethNt280Pitch circle radius of the tensile-cord layerR350.9296 103 (m)Coordinate of the center of the driver sprocketO1(0.48, 0.0) (m)Coordinate of the center of the driven sprocketO2(0.48, 0.0) (m)Thickness of the beltT h12.6 103 (m)Thickness of the tensile-cord layerT h20.2 103 (m)Thickness of the backing-rubber layerT h31.815 103 (m)Width of the beltWb30 103 (m)Youngs modulus of carbon-fiber materialE12.3 1011 (N/m2)Youngs modulus of rubber materialE2108 (N/m2)Density of carbon-fiber material12000 (kg/m3)Density of rubber material21250 (kg/m3)Poissons ratio of carbon-fiber materialV10.3Poissons ratio of rubber materialV20.4Dilatational stress dissipation factors0.6915Deviatoric stress dissipation factord2.3345 105Reducing coefficient of bending stiffness0.1The length of the synchronous belt pitch line is 2.240 m, and the central distance of two sprockets is 0.96 m. The module of arc tooth is 8m, and the pitch of the synchronous belt is 8 103 m. The synchronous belt has 280 teeth, and the sprocket has 40 teeth. The pitch circle radius of the tensile-cord layer is 50.9296 103 m, and the addendum circle radius of the sprocket is 50.245 103 m. The thickness of the belt is 2.6 103 m, and the width of the belt is 30 103 m. The thickness of the tensile-cord layer is 0.2 103 m, and the thickness of the backing- rubber layer is 1.815 103 m. The increment of the central distance of two sprockets is 1 103 m for the synchronous belt pretensioning. The density of backing-rubber layer material is 1250 kg/m3, and the density of tensile-cord layer material is 2000 kg/m3. The Poissons ratio of the backing-rubber layer material is 0.4, and the Poissons ratio of the tensile-cord layer material is 0.3. The dilatational stress dissipation factor is 0.6915, and the deviatoric stress dissipation factor is 2.3345 105. The Youngs modulus of the backing-rubber layer material is 108 N/m2, and the Youngs modulus of the tensile-cord layer material is 2.3 1011 N/m2. The acceleration of the driver sprocket is 102 rad/s2. The load torque of the driven sprocket is 50.25 N*m. The parameters of the arc-teeth synchronous belt drive system are shown in Table 1. The total formulation time of theoretical formulation is 0.15 s. The parameters used in software RecurDyn are the same with the theoretical-analysis parameters above. The total simulation time in software RecurDyn is 0.2 s and contains two processes. The time of pretensioning process is 0.05 s, and the time of the acceleration process is 0.15 s.The hexahedral elements in software RecurDyn can describe three-dimensional deformations and shear deformations of the layer axis. Beam elements without shear deformation can only describe axial deformation and bending deformation, and the bending deformation on the element axis is zero. Thus, the theoretical elastic dynamic formulation result mainly contains axial stress on the element axis. In this section, we focus on comparing the axial stress curves obtained from the theoretical elastic dynamic formulation with the Mises stress curves obtained from the software simulation.Elastic dynamic analysisFig. 6 Angle velocity and angle curves of sprocketsFig. 7 Force curves of sprocket joints+Fig. 8 Driving-torque curves of driver sprocketFig. 9 Stress curves for the backing-rubber layer axis in tight spanFig. 10 Stress curves for the tensile-cord layer axis in tight spanFig. 11 Stress curves for the backing-rubber layer axis in slack spanFig. 12 Stress curves for the tensile-cord layer axis in slack spanFig. 13 Stress curves for the backing-rubber layer axis at initial engaging-in position on driver sprocketFig. 14 Stress curves for the tensile-cord layer axis at initial engaging-in position on driver sprocketFig. 15 Stress curves for belt tooth root at initial engaging-in position on driver sprocketFig. 16 Stress curves of backing-rubber layer axis at initial engaging-in position on driven sprocketFig. 17 Stress curves of tensile-cord layer axis at initial engaging-in position on driven sprocketFig. 18 Stress curves for belt tooth root at initial engaging-in position on driven sprocketFigure 6 shows the angle velocity and angle curves of sprockets obtained from the software simulation. As shown in Fig. 6a, Angle Velocity1 is the angle- velocity curve of the driver sprocket, and Angle Veloc- ity2 is the angle-velocity curve of the driven sprocket. The Angle Velocity2 mainly fluctuates around Angle Velocity1, and the fluctuation amplitude at acceleration starting period is large. As shown in Fig. 6b, Angle1 is the angle curve of the driver sprocket, and Angle2 is the angle curve of the driven sprocket. The angle curve of the driven sprocket approaches the angle curve of the driver sprocket in the acceleration process. It is proved that the synchronous transmission effect is good when the tensile-cord material is forceful.Figure 7 shows the force curves of sprocket joints obtained from the software simulation. As shown in Fig. 7a, FX is the force curve of revolute joint of the driver sprocket in X direction, and FY and FZ are the force curves in X and Z directions. FX has the large impact at the simulation starting period and then gradually increases from 0 to 3000 N in the pretensioning process. FX fluctuates around 3000 N in the acceleration process, and the fluctuation amplitude is large. FYand FZ fluctuate around zero with small amplitudes. As shown in Fig. 7b, FX is the force curve of translational joint of the driven sprocket in X direction, and FY and FZ are the force curves in X and Y directions. FX also has the large impact at the simulation starting period and then gradually increases from 0 to 3000 N in the pretensioning process. FX fluctuates around 3000 N in the acceleration process, but the fluctuation amplitude is small. FY and FZ fluctuate around zero with small amplitudes.Based on Fig. 7, the bilateral tensile force of belt is about 3000 N, and the unilateral tensile force of belt is about 1500 N after the pretensioning process. After the driving torque and load torque are applied to synchronous belt, the tensile force of the tight span is about 2000 N and the tensile force of the slack span is about 1000 N. The tensile-force magnitudes of the tight span or the slack span can be obtained by the element stress curves of the tensile-cord layer and the backing-rubber layer in the following Figs. 9, 10, 11, and 12.Figure 8 shows the driving-torque curves of the driver sprocket. This driving torque consists of the initial driving-torque and the dynamic-torque incre ments. As shown in Fig. 8a, the driving-torque curve of the driver sprocket obtained from the software simulation fluctuates about zero in the pretensioning process and then fluctuates about 50 N*m in the acceleration process, and the magnitude of the fluctuation is about 50 N*m. As shown in Fig. 8b, the driving-torque curve of the driver sprocket obtained from the theoretical formulation gradually increases from 50 to 97 N*m. The driving-torque curve of the driver sprocket obtained from the theoretical formulation approaches the curve obtained from the software simulation.Figure 9 shows the stress curves for the backing- rubber layer axis, with the initial position of backing- rubber layer element at the midway of the tight span. As shown in Fig. 9a, SX is the nodal stress in X direction for the backing-rubber layer axis obtained from software simulation, SY is the nodal stress in Y direction, SZ is the nodal stress in Z direction, SXY, SYZ, and SZX are the shear stress components of element node, and SMISES is the Mises stress of element node. The stress SX gradually increases from 0 to 1.4 105 Pa in the pretensioning process and then fluctuates around 1.8 105 Pa in the acceleration process. The Mises stress SMISES approaches the SX stress. The other stress components mainly fluctuate around zero with small amplitudes.As shown in Fig. 9b, the curve is the axial stress of the backing-rubber layer obtained from the theoretical formulation in the acceleration process, and the axial stress fluctuates around 1.794 105 Pa. In the acceleration process, the axial stress curve in Fig. 9b approaches the Mises stress curve in Fig. 9a.Figure 10shows the stress curves for the tensile- cord layer axis, with the initial position of tensile-cord layer element at the midway of the tight span. As shown in Fig. 10a, the stress SX obtained from the software simulation gradually increases from 0 to 2.5 108 Pa in the pretensioning process and then fluctuates around 3.3 108 Pa in the acceleration process. The Mises stress SMISES approaches the SX stress. The other stress components mainly fluctuate around zero with small amplitudes.As shown in Fig. 10b, the curve is the axial stress of the tensile-cord layer obtained from the theoretical for- mulationintheaccelerationprocess,andtheaxialstress fluctuates around 3.452 108 Pa. In the acceleration process, the axial stress curve in Fig. 10b approaches the Mises stress curve in Fig. 10a.Figure 11 shows the stress curves of the backing- rubber layer axis, with the initial position of backing- rubber layer element at the midway of the slack span. As shown in Fig. 11a, the stress SX obtained from the software simulation gradually increases from 0 to 1.0 105 Pa in the pretensioning process and then fluctuates around 8 104 Pa in the acceleration process. The Mises stress SMISES approaches the SX stress. The other stress components mainly fluctuate around zero with small amplitudes.As shown in Fig. 11b, the curve is the axial stress of the backing-rubber layer obtained from the theoretical formulation in the acceleration process, and the axial stress fluctuates around 8.23 104 Pa. In the acceleration process, the axial stress curve in Fig. 11b approaches the Mises stress curve in Fig. 11a.Figure 12 shows the stress curves for the tensile-cord layer axis, with the initial position of tensile-cord layer element at the midway of the slack span. As shown in Fig. 12a, the stress SX obtained from the software simulation gradually increases from 0 to 2.28108 Pa in the pretensioning process and then fluctuates around 1.8 108 Pa in the acceleration process. The Mises stress SMISES approaches the SX stress. The other stress components mainly fluctuate around zero with small amplitudes.As shown in Fig. 12b, the curve is the axial stress of the tensile-cord layer obtained from the theoretical for- mulationintheaccelerationprocess,andtheaxialstress fluctuates around 1.728 108 Pa. In the acceleration process, the axial stress curve in Fig. 12b approaches the Mises stress curve in Fig. 12a.Figure 13 shows the stress curves for the backing- rubber layer axis, with the initial position of backing- rubber layer element at the engaging-in position on the driver sprocket. As shown in Fig. 13a, the stress SX obtained from the software simulation gradually increases from 0 to 1.8 105 Pa in the pretensioning process. And it increases to the maximum value of 1.03 106 Pa in the acceleration process and then gradually decreases to 1.0 105 Pa. The stress SY gradually increases to 1.2 106 Pa in the acceleration process. The stress SZ gradually increases to 5 105 Pa in the acceleration process. The shear stress SXY gradually increases to 6.1 105 Pa in the acceleration process and then gradually decreases to 3.5 105 Pa.The Mises stress SMISES gradually increases from 0 to 1.1 106 Pa in the simulation process. The other stress components mainly fluctuate around zero with small amplitudes. As shown in Fig. 13b, the curve is the axial stress of the backing-rubber layer obtained from the theoretical formulation in the acceleration process, and the axis stress gradually increases to 9.8 105 Pa from 2.2 105 Pa and then decreases to 7.2 105Pa.The axis stress in the theoretical formulation is a little smaller than the Mises stress in the software simulation. This is because that the beam elements used in theoretical formulation can only describe axial stress and bending stress, and cannot describe stresses in Y, Z directions and shear stress components. The backing- rubber layer axis is at the outside of the pitch circle of belt, and the backing-rubber layer axis on the sprocket is stretched before the pretensioning process. So the axial stress of backing-rubber layer increases in a large amplitude when engaging with the sprocket.Figure 14 shows the stress curves for the tensile- cord layer axis, with the initial position of tensile-cord layer element at the engaging-in position on the driver sprocket. As shown in Fig. 14a, the stress SX obtained from the software simulation gradually increases from0 to 1.9 108 Pa in the pretensioning process. And it increases to the maximum value of 2.9 108 Pa in the acceleration process and then gradually decreases to 0.26 108 Pa. The stress SY gradually increases to 0.95 108 Pa in the acceleration process. The shear stress SXY gradually increases to 0.5 108 Pa in the acceleration process. The shear stress SYZ gradually increases from 0 to 0.8 108 Pa in the pretensioning process, increases to the maximum value of 1.9108 Pa in the acceleration process, and then gradually decreases to 0.3 108 Pa. The shear stress SZX gradually decreases to 1.1108 Pa in the acceleration process. The Mises stress SMISES gradually increases from 0 to 4.1108 Pa in the pretensioning process. And it increases to the maximum value of 4.8 108 Pa in the acceleration process and then gradually decreases to 2.3 108 Pa. The amplitude of the shear stress SYZ fluctuation is large, which results in a large Mises stress fluctuation as well. The other stress components mainly fluctuate around zero with small amplitudes. If the fluctuation is neglected, the Mises stress mainly decreases from 3.3 108 to 2.4 108 Pa in the acceleration process.As shown in Fig. 14b, the curve is the axial stress for the tensile-cord layer obtained from the theoretical formulation in the acceleration process, and the axial stress gradually decreases from 3.32 108 to 2.29 108 Pa. The axial stress in the theoretical formulation approaches the Mises stress obtained from the software simulation, if the Mises stress does not have the large fluctuation amplitude.Figure 15 shows the stress curves for the root of the belt tooth, with the initial position of belt tooth element at the engaging-in position on the driver sprocket. As shown in Fig. 15a, the stress SX obtained from the software simulation gradually decreases from 0 to 1.5 106 Pa in the simulation process. The stress SY gradually decreases to 1.4 106 Pa in the acceleration process and then decreases to 0.42 106 Pa. The stress SZ gradually decreases to 1.2 106 Pa in the acceleration process and then decreases to 0.62 106 Pa. The shear stress SXY gradually increases to 1.6 106 Pa in the acceleration process and then decreases to 0.2 106 Pa. The shear stress SYZ gradually decreases to 2.1106 Pa in the acceleration process and then increases to 1.5105 Pa. The shear stress SZX fluctuates around zero in the acceleration process and has a larger amplitude of 2 106 Pa. The Mises stress SMISES gradually increases from 0 to 4.3 106 Pa in the pretensioning process. And it increases to the maximum value of 5.8 106 Pa in the acceleration process and then gradually decreases to 2.1 106 Pa. The amplitude of the shear stress SZX fluctuation is large, which results in a large Mises stress fluctuation as well.Figure 15b shows the theoretical bending stress for the root of the belt tooth in the acceleration process. The bending stress gradually increases from 0.9 106 to 3.35106 Pa and then gradually decreases to 1.9106 Pa. The bending stress in the theoretical formulation approaches the Mises stress in the software simulation, if the Mises stress does not have the large fluctuation amplitude. The belt tooth engaging with the driver sprocket in the theoretical formulation bears the pretensioning force, the driving force, and the dynamic force increment from the tensile-cord layer.Figure 16 shows the stress curves of the backing- rubber layer axis, with the initial position of backing- rubber layer element at the engaging-in position on the driven sprocket. As shown in Fig. 16a, the stress SX obtained from the software simulation gradually increases from 0 to 1.5 105 Pa in the pretensioning process. And it increases to the maximum value of 9.2105 Pa in the acceleration process and then gradually decreases to 9.8104 Pa. The stress SY gradually increases to 1106 Pa in the acceleration process. The stress SZ gradually increases to 3.1 105 Pa in the acceleration process. The shear stress SXY gradually increases to 5.7105 Pa in the acceleration process and then gradually decreases to 2.7 105 Pa. The Mises stress SMISES gradually increases to 1 106 Pa in the simulation process. The other stress components mainly fluctuate about zero with small amplitudes. Figure 16b shows the axial stress of the backing-rubber layer in the theoretical formulation. The axis stress gradually increases to 9.7105 Pa from 1.35105 Pa.Figure 17 shows the stress curves of the tensile- cord layer axis, with the initial position of tensile-cord layer element at the engaging-in position on the driven sprocket. As shown in Fig. 17a, the stress SX obtained from the software simulation gradually increases from 0 to 1.2 108 Pa in the pretensioning process, and it decreases to zero in the acceleration process. The shear stress SYZ gradually increases to 0.9 108 Pa in the pretensioning process. The shear stress SYZ gradually increases to 0.5 108 Pa in the acceleration process, increases to the maximum value of 2108 Pa, and then gradually decreases to 0.9 108 Pa. The shear stress SZX gradually decreases to 1.8108 Pa in the acceleration process. The Mises stress SMISES gradually increases from 0 to 2.1 108 Pa in the pretensioning process,increasestothemaximumvalueof3.6108 Pa in the acceleration process, then gradually decreases to 1.5108 Pa, and finally increases to 3.1108 Pa. The amplitude of the shear stress SYZ is large, which results in a large amplitude Mises stress fluctuation. The other stress components mainly fluctuate around zero with small amplitudes. If the fluctuation is neglected, the Mises stress increases essentially from 1.5 108 toFigure 17b shows the axial stress of the tensile- cord layer obtained from the theoretical formulation in the acceleration process, and the axial stress gradually increases from 1.62108 to 2.43108 Pa. The stress in the theoretical formulation approaches the Mises stress obtained from the software simulation, if the amplitude of the fluctuations is neglected.Figure 18 shows the stress curves for the root of the belt tooth, with the initial position of belt tooth element at the engaging-in position on the driven sprocket. As shown in Fig. 18a, the stress SX obtained from the software simulation gradually decreases from 0 to 5.4105 Pa in the simulation process. The stress SY gradually decreases to 7.8105 Pa in the acceleration process and then decreases to 1.9105 Pa. The stress SZ approaches the stress SX. The shear stress SYZ fluc- tuates about zero in the simulation process and has a larger amplitude of 2.1 106 Pa. The Mises stress SMISES gradually increases from 0 to 4.4 106 Pa in the pretensioning process and then gradually decreases to 2.2 106 Pa in the acceleration process. The other stress components mainly fluctuate around zero with small amplitudes. The amplitude of the shear stress SYZ fluctuation is large, which results in a large Mises stress fluctuation as well.Figure 18b shows the theoretical bending stress for the root of the belt tooth in the acceleration process, and the bending stress gradually increases from 9.958105 to 10.03 105 Pa. The belt tooth engaging with the driven sprocket in the theoretical formulation bears the loads from the driven sprocket and the dynamic force increment from the tensile-cord layer and does not bear the pretensioning force .5 ConclusionThe objective of this study is to research the inertial and damping force distributions of a synchronous belt by using the absolute node coordinate formulation, and the two-dimensional beam elements without shear deformation can meet the requirement. Before analyzing the elastic dynamics of synchronous belt drive system, the pretensioning force distribution in the synchronous belt has been analyzed. To verify the results of the elastic dynamic analysis, the rigidflexible coupling dynamic simulation of synchronous belt drive system with dynamic software RecurDyn is carried out.The driving-torque curve of the driver sprocket obtained from the theoretical formulation approaches the curve obtained from the software simulation, so it is generally proved that the precision of the theoretical formulation approaches that of the software simulation. Compared with the stress curves obtained from the software simulation, the axial stress curves of beam elements without shear deformation generally approach the Mises stress curves. Because of the fluctuations in shear stress, some simulated Mises stress curves have larger fluctuations. The results of a theoretical elastic dynamic formulation are not accurate as the results of the rigidflexible coupling dynamic simulation obtained with RecurDyn. This is because a beam element without shear deformation can only describe the axial deformation and bending deformation, whereas the hexahedral element in the dynamic software RecurDyn can describe three-dimensional deformations and shear deformations.Generally speaking, this paper applies the absolute nodal coordinate formulation to a new system: the synchronous belt drive. Little research has been carried out heretofore on the elastic dynamic analysis of synchronous belt drive systems. Owing to the complex composition of synchronous belts, analyzing the static forces and the elastic dynamics of synchronous belt drive systems is complicated. Beam element without shear deformation can describe the main deformations of synchronous belt, but cannot describe the shear deformations of the belt engaging with the sprockets. Thus, future analyses of the elastic dynamics of synchronous belts should use elements that can describe shear deformations.中文译文采用绝对节点坐标法对同步带传动系统进行弹性动力学分析摘要同步带与链轮啮合时，由于其复杂的组成，具有复杂的动力学行为。当同步带处于高加速度或高速传动状态时，同步带的惯性力和阻尼力是不可忽视的，这些因素对带型的选择和系统的总体设计有很大的影响。预紧力对同步带的应力状态和疲劳寿命也有显著影响，因此在进行弹性动力分析之前应进行预紧力分析。本文的目的是研究高加速度和高速同步带动态性能的精确影 响。为此，本文研究了基于绝对节点坐标的无剪切变形二维梁单元的同步带传动系统的预紧力和弹性动力分析。为验证弹性动力学分析结果，利用动力学软件RecurDyn对同步带传动系统进行了刚柔耦合动力学仿真的结果以同步带应力曲线的形式给出。关键词:同步带传动系统;梁单元;弹性动力学分析;绝对节点坐标;刚柔耦合动力学仿真1介绍同步带链轮图1典型的同步带传动图1显示了涉及旋转机械元件的动力传输设备的典型同步带传动。同步带通常由拉伸绳、形成带齿体和带背的橡胶化合物和覆盖带面和带齿面的饰面织物组成。拉力绳承担外部载荷，饰面织物增强皮带和齿的耐磨性。同步带传动比平带传动具有更高的传动精度和承载能力，比链传动产生更少的噪声。同步带传动广泛应用于汽车应用，生产线，工业机械，和其他工业应用。齿的形式在很大程度上决定了同步带的承载能力。圆弧齿的承载能力一般比梯形齿的承载能力大得多。随着拉绳材料的发展，同步带的速度得到了提高，同步带的瞬时切向速度现已达到50m/s。当连同步传动带在高速、高加速度或重载下运行，在设计传动带时必须考虑惯性和阻尼力，并为给定的应用选择合适的传动带类型。同步带与链轮之间的滑动摩擦影响同步带的疲劳寿命。目前对同步带传动的研究主要集中在静态分析、实验研究和有限元计算机辅助工程方面。本文提出了Nestor和Peter在参考文献1中提出的在带束力随时间变化的情况下计算带束和链轮齿所受载荷的方法，并且用若干相同的伸缩弹簧来模拟带束的张力。当链轮以16转/分的低速旋转时，使用安装在链轮齿上的一对半导体应变仪测量齿上的负载和摩擦力。Dalgarno等人2的实验表明，同步带的伸展刚度随着运行时间的增加逐渐减小，直到趋于恒定。同步带的刚度损失被认为是由加强同步带的索的裂纹的发展引起的。Ergin等人提出了一种基于结构神经网络的技术来估计用于精密机械的同步带传动的位置。小车的位置是通过一个结构化的神经网络拓扑结构来计算的，该拓扑结构接受同步带执行器侧位置传感器的输入。Kagotani和Ueda4研究了影响螺旋同步带传动误差的因素，确定了双向运行时误差发生在带侧端面。在文献5中，魏明和富雄对同步带传动中的噪声进行了实验和理论分析。同步带传动的多体建模和仿真可以用商业软件包来完成，如Callegari等人6所示。湛国等人7利用RecurDyn软件对汽车梯形同步带传动系统进行了动力学分析。基于绝对节点坐标公式的平面胶带传动弹性动力学分析得到了迅速的发展，这主要是由于这一理论的贡献，Berzeri和Shabana8提出了一种基于绝对坐标的无剪切变形的二维梁单元。扁平橡胶带传动采用Cepon9,10的无剪切变形的二维梁单元建模。此外,一个二维shear-deformable梁元素奥马尔提出的基于绝对坐标和Shabana11修改Kerkkanen et al。(12、13)获得一片带状元件平橡胶皮带传动配方,和修改shear-deformable梁元素降低了抗弯刚度。同步带的厚度较小，与扁平橡胶带相似，皮带变形主要通过厚度和节距分布。同步带的弯曲刚度也远低于轴向刚度，就像扁平橡胶带的情况一样。因此，同步带传动系统的力学模型可以简化为二维问题，如平面胶带传动系统。采用Berzeri和Shabana8提出的无剪切变形的二维梁单元技术对同步带传动系统进行建模，并将同步带齿简化为悬臂梁单元。但是，Berzeri和Shabana8提出的梁单元的曲率在轴向变形较小的情况下进行了简化，对于轴向变形较大的情况，这种简化并不准确。为了解决这一问题，我们使用了Gerstmayr和Irschik14提出的耦合弯曲和轴向应变的曲率表达式。这个曲率对于轴向变形大的情况是精确的。同步带传动系统采用Berzeri和Shabana提出的梁单元8和Gerstmayr和Irschik14提出的正确曲率表示进行计算，从而保证同步带的准确弯曲刚度。同步带的较低弯曲刚度详见文献12。2圆弧齿同步带传动系统的建模s = ls紧张的跨度Ys = 0 = 2 ls+ 2Rp 驱动s =司机ls +RpOX松弛的跨度s = 2 ls +Rpls图2同步带传动简化示意图如图2所示，假设同步带传动系统的主动链轮逆时针旋转。坐标的原点系统位于两个链轮的转动中心之间。坐标的X轴是水平的，而y轴是垂直的。s = 0为同步带在从动链轮上的啮合位置，s = ls 同步带在从动链轮上的啮合位置，s = ls +Rp 同步带在从动链轮上的啮合位置，s = 2ls +Rp 同步带与从动链轮的啮合位置在ls 两个链轮和R的中心距离是多少p 为同步带的节圆半径。2.1同步带的元素划分为了方便弧齿同步带传动公式的建模，将带的拉伸绳按等效体积原则简化为一层拉伸绳材料。基于这一假设，橡胶部分分为两部分:齿体和背衬带。将受拉材料层定义为受拉绳层，将带的背衬层定义为背衬橡胶层。贴面织物轻薄，被认为是传递力的中间层。只有分析皮带与链轮之间的摩擦时，才把衬布作为主要的动作对象。由于带齿和拉索层是同步带输送载荷的主要部件，所以弧齿最好固定在拉索层上。悬臂梁是固定在高强度物体上的梁;拉帘线层通常具有较高的强度，因此同步带的弧齿可视为悬臂梁。一个弧齿可以简化为一个梁单元。也考虑由带齿隔开的橡胶层传递力，并考虑这一层的惯性力。为保证计算精度，一个拉索层单元对应一个背胶层单元和一个齿单元。在圆弧齿同步带传动中，背胶层、拉绳层和与链轮啮合的带齿是研究的主要对象。从从动链轮的负载和从从动链轮的驱动力通过与链轮啮合的带齿传递到拉绳层和背胶层。同步带传动系统部件的惯性力，如带层、带齿、从动链轮，可以转化为适用于拉绳层。所述背胶层的阻尼力也可转化为适用于所述拉索层。如果带牙齿和tensile-cord层是强大到足以传递的驱动力驱动链轮驱动链轮,带的影响牙齿和皮带层同步皮带传动系统的传动精度的响应延迟从动链轮驱动链轮。如果带齿和拉绳层不够牢固，同步带传动系统的齿可能打滑或拉绳可能断裂。与平胶带传动相比，同步带传动由于平胶带在滑轮上不断打滑，传动精度更高，这种现象无法消除。2.2同步带传动系统的静力学分析预紧力是决定同步带承载能力和疲劳寿命的重要因素。如果预紧力小于要求，同步带不能承载所需的载荷和齿滑移将容易发生。如果预紧力大于要求，同步带可能会迅速磨损或过早开裂，因为疲劳因素。对于具有相同传动比的异步传动带传动，通常采用预张力法通过增加链轮之间的中心距离。在预张过程中，背胶层和拉索层的元件被拉长。假定每个单元的伸长是均匀的，并假定自由跨中的单元承受来自同一层的力。链轮通过与皮带的齿和槽接触将预紧力传递给皮带。由于链轮齿高于同步带齿，所以链轮齿的上表面可以始终与同步带槽的下表面保持接触。背胶层通过两层之间的接触面承受来自拉索层的外力。根据单元节点力的特点，定义了二维表面力分布系数来求解边界力。计算从从动链轮或从动链轮的紧跨到松跨的边界力。当一个节点与链轮啮合而另一个节点在皮带的自由跨度处时，由于自由跨度处的节点处于平衡状态，首先计算节点与链轮啮合时的边界力。然后一个节点接着一个节点执行相同的工作。图3两层间的表面力分布系数图3显示了与链轮啮合的同步带的一部分。backing-rubber层假定元素j沿链轮轴逆时针方向增加。定义了4个二维表面力分布系数来计算节点i的边界力。背胶层的边界力是来自拉索层的接触力。边界力计算公式为:Fb1 j+ Fb2 j+ I1GbSb + I2GbSb + I1Feb = 0j。从主链轮的驱动力和从从动链轮的负载以相对于带的切向作用于带，并且假定每个齿上的链轮的初始驱动力或负载是均匀的。预紧带由于链轮的驱动力和载荷的作用，具有紧跨和松弛跨，并通过链轮上的啮合跨距逐渐转变为松弛跨距。因此，紧跨的拉力逐渐向松弛跨的拉力转变，受拉索层和背胶层均遵循这一规律。对于各层梁单元，紧跨距和松跨距的单元拉伸力是均匀的，但与链轮啮合的各层单元的拉伸力随齿载大小的变化而逐渐变化。因此，一个拉索层单元对应一个齿单元和一个背胶层单元，使配方更加方便，并能清楚地描述每一层的拉力。当从动链轮的驱动力和从动链轮的负载作用于同步带上后，同步带上的张力分布清晰可见。2.3求解同步弹性动力学方程 皮带传动系统弹性动力学方程的求解主要涉及到背胶层、拉索层和带齿与链轮啮合的动力学问题。自由跨度处的带齿可视为刚体。对于每一个时间积分步骤，背衬橡胶层的弹性动力学方程首先求解，获得为背衬橡胶层提供动态增量的外力，如惯性力和阻尼力，这些外力来自拉索层。将背胶层的动力增量施加到受拉帘子层上，求解受拉帘子层的弹性动力方程。然后，对与链轮啮合的带齿施加张力绳层的动态增量，求解了带齿与链轮啮合的弹性动力学方程;整个弹性动力学方程为(34)M是整个质量矩阵，C是整个阻尼矩阵，K是整个刚度矩阵，et 是整个节点位移矩阵，Qt 是整个广义外力矢量矩阵。直接强度采用栅格法求解整个弹性动力方程(34):(35)其中Mv为等质量矩阵，由(36)(37)(38)(39)当节点加速et 得到t时刻的节点速度和节点位移可计算为:(40)(41)3.5 RecurDyn同步带传动系统刚柔耦合动力学仿真在同步带传动系统的刚柔耦合动力学仿真中，将弧齿同步带视为柔性体，链轮视为刚体。多体动力学软件RecurDyn不便于对同步带等复合材料体的单元网格进行计算，因此在其他有限元软件中对同步带进行网格计算。同步带在有限元软件Ansys中采用solid185单元进行网格化，不同层在接触面上有共同的节点。通过固定的数据库文件将柔性同步带导入动态软件RecurDyn中(如图8所示)，链轮齿与带齿通过刚柔耦合接触副连接。接触副的接触刚度为9 106 N/m，摩擦系数为0.5，阻尼系数为0.57。图4动态软件RecurDyn中刚柔耦合模型图5同步带内应力分布如图4所示，拉索层有两层元件，背衬橡胶层也是如此。单元节点的应力是相邻单元的应力的平均值，因此节点每一层轴上的应力近似于单元应力。在RecurDyn中，只模拟单元的节点应力，因此节点应力是主要描述的内容。为了方便地张紧同步带，采用滑块将地面与从动链轮连接起来(如图5 b)。滑块可以缓慢移动，增加从动链轮与从动链轮之间的距离。同步带预紧后，对从动链轮施加恒定的角加速度，对从动链轮施加恒定的扭矩。如图9所示，输送带中较大的应力出现在拉索层和带齿处。为了更清楚地显示皮带齿的应力分布，将鳞板的最大应力保持在一个合理的值3计算结果在这一节中，应用绝对节点坐标公式建模同步带传动是使用一个简单的双链轮系统具有相同的大小(对于一个相同的传动比)。将绝对节点坐标公式的计算结果与软件仿真结果进行了比较。同步带节距线长度为2.240 m，两个链轮中心距离为0.96 m。圆弧齿模量为8m，同步带间距为8 103 m。同步带有280齿，链轮有40齿。拉索层的节圆半径为50.9296 103 m，链轮齿顶圆半径为50.245 103 m。胶带的厚度为2.6 103m，胶带的宽度为30 103m。拉索层厚度为0.2 103 m，背胶层厚度为1.815 103 m。同步带预紧时，两个链轮的中心距离增量为1 103 m。背胶层材料密度为1250kg /m3，拉绳层材料密度为2000kg /m3。背胶层材料的泊松比为0.4，拉索层材料的泊松比为0.3。膨胀应力耗散系数为0.6915，偏应力耗散系数为2.3345 105。背衬橡胶层材料的杨氏模量为108 N / m2，拉伸索层材料的杨氏模量Ial是2。3 1011 N / m2。主链轮的加速度为102 rad /秒2。从动链轮的负载扭矩为50.25 N*m。弧齿同步带传动系统参数如表1所示。理论配方的总配方时间为0.15 s。软件RecurDyn所使用的参数与上述理论分析参数相同。在软件RecurDyn中，总模拟时间为0.2 s，包含两个进程。预张拉过程时间为0.05 s，加速过程时间为0.15 s。表1参数说明研究了同步带同步带传动参数象征分配值驱动系统主动链轮齿顶圆半径R150.245 103 (m)从动链轮齿顶圆半径R250.245 103 (m)跨度ls0.96(米)同步带节距线的长度Lp2.24(米)皮带齿总数Nt280拉索层的节圆半径R350.9296 103 (m)驱动链轮中心的坐标O1群(0.48,0.0)(米)从动链轮中心的坐标O2(0.48, 0.0)(米)输送带厚度T h12.6 103 (m)拉索层的厚度T h20.2 103 (m)背胶层厚度T h31.815 103 (m)皮带宽度白平衡30 103 (m)碳纤维材料的杨氏模量E12.31011 (N / m2)橡胶材料的杨氏模量E2108 (N / m2)碳纤维材料密度12000(公斤/米3)橡胶材料密度21250(公斤/米3)碳纤维材料的泊松比V10.3橡胶材料泊松比V20.4膨胀应力消散因子s0.6915偏应力消散因子d2.3345105弯曲刚度降低系数0.1软件RecurDyn中的六面体单元可以描述层轴的三维变形和剪切变形。没有剪切变形的梁单元只能描述轴向变形和弯曲变形，单元轴上的弯曲变形为零。因此，弹性动力理论公式结果主要包含单元轴上的轴向应力。在本节中，我们重点比较了从理论弹性动力公式得到的轴向应力曲线和从软件模拟得到的Mises应力曲线。图6链轮角速度和角曲线图7链轮结合部受力曲线图图8主动链轮驱动扭矩曲线图图9紧跨时背胶层轴线应力曲线图图10紧跨时拉索层轴的应力曲线图图11背胶层轴线松弛跨应力曲线图图12拉索层轴线松弛跨应力曲线图图13主链轮初始啮合时背胶层轴的应力曲线图图14主链
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