JX02-096@小功率机械摩擦式无级变速器结构设计
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附录：文献翻译 摘要 一个新的计算机数控 (CNC)机床与管架床上。这种结构将有足够的空间桁架之间的酒吧来解决空间问题 ,有足够的刚度对机床。因此 ,数控车床的框架由管道、对角括号了有足够的刚度和空间效用芯片疏散。从机床的角度使用 ,实时振动控制理论应用于控制刀架之间的相对位移和轴抑制特定的相对振动模式。 介绍 传统机床床或框架一般由铸铁或焊接钢板。这些结构适合于机床的动态特性的观点。尽管裁员机床的需求 (比如在台式机 )越来越多 ,有一个严格的体积限制来获得足够的空间足够的芯片疏散。在这种情况下 ,有热传导和热传递的问题。为了解决这些问题 ,我们提出一个传统但是新的框架结构机床 ,包括管道和连接部分。我们看着框架的灵活性设计使用一个管架结构。框架结构可以安排与标准管 (直径和长度 )和连接块。在车床的情况下 ,这种结构允许的变化大小根据主轴单元和轴表 ,设计成标准化的单位。 为开发利用这一结构的数控车床 1，被认为存在以下问题： （ 1）由主轴电机不平衡和切割力扰动产生结构振动， （ 2）由于低的热容量的热变形， 心轴和 X轴和 Z轴之间的表 （ 3）足够的刚性。 在这份报告中，我们专注于振动问题。已经有振动控制机床的应用几个以前的研究 2-7。在这些研究论文中，控制方法已被应用到车床的刀架。这些结果表明刀架位移振动控制的优越性，但并不认为会出现在低频率范围内与当地的振动模式，如镗杆和刀架相比，结构振动模式。 数控车床的开发 根据一个管架结构的构思独特的数控车床已经研制成功。的规格示于表 1，和的示意图示于图 1。 可以得到 10,000分， 1最大主轴转速。最小分辨率为 X轴和 Z轴被设定为 0.1 微米。机床重量只有 525公斤（包括封面）和 440的空间内适合宽 1450长 1050立方毫米，这是足够紧凑。数控控制器（三菱电机有限公司， M700V）设置在机身旁边。 振动控制方法 最重要的振动运动的数控车床是主轴和刀架之间的相对位移。加工工件通常由主轴和刀具之间的反相运动的影响 ;这项议案是由谐振频率产生的。在这种情况下，逆位相运动是针对工件的几何误差的最重要的特性。这个运动可通过测量主轴和刀具之间的相对振动进行评nts价。虽然该位移 应直接由位移传感器在切割过程中测得，它是非常困难的传感器的位置，因为该空间的限制，冷却剂，和芯片。图 2示出了主轴头和代表帧连接的块之间的典型的遵从传递函数。有几个峰。当我们试图用这个数据来施加振动控制，控制点是很难确定的。然后，两个小的加速度传感器用于检测所述相对振动模式。一个被设置在所述主轴单元壳体和测量作为信号 x 秒 ;另一种是设置在 X轴工作台的刀架和计量为信号 x 秒 。 在 45赫兹的频率峰可以观察到，并且它可以被认为是一个自由度的简单系统。我们可以使用该差分信号作为振动响应的输出。 然后，模态分析 ，可用于获得所述振动的模式，来决定上述目标振动模式，并且确定在何处设定所述致动器用于振动控制。在 45 赫兹的模式合成的形状矢量示于图 4，主轴和刀架之间的反相模式实际上是从向量图所示。在头部库存为主轴（ 和 ）的模式矢量是如此之小，这两个致动器的端部可以设置在这些点。根据这一结果，致动器的另一端上连接所述 X轴工作台下面的块来控制反相振动模式，如图所示设置。 1 ，这两个执行器产生控制力的振动主动控制方法来抑制振动运动。 的相对位移，速度和加速度被定义为 X， X ，且 x 分别。假设 XD且 x d表示所需的相对位移和相对速度，相对位移和相对速度被定义为 （ XD -X ），（ X -X ） 受控力可表示为如下： （ 2 ） Kv 值（倍 -X ） + KD（ XD -X ） 其中权重系数 Kd 和 Kv值被使用。运动方程描述为当谐力 faej t作用在这个系统上如下： （ 3 ） MX + CX + KX = faej t + Kv值（倍 -X ） + KD（ XD -X ） 因为所需的相对位移 XD和相对速度 x d分别为零，方程（ 3 ）被描述为如下： （ 4 ） MX + （ C + Kv值） + （ K + Kd值） X = faej t 在方程（ 2） ，替换 f = faej t ，下面的公式可以推导出： （ 5 ） X x = -c/m1-k/m0 X x + 1/m0 f 运动这个方程改写为如下状态方程： （ 6 ） = AX + BF 在施加控制的力成比例的状态向量 X的情况下，控制的力被描述为： F = -KX 。然后，代入公式这一点。 （ 6）下列状态方程获得： （ 7） X = （ A- BK ） 一旦可以得到这个方程，唯一要做的就是用 matlab （ MathWorks公司 - MATLAB和Simulink ）计算反馈增益 K 。计算这个系数 K ，模态参数 M， C，和 k必须首先确定。我们使用简单的“大规模反应法”，因为该振动运动可以被认为是一个自由度从结果在图计算这些值。 3 ，一旦已知质量设置在刀架，在该共振频率的变化被测量，并且该模态参数获nts得，如表 2所示，这些代入方程。 （ 7） ，且反馈增益 K根据线性二次（ LQ）控制理论进行计算。 控振系统 锆钛酸铅（ PZT）的装置被用于振动控制系统的致动源。 结构振动控制和机床运动控制使用，因为它的大产生的力和高频响应 8-11 此设备。 我们开发了一种致动器，它由一个压电陶瓷中，流体腔室与两个不同的横截面面积，并且连接杆，如图所示。 5，流体腔，用于从 PZT变换所产生的力。虽然 PZT的纵向位移是足够短的位移由该腔室面积减小。这种效应使得产生位移比亚微米级以下。两个执行器的连接块，撑杆之间。利用脉冲响应的方法的动态特性进行评价，其结果示于图 6，该致动器可用于低于 200赫兹。振动控制的效果在振动主动控制部分的影响进行评估。虽然撑杆的热伸长，必须考虑到当长期操作 中，我们不考虑由于短时加工的框架的热变形。 主动振动控制的影响 主轴和刀架之间的相对振动是由两个致动器来控制。在这种情况下，根据 LQ 控制理论，我们已经开发出了控制系统，以实现通过在图 3的框图所示的控制方法。 7和表 3中 ，该系统与 3000赫兹的采样速度运行。相对振动由主轴单元的不平衡产生的。由于最大位移在大约 2725分钟 -1（ 45.4赫兹）观察时，目标频率设定为 45 赫兹。 在实验过程中有空气及切割过程中被处决。从致动器的实验结果和频率响应，目标控制的频率范围设置为从 10到 100赫兹。一个带通实时滤波器，用于获得所述振动信号并抑制干扰噪声和高频信号。 为了评价有源振动控制的效果，由于安装空间的限制，二个加速度传感器被用于控制。的相对振动的数值积分获得的相对速度和相对位移。运动过程中的相对位移本身是由设置在刀架和检测主轴单元的相对位移的激光位移传感器测量。 在精切削的控制结果示于图 8 ，控制参数已在空中切割先前确定。这个条件并不总是适用于切割条件，因为在工件和工具彼此接触时，和在 接触点的切割力的作用。虽然模态参数并不总是恒定的，我们的控制系统是采用相同的条件空气切削应用，并在切割过程中的振动控制的效果进行了评价。 当开始控制，则相对位移减小其振幅。 FFT分析用于确定与控制和无控制之间在频元素的不同，如图所示。 9 ，实线表示其结果与对照组，且虚线示出的结果，而不控制。几乎 50 约 45赫兹的目标频率的振动幅度是由我们开发的系统抑制。通过安装为支撑所述致动器的增加刚度的影响可以看出，在较低的频率范围。另一方面，高于 100 赫兹的效果不能的，因为控制系统的过滤效果可见。这些结果 证明，在精切削中使用该振动控制系统适合nts于我们的管架的数控车床。以提高振动控制的效果，布局（包括致动器的方向） ，应考虑和进一步改进本致动器系统的需要，以更有效地抑制振动的振幅。 评价工件的振动控制 因为已经证实，该振动控制是即使在切削有效，我们评估了加工工件的精度。因为目标控制频率设定为 45赫兹，控制结果的影响表现为表面波纹度。然后，该表面轮廓是由表面粗糙度测量仪测量而一个高通滤波器。图 10显示了测量结果。上图显示的曲线没有振动控制，以及下图显示的结果与振动控制。表面波纹度显着提高。波状起伏的在该频率范围 内的改进是关联到出现在这个频率范围内的相对振动模式。 图 11显示了测得的圆度的结果的谐波分析。因为目标频率被设定为在主轴的旋转频率时，振动控制的效果一般出现在圆度与每一轮 2起伏。测量结果表明这种 良好的趋势。每一轮波动的幅度从幅度下降 50 ，无控制权。 这些结果表明在抑制主轴和刀架，并在加工工件的精度的提高之间的相对振动模式我们的控制系统的有效性。 总结 在本文中，实时振动控制的开发数控车床由单一自由度模型的有效性使用主轴和刀架之间的差分信号进行了评估。获得的主要结果如下： （ 1）近 50约 45赫兹的目标频率的振动幅度是由我们开发的系统抑制。 （ 2）每一轮起伏的圆度振幅的振幅下降 50，无控制权。 （ 3）获得的结果表明在抑制主轴和刀架之间的相对振动模式所施加的控制系统的有效性。需要进一步调查，以提高振动控制系统，包括布局（致动器的方向）和致动器的性能能够更有效地抑制振动的振幅。 感谢 作者表达自己的感谢和弘清水，一直支持我们的实验的协助。这项研究部分由新能源产业技术综合开发机构补助资金支持。 参考文献 1.Suzuki, N., 2010, “Development of Ultra Small Size CNC Lathe by Pipe Frame Structure,” Proceedings of 4th CIRP International Conference on High Performance Cutting, Nagoya, Japan, Vol. 2, p. 183. 2.Claesson, I., 1998, “Adaptive Active Control of Machine-Tool Vibration in a Lathe,” Int. J. Acoust. Vib., 3(4), pp. 155162. nts3.Choudhury, S. K., 1997, “On-Line Control of Machine Tool Vibration in Turning,” Int. J. Mach. Tools Manuf., 37(6), pp. 801811. CrossRef 4.Tarng, Y. S., 2000, “Chatter Suppression in Turning Operations With a Tuned Vibration Absorber,” J. Mater. Process. Technol., 105, pp. 5560. CrossRef 5.Smirnova, T., 2010, “Analysis, Modeling and Simulation of Machine Tool Parts Dynamics for Active Control of Tool Vibration,” Ph. D. thesis, Blekinge Institute of Technology, Karlskrona, Sweden. 6.Reza, M., 2011, “Modeling and Analysis of Frictional Damper Effect on Chatter Suppression in a Slender Endmill Tool,” J. Adv. Mech. Des. Syst. Manuf., 5(2), pp. 115128. CrossRef 7.Ast, A., 2009, “Active Vibration Control for a Machine Tool With Parallel Kinematics and Adaptronic Actuator,” J. Comput. Nonlinear Dyn., 4(3), p. 031004. CrossRef 8.Hwang, C. L., 2005, “Trajectory Tracking of Large-Displacement Piezoelectric Actuators Using a Nonlinear Observer-Based Variable Structure Control,” Trans. IEEE, 13(1), pp. 5666. CrossRef 9.Preumont, A., 2011, Vibration Control of Active Structures: An Introduction, 3rd ed., Springer, New York. ntsVibration Control of RelativeTool-Spindle Displacement forComputer Numerically ControlledLathe With Pipe Frame StructureYoshitaka MorimotoMem. ASMEProfessorDirector of Advanced Materials ProcessingResearch Laboratory,Kanazawa Institute of Technology,3-1 Yatsukaho, Hakusan,Ishikawa 924-0838, Japan;e-mail: mosandb1neptune.kanazawa-it.ac.jpNaohiko SuzukiTakamatsu Machinery Co., Ltd.,3-1 Asahigaoka, Hakusan,Ishikawa 924-8558, Japane-mail: suzukitakamaz.co.jpYoshiyuki KanekoTakamatsu Machinery Co., Ltd.,3-1 Asahigaoka, Hakusan,Ishikawa 924-8558, Japane-mail: kanekotakamaz.co.jpMinoru IsobeTakamatsu Machinery Co., Ltd.,3-1 Asahigaoka, Hakusan,Ishikawa 924-8558, Japane-mail: kanekotakamaz.co.jpA new computer numerically controlled (CNC) lathe with a pipeframe bed has been developed. This structure is expected to haveenough space between the truss bars to solve the space problemand have enough rigidity for machine tools. Therefore, a CNClathe whose frame consists of pipes, joints, and diagonal braceshas been developed with enough rigidity and space utility for chipevacuation. From the viewpoint of machine tool usage, real-timevibration control theory is applied to control the relative displace-ment between the tool post and the spindle to suppress specificrelative vibration modes. DOI: 10.1115/1.4027594IntroductionConventional beds or frames for machine tools generally con-sist of cast iron or welded steel plates. These structures are suita-ble from the viewpoint of the machine tools dynamiccharacteristics. Although the demand for downsizing machinetools (such as in desktop machines) is increasing, there is a strictvolumetric limitation to obtain enough space for sufficient chipevacuation. In this case, there are problems of heat conductionand heat transfer. To solve these problems, we propose a tradi-tional but new frame structure for machine tools that consists ofpipes and connecting parts. We looked at the flexibility of theframe design using a pipe frame structure. The frame structurecan be arranged with standardized pipe (diameter and length) anda connecting block. In the case of a lathe, this structure allows fora change in size based on the spindle unit and X-axis table, whichare designed as standardized units.For developing a CNC lathe using this structure 1, the follow-ing problems are assumed:(1) structural vibration generated by the spindle motor unbal-ance and the cutting force disturbance,(2) thermal deformation due to the low heat capacity,(3) insufficient rigidity between the spindle and the X- and Zaxis tables.In this report, we focus on the vibration problem. There havebeen several prior studies 27 on the applications of vibrationcontrol for machine tools. In these research papers, the controlmethods have been applied to the tool post of the lathe machine.These results demonstrated the superiority of vibration control ofthe tool post displacement but do not consider the structural vibra-tion modes that appear in the low-frequency range compared withthe local vibration modes, such as the boring bar and tool post.Developed CNC LatheA unique CNC lathe has been developed based on the idea of apipe frame structure. The specifications are shown in Table 1, anda schematic view is shown in Fig. 1.A maximum spindle speed of 10,000 minC01can be obtained.The smallest resolution for the X and Z axes is set to 0.1lm. Thelathe weighs only 525 kg (including the covers) and fits within aspace of 440 WC21450 LC21050 H mm3, which is sufficientlycompact. The CNC controller (Mitsubishi Electric Co., Ltd.,M700V) is set beside the machine body.Vibration Control MethodThe most important vibratory motion for the CNC lathe is therelative displacement between the spindle and the tool post. Themachined workpiece is often influenced by the antiphase motionbetween the spindle and the tool; this motion is generated by theresonant frequency. In this case, the antiphase motion is the mostimportant behavior for the workpieces geometric error. Thismotion can be evaluated by measuring the relative vibrationbetween the spindle and the tool. Although this displacementshould be measured directly by a displacement sensor during cut-ting, it is very difficult to position the sensor because of the spaceconstraints, coolant, and chips. Figure 2 shows the typical compli-ance transfer function between the spindle head and the represen-tative frame-connected block. There are several peaks. When wetry to apply vibration control using this data, the control point ishardly determined. Then, two small acceleration sensors are usedto detect the relative vibration mode. One is set on the spindle unitcase and is measured as the signal xs; the other is set on the toolpost of the X-axis table and is measured as the signal xt.The relative vibration x is expressed as follows:x xsC0 xt(1)Then, the difference between the two acceleration sensor meas-urements in Eq. (1) is calculated and used to extract the antiphasemotion between them. The result of the calculated transfer func-tion is shown in Fig. 3. A frequency peak at 45 Hz can beobserved, and it can be considered a simple one-degree-of-free-dom system. We can use this differential signal as the output ofthe vibration response.Then, modal analysis can be used to obtain the vibration modes,to decide the target vibration mode, and to determine where to setthe actuator for the vibration control. The synthesized mode shapevector at 45 Hz is shown in Fig. 4. The antiphase mode betweenthe spindle and the tool post is actually seen from the vectorContributed by the Manufacturing Engineering Division of ASME for publicationin the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedFebruary 20, 2013; final manuscript received April 11, 2014; published online May21, 2014. Assoc. Editor: Tony Schmitz.Journal of Manufacturing Science and Engineering AUGUST 2014, Vol. 136 / 044502-1CopyrightVC2014 by ASMEDownloaded From: / on 05/26/2014 Terms of Use: /termsntsfigure. The mode vectors at the head stock for the spindle ( andfi) are so small that the ends of the two actuators can be set atthese points. Based on this result, the other ends of the actuatorsare set on connecting blocks below the X-axis table to control theantiphase vibration mode, as shown in Fig. 1. These two actuatorsgenerate the control forces to suppress the vibratory motion by theactive vibration control method.The relative displacement, velocity, and acceleration aredefined as x, _x, and x, respectively. Assuming that xdand _xddenote the desired relative displacement and the relative velocity,the relative displacement and the relative velocity are defined asxdC0x;_xdC0 _xThe controlled force can be expressed as follows:Kv_xdC0 _xKdxdC0x (2)where the weight coefficients Kdand Kvare used. The equation ofmotion is described as follows when a harmonic force faejxtactson this system:mxc_xkx faejxtKv_xdC0 _xKdxdC0x (3)Because the desired relative displacement xdand relativevelocity _xdare to be zero, Eq. (3) is described as follows:mx cKv_x k Kdx faejxt(4)In Eq. (2), substituting f faejxt, the following equation can bederived:x_xC26C27C0c=m C0k=m10C20C21_xxC26C271=m0C20C21f (5)This equation of motion is rewritten as the following stateequation:_X AXBF (6)In the case of applying a control force that is proportional to thestate vector X, the control force is described as F C0KX.Then,substituting this into Eq. (6), the following state equation is obtained:_X AC0BKX (7)Once this equation can be obtained, the only thing to dois calculate the feedback gain K using MATLAB (MathWorksMATLAB and Simulink). To calculate this coefficient K, the modalparameters m, c, and k must first be determined. We calculatedthese values using the simple “mass response method” becausethe vibratory motion can be assumed to be one-degree-of-freedomfrom the result in Fig. 3. Once the known mass is set on the toolFig. 1 Schematic view of developed CNC latheFig. 2 Typical measurement result of compliance transferfunction by conventional impulse excitation method on a repre-sentative pointFig. 3 Transfer function between input force and relative dis-placement (tool post and spindle)Fig. 4 Mode shape vector (top view)Table 2 Identified modal parameterItems Value UnitMass, m 214 kgDamping factor, c 8.01 N/(m/s)Spring constant, k 1.62C21007N/mKv3.66C21003N/m/sKd5.73C21002N/(m/s)Table 1 Specifications of CNC latheItems SpecificationsHead stock Chuck Collet chuckMaximum speed 10000 minC01Tool post type Horizontal linearMinimum resolution 0.0001 mmSize 440 WC21450 LC21050 H mmMachine weight 525 kgCNC controller Mitsubishi Electric Co., Ltd. M700V044502-2 / Vol. 136, AUGUST 2014 Transactions of the ASMEDownloaded From: / on 05/26/2014 Terms of Use: /termsntspost, the change in the resonant frequency is measured, and themodal parameters are obtained, as shown in Table 2. These aresubstituted into Eq. (7), and the feedback gain K is calculatedaccording to linear-quadratic (LQ) control theory.Vibration Control SystemLead zirconate titanate (PZT) devices are used in the vibrationcontrol system as the actuator source. Structural vibration controland motion control for machine tools use this device because ofits large generated force and high-frequency response 811.We developed an actuator that consists of a PZT, a fluid cham-ber with two different cross-sectional areas, and connecting rods,as shown in Fig. 5. The fluid chamber is used to transform thegenerated force from the PZT. Although the longitudinal displace-ment of the PZT is short enough, the displacement is reduced bythe chamber area. This effect makes the generated displacementless than a submicron order. Two actuators are set between theconnecting blocks as brace bars. The dynamic characteristics areevaluated using the impulse response method, and the result isshown in Fig. 6. This actuator can be used below 200 Hz. Theeffects of the vibration control are evaluated in Effects of ActiveVibration Control section. Although the thermal elongation of thebrace bars has to be considered when the long run operation, wedo not consider the thermal deformation of the frame because ofthe short-time machining.Effects of Active Vibration ControlThe relative vibration between the spindle and the tool post iscontrolled by two actuators. In this case, according to LQ controltheory, we have developed the control system to realize thecontrol method represented by the block diagram in Fig. 7 and inTable 3. This system runs with a sampling speed of 3000 Hz. Therelative vibration is generated by the unbalance of the spindleunit. Since the maximum displacement is observed at around2725 minC01(45.4 Hz), the target frequency is set to 45 Hz.The experimental procedure has been executed during air cut-ting and during cutting. From the experimental result and fre-quency response of the actuator, the target-controlled frequencyrange is set from 10 to 100 Hz. A band-pass real-time filter isadapted to obtain the vibration signal and to suppress the disturb-ance noise and high-frequency signals.To evaluate the effect of the active vibration control, two accel-eration sensors due to the limitation of the installation space areused for the control. The relative vibration is numerically inte-grated to obtain the relative velocity and the relative displace-ment. The relative displacement itself during motion is measuredby a laser displacement sensor that is set on the tool post anddetects the relative displacement of the spindle unit.The control result during finish cutting is shown in Fig. 8. Thecontrol parameters have been determined previously during aircutting. This condition is not always applicable to the conditionduring cutting because the workpiece and the tool contact eachother, and the cutting force acts on the contact point. Although themodal parameters are not always constant, our control system isapplied using the same conditions as air cutting, and the effect ofthe vibration control is evaluated during cutting.When the control is started, the relative displacement decreasesits amplitude. FFT analysis is used to determine the difference inthe frequency element between with control and without control,as shown in Fig. 9. The solid line shows the result with control,and the dotted line shows the result without control. Almost 50%of the vibration amplitude around the target frequency of 45 Hz issuppressed by our developed system. The effect of an increase inrigidity by the actuator installed as a brace can be seen in thelower frequency range. On the other hand, the effect above100 Hz cannot be seen because of the filter effect of the controlFig. 5 Schematic diagram of PZTactuator for vibration controlFig. 6 Transfer function between input voltage of actuator andtable displacement by impulse responseFig. 7 Block diagram for vibration controlTable 3 Experimental setup for vibration controlItems Value UnitSpindle speed 2700 minC01Target frequency 45 HzSampling frequency 3000 HzDepth of cut 0.1 MmFeed 0.02 mm/revWorkpiece material C3604BDFig. 8 Comparison of relative motion during cuttingJournal of Manufacturing Science and Engineering AUGUST 2014, Vol. 136 / 044502-3Downloaded From: / on 05/26/2014 Terms of Use: /termsntssystem. These results demonstrate that this vibration control sys-tem used during finish cutting is suitable for our pipe frame CNClathe. To improve the effect of the vibration control, the layout(including the direction of the actuator) should be considered, andfurther improvement of this actuator system is needed to suppressthe vibration amplitude more efficiently.Evaluation of Workpiece by Vibration ControlSince it has been confirmed that the vibration control is effec-tive even during cutting, we evaluated the accuracy of themachined workpiece. Because the target control frequency wasset to 45 Hz, the effect of the control result appears as surfacewaviness. Then, the surface profile was measured by the surfaceroughness tester without a high-pass filter. Figure 10 shows themeasured results. The upper figure shows the profile withoutvibration control, and the lower figure shows the result with vibra-tion control. The surface waviness is remarkably improved. Theimprovement of the waviness at this frequency range is related tothe relative vibration mode that appears in this frequency range.Figure 11 shows the harmonic analysis of the measured round-ness result. Since the target frequency is set to the spindle rota-tional frequency, the effect of the vibration control generallyappears on the roundness with two undulations per round. Themeasured result shows this tendency well. The amplitude of theundulations per round decreased by 50% from the amplitude with-out control.These results show the effectiveness of our control system atsuppressing the relative vibration mode between the spindle andthe tool post and the improvement in the accuracy of the machinedworkpiece.ConclusionsIn this paper, the effectiveness of the real-time vibration controlfor the developed CNC lathe by a single degrees of freedommodel has been evaluated using the differential signal between thespindle and the tool post. The main results obtained as follows:(1) Almost 50% of the vibration amplitude around the targetfrequency of 45 Hz is suppressed by our developed system.(2) The roundness amplitude of the undulations per rounddecreased by 50% from the amplitude without control.(3) Results obtained show the effectiveness of the applied con-trol system at suppressing the relative vibration modebetween the spindle and the tool post.Further investigations are required to improve the vibrationcontrol system, including the layout (the direction of the actuator)and actuator performance to suppress the vibration amplitudemore efficiently.AcknowledgmentThe authors express their gratitude for the assistance of Kazu-hiro Shimizu, who has supported our experiments. This stud
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