- Email: [email protected]
Structural damper for auto-damping mechanical components
Structures 24 (2020) 864–868
Contents lists available at ScienceDirect
Structures journal homepage: www.elsevier.com/locate/structures
Structural damper for auto-damping mechanical components a,⁎
Yasunori Sakai , Tomohisa Tanaka a b
T
b
Department of Machinery & Control Systems, College of Systems Engineering & Science, Shibaura Institute of Technology, Japan Department of Mechanical Engineering, School of Engineering, Tokyo Institute of Technology, Japan
A R T I C LE I N FO
A B S T R A C T
Keywords: Passive damper Structural damper Composite Vibration absorption Vibration mode control
This paper describes a direct passive vibration control approach for mechanical components. The passive damper with anisotropic composite structure (P-DACS) is a composite of metal and polymer, and it can be directly integrated into mechanical components. P-DACS can control static and dynamic characteristics with correctly tuned design parameters and material characteristics. Moreover, the damping can be increased without decreasing static stiffness when optimal design parameters and material are selected. Thus, P-DACS can add autodamping of static and dynamic vibrations to a structural component. Furthermore, the damping ability can be increased simultaneously for multiple elastic vibration modes. Simulations clarifying the characteristics of PDACS were conducted to indicate the feasibility of integrating P-DACS into structural components.
1. Introduction In many industrial fields, unwanted vibration of mechanical components deteriorates the performance of mechanical systems. For instance, in the machine tool industry, structural vibration of components, such as spindles, tables, columns, beds, and joints, degrades manufacturing accuracy and causes chatter vibrations [1]. Thus, the reduction of structural vibration is a very important challenge in precise mechanical systems. Conventional vibration reduction methods include passive and active dampers [2–7], which are added to mechanical structures and machine elements. These conventional methods, however, have the following drawbacks: 1. The damper needs to anticipate the dynamic characteristics of the isolated object, so the damper design must be optimized and tailored to specific isolated objects. 2. The damper needs to be attached to the antinode of the vibration mode precisely for obtaining its optimized performance. In addition, a different damper is necessary for each vibration mode. These problems are caused by the need to add conventional dampers to structural components. Hence, this study proposes a novel approach for directly controlling vibration by integrating a multimode damper into mechanical components. In this approach, many passive damper cells, or structural dampers (SDs), are directly integrated into mechanical components. This paper
⁎
first provides an overview of SDs and describes their advantages. Then, the developed SD, the passive damper with anisotropic structure (PDACS), is introduced. The control of static and dynamic vibration characteristics by a single P-DACS integrated into a mechanical component is discussed. Finally, several ideas for future work are provided. 2. Structural dampers 2.1. Application of SDs At first, an example application of SDs to linear rolling guideways, which are widely used for machine tools, robots, and medical instruments is discussed. When an external force is exerted on a linear guideway, the carriage vibrates with some elastic vibration modes. Since this vibration can cause posture errors in mechanical systems, such as feed drives, it reduces motion stability and system performance. Especially in machine tools, this vibration seriously deteriorates the cutting performance and positioning accuracy. Hence, increasing the damping capacity of the carriage would be very beneficial for improving the performance of such mechanical systems. Fig. 1 illustrates the linear rolling guideway in our example. The conventional carriage without a SD is made of a solid material, such as chromium molybdenum steel. Thus, material damping is extremely low. Additionally, conventional dampers cannot be attached because the available space is too narrow. In contrast, SDs integrated into the carriage improves its ability to damp itself when they are optimally placed into the design volume of the carriage. Integrated SD
Corresponding author. E-mail address: [email protected] (Y. Sakai).
https://doi.org/10.1016/j.istruc.2020.02.012 Received 14 September 2019; Received in revised form 9 February 2020; Accepted 10 February 2020 2352-0124/ © 2020 The Author(s). Published by Elsevier Ltd on behalf of Institution of Structural Engineers. This is an open access article under the CC BY license (http://creativecommons.org/licenses/BY/4.0/).
Structures 24 (2020) 864–868
Y. Sakai and T. Tanaka
Fig. 1. Application of structural damper in linear rolling guideway.
by both geometrical design parameters and material properties. The geometrical definition of the metal structure is shown in Fig. 2(d). To clarify the influence of these parameters and material properties, the static and dynamic characteristics were investigated by finite element analysis using COMSOL Multiphysics® software. In the analysis, the material of the metal part was assumed to be maraging steel (Young’s modulus: 189 GPa, Poisson’s ratio ν = 0.33, Loss factor η = 0.05), which is widely used in additive manufacturing. These polymers have the same Poisson’s ratio (ν = 0.33) and density (ρ = 30 kg/m3) but have different Young’s modulus E and loss factors η. These values are E = 30 MPa and η = 1 for material A, E = 300 MPa and η = 0.1 for material B, E = 3000 MPa and η = 0.01 for material C. Additionally, the following assumptions were made to simplify the analytical model:
components do not need additional spaces and apparatuses for reducing vibration for multiple modes of the carriage. 2.2. Proposed P-DACS If an SD is made as a composite of materials, one with high stiffness and the other with high visco-elasticity, it attains high damping without decreasing static stiffness. Thus, an SD as a composite of metal and polymer is developed. Fig. 2 illustrates the SD developed in this study. The metal part can be uniformly fabricated inside of a mechanical component by additive manufacturing technology [8]. Since the damping capacity of the SD is mainly generated by the damping properties of the polymer, increasing elastic strain energy generated by elastic deformation in the polymer is important. Thus, the lattices in the metal component are designed to generate large deformation of the SD by anisotropic static stiffness. Translational motion of the metal part and rotational motion of the tuning mass are synchronously generated. This motion is called “squeeze motion”, and it gives polymers the ability to undergo complex deformations, such as tensile/compressive, torsional, and shear.
1. Friction damping of the contact surface between metal and polymer parts was ignored 2. All materials were assumed to be linearly isotropic elastic materials As the boundary conditions, the bottom surface of the metal part was perfectly fixed, and the top surface is not fixed. An external force of 100 N was applied to the top surface of the metal part in a vertical direction. This force was compressive and sinusoidal for static and dynamic analysis, respectively. In the dynamic analysis, the half-amplitude was set to 100 N. The static characteristic was evaluated by specific static stiffness Ks
3. Controllability of static and dynamic characteristics 3.1. Design parameters and evaluation method The static and dynamic characteristics of P-DACS can be controlled
Fig. 2. Construction of passive damper with anisotropic composite structure (P-DACS). (a) Metal part; (b) Polymer part; (c) P-DACS; (d) Definition of design parameters. 865
Structures 24 (2020) 864–868
Y. Sakai and T. Tanaka
Fig. 4. Controllable range of stiffness and damping with single P-DACS. The plots for each material indicate the result for a different inclination angle θ and polymers (α = 1.0, n = 20, γ = 0.4).
Fig. 3. Influence of inclination angle θ on frequency response function of single P-DACS in the vertical direction (polymer is material A with low stiffness, and other geometrical parameters are fixed).
These results indicate the combination of the material and geometrical parameter is important for optimizing the performance of proposed damper. The influence of combination of material and the geometrical parameters will be indicated in our future work. In Fig. 4, the calculation results obtaining the same static stiffness are indicated by a vertical yellow bar. As can be seen, the damping ratio increased without a decrease in the static stiffness for polymer material without any geometrical dimension change. Controllability of the dynamic characteristics of the damper without any geometrical dimension changing is one of the main advantages of P-DACS. According to the same evaluation for the metal structure without polymer material, the damping ratio is 0.05, specific stiffness is almost same value as for Material A. These results indicate the damping capacity of the polymer material dedicates the damping capacity of PDACS. However, the specific stiffness of P-DACS is not changed if the Young’s modulus of polymer material is low. Next, the controllability of the dynamic characteristics of a table supported by two P-DACS, as shown in the upper right corner of Fig. 5, were investigated to evaluate the feasibility of integrating the SD into mechanical components. In the finite element analysis, the bottom surface of each P-DACSs was perfectly fixed, and the table and top surface of the P-DACS were glued together. An excitation force was applied to the center of the table. Fig. 5 shows the FRFs at the center of the table for different lattice inclination angles θ. Inclination angle of the lattice is defined in the Fig. 2(d). The direction of inclination is same for two P-DACSs used in this paper. Two resonance peaks caused by bending vibration modes in the table were observed in the FRF. Both the frequencies and peaks for each resonance varied with θ. The compliance at first resonance peak is not decrease by the P-DACS, while the compliance at the second vibration mode decreases by P-DACS that has large inclination angle θ. These results indicated that the damping effect by P-DACS is different depending on the vibration mode. The compliance of the first bending mode does not increase when the resonance frequency becomes low. In general, the lower resonance frequency, the larger the compliance amplitude in the linear vibration system. However, even when the resonance frequency in the result is low (when the inclination angle is large), the compliance of the first elastic vibration mode has not been increased. It is caused by the increase of the amount of elastic strain energy generated in the polymer material as the response amplitude increases. These results mean that the damping characteristics for each elastic vibration mode increase. However, the damping effect of P-DACS is different in each vibration mode. For this important problem, authors
defined by Eq. (1)
Ks =
F0 mxst
(1)
where xst is displacement of the upper side plate in the vertical direction, F0 is static compressive load, and m is the mass of P-DACS. The dynamic characteristics were evaluated using a frequency response function (FRF) and damping ratio ζ calculated by the modal strain energy method [9]. 3.2. Evaluation results In this study, four parameters were chosen: number of lattices n, lattice inclination angle θ, cross-sectional shape of the lattice (γ = Ll/ rm), and height ratio between tuning mass and lattice (α = Hm/Hl). Fig. 3 shows the influence of θ on the FRF with material A used for the polymer part. A resonance peak was observed in the FRF. This resonance is caused by the natural vibration that P-DACS exhibits with the squeeze motion shown in Fig. 3. In addition, the frequency and amplitude at resonance were drastically changed by θ. Similarly, other design parameters also changed the frequency and amplitude at resonance over a wide range. These results indicated that the design parameters can control the dynamic characteristics of P-DACS. The controllable range of specific stiffness and damping ratio at resonance in the vertical direction is shown in Fig. 4. In this figure, calculated results for different inclination angle θ (5°–40°) and polymer materials (A, B and C) are indicated. Geometrical parameters excluding θ is fixed as γ = 0.4, n = 20, α = 1.0. When the polymer material was changed, the range of specific stiffness and damping ratio controllable by geometrical parameters also changed. If a polymer material of low stiffness was used, the range of stiffness controllable by design parameters became narrow, but the damping ratio increased. In addition, the stiffness decrease, and the damping ratio decreases along with the increase of the inclination angle θ. When θ is large, the elastic deformation of the polymer material increases because of the large deformation of damper. Thus, the damping ratio increase with increase of θ. The controllable range of static stiffness is more sensitive to the inclination angle when the softer material (Material C) was used. On the other hand, the controllable range of damping ratio is more sensitive to the inclination angle when the harder material (Material A) was used. 866
Structures 24 (2020) 864–868
Y. Sakai and T. Tanaka
Fig. 5. Influence of inclination angle θ of P-DACS metal part on frequency response function of midpoint of table structure in vertical direction (polymer is material C with higher stiffness, and other geometrical parameters are fixed).
DACS. These results indicate the possibility of integrating an SD into mechanical components. The numerical consideration of this paper was performed on the assumption that the deformation direction of the damper and the deformation direction of the structure were the same. However, since the actual structure has many vibration modes, the deformation directions are various. Therefore, above mentioned assumption will be broken. As the solution for the problem, the optimization of the orientation angle of the damper cell according to the vibration mode of structures should be considered. If the small damper cells are stacked in the structure with various orientation, the damping will be improved in many vibration modes because the damper that obtains the damping effect changes according to the vibration mode of the structure. In the future work, a method for optimizing the orientation angle of the damper cell will investigate theoretically and experimentally according to the dynamic deformation of the structure. In addition, the actual dynamic and static mechanical characteristics of the structure incorporating the damper will be clarified in more detail.
will try to embed the small size P-DACS into whole volume of structure part in the future work. Since a specific small P-DACSs embedded in the structure part are deformed according to the vibration mode, the effective damper is selected by the self-selective manner depends on the vibration mode. Finally, high damping effect will be obtained in all elastic vibration modes of structure. The damping effect can be controlled by optimizing the P-DACS design and placing position. Controlling the damping capacity of the small amplitude vibration in structural component is difficult because the energy dissipation caused by the elastic deformation of damping material is less. The large elastic deformation in tensile/compress and shear direction of the damping material (polymer material in the paper) can be obtained by the rotation of the tuning mass when the inclination angle is large. Thus, the compliance of the second bending vibration mode becomes small with high inclination angle. If there is no damper (if using a metal block with same outer dimensions of the damper), the stiffness becomes higher than that of PDACS. In addition, the damping is almost the same as the damping capacity of the material damping of metal if the friction at the joint part is not in consideration. The damping becomes too small than that of PDACS. Therefore, the natural frequency without the damper is higher than that with damper, and the resonance amplitude is higher. The structure damper, P-DACS can increase the damping of structures instead of little lowering the static stiffness. The one of important advantages of P-DACS is that the dynamic characteristics of damper and damper embedded structure can be adjusted without changing the external dimensions by adjusting the internal geometry. In this paper, the numerical analysis assuming there is no friction between jointed parts are performed. Although the friction damping of the joint parts is important for evaluating the vibration, it is generally difficult to modeling in finite element analysis. Therefore, the comparison of the structure's dynamic characteristics with/without P-DACS will be examined in detail in future research by experiments.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by Grants-in-Aid from NSK Foundation for the Advancement of Mechatronics. In addition, the authors especially thank NSK Ltd. and Matsuura Machinery Corp. References [1] Hosseinabadi AHH, Altintas Y. Modeling and active damping of structural vibrations in machine tools. CIRP J Manuf Sci Technol 2014;7:246–57. [2] Brecher C, Fey M, Bäumler S. Damping models for machine tool components of linear axes. CIRP Ann 2013;62:399–402. [3] Zhang SJ, Tob S, Zhang GQ, Zhu ZW. A review of machine-tool vibration and its influence upon surface generation in ultra-precision machining. Int J Mach Tools Manuf 2015;91:34–42. [4] Ema S, Marui E. Suppression of chatter vibration of boring tools using impact dampers. Int J Mach Tools Manuf 2000;40:1141–56. [5] Slocum AH, Marsh ER, Smith DH. A new damper design for machine tool structures: the replicated internal viscous damper. Precis Eng 1994;16:174–83. [6] Fei J, Lin B, Yan S, Ding M, Xiao J, Zhang J, et al. Chatter mitigation using moving damper. J Sound Vib 2017;410:49–63. [7] Wang M. Feasibility study of nonlinear tuned mass damper for machining chatter suppression. J Sound Vib 2011;330:1917–30.
4. Conclusion In this study, a direct damping approach for controlling static and dynamic characteristics of mechanical components, P-DACS, which can be integrated into mechanical components is proposed. P-DACS is a composite of metal and polymer, and it controls static and dynamic vibration characteristics using the geometrical design parameters of the metal part and material characteristics of the polymer part. Additionally, it is clarified through simulation that a vibration response in a table supported by P-DACS could be controlled by the design of P867
Structures 24 (2020) 864–868
Y. Sakai and T. Tanaka
[8] Tan XP, Tan YJ, Chow CSL, Tor SB, Yeong WY. Metallic powder-bed based 3D printing of cellular scaffolds for orthopaedic implants: a state-of-the-art review on manufacturing, topological design, mechanical properties and biocompatibility. Mater Sci Eng, C 2017;76:1328–43.
[9] Davis CL, Lesieutre GA. A modal strain energy approach to the prediction of resistively shunted piezoceramic damping. J Sound Vib 1995;184:129–39.
868
Contents lists available at ScienceDirect
Structures journal homepage: www.elsevier.com/locate/structures
Structural damper for auto-damping mechanical components a,⁎
Yasunori Sakai , Tomohisa Tanaka a b
T
b
Department of Machinery & Control Systems, College of Systems Engineering & Science, Shibaura Institute of Technology, Japan Department of Mechanical Engineering, School of Engineering, Tokyo Institute of Technology, Japan
A R T I C LE I N FO
A B S T R A C T
Keywords: Passive damper Structural damper Composite Vibration absorption Vibration mode control
This paper describes a direct passive vibration control approach for mechanical components. The passive damper with anisotropic composite structure (P-DACS) is a composite of metal and polymer, and it can be directly integrated into mechanical components. P-DACS can control static and dynamic characteristics with correctly tuned design parameters and material characteristics. Moreover, the damping can be increased without decreasing static stiffness when optimal design parameters and material are selected. Thus, P-DACS can add autodamping of static and dynamic vibrations to a structural component. Furthermore, the damping ability can be increased simultaneously for multiple elastic vibration modes. Simulations clarifying the characteristics of PDACS were conducted to indicate the feasibility of integrating P-DACS into structural components.
1. Introduction In many industrial fields, unwanted vibration of mechanical components deteriorates the performance of mechanical systems. For instance, in the machine tool industry, structural vibration of components, such as spindles, tables, columns, beds, and joints, degrades manufacturing accuracy and causes chatter vibrations [1]. Thus, the reduction of structural vibration is a very important challenge in precise mechanical systems. Conventional vibration reduction methods include passive and active dampers [2–7], which are added to mechanical structures and machine elements. These conventional methods, however, have the following drawbacks: 1. The damper needs to anticipate the dynamic characteristics of the isolated object, so the damper design must be optimized and tailored to specific isolated objects. 2. The damper needs to be attached to the antinode of the vibration mode precisely for obtaining its optimized performance. In addition, a different damper is necessary for each vibration mode. These problems are caused by the need to add conventional dampers to structural components. Hence, this study proposes a novel approach for directly controlling vibration by integrating a multimode damper into mechanical components. In this approach, many passive damper cells, or structural dampers (SDs), are directly integrated into mechanical components. This paper
⁎
first provides an overview of SDs and describes their advantages. Then, the developed SD, the passive damper with anisotropic structure (PDACS), is introduced. The control of static and dynamic vibration characteristics by a single P-DACS integrated into a mechanical component is discussed. Finally, several ideas for future work are provided. 2. Structural dampers 2.1. Application of SDs At first, an example application of SDs to linear rolling guideways, which are widely used for machine tools, robots, and medical instruments is discussed. When an external force is exerted on a linear guideway, the carriage vibrates with some elastic vibration modes. Since this vibration can cause posture errors in mechanical systems, such as feed drives, it reduces motion stability and system performance. Especially in machine tools, this vibration seriously deteriorates the cutting performance and positioning accuracy. Hence, increasing the damping capacity of the carriage would be very beneficial for improving the performance of such mechanical systems. Fig. 1 illustrates the linear rolling guideway in our example. The conventional carriage without a SD is made of a solid material, such as chromium molybdenum steel. Thus, material damping is extremely low. Additionally, conventional dampers cannot be attached because the available space is too narrow. In contrast, SDs integrated into the carriage improves its ability to damp itself when they are optimally placed into the design volume of the carriage. Integrated SD
Corresponding author. E-mail address: [email protected] (Y. Sakai).
https://doi.org/10.1016/j.istruc.2020.02.012 Received 14 September 2019; Received in revised form 9 February 2020; Accepted 10 February 2020 2352-0124/ © 2020 The Author(s). Published by Elsevier Ltd on behalf of Institution of Structural Engineers. This is an open access article under the CC BY license (http://creativecommons.org/licenses/BY/4.0/).
Structures 24 (2020) 864–868
Y. Sakai and T. Tanaka
Fig. 1. Application of structural damper in linear rolling guideway.
by both geometrical design parameters and material properties. The geometrical definition of the metal structure is shown in Fig. 2(d). To clarify the influence of these parameters and material properties, the static and dynamic characteristics were investigated by finite element analysis using COMSOL Multiphysics® software. In the analysis, the material of the metal part was assumed to be maraging steel (Young’s modulus: 189 GPa, Poisson’s ratio ν = 0.33, Loss factor η = 0.05), which is widely used in additive manufacturing. These polymers have the same Poisson’s ratio (ν = 0.33) and density (ρ = 30 kg/m3) but have different Young’s modulus E and loss factors η. These values are E = 30 MPa and η = 1 for material A, E = 300 MPa and η = 0.1 for material B, E = 3000 MPa and η = 0.01 for material C. Additionally, the following assumptions were made to simplify the analytical model:
components do not need additional spaces and apparatuses for reducing vibration for multiple modes of the carriage. 2.2. Proposed P-DACS If an SD is made as a composite of materials, one with high stiffness and the other with high visco-elasticity, it attains high damping without decreasing static stiffness. Thus, an SD as a composite of metal and polymer is developed. Fig. 2 illustrates the SD developed in this study. The metal part can be uniformly fabricated inside of a mechanical component by additive manufacturing technology [8]. Since the damping capacity of the SD is mainly generated by the damping properties of the polymer, increasing elastic strain energy generated by elastic deformation in the polymer is important. Thus, the lattices in the metal component are designed to generate large deformation of the SD by anisotropic static stiffness. Translational motion of the metal part and rotational motion of the tuning mass are synchronously generated. This motion is called “squeeze motion”, and it gives polymers the ability to undergo complex deformations, such as tensile/compressive, torsional, and shear.
1. Friction damping of the contact surface between metal and polymer parts was ignored 2. All materials were assumed to be linearly isotropic elastic materials As the boundary conditions, the bottom surface of the metal part was perfectly fixed, and the top surface is not fixed. An external force of 100 N was applied to the top surface of the metal part in a vertical direction. This force was compressive and sinusoidal for static and dynamic analysis, respectively. In the dynamic analysis, the half-amplitude was set to 100 N. The static characteristic was evaluated by specific static stiffness Ks
3. Controllability of static and dynamic characteristics 3.1. Design parameters and evaluation method The static and dynamic characteristics of P-DACS can be controlled
Fig. 2. Construction of passive damper with anisotropic composite structure (P-DACS). (a) Metal part; (b) Polymer part; (c) P-DACS; (d) Definition of design parameters. 865
Structures 24 (2020) 864–868
Y. Sakai and T. Tanaka
Fig. 4. Controllable range of stiffness and damping with single P-DACS. The plots for each material indicate the result for a different inclination angle θ and polymers (α = 1.0, n = 20, γ = 0.4).
Fig. 3. Influence of inclination angle θ on frequency response function of single P-DACS in the vertical direction (polymer is material A with low stiffness, and other geometrical parameters are fixed).
These results indicate the combination of the material and geometrical parameter is important for optimizing the performance of proposed damper. The influence of combination of material and the geometrical parameters will be indicated in our future work. In Fig. 4, the calculation results obtaining the same static stiffness are indicated by a vertical yellow bar. As can be seen, the damping ratio increased without a decrease in the static stiffness for polymer material without any geometrical dimension change. Controllability of the dynamic characteristics of the damper without any geometrical dimension changing is one of the main advantages of P-DACS. According to the same evaluation for the metal structure without polymer material, the damping ratio is 0.05, specific stiffness is almost same value as for Material A. These results indicate the damping capacity of the polymer material dedicates the damping capacity of PDACS. However, the specific stiffness of P-DACS is not changed if the Young’s modulus of polymer material is low. Next, the controllability of the dynamic characteristics of a table supported by two P-DACS, as shown in the upper right corner of Fig. 5, were investigated to evaluate the feasibility of integrating the SD into mechanical components. In the finite element analysis, the bottom surface of each P-DACSs was perfectly fixed, and the table and top surface of the P-DACS were glued together. An excitation force was applied to the center of the table. Fig. 5 shows the FRFs at the center of the table for different lattice inclination angles θ. Inclination angle of the lattice is defined in the Fig. 2(d). The direction of inclination is same for two P-DACSs used in this paper. Two resonance peaks caused by bending vibration modes in the table were observed in the FRF. Both the frequencies and peaks for each resonance varied with θ. The compliance at first resonance peak is not decrease by the P-DACS, while the compliance at the second vibration mode decreases by P-DACS that has large inclination angle θ. These results indicated that the damping effect by P-DACS is different depending on the vibration mode. The compliance of the first bending mode does not increase when the resonance frequency becomes low. In general, the lower resonance frequency, the larger the compliance amplitude in the linear vibration system. However, even when the resonance frequency in the result is low (when the inclination angle is large), the compliance of the first elastic vibration mode has not been increased. It is caused by the increase of the amount of elastic strain energy generated in the polymer material as the response amplitude increases. These results mean that the damping characteristics for each elastic vibration mode increase. However, the damping effect of P-DACS is different in each vibration mode. For this important problem, authors
defined by Eq. (1)
Ks =
F0 mxst
(1)
where xst is displacement of the upper side plate in the vertical direction, F0 is static compressive load, and m is the mass of P-DACS. The dynamic characteristics were evaluated using a frequency response function (FRF) and damping ratio ζ calculated by the modal strain energy method [9]. 3.2. Evaluation results In this study, four parameters were chosen: number of lattices n, lattice inclination angle θ, cross-sectional shape of the lattice (γ = Ll/ rm), and height ratio between tuning mass and lattice (α = Hm/Hl). Fig. 3 shows the influence of θ on the FRF with material A used for the polymer part. A resonance peak was observed in the FRF. This resonance is caused by the natural vibration that P-DACS exhibits with the squeeze motion shown in Fig. 3. In addition, the frequency and amplitude at resonance were drastically changed by θ. Similarly, other design parameters also changed the frequency and amplitude at resonance over a wide range. These results indicated that the design parameters can control the dynamic characteristics of P-DACS. The controllable range of specific stiffness and damping ratio at resonance in the vertical direction is shown in Fig. 4. In this figure, calculated results for different inclination angle θ (5°–40°) and polymer materials (A, B and C) are indicated. Geometrical parameters excluding θ is fixed as γ = 0.4, n = 20, α = 1.0. When the polymer material was changed, the range of specific stiffness and damping ratio controllable by geometrical parameters also changed. If a polymer material of low stiffness was used, the range of stiffness controllable by design parameters became narrow, but the damping ratio increased. In addition, the stiffness decrease, and the damping ratio decreases along with the increase of the inclination angle θ. When θ is large, the elastic deformation of the polymer material increases because of the large deformation of damper. Thus, the damping ratio increase with increase of θ. The controllable range of static stiffness is more sensitive to the inclination angle when the softer material (Material C) was used. On the other hand, the controllable range of damping ratio is more sensitive to the inclination angle when the harder material (Material A) was used. 866
Structures 24 (2020) 864–868
Y. Sakai and T. Tanaka
Fig. 5. Influence of inclination angle θ of P-DACS metal part on frequency response function of midpoint of table structure in vertical direction (polymer is material C with higher stiffness, and other geometrical parameters are fixed).
DACS. These results indicate the possibility of integrating an SD into mechanical components. The numerical consideration of this paper was performed on the assumption that the deformation direction of the damper and the deformation direction of the structure were the same. However, since the actual structure has many vibration modes, the deformation directions are various. Therefore, above mentioned assumption will be broken. As the solution for the problem, the optimization of the orientation angle of the damper cell according to the vibration mode of structures should be considered. If the small damper cells are stacked in the structure with various orientation, the damping will be improved in many vibration modes because the damper that obtains the damping effect changes according to the vibration mode of the structure. In the future work, a method for optimizing the orientation angle of the damper cell will investigate theoretically and experimentally according to the dynamic deformation of the structure. In addition, the actual dynamic and static mechanical characteristics of the structure incorporating the damper will be clarified in more detail.
will try to embed the small size P-DACS into whole volume of structure part in the future work. Since a specific small P-DACSs embedded in the structure part are deformed according to the vibration mode, the effective damper is selected by the self-selective manner depends on the vibration mode. Finally, high damping effect will be obtained in all elastic vibration modes of structure. The damping effect can be controlled by optimizing the P-DACS design and placing position. Controlling the damping capacity of the small amplitude vibration in structural component is difficult because the energy dissipation caused by the elastic deformation of damping material is less. The large elastic deformation in tensile/compress and shear direction of the damping material (polymer material in the paper) can be obtained by the rotation of the tuning mass when the inclination angle is large. Thus, the compliance of the second bending vibration mode becomes small with high inclination angle. If there is no damper (if using a metal block with same outer dimensions of the damper), the stiffness becomes higher than that of PDACS. In addition, the damping is almost the same as the damping capacity of the material damping of metal if the friction at the joint part is not in consideration. The damping becomes too small than that of PDACS. Therefore, the natural frequency without the damper is higher than that with damper, and the resonance amplitude is higher. The structure damper, P-DACS can increase the damping of structures instead of little lowering the static stiffness. The one of important advantages of P-DACS is that the dynamic characteristics of damper and damper embedded structure can be adjusted without changing the external dimensions by adjusting the internal geometry. In this paper, the numerical analysis assuming there is no friction between jointed parts are performed. Although the friction damping of the joint parts is important for evaluating the vibration, it is generally difficult to modeling in finite element analysis. Therefore, the comparison of the structure's dynamic characteristics with/without P-DACS will be examined in detail in future research by experiments.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by Grants-in-Aid from NSK Foundation for the Advancement of Mechatronics. In addition, the authors especially thank NSK Ltd. and Matsuura Machinery Corp. References [1] Hosseinabadi AHH, Altintas Y. Modeling and active damping of structural vibrations in machine tools. CIRP J Manuf Sci Technol 2014;7:246–57. [2] Brecher C, Fey M, Bäumler S. Damping models for machine tool components of linear axes. CIRP Ann 2013;62:399–402. [3] Zhang SJ, Tob S, Zhang GQ, Zhu ZW. A review of machine-tool vibration and its influence upon surface generation in ultra-precision machining. Int J Mach Tools Manuf 2015;91:34–42. [4] Ema S, Marui E. Suppression of chatter vibration of boring tools using impact dampers. Int J Mach Tools Manuf 2000;40:1141–56. [5] Slocum AH, Marsh ER, Smith DH. A new damper design for machine tool structures: the replicated internal viscous damper. Precis Eng 1994;16:174–83. [6] Fei J, Lin B, Yan S, Ding M, Xiao J, Zhang J, et al. Chatter mitigation using moving damper. J Sound Vib 2017;410:49–63. [7] Wang M. Feasibility study of nonlinear tuned mass damper for machining chatter suppression. J Sound Vib 2011;330:1917–30.
4. Conclusion In this study, a direct damping approach for controlling static and dynamic characteristics of mechanical components, P-DACS, which can be integrated into mechanical components is proposed. P-DACS is a composite of metal and polymer, and it controls static and dynamic vibration characteristics using the geometrical design parameters of the metal part and material characteristics of the polymer part. Additionally, it is clarified through simulation that a vibration response in a table supported by P-DACS could be controlled by the design of P867
Structures 24 (2020) 864–868
Y. Sakai and T. Tanaka
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