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SCIENCE CHINA Technological Sciences, Volume 60 , Issue 8 : 1175-1187(2017) https://doi.org/10.1007/s11431-016-9060-y

Deformation analysis of the flexspline of harmonic drive gears considering the driving speed effect using laser sensors

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  • ReceivedDec 22, 2016
  • AcceptedApr 28, 2017
  • PublishedJun 15, 2017

Abstract

Accurate description of the elastic deformation of the flexspline is the foundation for optimization design of the structure and conjugate profiles of the harmonic drive gear. This paper proposed an experimental method to investigate the effect of the driving speed on the deformation characteristics of the flexspline. First, an experimental apparatus that integrates a special-fabricated micro-displacement platform and a pair of laser displacement sensors is developed, and the radial displacement of the flexspline is measured in vertical and horizontal directions. Next, the deformation analyses of the flexspline at different driving speeds are performed with our method and the conventional method, and the comparison results reveal that the radial displacement of the flexspline is actually composed of both harmonic and random components, and the amplitude decreases and tends to zero with the increase of the driving speed, especially near the closed end of the flexspline. Last, the mechanisms of the inherent multi-frequency and amplitude attenuation characteristics of the radial displacement of the flexspline are discussed. It is indicated that the impact and friction existing in the flexible bearing of the wave generator is likely responsible for the existence of the random component, and the assumption of linear distribution of the flexspline deformation along the rotating axis is invalid under high speed condition. Our research promotes the further study on the contact-impact problem of the flexible bearing of the wave generator and the transfer characteristic of the elastic deformation of the flexspline.


Funded by

Beijing Natural Science Foundation(3172017)

National Natural Science Foundation of China(11272171)

Education Ministry Doctoral Fund of China(20120002110070)


Acknowledgment

This work was supported by the Beijing Natural Science Foundation (Grant No. 3172017), the National Natural Science Foundation of China (Grant No. 11272171), and Education Ministry Doctoral Fund of China (Grant No. 20120002110070).


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  • Figure 1

    (Color online) Overview of a normal HD. (a) Basic components of the HD; (b) schematic diagram of the force condition of the FS in transmission state.

  • Figure 2

    (Color online) Description of the elastic deformation of the FS in assembly state. (a) Deformation of the FS neutral layer; (b) deformation of the FS neutral line and the geometry model of the radial displacement; (c) deformation of the FS neutral line element. The dashed and solid lines denote the FS before and after deformation, respectively. z is measured from the open end to the cross section along the rotating axis of the FS, and θ is measured from YW-axis to YF-axis. NF and Nf are normal lines of the neutral line element at points PF and Pf, respectively.

  • Figure 3

    (Color online) Experimental set-up for the measurement of the radial displacement of the FS.

  • Figure 4

    (Color online) Schematic diagram of the proposed non-contact measurement method for the radial displacement of the FS. (a) The measuring principle. The light paths of the sensors in horizontal and vertical directions are represented by two arrows, and the corresponding test points on the FS are represented by two points. (b) The configuration of the test harmonic reducer. L, L2 and L4 are constant, and L3 varies with L1. L3 is needed to be measured at different test positions to obtain L1. (c) The configuration of the micro-displacement platform and laser displacement sensors. (d) The illustration of the data acquisition at a certain driving speed.

  • Figure 5

    (Color online) Results of the conventional model of the radial displacement of the FS. (a) The calculated w of the neutral layer of the FS and its projection on w-z plane. (b), (c) Waveform and spectrum of the calculated w of the neutral line of the FS at the test cross section (z=L1=21 mm). K=(LL1)w0/(LL2), and X=nh/60.

  • Figure 6

    (Color online) Waveforms and spectra of the measured w of the FS at the test cross section as nh changes. (a) nh=1 r/min; (b) nh=10 r/min; (c) nh=100 r/min; (d) nh=300 r/min; (e) nh=1000 r/min. The abscissas in the waveforms and spectra show the time range t (From 0 to 1.5P) and the frequency range f (From 0 to 10X), respectively. The ordinates in the waveforms and spectra show the measured w in mm and the amplitude A in mm, respectively. The rotating period and frequency of the input WG are denoted by P and X, respectively.

  • Figure 7

    (Color online) Effects of nh on the amplitude characteristics of the measured w. (a) Vertical direction: Spectra of the measured w at different nh; (b) horizontal direction: Spectra of the measured w at different nh; (c) 4th order polynomial fit of the relationship between A2X and nh; (d) the corresponding fitting results; (e) the minima of A2X (obtained at nh=1000 r/min) at different z.

  • Figure 8

    (Color online) Effects of nh on the frequency characteristics of the measured w. (a) Waveforms of the harmonic component of the measured w at different nh; (b) waveforms of the random component of the measured w at different nh; (c) the ratio r of AiX to A2X at different nh.

  • Table 1   Parameters of the main equipment of the experimental set-up

    Equipment

    Parameter

    Content

    Test harmonic

    reducer

    WG type

    Elliptical cam, double waves

    Tooth profile

    Involute

    Reduction ratio

    100

    Lubrication mode

    Grease lubrication

    Load condition

    No-load

    Temperature condition

    Room temperature

    AC motor

    Rotating speed

    1–1000 r/min

    Laser displacement

    sensors

    Sampling interval

    200 μs

    Mounting mode

    Diffuse reflective

    Reference distance

    20 mm

    Measuring range

    ±3 mm

    Linearity

    ±0.02% of F.S.

    (F.S.=6 mm)

    Repeatability

    0.02 μm

    Temperature characteristic

    0.01% of F.S. /°C

    (F.S.=6 mm)

  • Table 2   Parameters of the test harmonic reducer used for the simulation

    Parameter

    Notation

    Value

    Unite

    Module of the FS and the CS

    m

    0.2

    mm

    Teeth number of the FS

    ZF

    200

    Teeth number of the CS

    ZC

    202

    Diameter of the neutral line of the FS before deformation

    dm

    40.350

    mm

    Length from the open end to the closed end of the FS

    L

    35

    mm

    Length from the open end to the test position of the FS

    L1

    21

    mm

    Length from the open end to the tooth rim midsection of the FS

    L2

    6

    mm

    Length measured from the outer box of the test HD to the test position of the FS

    L3

    32

    mm

    Length measured from the outer box of the test HD to the closed end of the FS

    L4

    46

    mm

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