An Experimental Analysis of the Dynamics of Lightly Damped Subcritically Excited Gear Pairs

1
An Experimental Analysis of the Dynamics of
Lightly Damped Subcritically Excited Gear
Pairs Department of Mechanical Engineering
Researcher James R. Ottewill
Supervisors Simon A. Neild, R. Eddie Wilson

• Mathematical Model
• The equations of motion for a geared system are
  rendered non-linear by the backlash.
• A simplified mathematical model has been derived
  considering a displacement input into a Drive
  Gear
• where is the displacement of the drive gear,
  is the size of a fluctuation and is the time.
• The relative motion of the two gears, , is
  described by

• Introduction
• Gear teeth are specially designed so that a
  constant angular velocity ratio should be
  transmitted between two shafts.
• However, there is a necessity to design a small
  gap between the meshing gear teeth, known as the
  backlash, in order to allow for lubrication and
  manufacturing errors.
• This backlash gives rise to undesirable modes of
  operation with vibration, known as Noise,
  Vibration and Harshness (NVH) problems.
• This is an issue in any geared system, and is of
  particular concern in the automotive and
  aeronautical industries.

where is damping, is stiffness, is a
function describing the interaction at the tooth
meshing interface and is time. These parameters
are all non-dimensionalised.

• Experimentation
• It is difficult to obtain reliable experimental
  data with a high enough resolution to fully
  record the gear motions as they traverse the very
  small backlash gap.
• An experimental rig has been designed
  incorporating high resolution encoders and high
  module gears that have been separated so as to
  increase backlash to a measurable size.
• Experiments were carried out using a controlled,
  displacement input investigating the effect of
  varying the amplitude of the fluctuation.
• Multiple tests were carried out at each
  fluctuation amplitude, at different phases.
  According to the derived model, this phase should
  have no effect on the end response
• At 0, a sinusoidal motion was observed due
  to small eccentricities in the gear mountings. To
  observe the amplitude of relative motions away
  from contact this sinusoid was subtracted from
  all tests.

• It was found that the phase of the fluctuation
  did in fact have an effect on the amplitude of
  the steady state response. Here we see the red
  dotted lines indicating the backlash boundaries
  and three different responses, all at 0.16
  radians, but at different phase.

• Plotting the maximum and minimum amplitudes
  found at each input fluctuation amplitude
  produces bands of possible solutions (shown here
  in green). The red curve shows the single line
  predicted by the mathematical model.

• Phase of any fluctuation input IS IMPORTANT.
  This could explain some of the multiple solutions
  that some gear pairs display, which have
  previously been attributed to the nonlinearity of
  the system.
• The first hypothesis of the cause of these input
  phase dependent solutions is that the
  eccentricity is interacting with the input.
• A highly versatile gear dynamics rig has been
  developed, overcoming the difficulties associated
  in measuring gear motions and allowing the future
  investigation of many of the parameters generally
  considered important in gear mechanics.

• Future Work
• A new mathematical model incorporating
  eccentricity has been developed and is currently
  being analysed.
• The driven gear is split into two one gear disc
  can be moved and fixed relative to the other to
  allow future investigations of the effect of
  changing the backlash size.
• Effective force Real time testing, using a
  second servomotor on the driven shaft to provide
  a load. This can be used to investigate the
  effect of changing the stiffness and damping of a
  gear system.

Email james.ottewill_at_bristol.ac.uk Tel44
(0)117 331 7601 Fax[phone]