Title: An Experimental Analysis of the Dynamics of Lightly Damped Subcritically Excited Gear Pairs
1An 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 Fax44 (0)117 929 4423