The Inertia Torque of The Hooke Joint
The Inertia Torque of The Hooke Joint
The Inertia Torque of The Hooke Joint
History
Already described by Philo of
Byzantium in the III century B.C.
Gerolamo Cardano in the XVI century
proposed it as motive power
transmission joint.
Common Applications
Cardan Joint
Common Applications
Common Applications
Common Applications
U-Joints
Common Applications
Schematic
In order to study the
kinematics and the dynamics
of the joint, spherical
geometry is used
reference frame
Nomenclature
Z
z
4
Y
1
is the angle
y
Nomenclature
Ixx, Iyy, Izz are the mass moments of inertia of the floating link about x,
y and z axes
Mx, My, Mz are the torque components acting on the floating link
Nomenclature
T1H, T1V
T4H, T4V
Tx, Ty, Tz are the inertia torque components of floating link along x,y
and z axes
TX, T Y, TZ are the inertia torque components of floating link along X,Y
and Z axes
Kinematics
y and z axes must be always perpendicular
Transmission Ratio
follow:
To find the inertia torque Eulers moment equations for motion with a
Where I and are constant and the three functions u (1), v (1), w (1)
are the following:
Where:
And hence:
Differentiating by time:
Approximated Equations
An approximated solution can
be evaluated doing the
following assumptions:
Approximated Equations
With these assumptions the approximate equations for the inertia
torques with terms of first, second and third order in , are the
following, being J = (Iyy + Izz Ixx)/Izz = 2 (1 ) :
Approximated Equations
Approximated Equtions
Approximated Equations
Where
is the vector
Using the approxmate equations for inertia torques the results are:
Conclusions
The inertia torques are of even order, the predominant harmonic being
of order 2
The inertia torque component along the direction of the input shaft
axis is of order 2. The other two components are of order
The dominant terms of the inertia torque components, T Y and TZ,
vanish if the mass distribution were such that J vanishes. This can be
achieved by locating the mass of the floating link as close as possible to
the plane of the pin joints
The approximate solutions are precise for common values of and
allow a deep insight into the dynamics of the system
The approximate solutions for the rocking torques provide a steady
torque and a rotating component, equal in magnitude for both input
and output shaft
Conclusions
The magnitude of the statically induced rocking torques on either shaft