~OT: 3-axis outputs on a pendulum

Started by Tom Becker June 4, 2004
I've been challenged to predict the outputs of a 3-axis accelerometer
mounted on the end of a pendulum that swings through a cone, on Earth.
Assuming the x and y axes are fixed north-south (i.e. the pendulum does
not rotate in its suspension bearing), I expect that x and y will be
sines in quadrature, the response to gravity's apparent motion. What of
z, though? Will it be constant?

And, will an unconstrained pendulum that is started in a conic swing
continue to swing in a cone or will it settle to an arc, ala Foucault,
anywhere but the Poles? Tom
Tom Becker
--... ...--
www.RighTime.com
The RighTime Clock Company, Inc., Cape Coral, Florida USA
+1239 540 5700



Hi Tom,

Indeed, the z-axis would be a constant reading as long as the motion stayed
perfectly conical and did not decelerate. As soon as it starts to go into an
"oval" cone, there will be a moment towards the outside of the ellipse where
the pendulum will slow down compared to the rest of the motion, and thus the
z-axis reading will change. The z-axis component will be a compound of g
(towards the Earth's center) and the centrifugal/centripetal force in the
pendulu's axis.

Regarding the free-motion pendulum, over time it will set itself into a
Focault-mode (so-to-speak), as the earth's motion does not provide the
acceleration needed to keep two axis in motion. In a science museum in
Barcelona we have a Focault pendulum, it's quite impressive as the
suspension rod is some 30 meters high - it operates as a clock, slowly
rotating the swing axis during the day and knocking down a small pole each
hour.

Regards,

Mike

----- Original Message -----
From: "Tom Becker" <>
To: <>
Sent: Friday, June 04, 2004 5:42 PM
Subject: [BasicX] ~OT: 3-axis outputs on a pendulum > I've been challenged to predict the outputs of a 3-axis accelerometer
> mounted on the end of a pendulum that swings through a cone, on Earth.
> Assuming the x and y axes are fixed north-south (i.e. the pendulum does
> not rotate in its suspension bearing), I expect that x and y will be
> sines in quadrature, the response to gravity's apparent motion. What of
> z, though? Will it be constant?
>
> And, will an unconstrained pendulum that is started in a conic swing
> continue to swing in a cone or will it settle to an arc, ala Foucault,
> anywhere but the Poles? > Tom >
> Tom Becker
> --... ...--
> www.RighTime.com
> The RighTime Clock Company, Inc., Cape Coral, Florida USA
> +1239 540 5700 > Yahoo! Groups Links >





Hmm, I don't know but I think the pendulum would slowly spiral inwards due to
loss of energy (at the suspension point, and air resistance). In a
Foucault pendulum
the earth rotates under the pendulum's plane of vibration, which is sensibly
fixed in inertial space (given no outside torques). So I would think that
the pendulum
would continue move with a conical motion of decreasing amplitude. If this is
the case the z component would gradually grow and the amplitude of the x-y
components would gradually decrease, but remain sinusoidal in quadrature.
Rotating about the bearing point would have no bearing on the x,y,z motion,
the
pendulum body would just rotate about the line of the suspension cable (you
can observe this with a small, hand held pendulum).

John-

>Indeed, the z-axis would be a constant reading as long as the motion stayed
>perfectly conical and did not decelerate. As soon as it starts to go into an
>"oval" cone, there will be a moment towards the outside of the ellipse where
>the pendulum will slow down compared to the rest of the motion, and thus the
>z-axis reading will change. The z-axis component will be a compound of g
>(towards the Earth's center) and the centrifugal/centripetal force in the
>pendulu's axis.
>
>Regarding the free-motion pendulum, over time it will set itself into a
>Focault-mode (so-to-speak), as the earth's motion does not provide the
>acceleration needed to keep two axis in motion. In a science museum in
>Barcelona we have a Focault pendulum, it's quite impressive as the
>suspension rod is some 30 meters high - it operates as a clock, slowly
>rotating the swing axis during the day and knocking down a small pole each
>hour.
>
>Regards,
>
>Mike
>
>----- Original Message -----
>From: "Tom Becker" <>
>To: <>
>Sent: Friday, June 04, 2004 5:42 PM
>Subject: [BasicX] ~OT: 3-axis outputs on a pendulum > > I've been challenged to predict the outputs of a 3-axis accelerometer
> > mounted on the end of a pendulum that swings through a cone, on Earth.
> > Assuming the x and y axes are fixed north-south (i.e. the pendulum does
> > not rotate in its suspension bearing), I expect that x and y will be
> > sines in quadrature, the response to gravity's apparent motion. What of
> > z, though? Will it be constant?
> >
> > And, will an unconstrained pendulum that is started in a conic swing
> > continue to swing in a cone or will it settle to an arc, ala Foucault,
> > anywhere but the Poles?
> >
> >
> > Tom
> >





> ... Rotating about the bearing point would have no bearing on the
x,y,z motion...

No, but it would change the accelerometer's x,y phases. If the pendulum
rotates 360 degrees about its axis the accelerometer outputs will have
advanced or retarded by the same angle. If one were to use the
quadrature signals to count the conic swings, the count would be
incorrect by four edges (one rotation) if the pendulum rotated one turn
in its mount. Tom
Tom Becker
--... ...--
www.RighTime.com
The RighTime Clock Company, Inc., Cape Coral, Florida USA
+1239 540 5700