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Many thanks to respondents to my recent query concerning horizontal and
vertical components of leg extension. Much food for thought. The idea
of the INTERNALITY of the forces vs. the idea of exerting an EXTERNAL
force on the knee (something I have considered) is esp. interesting.
But I wonder if some discussion of toggle mechanisms (as defined by
engineers) might be in order here to provide a little background to my query.
The toggle we're probably most familiar with is the toggle bolt we use
to put picture hanging hooks in wall board or sheetrock when we
have no other way of securing a good anchor. Those toggles have spring
loaded "wings" that expand after we have compressed them to slip them
into the hole we drill in the wallboard. But suppose we had only the
"wings" and no bolt at the joint. Let's review the formula for the
forces here (in case some biomechanics folks didn't take
"Statics & Dynamics"). If we fix the end of one of the wings and exert
force P downward at the joint, the end of the other wing will move
horizontally with resultant force Q. (I've wondered whether we should
consider a trampoline an inverted toggle--with elastic wings.)
In any case, the formula is: Q = 0.5 P (cotan theta) where theta is
the angle each
wing makes with the horizontal. So if the angle at the joint is 90 degress,
each wing of the toggle makes a 45 degree angle with the horiz., cot = 1.0,
and the formula tells us that the resultant Q is only half of the downward
force P. Theoretically, when theta goes to 0 the cotangent goes to
infinity and so does resultant Q. Doesn't happen in reality, but we
get the picture of why toggle mechanisms are used to generate large
forces (as in devices to crush big rocks into small gravel). When
theta gets beyond 25 degrees or so we start to enjoy increasing
mechanical advantage. And I don't think there are too many lever
systems in the human body (the jaw? doing heel lifts?) where we get
much mechanical advantage. Acceleration advantage, yes. But not much
m.e.
Now the human knee is not a pin joint so the applicability of the
formula to an understanding of lifting forces during leg extension
might well be questioned--hence my original query. But I do think this
whole discussion helps us understand why we're stronger nearer full
extension. (Something we should all remember if three of us ever had
to lift the front of a big automobile three cm. so that a victim
trapped underneath could be pulled free.) Or put another way, we
do the same amount of work lifting our torso one inch at the start of
a pushup that we do during the last inch. But we can all do more "one
inch" pushups if we do final inch ones rather than initial inch ones.
But the bottom line here (and why I presume to take up time on the
biomch network) is that these questions have potentially great
consequences for the design of bicycles, wheelchairs and other human
powered vehicles, which is what our past and planned work is all
about. Simply put: linear power permits you to take advantage of the
leg or arm up to and including full extension; rotary power means the
crank arms of the bicycle are approaching dead center when you have
the most strength to exert (unless you're standing up and "pumping"--
with consequent aerodynamic losses). This is hardly the whole story
of HPV design and fabrication so we'd like to know whether that last
little push of full extension with its considerable mechanical
advantage compared to a more flexed limb, is worth going after
considering all the design modifications necessary in the vehicle?
And if you've followed me this far--thanks for your patience. And if
you find this interesting, I welcome your comments.
John Martinson Univ. of Nevada, Reno
vertical components of leg extension. Much food for thought. The idea
of the INTERNALITY of the forces vs. the idea of exerting an EXTERNAL
force on the knee (something I have considered) is esp. interesting.
But I wonder if some discussion of toggle mechanisms (as defined by
engineers) might be in order here to provide a little background to my query.
The toggle we're probably most familiar with is the toggle bolt we use
to put picture hanging hooks in wall board or sheetrock when we
have no other way of securing a good anchor. Those toggles have spring
loaded "wings" that expand after we have compressed them to slip them
into the hole we drill in the wallboard. But suppose we had only the
"wings" and no bolt at the joint. Let's review the formula for the
forces here (in case some biomechanics folks didn't take
"Statics & Dynamics"). If we fix the end of one of the wings and exert
force P downward at the joint, the end of the other wing will move
horizontally with resultant force Q. (I've wondered whether we should
consider a trampoline an inverted toggle--with elastic wings.)
In any case, the formula is: Q = 0.5 P (cotan theta) where theta is
the angle each
wing makes with the horizontal. So if the angle at the joint is 90 degress,
each wing of the toggle makes a 45 degree angle with the horiz., cot = 1.0,
and the formula tells us that the resultant Q is only half of the downward
force P. Theoretically, when theta goes to 0 the cotangent goes to
infinity and so does resultant Q. Doesn't happen in reality, but we
get the picture of why toggle mechanisms are used to generate large
forces (as in devices to crush big rocks into small gravel). When
theta gets beyond 25 degrees or so we start to enjoy increasing
mechanical advantage. And I don't think there are too many lever
systems in the human body (the jaw? doing heel lifts?) where we get
much mechanical advantage. Acceleration advantage, yes. But not much
m.e.
Now the human knee is not a pin joint so the applicability of the
formula to an understanding of lifting forces during leg extension
might well be questioned--hence my original query. But I do think this
whole discussion helps us understand why we're stronger nearer full
extension. (Something we should all remember if three of us ever had
to lift the front of a big automobile three cm. so that a victim
trapped underneath could be pulled free.) Or put another way, we
do the same amount of work lifting our torso one inch at the start of
a pushup that we do during the last inch. But we can all do more "one
inch" pushups if we do final inch ones rather than initial inch ones.
But the bottom line here (and why I presume to take up time on the
biomch network) is that these questions have potentially great
consequences for the design of bicycles, wheelchairs and other human
powered vehicles, which is what our past and planned work is all
about. Simply put: linear power permits you to take advantage of the
leg or arm up to and including full extension; rotary power means the
crank arms of the bicycle are approaching dead center when you have
the most strength to exert (unless you're standing up and "pumping"--
with consequent aerodynamic losses). This is hardly the whole story
of HPV design and fabrication so we'd like to know whether that last
little push of full extension with its considerable mechanical
advantage compared to a more flexed limb, is worth going after
considering all the design modifications necessary in the vehicle?
And if you've followed me this far--thanks for your patience. And if
you find this interesting, I welcome your comments.
John Martinson Univ. of Nevada, Reno