Pedro_The_Swift
23rd April 2018, 07:06 PM
This should be of particular interest to our members with older vans,,,
I know we have some very clever people as members,,
I would welcome any comment...
How does the 350 mm A-frame extension affect the dynamic forces on the ball coupling? As explained, the dynamic forces exerted by the ball on the coupling are much greater than the static weight force on the ball, and are as a result of large vertical accelerations of the ball, when the tow vehicle drives over bumps. I can’t reliably calculate the actual dynamic forces, because that depends on the size of the bump, and the moment of inertia of the van, both unknown quantities. However, we can calculate the percentage change in the dynamic forces, as a result of increasing drawbar length, which is what we want to know.
Need some background theory. Moment of inertia (I) can be thought of as ‘rotational mass’. A flywheel has a large moment of inertia because it is heavy, and because the mass is deliberately placed around the circumference, at the maximum radius. A caravan has a moment of inertia about it’s axle, and for a given overall weight, the moment of inertia is greatest if the weight is concentrated at the ends, meaning at the greatest radius. For many reasons, we want the moment of inertia of the van to be as small as possible.
When the ball on the tow vehicle rapidly accelerates upward over a bump, then this linear upward motion is converted to rotational motion of the van about it’s axle. For a given upward acceleration at the ball, the dynamic upward force exerted on the coupling is proportional to the moment of inertia of the van about it’s axle. The following formula relates the linear upward acceleration at the coupling, to the rotational acceleration of the van :-
A = a/R (Equation 1) where
A is the rotational acceleration, in radian/s/s (degree/s/s of you prefer)
a is the linear upward acceleration of the ball, in m/s/s
R is the distance from coupling to van axle in meters
This is not rocket science. All this formula is saying, is that the rotational acceleration of the van is proportional to the linear upward acceleration of the ball, and inversely proportional to the length of the lever arm about which the van rotates. Intuitive common sense.
Many would be familiar with the formula F=ma. There is a directly analogous rotation form of this formula, also very intuitive :-
T = IA (Equation 2) where
T is the Torque, in Nm
I is the moment of inertia, in kg-m^2
A is the angular acceleration, in radians/s/s
The final equation, that should be second nature by now, that ties it all together is :-
T = FR (Equation 3) where
T is the torque applied to the coupling, in Nm
F is the upward force exerted on the coupling by the ball (N)
R is the lever arm length, from coupling to axle (m)
Combining these three equations gives us the equation we want :-
F = Ia / (L^2) (Equation 4)
Thus we see that the dynamic force exerted upward on the coupling by the ball is proportional to the linear upward acceleration of the ball and the moment of inertia of the van – no surprises there.
The interesting part though, is that for a given size of bump (given value of a), and a given van having a particular moment of inertia, the dynamic force exerted on the coupling by the ball is inversely proportional to the square of the distance from coupling to axle.
How very interesting! And how very desirable. The percentage increase in total drawbar length is an increase of 350mm in an original length of 4840m, so the percent increase is 350/4840 = 7.2% The L-squared term in Equation 4 means that if we increase total drawbar length (from coupling to axle) by 7.2%, then the dynamic forces exerted on the coupling decrease by 14.4%
This is a beautiful result in every way. It means that there is no structural problem with the increased-length A-frame, because although the static bending moment is increased, the much-more-important dynamic loading is very substantially reduced, though I need to fully tie up the dynamic bending moment in the next posting.
Even more importantly, it means that in general, the dynamic forces exerted by the van coupling onto the tow vehicle, are reduced by twice as much as you might think, on account of the L-squared term in equation 4, namely a 14% decrease rather than the 7% decrease you might expect.
In a similar way, having excessive weight at the ends of the van has a bigger dynamic effect that you might expect, because the formula for moment-of-inertia is mR^2, so if the mass that you are packing is placed at double the radius (distance from axle), then it’s moment of inertia and thus it’s dynamic influence, is increased by a factor of four. I can hear Collyn Rivers nodding his head.
One more short post to tie up the exact effect that increasing A-frame length has on dynamic bending moment, and I’ll be done.
I know we have some very clever people as members,,
I would welcome any comment...
How does the 350 mm A-frame extension affect the dynamic forces on the ball coupling? As explained, the dynamic forces exerted by the ball on the coupling are much greater than the static weight force on the ball, and are as a result of large vertical accelerations of the ball, when the tow vehicle drives over bumps. I can’t reliably calculate the actual dynamic forces, because that depends on the size of the bump, and the moment of inertia of the van, both unknown quantities. However, we can calculate the percentage change in the dynamic forces, as a result of increasing drawbar length, which is what we want to know.
Need some background theory. Moment of inertia (I) can be thought of as ‘rotational mass’. A flywheel has a large moment of inertia because it is heavy, and because the mass is deliberately placed around the circumference, at the maximum radius. A caravan has a moment of inertia about it’s axle, and for a given overall weight, the moment of inertia is greatest if the weight is concentrated at the ends, meaning at the greatest radius. For many reasons, we want the moment of inertia of the van to be as small as possible.
When the ball on the tow vehicle rapidly accelerates upward over a bump, then this linear upward motion is converted to rotational motion of the van about it’s axle. For a given upward acceleration at the ball, the dynamic upward force exerted on the coupling is proportional to the moment of inertia of the van about it’s axle. The following formula relates the linear upward acceleration at the coupling, to the rotational acceleration of the van :-
A = a/R (Equation 1) where
A is the rotational acceleration, in radian/s/s (degree/s/s of you prefer)
a is the linear upward acceleration of the ball, in m/s/s
R is the distance from coupling to van axle in meters
This is not rocket science. All this formula is saying, is that the rotational acceleration of the van is proportional to the linear upward acceleration of the ball, and inversely proportional to the length of the lever arm about which the van rotates. Intuitive common sense.
Many would be familiar with the formula F=ma. There is a directly analogous rotation form of this formula, also very intuitive :-
T = IA (Equation 2) where
T is the Torque, in Nm
I is the moment of inertia, in kg-m^2
A is the angular acceleration, in radians/s/s
The final equation, that should be second nature by now, that ties it all together is :-
T = FR (Equation 3) where
T is the torque applied to the coupling, in Nm
F is the upward force exerted on the coupling by the ball (N)
R is the lever arm length, from coupling to axle (m)
Combining these three equations gives us the equation we want :-
F = Ia / (L^2) (Equation 4)
Thus we see that the dynamic force exerted upward on the coupling by the ball is proportional to the linear upward acceleration of the ball and the moment of inertia of the van – no surprises there.
The interesting part though, is that for a given size of bump (given value of a), and a given van having a particular moment of inertia, the dynamic force exerted on the coupling by the ball is inversely proportional to the square of the distance from coupling to axle.
How very interesting! And how very desirable. The percentage increase in total drawbar length is an increase of 350mm in an original length of 4840m, so the percent increase is 350/4840 = 7.2% The L-squared term in Equation 4 means that if we increase total drawbar length (from coupling to axle) by 7.2%, then the dynamic forces exerted on the coupling decrease by 14.4%
This is a beautiful result in every way. It means that there is no structural problem with the increased-length A-frame, because although the static bending moment is increased, the much-more-important dynamic loading is very substantially reduced, though I need to fully tie up the dynamic bending moment in the next posting.
Even more importantly, it means that in general, the dynamic forces exerted by the van coupling onto the tow vehicle, are reduced by twice as much as you might think, on account of the L-squared term in equation 4, namely a 14% decrease rather than the 7% decrease you might expect.
In a similar way, having excessive weight at the ends of the van has a bigger dynamic effect that you might expect, because the formula for moment-of-inertia is mR^2, so if the mass that you are packing is placed at double the radius (distance from axle), then it’s moment of inertia and thus it’s dynamic influence, is increased by a factor of four. I can hear Collyn Rivers nodding his head.
One more short post to tie up the exact effect that increasing A-frame length has on dynamic bending moment, and I’ll be done.