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enkii

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I am working on an undergraduate design project to design a wood chipper. This involves a large rotating disk with blades attached, which cut the wood by means of an impact force. My problem is that I can’t figure out the force which will act tangentially to the circle due to the angular momentum of the disk. Obviously, a force will act tangentially due to the torque, and I have calculated this, but intuitively I know that a larger force can act due to the angular momentum of the disk.

Disk Properties:

Diameter of disk: 1m

Thickness: 20mm

Mass: 122kg

Angular Velocity: 1000rpm

Torque: Undecided as of yet

Inertia: 15.24 kgm^2

Kinetic Energy of Disk at 1000rpm: ~ 83.6kJ

The disk can be modeled as having uniform density for simplicity, though obviously the presence of the blades will mean that in reality this is not the case. Furthermore, the disk will rotate steadily about its centre as a number of blades will be evenly placed so that the disk is balanced. Also, sufficient torque is provided to keep the disk spinning at constant angular velocity in the absence of the resistive force.

The resistive force, shown in the attached diagram, has been calculated as having magnitude of 610N (This is the force needed to cut the timber). I’m assuming that due to the torque, the disk will have the greatest difficulty in cutting the material at the furthest point from the centre, i.e. at max radius and so I am assuming that the resistive force acts at max radius, as a point force.

Inertia = 0.5MR^2

Kinetic Energy = 0.5Iω^2

Torque= Force x Distance

I’ve tried using the

Also tried to figure out the force by using

Where k is a constant (I think) and r is the radius at which the force acts. This seems more promising as I feel it makes sense, however as I’ve no experimental data, I can’t see a way to find what k equals.

To restate: I am aware that the disk will provide a certain amount of force due to the torque of the disk, but think that a far greater cutting force will be applied by the angular momentum of the disk. It is thus the force due to the angular momentum that I think I need to find.

Finally, if any info is missing, I’m sure I probably have it, though I hope I’ve included all the relevant info. I reckon this problem just needs to be looked at in a different way so any help would be greatly appreciated.

Many thanks

enkii

__Some specifics:__Disk Properties:

Diameter of disk: 1m

Thickness: 20mm

Mass: 122kg

Angular Velocity: 1000rpm

Torque: Undecided as of yet

Inertia: 15.24 kgm^2

Kinetic Energy of Disk at 1000rpm: ~ 83.6kJ

The disk can be modeled as having uniform density for simplicity, though obviously the presence of the blades will mean that in reality this is not the case. Furthermore, the disk will rotate steadily about its centre as a number of blades will be evenly placed so that the disk is balanced. Also, sufficient torque is provided to keep the disk spinning at constant angular velocity in the absence of the resistive force.

The resistive force, shown in the attached diagram, has been calculated as having magnitude of 610N (This is the force needed to cut the timber). I’m assuming that due to the torque, the disk will have the greatest difficulty in cutting the material at the furthest point from the centre, i.e. at max radius and so I am assuming that the resistive force acts at max radius, as a point force.

Inertia = 0.5MR^2

Kinetic Energy = 0.5Iω^2

Torque= Force x Distance

__Methods Tried:__I’ve tried using the

**kinetic energy**of the disk to see what effect the angular momentum of the disk will have. I calculated that it will take approx 26J to cut the wood and the disk has a kinetic energy many times larger than this. In effect this only told me what I already knew; that the angular momentum has a large effect on the cutting force. Also this does not take into account the radius at which the resistive force acts so ultimately isn’t much use.Also tried to figure out the force by using

**dimensional analysis.**Found that the force due to the angular momentum is :*Force =(k x Kinetic Energy of Disk)/r*Where k is a constant (I think) and r is the radius at which the force acts. This seems more promising as I feel it makes sense, however as I’ve no experimental data, I can’t see a way to find what k equals.

To restate: I am aware that the disk will provide a certain amount of force due to the torque of the disk, but think that a far greater cutting force will be applied by the angular momentum of the disk. It is thus the force due to the angular momentum that I think I need to find.

Finally, if any info is missing, I’m sure I probably have it, though I hope I’ve included all the relevant info. I reckon this problem just needs to be looked at in a different way so any help would be greatly appreciated.

Many thanks

enkii