Under what conditions a rigid body on axle is balanced, i.e. rotation does not cause vibration?

Answer 1

The inertia matrix (ref.) of the wheel taken about the axle must not contain any off-diagonal elements ("products of inertia"). The products can leave the CM unchanged but produce a torque perpendicular to the rotation axis. The axis of this torque rotates with the wheel causing the familiar shimmy, and its magnitude is proportional to omega^2. (Specifically this torque = omega X I * omega, X being the cross-product). Another way to look at it is that the products change the mass distribution such that the axle no longer aligns with a principal axis. A disk whose plane is not perpendicular to the axle is an example.
There is a significant difference between dynamic and static wheel balancing. Static balancing only ensures that the CM lies on the rotation axis so no perpendicular forces are generated, but dynamic balancing also eliminates the perpendicular torques.

Answer 2

The axle should go through the center of mass. In that case whole system does not experience any acceleration (read force). If center of mass is off the axle, than it has to rotate around the axle, so force has to be applied and you have a vibration.

You can easily prove this.
Consider small portion of the body with mass m and on distance r from the axle (r here and later is vector). Then centripetal force is -mw^2r. The sum of the forces on all the parts of your body is the sum of -mw^2r over all parts or -w^2 multiplied by the sum of mr. But by definition sum(mr) = MR, where M is the body's mass and R is radius vector to its center of mass. So total force is -Mw^2R and this force is equal to 0 when R=0, i.e. axle goes through center of mass.

Answer 3

The mass has to be distributed concentrically in an even manner. In other words, take a car wheel and tire. As you go outward from the hub, each "ring" section has its mass distributed evenly around the circumference - the part nearest the hub is solid, the same thickness all the way around; the part with the holes has the same size holes distributed evenly all the way around, the wheel rim is solid, evenly distributed, and the solid part of the tire is evenly distributed all the way around. The only imbalance should be the valve stem. To correct any imbalance in a car wheel and tire assembly, mechanics add small weights to the wheel rim opposite any extra weight detected by the balancing machine.

Answer 4

A rigid body is said to be in equilibrium when the sum of external forces (active and reactive too) acting on it forms a system equivalent to zero. For a body this generally means that


= , = (×) = (3.28)




These two vector equations may be reduced in the planar case (the basic plane being x, y) to the following three scalar equations written in rectangular components of each force and each moment:

Fix = 0, Fiy = 0, Miz = 0

The equations may be used to determine unknown forces applied to the rigid body in plane or unknown reactions exerted by its support.
These equations may be solved for just three unknowns. If they involve more than three unknowns the body is said to be statically indeterminate. If they involve fewer than three unknowns, the body is said to be partially constrained.

The statement above is not valid absolutely. The solvability of the three equations depends on the properties of the system matrix.

Generally speaking the problem of the equilibrium of a body is always transformed to the problem of the equilibrium of the system of forces that act on the body. To identify all such forces the free-body diagram is essential.

Answer 5

Wait a 2d whilst I purely examine this tale....... "A Russian guy has tattooed his call on the face of his lady pal, after understanding her for purely 24 hours. The pair reportedly met in a internet chat room and now are engaged." I purely...what the ......i do no longer comprehend what to declare different than rejoice with your tattoo while your 70 this is purely stupid...

Answer 6

If the forces acting clockwise and anti-clockwise are balanced or net force is 0.