2-3 Static balance of a four-wheeled tractor     return 

(Three-dimensional system)

A stationary four-wheel tractor with pivot support on the front wheels (Figure 2-3) has the following force and moment

equations:

(24)

(25)

(26)

With the pivot on the front wheels:

(27)

Using these equations:

The above general equations can be solved in terms of relative position of points:

Sample problem:

In Figure 2-3, point O' is the pivot of the front wheel of the four-wheel tractor whose weight W = 1000 kg f, l = 100 cm, a= 30 cm, m = 80 cm, n = 60 cm, and c = 30 cm. Determine the reaction forces R1, R2, R3 and R4 at each of the wheel.

Figure 2-3 Four-wheel tractor

Solutions:

2-4 Balance of a two-wheel tractor     return 

A two-wheel tractor without any attachment and not supported at the handle, will fall out of balance.

Figure 2-4 A two-wheel tractor

In Figure 2-4, when the handle is supported by force H, the position can be maintained if:

(1)

(2)

In these equations, the weight vector is (0,0,Wz), the position vector representing the center of gravity is (Gx, 0, Gz), the position vector of the contact point of the right wheel with the ground is (0, m/2, 0), that of the left wheel is (0, -m/2, 0), and the combined supporting force of both hands is (0,0,Hz) with its position represented by vector (hx, 0, hz). For a symmetric tractor:

(3)

By performing the calculation using equation (2),

(4)

The unknown quantities R1z, R2z and H can be solved from

equations (1), (3) and (4). With equation (4):

(5)

Substituting (3) and (5) into (1):

R1z = R2z = -( Wz + Hz ) / 2 = Wz * ( Gx - hx ) / ( 2 * hx )     (6)

From the above, when the position vector of the center of gravity G has zero component along the X-axis, (Gx=0), the supporting force H is equal to zero, and the rection forces on each wheel is equal to W/2.

Depending on whether a(a=-Gx) is in front or to the rear of the wheel axis, the required supporting force is equal to a value of a/h (h = |hx|) of W directed either upward or downward.

The reaction forces R1z and R2z is either decreased or increased by one half of the value of this supporting force.

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