Mechanics of Tillage and Traction 3(2+1)

LESSON 15. Development of prediction equations

σ = f (d, γ) = Cd^C₁ γ^C₂                         ------------------------------------ (2.1)                                          

or, in terms of dimensions:

\[\sigma \dot=F{L^{ - 2}},d\dot=L,\gamma \dot=F{L^{ - 3}}\]               ------------------------------------ (2.2)

where C is a dimensionless constant, and  \[\dot =\] represents dimensional equivalence and not necessarily numerical equivalence. Therefore:

\[\left( {F{L^{ - 2}}} \right)\dot=\left( {{L^{{C_1}}}} \right){\left( {F{L^{ - 3}}} \right)^{{C_2}}}\] ..........(2.3)

\[F{L^{ - 2}}\dot=\left( {{L^{{C_1} - 3{C_2}}}} \right){\left( F \right)^{{C_2}}}\] ..............(2.3)

Since the dimensions must be consistent for this equation to be general, exponents of each dimension should match, i.e.:

1  = C2                                                ------------------------------------- (2.4a)

-2 = C1 - 3C2                                     ------------------------------------- (2.4b)

From equations 2.4a and 2.4b, we have C1 = 1 and C2 = 1. Therefore, equation 2.1 becomes:

σ = Cγ¹d¹ = Cγd                                  -------------------------------------- (2.5)

In this case, dimensional analysis yielded the form of the equation completely, except for the dimensionless constant C. This constant, which is equal to unity (i.e., C = 1), can be determined by conducting limited experiments.

Now consider the case of a wide vertical blade (i.e., rake angle is 90°) of width w, operating at depth d, in a cohesionless, homogeneous soil in the absence of any surcharge. The draft (D) needed to overcome the gravitational component of the soil cutting resistance (i.e., the soil cutting force necessary to resist the passive pressure on the tillage tool due to the soil weight) under quasistatic conditions (i.e., very low ground speed) is dependent on the specific weight of the soil (γ), operating depth (d), blade width (w), and soil internal friction angle (φ) in the absence of soil metal friction. Therefore, we have:

D = f(d,w, γ, φ)= Cd^C₁w^C₂ γ^C₃ f^C₄       ------------------------------------ (2.6)

or, in terms of dimensions:

D \[\dot =\] F

d \[\dot =\] L

w \[\dot =\] L

γ \[\dot =\] FL⁻³

 \[\dot =\] dimensionless

Therefore, in terms of dimensions, equation (2.6) becomes:

                        F \[\dot =\]  (L)^C₁(L)^C₂(FL⁻³)^C₃                        ----------------------------------- (2.7)

Or,

                        F \[\dot =\] (L)^C₁^+C₂^-3C₃(F)^C₃                           ----------------------------------- (2.8)

Therefore,

C₁ + C₂ - 3C₃ = 0                               -----------------------------------    (2.9a)

C₃ = 1                                                -----------------------------------    (2.9b)

From equations (2.9a) and (2.9b), we get:

C₁ + C₂ = 3 or C₂ = 3 - C₁                 ----------------------------------- (2.9c)

Therefore, from equation (2.6), we have:

D = Cd^C₁w^(3−C₁ ⁾γf^C₄                         ---------------------------------- (2.10)

Simplifying and rearranging equation (2.10), we obtain:

                       

Although equation (2.11) provides the form, constants C, C₁, and C₄ are unknown and should be determined experimentally. Note that (D/γw³), (d/W), and  are all dimensionless terms and are often called Pi terms. A more general way of writing equation (2.11) is to express the dependent Pi term (D/γw³) as a function of independent Pi terms (d/w) and (Murphy, 1950). Thus, equation (2.11) can be written in a more general form as:

In going from equation (2.6) to equation (2.11), a five-variable problem has been reduced to a three-variable or Pi terms problem. This reduction in the number of variables is a major advantage provided by dimensional analysis. This reduction in the number of variables is easily determined by the Buckingham Pi theorem.