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MODULE 1. Introduction to mechanics of tillage tools
MODULE 2. Engineering properties of soil, principl...
MODULE 3. Design of tillage tools, principles of s...
MODULE 4. Deign equation, Force analysis
MODULE 5. Application of dimensional analysis in s...
Module 6. Introduction to traction and mechanics, ...
Module 7. Traction model, traction improvement, tr...
Module 8.Soil compaction and plant growth, variabi...
LESSON 17. Application of dimensional analysis and similitude to soil mechanics
Bames et al, 
Similitude studies in 
Considered two approaches to representing soil properties. One model for soil considered specific weight of soil (FL^{}). moisture content (), clay content (), and soilmetal friction (). This led to an undistorted model, but did not lead to good prediction of prototype disk draft (F). The second model considered cohesion (FL^{2}), soil internal friction angle (), adhesion (FL^{2}), and soilmetal friction (). This led to a distorted model .When the prediction factor was estimated from appropriate consideration of the distort ion. the model predicted prototype behavior reasonably well. The other variables considered were: disk diameter (L), other pertinent lengths (L), angle of inclination (), angle of approach(), implement speed (LT^{1}), and acceleration due to gravity (LT^{2}) 
Hicks (1961) 
Similitude studies in 
Use of sinkage parameters in predicting the rolling resistance and sinkage of wheels was investigated. The author considered two dependent variables and 11 independent variables. Independent variables considered were: wheel width (L), wheel diameter (L), soil depth (L), wheel aspect ratio (), cohesive sinkage modulus (FL^{(n+2)}, frictional sinkage modulus n(FL^{(n+2)}), sinkage exponent (), coefficient of soiltire friction (), specific weight of soil (FL^{3}), and acceleration due to gravity (LT^{2}). Dependent Variables were: rolling resistance (F)and wheel sinkage (L). 
Nuttall and 
Similitude studies, in 
Reviewed the principles of dimensional analysis, Described similitude studies aimed at determining rolling resistance and sinkage of wheels. Considered two dependent and six independent variables. Independent variables considered were: wheel diameter (L), section height (L), slip (), cohesion (FL^{2}) internal frictional angle (), and axle load (F). Dependent variables of interest were rolling resistance (F)and wheel sinkage (1). 
Sullivan (1964) 
Soil bin facilities for scale model studies at Caterpillar Tractor, Inc 
Modeled the weight of the soil in the scraper bowl as a function often indepe1ldent variables. Independent variables considered were: size of the bowl (L), velocity of the bowl (LT^{2}), cutting depth (L). duration of cutting (T), soil cohesion (FL^{2}), internal angle of friction of soil (), coefficient of soil metal friction (), specific weight of soil (FL'^{J}), and acceleration due to gravity (LT^{2}). The dependent variable was horizontal pushing force (F). Use of artificial soil, which was a mixture of clay, silt, and sand, for better control of soil profit1ies and quicker preparation of test soil was emphasized. 
Clark et al. 
Scale model studies of off 
Considered two dependent and 14 independent variables to investigate the behavior of scale models of offroad vehicles to predict the performance of fullscale models in cohesive soils. Independent variables considered were: wheel diameter (L), tire width (L), tire deflection (L), forward speed of the vehicle (LT^{1}). slip (), precollapse structural cohesion (FL^{2}), postcollapse structural cohesion (FL^{2}), apparent structural cohesion (FL^{2}), angle of internal friction (), coefficient of friction between soil and the traction device (), plastic kinematic viscosity of soil (FL^{2}T). specific weigh of soil (FL^{3}), soil moisture content () and vehicle weight (F) Dependent variables were: sinkage (L) and net traction 
Hegedus (1965) 
Plate sinkage test and 
Considered one dependent and six Independent variables, Independent variables considered were: characteristic length of plate (L). circumference of plate (L), specific weight of soil (FL^{2}), cohesion (FL^{2}), internal angle of friction (). and applied pressure (FL^{l}). The dependent variable was depth of sinkage (L). If cohesion was low or negligible then distortion was minimal and we sinkage constant could be determined with the use of a single plate. 
Cohron (1966) 
Artificial soils and soil 
Described the soil bin test facilities at Caterpillar Tractor, Inc., and use of artificial soils for model studies. Artificial soils were made by mixing fire clay, sand, and low viscosity oil to provide consistent engineering properties of soils (i.e., cohesion, adhesion , angle of internal friction, specific weight). 
Freitag (1966) 
Dimensional analysis of 
Considered 11 independent and four dependent variables to investigate the performance of treadless pneumatic tires in clay. Independent variables selected were: tire diameter (L), section height (L), section width (L). deflection (L), cone index (FL^{2}). spissitude (FL^{1}T), load (F), wheel speed (LT^{1}), slip (), tiresoil friction (), and acceleration due to gravity (LT^{1}) Dependent variable; were: pull (F), towed force (F), torque (FL), and sinkage (L). The concept of clay mobility number was introduced;
where C is the cone index value, W is axle load, b is we section width of tire, d is the overall diameter, h is the section height, and the tire deflection. 
Korayem(1966) 
Artificial soils for tillage 
Used of a mixture of day, sand, and ethylene glycol to simulate natural soils with respect to cohesion and internal angle of friction.

Reaves (1966) 
Artificial soils for tillage 
A mixture of airdried Houston clay and either spindle oil or ethylene glycol was used for making artificial soils of different specific weights. The effect of liquid content on cohesion, angle of internal friction, and soilmetal friction was investigated. A distorted model and a prototype plane chisel were tested in the artificial soil that used ethylene glycol as the fluid, which resulted in acceptable draft force prediction.

Young (1966. 
Similitude of dynamic 
Considered the application of dimensional analysis and characteristic equation in developing modeling laws. The formulation was done in terms of general variables. Applications of the technique to tillage 
Garrity et al. 
Prediction of prototype 
For geometrically similar bulldozer blades operating a low speed and same height of cut in similar soil type and condition, draft requirements were found to be functions f length scale only. Reaves et at. (1969) employed a more rigorous treatment such data using similitude principle 
Freitag (1968) 
Dimensional analysis of soil treadless pneumatic tires in clay 
Considered 13 independent and four dependent variables to investigate the performance of treadless pneumatic tires in sand. The independent variables were, tire diameter (L), section height (L) section width (L), deflection (L) cohesion (FL^{2}) internal friction angle (), specific weight (FL^{3}), spissitude (FL^{2}T), load (F), wheel speed (LT^{1}), slip (), tiresoil friction (), and acceleration due to gravity (LT^{2}), Dependent variables were: pull (F), towed force (F), torque (FL), and sinkage (L), For sandy soils, cone index gradient (FL^{3}) was used to replace specific weight of soil, The concept of sand number was discussed: G(bd) ^{3/2} W Where G is cone index gradient with depth b is section width. d is diameter, and W is axle load, 
Larson et al. (1968) 
Similitude studies in tillage

Considered one dependent and 11 independent variables in the similitude study of moldboard plows. The independent variables were: plow width (L) , all other pertinent lengths (L), lateral angle of plow surface (), a design parameter related to the shape of the moldboard (), cohesion (FL^{2}). internal friction angle (), specific weight of soil (FL^{3}), adhesion (FL^{2}), soilmetal friction angle (), implement speed (LT^{1}), and acceleration due to gravity (LT^{2}). The dependent variable was draft force (F), Distortion arising from cohesion and internal friction angle related Pi terms were found to be important, and prediction fact were developed for overcoming distortion caused by these variables, Effects of disorientation due to Pi terms related to adhesion and soilmetal friction were found to be negligible. 
Pierrot and Buchele (1968) 
Similitude studies of unpowered pneumatic tire

Considered one dependent and ten independent variables in studying unpowered pneumatic tire performance. The independent variables were: tire diameter (L), width (L), tread configuration (ƛ), tire stiffness (FL^{1}) wheel mass density (FL^{4}T^{2}), Sinkage plate penetration pressure (FL^{2}), soil mass density (FL^{4}T^{2}), speed (LT^{2}), vertical lead (F), and acceleration due to gravity (LT^{2}) The dependent variable was rolling resistance (F). 
Reaves et al. (1968)
Note: Used (MLT) instead of (FLT) system 
Similitude studies in tillage

Considered one dependent and 12 independent variables in the similitude study of vertical chisels. The independent variables were: chisel width (L), depth (L), surface roughness (L), leading apex angle (), angle of inclination(), cohesion (ML^{1}T^{2}). interna1 friction angle (), bulk density (ML^{2}) adhesion (ML^{1}T^{2}), cone index (ML^{1}T^{2}), chisel speed (LT^{1}), and acceleration due to gravity (LT^{2}), The dependent variable was draft force(ML^{1}T^{2}), Distortion due to soil properties related Pi term was considered, and the prediction factor was experimentally evaluated 
Scafer et al.(1968) 
Similitude in tillage studies 
Considered one dependent and nine independent variables in similitude studies of disktype implements. The independent variables were: disk diameter (L), all other pertinent lengths (L), angle of approach (), tilt angle (), specific weight of soil (FL^{3}), soil moisture content (), other pertinent soil properties (F^{a}L^{b}T^{c}), disk speed (LT^{2}) and acceleration due to gravity (LT^{2}). The dependent variable was draft force (F). The models and the prototype were operated in the same soil condition leading to distortion in the soil properties related Pi term, Cone index values and soil moisture content and its history were used to create similar soil conditions with the hope that all pertinent soil 
Reaves et al. (1969)

Similitude in bulldozer blades 
Considered one dependent and five independent variables With the assumption that all relevant soil properties were held constant between models and the prototype during experiments, and blades had similar geometrical shape. The independent variables were: blade width (L), cutting depth (L), blade loading distance (L), operating speed (LT^{1}), and acceleration due to gravity LT^{2}), The dependent variable was draft force (F). Concept of loadgrowth curve was used to accurately predict draft requirements of bulldozer blades, 
Scafer et al.(1968) 
Similitude in soilmachine systems and distortion due to inability in scaling properties 
Considered a simplified case in which only one dependent and five independent variables were considered for a soilmachine system that involved the operation of both models and prototype in the same soil type and condition at a low ground speed, The variables considered were: characteristic length (L), other pertinent lengths (L), desired force (F), other relevant forces (F), characteristic soilproperty (F^{a}L^{b}) and other pertinent soil properties (F^{a}L^{b}). The authors showed that the prediction factor was a simple power function of the length scale used. The existence of this relationship makes the use of distorted models a valuable research tool in model studies 
Freitag et al. (1970a, 1970b, 1977) 
Similitude studies of soilmachine system 
Stateofthe art review paper related to similitude in soilmachine system. Detailed accounts of relevant soil properties, their measurement, inconsistencies in measured values due to different techniques, desirability of using analog devices to measure soil properties. difficult in scaling soil properties, problems in maintaining same soil properties for model and prototype even when soil preparation techniques were the same, and use of artificial soils were given. 
Sprinkle et al. (1970) 
Similitude study with static and dynamic properties of artificial soils 
Considered different combination of variables to predict the draft of rectangular blades using scale models, use of artificial (Goose lake fire clay and SAE 5W) to scale soil properties was investigated moreover, scaling for speed effect was also found to be important, Inclusion of viscous effects (speed effect improved the prediction at higher speeds. At lower speeds (<1.6kph or mph), 
Wang and Liang(1970) 
Tillage tool draft 
Outlined a technique to predict draft of a tillage tool in any soil and at any speed using a geometrically similar model. However, the technique required selecting length scale based on soil properties (cohesion and specific weight of soil). The investigators considered higher variables. Independent variables were: characteristic length (L), cohesion (FL^{1}), internal friction angle (), specific weight (FL) Minge tool speed (LT^{1}), apparent Soiltool friction angle (), and acceleration due to gravity (LT^{2}), The Dependent variable was draft force (F). 
Luth and Wismer

Development of prediction

Considered two dependent and seven independent variables to describe soil cutting by blades since soil metal friction and soil internal friction angle were constant for sandy sails used in these studies. Independent variables were: blade width (L), blade length (L), operating depth (L), blade angle (), specific weight of soil (FL^{3}) operating speed (LT^{1}), and acceleration due to gravity (LT^{2}) Dependent variables were: draft force (F)and vertical reaction (F).

Verma and


Developed a general technique to compensate for distortion due to soil properties (i.e., operating model and prototype in the same soil), The procedure was demonstrated by considering a soilchisel system consisting of one dependent variable and six independent variables. Independent variables were: chisel width (L), leading apex angle (), angle of inclination (), depth of cut (L), a characteristic soil property (FL^{1}), and other pertinent soil properties (F_{f }^{0 }L_{f}.). The dependent variable was draft force 
Verma and

Distorted models in non uniform soil profiles

Effect of distortion due to the operation of prototype and models at the same depth in same soil type and condition with nonuniformity with soil profile was discussed. The methodology was demonstrated using A soilchisel system. The variables considered were same as in Verma and Schafer (197la). They showed that operation of the models and prototype at the same depth an soils with non uniform profiles was a useful technique to deal with nonuniformity of soil profile with depth. 
Schafer et a1. 
Distortion in the similitude

Stateoftheart review paper that addresses the issue of distortion in similitude studies of soilmachine systems, It lists various sources of distortion and the key role played by soil properties in inducing distortion. Various cases involving purely frictional, purely cohesive, and cohesivefunctional soil were considered. Techniques for dealing with distorted models were described in detail 
Wismer and Luth (1972)

Development of prediction

Considered two dependent and nine independent variables to describe soil cutting by blades in almost fully saturated clay. Independent variables were: blade width (L), blade length (L), operating depth (L),blade angle (), specific weight of soil (FL^{3}), cohesion (FL^{2}), shear rate factor (), operating speed (LT ^{1}), and acceleration due to gravity (LT^{2}), Dependent variables were: draft force (F) and vertical reaction (F). The soil internal eagle of function was negligible, and soilmetal friction was constant. Soil cohesion and shear rate effect were incorporated using a dimensionless term developed from cone index values obtained using a standard cone, blade speed, and blade width. 
Swanson (1973a) 
Dimensional analysis to 
Considered two dependent and seven independent variables in the study of performance of dual and tandem wheels in sand. Independent variables were: wheel diameter (L), wheel width (L), dual wheel spacing (L), frictional soil sinkage parameter (FL^{n2}), soil sinkage exponent (), and load (F), Dependent variables were: motion resistance (F) and wheel sinkage (L). Note that n is the exponent in the plate sinkage equation. 
Swanson (1973b) 
Scale model studies of treadless tire tests in clay 
Tests with scale model treadless tires were conducred to verify the use of a slightly modified dimensionless clay mobility number (N_{C}) in predicting the performance of treadless pneumatic tires in clay:
where C is cone index (FL^{1}), b is tire section width (L), h is section height (L), d is overall diameter (L), S is tire deflection (L), and W is axle load (F), 
Wismer and Luth (1973, 1974)

Offroad vehicle traction

A set of simple. widely used traction prediction equations for offroad vehicles was developed using three dependent and six independent variables. Independent variables were: tire section width (L), overall tire diameter (L), rolling radius (L), soil cone index value (FL^{1}), axle load (F), and wheel slip (). Dependent variables were: towed force (F), net traction (F), and input torque (FL). The concept of wheel numeric was used to represent soil type and condition 


Considered one dependent and seven independent variables to predict the dynamic net traction of smooth, rigid wheels operating at relatively low speed. Independent variables were: wheel diameter (L), wheel width (L), other pertinent length parameters (L), slip (). characteristic soil property (FL_{a}), other pertinent soil properties (FL_{b},), and dimensionless soil properties (), The dependent variable was dynamic net traction (F). The authors did not specify the soil properties but wanted to determine the dimension of the relevant soil property or properties. based on the study of the distortion factor. They found fundamental differences between large prototype and small mode1s that made it impossible to determine the dimensions of the relevant soil property or properties. 
Burt et al. (1974b) 
Similitude studies related

Considered one dependent and seven independent variables to predict the sinkage of smooth, rigid wheels operating at relatively low speed. The independent variables were: wheel diameter (L), wheel width (L), other pertinent length parameters (L), slip (), characteristic soil property) (FL_{a}), other pertinent soil properties (FL_{b}), and dimensionless soil properties (). The dependent variable was wheel sinkage (L), Similar to Bun et al, (1974a), the authors did not specify the soil properties but wanted to determine the dimension of the relevant soil property properties based on the study of the distortion factor, They found fundamental differences between large prototype and s:ma11 models. However, the difference in the fundamental behavior was not as significant as it was for the prediction of dynamic net traction. The dimension of the relevant soil property was found to be FL ^{223} 
Wismer et al (1976) and Wismer et al. (1977)

Application of similitude 
Provides a stateoftheart review of dimensional analysis and similitude, and discusses the application of these principle to pneumatic tires in soft soils, soil cutting, tillage implements, and bulldozer blade. Traction studies: Considered 13 independent and four dependent variables. Independent variables were: tire diameter (L section height (L), section width (L), deflection (L), cohesion (FL^{2}), internal friction angle (), specific weight (FL3), spissitude (FL^{2}T),load (F), wheel speed (LT^{1}), slip (), tire soil friction (), and acceleration due to gravity (LT^{2}) Dependent variables were: pull (F), towed force (F), torque (FL) and sinkage (L). Soil cutting: Considered II independent variable; and two dependent variables. Independent variables were: blade width (L), length (L), depth (L), angle (), cohesion (FL·^{2}), internal friction angle (), Tillage implements: Considered ten independent and one dependent variables for a disk type implement. Independent variables were' disk diameter (L), other pertinent lengths (L). angle of inclination (), approach angle (), moisture cent (), clay content (), specific weight (FL^{3}), shear rate factor (), tool speed (LT^{1}), disksoil faction (). and acceleration due to gravity (LT^{2}). The dependent variable was waft force (F). Bulldozer blades: Considered I7 independent and one dependent variables. Independent variables were: blade width (L). height (L), radius .of curvature (L), angle of curvature of moldboard (), lift 
Gee Clough et al. (1978) 
Development of prediction 
Considered one dependent and seven independent variables to develop a prediction equation for draft of a particular plow working in different soils. The independent variables were: depth of cut (L), width of cut (L), speed (LT^{1}). acceleration due to gravity (LT^{2}), specific weight (FL^{3}), soil stress (FL^{2}). Three different stress terms were tried: I) cone index, (2) shear stress ґ=c+p tan(Φ) where p is passive earth pressure and µ is the angle of internal function, and (3)f=C_{a}+pµ where c_{a} is adhesion and µ is the soilmetal friction coefficient. Only the first representation of soil stress (i.e., cone index value) was found to be useful in predicting plow draft. 
Pandey and Ojha

Development of prediction 
The effect of lug height, lug angle, lug spacing, and rim width on the performance of a 685 mm rigid wheel in puddled, lateritic sandy clay loam soil operating at a slow speed was investigated using three dependent and eight independent variables. Independent variables considered were: wheel diameter (L), run width (L),lug height (L), lug angle (), lug spacing (), normal load (F), slip (), and specific weight of soil (FL^{3}). Dependent variables were: net traction (F), input torque (FL), and sinkage (L). A multiple linear regression technique was used to develop the prediction equation. 
Wadhwa (1980) 
Soil cutting by rectangular 
Developed soil cutting force predict ion equations for rectangular blades and ang1e tools using a distorted model that was operated in the same soil and depth as the prototype. A composite parameter, quasifluidity (F^{1}L^{2}T), that is related to operating depth and quasishear rate to represent soil strength was used. 
Ehrlich (1985) 
Similitude 
Address by the president of the International Society of Terrain Vehicle System that outlined general principles of dimensional analysis, prediction equation development. and scale modeling. The author suggested a unique way to deal with distortion in model studies. 
Evans et al. (1985)

Similitude of soil tillage

Considered one dependent and nine independent variables for the draft requirements of a three chisel system, Independent variables were: chisel width (L), distance of center chisel from the side chisels (L), angle subtended by the center chisel to the line that connects the outer chisels (), depth of operation (L), tool speed (LT^{1}), acceleration due to gravity (LT^{2}) dimensionless soil properties such as internal friction and soil metal friction (), soil properties that contain and L dimension such as cohesion and adhesion (FL^{a}), and soil properties that contain the dimensions of F, L, and T (F^{b}LT^{c}) The dependent variable was draft force (F). A concept of draft ratio between the center chisel and the whole tool was used to study interaction effect between chisels. 
Brixius(1987) 
Traction prediction

Extended the traction prediction equations developed by Wismer and Luth (1973, 1974) by replacing wheel numeric with a more detailed mobility number. The equations were based on three dependent and eight independent variables. Independent variables were: tire section width (L), overall tire diameter (L), rolling radius (L), tire deflection (L), section height (L), soil cone index value (FL^{2}), axle load (F), aud wheel slip (). Dependent variables were: towed force (F), net traction (F). and input torque (FL). The compressive mobility number (B_{n}) used to represent soil was given by:
where C is the cone index value, b is the section width, d is the wheel diameter, W is the axle load, S is the tire deflection. and h is the section height.

Salokhe and Ninh

Prediction equation 
Used dimensional analysis to develop a soil compaction prediction equation under a pneumatic tire in a clay soil. The authors investigated 11 variables: dry density (ML^{3}) tire section width (L), tire diameter (L), inflation pressure ( ML^{1}T^{2}), cone index(ML^{1}T^{2})Moisture content (), initial dry density (ML^{3}), travel speed (LT^{1}), axle load (MLT^{2}), number of passes (), and gravitational accelation. 
Canillas and 
Prediction equation 
Used dimensional analysis to develop a soil compaction prediction equation under a pneumatic tire in three different soils: clay, silty clay loam, and silty loam. The authors investigated eleven variables: bulk density (ML^{3}) and cone index(ML^{2}T^{2}) as dependent variables, and tire section width (L), tire diameter (L), inflation pressure (ML^{1}T^{2}), initial cone index (ML^{1}T^{2}) , moisture content (), initial dry density (ML^{3}) travel speed (LT^{1}), axle load (MLT^{2}), and number of passes() as independent variables.
