Mechanics of Orthogonal Cutting, Force Relationships

Mechanics of Orthogonal Cutting

Introduction to Orthogonal Cutting

• The lecture introduces the mechanics of orthogonal cutting, building on previous discussions about chip types and cutting tools.

• Key angles in cutting tools are highlighted: rake angle (gamma) and relief angle, with a focus on their roles in machining.

• The rake angle is defined as the inclination of the tool's rake face relative to a vertical plane, facilitating chip flow.

Characteristics of Orthogonal Cutting

• In orthogonal cutting, the cutting edge is perpendicular to the cutting velocity, creating a two-dimensional operation.

• The uncut chip thickness (t₀) and actual chip thickness (t) are discussed; typically, w₀ equals w during operations.

• The width of the workpiece is significantly greater than the depth of cut, leading to a plane strain condition.

Chip Formation Mechanics

• Chips flow along the rake face with velocities perpendicular to the cutting edge; forces act only in x and z directions.

• A single shear plane model is introduced for understanding chip formation dynamics during machining processes.

Shear Angle and Cutting Ratio

• The shear angle (phi), rake angle (alpha), and their relationship through geometric considerations are explained.

• A formula relating uncut chip thickness (t₁), actual chip thickness (t₂), shear angle (phi), and rake angle (alpha): t_1/t_2 = sin(phi)/cos(phi - alpha) .

Merchant’s Circle Diagram

• Ernst and Merchant's analysis using a single shear plane model illustrates forces acting on chips during machining.

• Forces such as friction force (F), normal reaction force (N), shear force ( F_s ), and resultant forces are discussed within Merchant's circle framework.

Force Relationships in Machining

• Relationships between various forces including active cutting force ( F_C ) and thrust force ( F_T ) are derived from geometrical representations.

• Merchant’s circle aids in visualizing relationships among these forces while ensuring they remain orthogonal.

Minimization of Power Consumption

• To minimize energy consumption during machining, an expression for power consumption based on F_C , v , and other parameters is established.

• Final expressions show that F_C 's dependence on factors like uncut chip thickness, ultimate shear strength ( tau_s ), and friction coefficient can be optimized for efficiency.

This structured summary captures key insights from each segment of the lecture while providing timestamps for easy reference.

Understanding Shear Strength and Force Relations in Machining

Ultimate Shear Strength and Force Equations

• The ultimate shear strength is constant when k_1 = 0 , indicating it does not depend on normal stress ( sigma ) .

• The force relation is expressed as F_s = w t_1 tau_0 sin(phi)(1 - k_1 tan(phi) + lambda - alpha) = tau_0 after rearranging terms, where F_s represents the cutting force .

• From triangle ABC, the relationship between forces leads to R = F_s / (cos(phi) + lambda - alpha), allowing for further calculations of cutting forces .

Deriving Expressions for Cutting Forces

• The expression for cutting force F_C = w t_1tau_0cos(lambda - alpha)sin(phi) incorporates ultimate shear stress without normal stress .

• By applying the principle of minimum energy consumption, differentiating with respect to shear angle yields a new relation: 2phi + lambda - alpha = C, where C = cot^-1(k_1). This indicates material dependency .

Experimental Validation and Material Dependency

• Experiments show that plotting phi versus lambda - alpha results in a straight line with varying intercepts based on material properties, confirming different relations across materials .

• Observations indicate that increasing cutting speed raises cutting force due to reduced friction, while the cutting ratio approaches a maximum of 1 .

Effects of Rake Angle on Cutting Forces

Influence of Rake Angle

• An increase in rake angle reduces cutting force while increasing the cutting ratio, demonstrating similar effects as increased cutting velocity .

• Various analyses have established relationships such as Ernst and Merchant's first solution: 2phi + lambda - alpha = pi/2, alongside Merchant’s second solution which introduces material dependency through constant C .

Energy Consumption in Cutting Processes

• Specific energy consumption per unit volume can be calculated using main cutting force divided by chip width times uncut chip thickness. This reflects power usage during machining operations .

• Empirical relations suggest that specific energy increases as uncut chip thickness decreases, highlighting challenges in micro-cutting scenarios due to size effects .

Numerical Examples in Orthogonal Machining

Example Calculations with Zero Rake Angle

• In an orthogonal machining scenario with zero rake angle, given forces are used to calculate friction and normal forces on the rake surface. Here, thrust equals friction due to equal directional components [].

• Using provided values leads to determining coefficient of friction at approximately 0.667 and calculating shear angles using Lee and Shaffer's relation yielding a value around 11.3 degrees [].

Adjusting Parameters for New Conditions

• A subsequent problem involves adjusting parameters like rake angle from 10 degrees to zero while maintaining other conditions constant; this results in recalculating thrust and cutting forces using Merchant’s relations [].

Friction Coefficient Variability with Rake Angle

Analysis of Friction Coefficients

• Data from Widia India Limited illustrates how coefficients vary significantly with changes in rake angles; higher angles generally lead to increased frictional resistance [].

Relationship Between Shear Angles and Rake Angles

• Utilizing Merchant's first solution allows tracking how shear angles change relative to rake angles; findings confirm that higher rake angles correlate positively with increased shear angles [].

Practical Applications in Turning Operations

Transitioning from Orthogonal Analysis

• While turning operations deviate from pure orthogonal cuts, approximations can still yield useful insights into expected behavior under various conditions including side edge inclination adjustments [].

Estimating Forces During Turning

• Formulas derived earlier can be adapted for turning processes by considering effective chip thicknesses related directly to feed rates adjusted by side edge angles [], ensuring accurate predictions of thrust components during machining operations.

This structured markdown file encapsulates key concepts discussed within the transcript while providing clear timestamps for reference.