Cutting Analysis - Mechanics of Machining

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Material removal by traditional machining operations has always been the workhorse of manufacturing industry. In this video lecture we will learn how to analyse a single point orthogonal cutting operation.

Detailed description of above video lecture is given below.

Cutting Force and Thrust Force During Cutting

During an orthogonal cutting operation material is removed from the work piece with help of cutting force. Along with that, a thrust force, which is normal to cutting direction is also induced. The thrust force could be either upward, or downward.

Fig.1 Cutting force,Fc and thrust force,Ft induced in an orthogonal cutting operation
Please refer article on Cutting Force Analysis, to learn how to determine this forces

Material Removal by Shearing - Shear Plane Theory

If you take a close look at cutting operation it will be evident that, material is getting removed from work piece by shearing action. Where different layers of atoms slip one another to form chip. This is why there is a change in chip thickness from uncut to cut chip. In actual case this shearing mechanism happens in a region called shear zone.

Fig.2 Material removal happens because of shearing action happens in shear zone
But analysis of such a case will be too complex for this lecture. So we will assume instead of zone the shearing happens in a single plane. This is known as shear plane theory. According to this theory, material from uncut region undergoes, a sudden shear transformation across this plane. As you can see in Fig.3 thin layers of material slip across this plane.
Fig.3 According to shear plane theory shearing action occurs all of a sudden at shear plane
If you know orientation of this plane or shear angle, you can easily predict chip thickness using following equation.
Which is derived from trigonometric analysis by equating length of common edge from cut and uncut side.

Determination of Shear Angle - Merchant Analysis

But what is the orientation of shear plane ?. That is the big question in machining analysis. It could be at any angle during a particular machining operation. As explained in this figures. As shear angle decreases thickness of the chip increases.

Fig.4 Increase of chip thickness with decrease in sear angle for same cutting operation
One way to predict this shear angle is assume shear plane adjust itself to reduce energy required for cutting operation. Or shearing will take place in a plane where there is maximum shear stress. So that force and work required for cutting is minimum. Shear stress at an angle phi is given by following equation.
By differentiating this equation with respect to phi and setting it to zero, one can obtain the plane on which shear stress is maximum.It is given as follows
Where, beta is the friction angle between tool and work piece interface. This is known as, Merchant analysis. Even though result given by Merchant analysis, does not always match with experimental results, this is a fairly good assumption. There are various other models also available to do this.

Since material flow before, and after cutting operation is same we can write.

This is assuming, width of the material does not increase much during machining.

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Working of Gas Turbine Engine

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In this lecture we will understand working principles behind a gas turbine engine. Gas turbines are used mainly for two purposes. First, for power production. Second, for generating thrust force in an aircraft. Even though functions are different working principle behind each case is the same. Here we will understand how gas turbine engine is used for generating thrust force in an aircraft.

Detailed description above lecture is given below.

Gas Turbine Engine for Jet Propulsion

Following figure shows of gas turbine engine of an aircraft. In order to make the flight move forward this engine should produce a force in forward direction.

Fig.1 Jet force required by aircraft is produced by a gas turbine engine
This force is produced by jet effect of this exit fluid. When a high velocity fluid is ejected from aircraft engine it will produce a reaction force which will power the aircraft flight. This force is known as jet force. By applying Newton’s 2nd law of motion to jet engine control volume, one can easily deduce magnitude this jet force as follows.
It is momentum out minus momentum in. So if jet velocity is high it means high thrust force! This is why exit portion of a jet engine has got decreasing area, assuming the flow is subsonic. Or the exit portion acts like nozzle which increase jet velocity.

Continuous Production of High Velocity Jet

If we can produce high velocity jet continuously, the engine will be continuous jet force. We will produce this by a combustion process, by injecting fuel into air. This will produce flames with very high velocity.

Fig.2 Combustion produces flames at high energy, which is then transformed to high velocity flames
But for a sustainable combustion process we need the inlet air to the combustion chamber to be at high temperature and pressure.

Use of Compressor

Surrounding air is brought to high temperature and pressure state with help of diffuser plus compressor arrangement. Air gets into the engine by forward motion of engine and sucking effect of compressor. Diffuser increases pressure and temperature of the fluid to some extent by converting some part of kinetic energy. After that compressor comes where both pressure and temperature of the air is raised by by energy supply from compressor.

Fig.3 Diffuser and compressor together raise pressure and temperature of the incoming air
So at outlet of the compressor we will have air at high pressure and temperature. But compressor requires some power input to do this compression process.

Turbine - A Source of Power to Compressor

Power required compressor is given by a turbine which is situated right after the combustion chamber. The turbine absorbs some amount of energy from the high energy fluid and transmits it to the compressor.

Fig.4 The complete gas turbine, which is self sustainable in operation

Nozzle - Production of High Velocity Jet

Now the fluid with high energy can be expanded in a nozzle section to produce a high velocity jet.In nozzle air will expand to surrounding pressure. Thus the process of producing high velocity jet at outlet has become a self sustainable. We will get continuous supply of high velocity jet and thrust force to this aircraft, thanks to synchronized working of all these components.

Thermal Cycle of Gas Turbine - Brayton Cycle

Variation of state of fluid from inlet to exit of gas turbine engine is shown in Fig.5 in a T-s diagram. Point 1 is the inlet condition of a gas turbine engine, which is same as state of surrounding air. Due to diffuser effect pressure and temperature of the fluid increases slightly, entropy remains same assuming this is an adiabatic reversible process (1-2). Next in compressor stage also same process continues, temperature and pressure rise to a level where combustion process is sustainable (2-3). Now fuel injection and heat addition to the fluid, this process happens almost at constant pressure, here pressure raises to very high level (3-4). Right after that, turbine will absorb some amount energy which is required by the compressor. So here temperature and pressure of the fluid comes down (4-5). Now the last section, which produces high velocity jet. This is again a constant entropy process, where internal energy of the fluid gets converted into kinetic energy. Here pressure expands to the surrounding pressure (5-6).

Fig.5 Variation of state of fluid as it executes Brayton cycle

It should be noted here that the exit stream never go back to the inlet condition, at inlet it sucks fresh steam of air in. So this is an open cycle process, but since both this points are having same pressure we can assume pseudo constant pressure process (6-1) in between in order to complete the cycle.

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Working of Steam Turbine

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Steam turbines are the hearts of the power plants, they are the devices which transform thermal energy in fluid to mechanical energy. In this video lecture working of steam turbine is explained in a logical manner.

A detailed webpage version of the video is given below.

Energy Absorption from fluid - Role of Rotor Blades

When high energy fluid (high pressure and high temperature) passes through series of rotor blades, it absorbs energy from fluid and starts rotating, thus it transforms thermal energy in fluid to mechanical energy.

Fig.1 Rotating blades of turbine helps in transforming thermal in fluid to mechanical energy
So series of such blade which eventually transform thermal energy are the most vital part of a steam turbine. One of such rotor set is shown in figure below.
Fig.2 A typical steam turbine rotor
If you take a close look at one of the blade, it would be clear that a blade is a collection of airfoil cross sections from bottom to top. When flow passes through such airfoils it induces a low pressure on bottom surface and high pressure on top surface of airfoil as shown in figure below.
Fig.3 Fluid flow around airfoil cross sectioned blade induces a high pressure (P) and low pressure(P) on blade surfaces

This pressure difference will induce a resultant force in upward direction, thus making the blade rotate. So some part of fluid energy will get transformed to mechanical energy of blade. Before analyzing energy transfer from fluid to blade, we will have a close look at energy associated with a fluid.

Energy Associated with a Fluid

A flowing fluid can have 3 components of energy components

  • Kinetic energy - Virtue of its velocity
  • Pressure Energy - Virtue of its pressure
  • Internal Energy - Virtue of its temperature
Last 2 components of energy together known as enthalpy. So total energy in a fluid can be represented as sum of kinetic energy and enthalpy.

Energy Transfer to Rotors

When fluid passes through rotor blades it loses some amount of energy to the rotor blades. Due to this both kinetic and enthalpy energy of fluid come down for a typical rotor. As kinetic energy comes down velocity of flow decreases. If we directly pass this stream to next stage of rotor blades it will not transfer much energy because of low velocity of flow stream. So before passing the stream to next rotor stage we have to increase the velocity first. This is achieved with use of a set of stationary nozzle blades, also known as stator. When fluid passes through stator blades velocity of fluid increase due to its special shape thus one part of enthalpy energy will get converted into kinetic energy. Thus enthalpy of stream reduces and kinetic energy of stream increase. It is to be noted that here there is no energy addition or removal from flow, what happens here is conversion of enthaply energy into kinetic energy. Now this steam of fluid can be passed to next rotor blades and process can be repeated. Velocity and enthalpy variation of flow is shown in following figure.

Fig.4 Velocity and enthalpy variations across rotor and stator stages of a typical steam turbine

Degree of Energy Transfer

Total energy transfer to the rotor blade is sum of decrease in kinetic energy and decrease in enthalpy. Degree of contribution of each term is an important parameter in axial flow machines. This is represented by a term called of degree of reaction, which is defined as

Where both enthalpy change and kinetic energy changes are defined across the rotor blade.

0 % Reaction - Impulse Turbines

When D.O.R = 0 there will not be any enthalpy change across the rotor, such a turbine is known as impulse turbine. Blades of such a turbine would like as shown below.

Fig.5 A typical impulse turbine rotor cross section and flow pattern
Here incoming flow stream hits the blade and produces and impulse force on it. Since enthalpy across the blade does not change temperature will also remain same. There will be minor pressure drop across the rotor, but this is almost negligible. Here energy transfer to the blade is purely due to decrease in kinetic energy of fluid.

100 % Reaction Turbines

When D.O.R = 1 kinetic energy change across the rotor will be zero, energy transfer will be purely due to decrease in enthalpy. Since kinetic energy is same across the rotor absolute value of velocities remain same. This is shown in figure below.

Fig.6 A typical reaction turbine rotor cross section and flow pattern
Usually people use compromise of above two discussed cases,that is 50% D.O.R . Such turbines are known as Parson turbines, where both kinetic and enthalpy energy transfer contribute equally to power transfer to rotor.

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Spur Gear Design

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Mechanical engineers working in transmission field would often have to decide upon kind of gears they have to use. Although this task has become a matter of selection of gear based on standards, it is also important to know what goes behind this. In this video tutorial you will learn how to design a pair of spur gears for mechanical strength, surface resistance and fluctuating load.

The video lecture is described below in detail

AGMA Standard of Gear Design

A designed gear should meet following design criteria conforming to AGMA standards. It should have

  1. Enough mechanical strength to withstand force transmitted
  2. Enough surface resistance to overcome pitting failure
  3. Enough dynamic resistance to carry fluctuating loads

Design Inputs and Outputs in Gear Design

Following figure shows design inputs and outputs of a gear design

Fig.1 Input and output parameters for a gear design
Various design output parameters are pictorially represented in following figure.
Fig.2 A general spur gear nomenclatures

Design for space constrains

The designed gear system should fit within a space limit. So the designer could say if he sums pitch diameters of the mating gears, it should be less than or equal to allowable space limit as shown in figure below.

Fig.3 Space constrain of gear design
The blue rectangle represents space on which gear should get fit. One can take 80% of width of this space as allowable width for gear design. So following is the relation obtained by this condition.
We also know speed ratio of gears, this will lead to one more relation in terms of pitch circle diameters.
By solving above 2 equations simultaneously we can obtain pitch circle diameters of both the gears.

Determination of Number of Teeth - Interference

Here we will understand how to determine number of teeth on both the gears. To do this we have to assume number of teeth on one gear(T1), say the smaller gear. Now using the relation given below we can determine number of teeth on other gear,T2.

So we got number of teeth on both the gears, but one should also check for a phenomenon called interference if gear system has to have a smooth operation. Interference happens when gear teeth has got profile below base circle. This will result high noise and material removal problem. This phenomenon is shown in following figure.
Fig.4 A pair of gear teeth under interference
If one has to remove interference , the pinion should have a minimum number of teeth specified by following relation.
Where aw represents addendum of tooth. For 20 degree pressure angle(which is normally taken by designers) aw = 1 m and bw = 1.2 m. Module m, and pitch circle diameter Pd are defined as follows.
If this relation does not hold for a given case, then one has to increase number of teeth T1, and redo the calculation. The algorithm for deciding number of teeth T1 and T2 is shown below.
Fig.5 Flow chart to determine number of teeth on each gears

Design for Mechanical Strength - Lewis Equation

Now the major parameter remaining in gear design is width of the gear teeth, b. This is determined by checking whether maximum bending stress induced by tangential component of transmitted load, Ft at the root of gear is greater than allowable stress. As we know power transmitted,P and pitch line velocity V of the gear Ft can be determined using following relation.

Here we consider gear tooth like a cantilever which is under static equilibrium. Gear forces and detailed geometry of the tooth is shown in figure below.
Fig.6 Gear tooth under load
One can easily find out maximum value of bending stress induced if all geometrical parameters shown in above figure are known. But the quantities t and l are not easy to determine, so we use an alternate approach to find out maximum bending stress value using Lewis approach. Maximum bending stress induced is given by Lewis bending equation as follows.
Where Y is Lewis form factor, which is a function of pressure angle, number of teeth and addendum and dedendum. Value of Y is available as in form of table or graph. Using above relation one can determine value of b, by substituting maximum allowable stress value of material in LHS of equation. But a gear design obtained so will be so unrealistic, because in this design we are considering gear tooth like a cantilever which is under static equilibrium. But that's not the actual case. In next session we will incorporate many other parameters which will affect mechanical strength of the gear in order to get more realistic design.

A More Realistic Approach - AGMA Strength Equation

When a pair of gear rotates we often hear noise from this, this is due to collision happening between gear teeth due to small clearance in between them. Such collisions will raise load on the gear more than the previously calculated value. This effect is incorporated in dynamic loading loading factor, Kv value of which is a function of pitch line velocity.

At root of the gear there could be fatigue failure due to stress concentration effect. Effect of which is incorporated in a factor called Kf value of which is more than 1.

There will be factors to check for overload (Ko) and load distribution on gear tooth (Km). While incorporating all these factors Lewis stregth equation will be modified like this

The above equation can also be represented in an alternating form (AGMA Strength equation) like shown below
Where J is
Using above equation we can solve for value of b, so we have obtained all the output parameters required for gear design. But such a gear does not guarantee a peacefull operation unless it does not a have enough surface resistance.

Design for surface resistance

Usually failure happens in gears due to lack of surface resistance, this is also known as pitting failure. Here when 2 mating surfaces come in contact under a specified load a contact stress is developed at contact area and surfaces get deformed. A simple case of contact stress development is depicted below, where 2 cylinders come in contact under a load F.

Fig.7 Surface deformation and development of surface stress due to load applied
For a gear tooth problem one can determine contact stress as function of following parameters
If contact stress developed in a gear interface is more than a critical value(specified by AGMA standard), then pitting failure occurs. So designer has to make sure that this condition does not arise.

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