The exact determination of contact stresses for complicated surfaces is a difficult process. It depends strongly on the geometry of the contacting surfaces. Point/line contacts are used to describe ideal situations whereby the contacting bodies do not undergo any deformation, that is, theoretically;

- For 2 spheres pressed against each other, the contact surface is a point
- For 2 parallel cylinders pressed against each other, the contact surface is a line

However in practice, a real area of contact is formed due to the immediate deformation of the mating surfaces. Thus, when two bodies having curved surfaces are pressed together, the point or line contact changes to area contact, and the stresses developed in the two bodies are three dimensional. The size of the contact area depends on the load and the materials from which the objects which are in contact are made of. The deformation is elastic in general, although some plastic deformation is also possible. These stresses were first studied by the German physicist Heinrich Hertz in 1881.

Hertz’s analysis of contact stress involves the following set of assumptions:

- The surfaces are smooth and frictionless.
- The contact area is small compared to the size of the bodies.
- The bodies are isotropic and elastic.

The preliminary steps towards the solution to contact problems basically entail determining the size and shape of the contact area, the normal pressure distribution, resulting stresses and eventually the deformations. The common classical solutions employed in elastic contact problems are illustrated below.

Suppose two spheres are pressed against each other with a collinear pair of forces F (Figure 1).

The contact surface is ideally a point. However, from a practical standpoint, the area of contact is circular in shape, with its radius given by the following equation, where E and v are the elastic constants:

The normal pressure distribution within the contact area is hemispherical as shown in Figure 1.The maximum contact pressure, which is also the maximum principal stress, occurs at the centre of the contact area and is given by:

The principal stresses occur on the z-axis and can be determined from the following equations:

Equation 3: \( \sigma_x = \sigma_y = -P_{max} \Bigg[ 1 – {\left| \frac{z}{a} \right|}\taninv{\frac{1}{z/a} } \Bigg] \)

The maximum shear stress in any of the two spheres can be determined from:

The relative displacement of the centres of the spheres (deformation or more precisely indentation) due to the contact force is given by:

Equations 1, 2 and 6 are perfectly general. They can be employed for determining the contact area and the contact pressure respectively for two scenarios; one scenario entailing a sphere pressed against a flat surface and the other, involving a sphere pressed against an internal spherical surface.

For a plane surface, the radius of curvature = ∞, and by extension, the diameter = ∞.

For an internal surface, the radius of curvature ∞, and by extension, the diameter < 0.

The corresponding parameters (a, Pmax and ∂) can be determined by substituting the above conditions in Equations 1, 2 and 6.

When two cylinders are pressed against each other (Figure 2) with a collinear pair of forces F, a rectangular area of contact, of width b and length l, is formed. Theoretically, this contact surface is supposed to be a line.

The half width of the rectangular area is given by:

The pressure distribution is elliptical and the maximum contact stress can be determined from:

For a special case, with similar elastic constants for both cylinders, i.e. v_{1}=v_{2}=v and E_{1}=E_{2}=E, the deformation caused by the contact force can be determined from:

The principal stresses on the z-axis can be determined using the following equations:

Another contact case entails friction that causes shear stresses on the contact zone. These elevated stresses are beyond the Hertzian field.

Equations 7 and 8 can be used to determine the contact parameters between a cylinder and an internal spherical surface or a flat surface. The aforementioned conventions pertaining to the diameters in 1.1 must be used.

For surfaces with a minimum and maximum radius of curvature, the contact pressure may be determined using:

Where a and b are the lengths of the semi-axes of the elliptical area of contact

The angle between the normal planes in which the minimum radii of the contacting surfaces lie, dictates the values of a and b.

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]]>There are three general cases of linear motion characterised by the type of acceleration it is undergoing.

If the object’s speed is fixed at all times, then it does not experience any acceleration or deceleration at all. Its speed will remain constant as time changes. Remember that there may or may not be forces acting on it but there is **no net force** causing it to accelerate. The equation of motion used in this case is simple, in the form of:

\(v=s/t\)

where,

\(v\): speed or velocity (in m/s or ft/s) \(s\): distance travelled or displacement (in m or ft) \(t\): time taken to travel that distance (in s)

Example 1

Answer: First, the time taken needs to be converted to seconds to remain consistent with units. By plugging in the values into the equation, we then get a distance of 900 m.

In this case, the body accelerates at a constant rate throughout its motion. The speed it travels will therefore change as time advances.

Typical graphs for this type of motion are as shown below.

The following formula of motion are then used:

\(v=u+at\)

\(s=ut+\frac{1}{2}at^2\)

\(v^2=u^2+2as\)

where,

\(v\): final velocity (m/s) \(u\): initial velocity (m/s) \(a\): rate of acceleration. If body is in deceleration, then use a minus sign (\(m/s^2\)) \(t\): time taken (s) \(s\): distance travelled (m)

Example 2

*Answer:* Using the third formula of motion (\(v^2=u^2+2as\)) and the quantities \(a=9.81 m/s^2 \), \(s=6m\), \(u=0 m/s\); we then get an answer of 10.85 m/s.

Each object undergoing this type of motion will have a specific equation describing its displacement, velocity and acceleration at specific points in time. By knowing one of these equations, it is possible to determine the others by integrating or differentiating with respect to time. For this type of motion, it is important to keep in mind the following:

- Speed (\(v\)) is the rate of change of distance (\(s\)) with time (\(t\)) or in calculus terms: \(\frac{ds}{dt}\)
- Acceleration (\(a\)) is the rate of change of speed (\(v\)) with time (\(t\)) or in calculus terms: \(\frac{{d^2}s}{d{t^2}}\)

Example 3

*Answer:* The speed can be found easily by plugging t = 5 s in the above equation, giving \(v=506 m/s\). The acceleration can also be found by differentiating \(v\) and using t = 5 s. To find the distance travelled, we need to integrate \(v\) using limits of t = 0s and t = 5 s.

Consider a body that is moving in circular motion about point O as shown below.

If the body takes a time \(t\) to make an angle \(\theta\) about point O, then we can find its angular speed using:

\(\omega = \theta/t\)

The period \(T\) for the object to complete a full revolution is given by:

\(T = \frac{2\pi}{\omega}\)

The distance travelled \(s\) is calculated using:

\(s = r\theta\)

The linear speed can also be found by dividing the last equation by time taken \(t\):

\(\frac{s}{t} = r\frac{\theta}{t}\)

\(v = r\omega\)

For the body to remain in circular motion, there needs to be a force pulling it towards the centre. Otherwise, the object will move out of orbit in a straight line according to Newton’s First law of motion. This force is also known as the centripetal force and it induces an acceleration on the body. The direction of the acceleration is towards the centre which is the same as that of the centripetal force. The centripetal acceleration can be calculated as:

\(a = \frac{v^2}{r}\)

or

\(a = r\omega^2\)

The motion of a projectile thrown at an angle \(x\) degrees relative to the ground is shown below.

The velocity vector \(u\) can be decomposed to its 2 components \(u_x\) and \(u_y\).

\(u_x = u\cos{x}\)

\(u_y = u\sin{x}\)

As soon as a projectile is released, its trajectory will depend on two things: its initial velocity and the acceleration due to gravity.

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]]>The post Powder Metallurgy appeared first on Engineering Notes.

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In summary, powder metallurgy involves, firstly, the transformation of solid metal into metal powder. Different powders can then be blended together to form a mixture of two or more metals according to what is desired. However, blending several metal powders is optional as we will see that the advantage of PM goes beyond the ability to mix several powders together. Once lubricant has been added to the mixture, it is then allowed to flow into a die in which it is compressed to form a green compact. Subsequent sintering operation allows the compact to gain strength.

There are several techniques that can be used for the production of metal powders.

One of the most popular methods is atomisation which is the disintegration of liquid metal into powder. It can be subdivided into 3 branches:

**Centrifugal atomisation**wherein droplets of liquid metal is ejected from a plate or disk which is made to rotate at high speeds.**Gas atomisation**makes use of a jet of air or noble gas such as nitrogen to turn metal into powder.- In
**Water atomisation**, the water is forced onto a nozzle at high pressure. The impact of water droplets on the metal causes it to disintegrate into powder.

Each of the three above-mentioned methods have their own relative advantage. For example gas and water atomisation can lead to contamination of the metal or compound. The centrifugal method, however, can produce uncontaminated metal powder. Other factors which differ include the energy consumption and the physical characteristics of the powder. Atomisation can be used to produce both metallic and non-metallic powders.

Metallic powders can also be produced from their ores. Several processes have been commercially developed for this purpose including the Hoganas Sponge Iron Process and the Pyron Process. One of the main differences between the two processes is the choice of reducing agent. The Hoganas process uses carbon while hydrygen is used in Pyron.

Electrolysis is also a very popular method for producing metallic powders. It is widely used for producing copper, nickel, silver and iron powders. One of the disadvantages of electrolysis is the difficulty to produce fine powders cheaply and quickly. For this reason, electrolysis is sometimes used in conjunction with other methods such as pulverisation.

This technique involves crushing pieces of metal through shear, compressive or impact mechanical forces to produce fine powders. It may be used on metals that are hard and brittle. Ball mills and vortex mills are common apparatus used for this process.

Specific quantities of different metals may be blended together to form a mixture that will give the desired properties. Additives and lubricants are also added to the mixture. The lubricant will allow for the flow of the powder into the mould and also help during the removal of the object from the mould.

In this process the powder is compressed and made into the part’s final or near final shape. Two basic methods exists for this:

A die is used to compress the powder to the desired shaped with forces going as high as 1 GPa may be used. Sufficient force is used so as to give enough *green strength* which will maintain the shape.

It is the same as hot compaction except that the powder is heated and the die needs to be resistant to heat. The next operation which is sintering, may or may not be required for hot compacted parts.

The sintering process improves the mechanical and physical properties of the *green compact.* The heat applied is around 0.7 of the absolute melting temperature of the powder. The process will cause the metal powder compact to shrink and bond by diffusion. The shrinkage will cause an increase in the density of the part.

PM is used widely in the automotive industry for making various engine parts such as porous bushings and connecting rods. It is also used for making filament lamps, turbine discs and self-lubricating bearings.

- Less energy required for processing compared to forging.
- Negligible tool wear compared to machining.
- Alloys can be created from compounds that are difficult to mix otherwise, such as due to having different melting points.
- Porous parts can be created with PM that can contain lubricants such as in self-lubricating bearings.

- There exists a variation in density throughout the part because of compaction. Inner core of part is normally less dense.
- It can be hard to create all types of shapes due to compaction process requirements.

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]]>The post Drilling appeared first on Engineering Notes.

]]>Drills come in various shapes and sizes and their use depend on the operation being performed. For example, twist drills with two flutes and two cutting edges, are used for drilling holes. Other types of tools exists for making counter-bores and counter-sinks.

Twist drills can have more than one cutting edge and flutes. The flutes may be straight or helical themselves. The shank may be straight or tapered depending on the machine chuck onto which it is to be mounted.

The flutes serves as an opening for lubricants to get into the drill region and also as a path for chips to move out of the hole being made.

There are two speeds involved in a drilling operation, namely the cutting speed and the rotation speed of the drill bit. They are related by the following equation:

\(v = \pi N D\)

where,

\(N\) is the spindle speed (rev/min),

\(v\) is the cutting speed (mm/min or in/min),

\(D\) is the drill bit diameter (mm or in).

By using the concentrated energy in a laser beam, a material can be made to melt at localised points. This principle can be used for making holes with very high accuracy. There are several methods used in practice such as direct, percussive or trepanning drilling when using lasers.

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]]>The post Introduction to Machining Operations appeared first on Engineering Notes.

]]>Often, processes such as polishing are performed after other operations such as casting. This setup helps to speed up the whole operation and also to minimise waste.

There are several **advantages of using machining** operations to build a part:

- A large variety of materials can be machined into final shape.
- Numerous tool shapes are available for machining parts of any geometry.
- Parts can be machined to very close tolerances compared to other processes of fabrication such as casting and forging.
- Surface finish of the part can be controlled to very small surface roughness values.

However, there are some **disadvantages** also:

- There is considerably more wastage of materials involved than in casting.
- It is time consuming and might not be appropriate where time is an issue.

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]]>The post Resistor Combination appeared first on Engineering Notes.

]]>Resistors are combined in series as shown in Figure 1.

The formula for calculating the effective resistance \(R_{eff}\) is

\(R_{eff} = R_1 + R_2 \)

The above diagram can then be simplified with only one resistor with resistance \(R_{eff}\) instead of two.

For a circuit with more than two resistors in series, the same addition principle applies. Say the circuit has n resistors, effective resistance becomes

\(R_{eff} = R_1 + R_2 + R_3 + \dots + R_n \)

Resistors can also be combined in parallel circuits as shown in Figure 2.

The effective resistance \(R_{eff}\) is given by

\(\large{\frac{1}{R_{eff}} = \frac{1}{R_1} + \frac{1}{R_2}}\)

The above equation is can be rewritten as

\(R_{eff} = \frac{R_1 R_2}{R_1 + R_2}\)

In the case of circuits with more than two resistors, the general equation is

\(\large{\frac{1}{R_{eff}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots + \frac{1}{R_n}}\)

Consider the circuit shown in Figure 3 below.

Resistors \(R_1\) and \(R_2\) are in parallel and can be combined using the above equation

\(R_{1,2} = \frac{R_1 R_2}{R_1 + R_2}\)

Next, the overall effective resistance can be found by combining \(R_{1,2}\) with \(R_3\). Since they are in series, the resistances are added.

\(R_{eff} = R_{1,2} + R_3 \)

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]]>The post Stress Transformations appeared first on Engineering Notes.

]]>Consider a body on which there are several forces acting. We can isolate an element of unit area around a point P and measure the stresses \(\sigma\) and \(\tau\) along Cartesian axes as shown.

Notice that \(\sigma_x\) is acting parallel to the x-axis and \(\sigma_y\) is acting parallel to the y-axis.

Let us now cut another element around the same point P but at an angle \(\theta\) to the x-axis. The normal and shear stresses acting on the element will be different. Let the x prime axis be at an angle to the x axis.

Therefore the stress we measure on an element around a point is dependent on its orientation. Since stress is a tensor, matrices are used to fully express the stress state in a material. For a 2D element, the matrix has the following form:

\(\large{\begin{bmatrix}

\sigma_{x} & \tau_{xy} \\

\tau_{xy} & \sigma_{y}

\end{bmatrix}}

\)

The following 3 stress transformation equations can be used for finding the stresses at other orientations than the one in which we are given.

\(\displaystyle{\sigma_{x’} = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x – \sigma_y}{2} \cos{2\theta} + \tau_{xy}\sin{2\theta}}\)

\(\displaystyle{\sigma_{y’} = \frac{\sigma_x + \sigma_y}{2} – \frac{\sigma_x – \sigma_y}{2} \cos{2\theta} – \tau_{xy}\sin{2\theta}}\)

\(\displaystyle{\tau_{{x’}{y’}} = – \frac{\sigma_x – \sigma_y}{2} \sin{2\theta} + \tau_{xy}\cos{2\theta}}\)

The above equations can be used as simple ‘plug and chug’ in most cases of 2D stress transformations. In fact, these equations can be applied to any 2 dimensional tensor other than stress, including strain.

Christian Otto Mohr was a German Engineer who developed a way to graphically express the above stress transformation equations. His method makes it simple to understand stress variations on the element with changes in its orientation. His diagram is known as the Mohr’s Circle.

The axes of the circle is made up by the normal \((\sigma)\) and shear stresses \((\tau)\). The cicle is drawn for 1 face of the element (normally the y face). The x face is found at a -90 degree rotation from the y face using the cartesian axes. But on the Mohr’s circle, **rotations need to be doubled**. This means that the x face is 180 degrees from the y face. Values for the x face is therefore found by making a 180 degree rotation from the point at which the stress for the y face is.

The circle itself is drawn by making use of 2 quantities, the average stress \((\sigma_{avg})\) and the radius \((R)\). Once the circle has been drawn, it becomes possible to extract additional information such as the principle stresses and the location of the principle axes from the plot.

Since the stress values vary depending on the angle \(\theta\), there is a special orientation at which the normal stresses \(\sigma\) will be maximum and minimum. These are known as the principal stresses. They are denoted by \(\sigma_1\) and \(\sigma_2\) with \(\sigma_1\) having the highest value.

There is a special orientation at which the normal stress is maximum while the shear stress is zero. These are called the principal axes and are denoted by 1 and 2.

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]]>The post Stress Concentration (Stress Riser) appeared first on Engineering Notes.

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The stresses at regions where stress concentrations occur are much higher than at other regions. If these are not taken into consideration during the design stage, the material may yield and fail prematurely.

Stress concentrations can occur upon application of different types of loads whether axial, bending or shear loads.

The theoretical stress concentration factor \(k_t\) can be used as a measure of stress concentration. It is also known as the form factor and is given by the following equation

\(k_t = \sigma_{max}/\sigma_{nom}\)
or in the case of shear loads,

\(k_{ts} = \tau_{max}/\tau_{nom}\)

Where,

\(\sigma_{max}\) and \(\tau_{max}\) are the maximum stresses in the part

\(\sigma_{nom}\) and \(\tau_{nom}\) are the reference stresses in the part

**Approximations**

In some cases where cracks or defects are present in a part, it might be reasonable to use an approximation for calculating the stresses. For example, a crack that is very similar to a hole can be approximated by the latter of a chosen radius.

Peterson’s Stress Concentration Factors is a great resource that contains K values for various shaped elements and is recommended for those wishing to have a closer look at stress concentrations phenomena.

Stress concentration factor

Stress concentration kt

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]]>The post Abbe’s Principle of Alignment appeared first on Engineering Notes.

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His principle consists of the following 3 points :

**1.** For best results, a linear reading should be taken either inline or sideways of the object.

**2.** In case the above is not possible, the measurement can be taken at a distance parallel to the line being measured. In this case, the distance separating the object and the scale is known as the **Abbe Offset**. The Abbe Offset introduces no more than a second order error and is negligible.

**3.** If the parallelism between the object and the measuring instrument is not respected, a first order error will be introduced. The error will be a function of the angle the scale makes with the object and the distance separating the two. This error is known as **Abbe Error** which is a subset of **Cosine Errors**.

The error introduced can be calculated using:

\( {\epsilon} = d({\sin}{\theta}) \)

Where, d is the distance between scale and line.

It is important to note that the error amplifies with both the distance and the angle.

By design, Vernier calipers does not conform to the Abbe’s rule of alignment. It is therefore possible to introduce Abbe errors when taking measurements with one.

On the other hand, Micrometers follow the principle. This means that no error of this type can be introduced when using it.

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]]>The post Airy Points and Bessel Points appeared first on Engineering Notes.

]]>Sir George Biddell Airy showed that the ends of the bar can be made completely horizontal when a beam or bar is supported at a special set of points. These points are known as airy points and following equation describes their location in a system with two supports:

\( s =\frac{L}{\sqrt{3}} \)

where,

s : Distance separating the supports

L : Length of the bar

Since \( \frac{1}{\sqrt{3}}=0.577 \), the above relationship is sometimes abbreviated as:

\( s =0.577L \)

The relationship was derived [PDF] using the Euler-Bernouilli beam bending equation and it can be used provided the following set of assumptions is applicable to the beam:

- It is homogeneous and isotropic
- Has uniform cross-sectional area
- It is linear-elastic

Unless these assumptions are true, the calculated airy point might not coincide with the actual one.

In case we are trying to minimise the maximum extent the member deflects from the centre line, then the bessel point should be used for support instead. Bessel Points can be calculated using the following formula.

\( s =0.559L \)

Sufficient care has to be taken when length standards, such as gauge blocks, are used for comparison purposes. Length standards of small size (< 3 inches) are quasi-non affected by any change in length no matter the method of support. However, longer standards can experience appreciable changes in length due to gravity. For example if the support method is vertical a reduction in length will occur because of the compression due to gravity. The change in length can be found using:

\( {\Delta}{L} = \frac{DL^2}{2E} \)

where,

ΔL : Change in Length

D : Density

L : Original Length

E : Young’s Modulus

However, if the standard is supported horizontally at its airy points, the effect of gravity on length can be almost neglected. This method of support can also be used for end standards.

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