Stress and strain

Symbols and units

The table below identifies the symbols and units used in the calculation of stress and strain.

Description	Symbol	Name	Units
Direct stress	σ	Sigma	N/m² and N/mm²
Direct strain	ε	Epsilon	None
Shear stress	τ	Tau	N/m² and N/mm²
Young’s modulus of elasticity	E		N/m² and N/mm²

 

Note: 1N/mm²  =  10⁶N/m² = 1MN/m²
And 1kN/mm² = 1GN/m²
The alternative for stress is the pascal (pa)which equals 1 N/m²

Two effects may be identified, when the force acts on a solid material which remains stationary.

The material will:-

• Exert an internal resisting force known as a state of stress;
• Experience dimensional changes

This behaviour is typical of stressed engineering components but the dimensional changes are usually small are not normally seen with the naked eye.

Direct forces

 

One end of a bar may be subjected to push or pull.  Now, if a bar remains stationary, a pull on one end must result in an equal and opposite pull on the other end, and the bar is said to be in tension.  Similarly, a push on one end is accompanied by a push on the other end, and the bar is in compression.

The forces which are producing a tension or compression are called direct forces.

Also, direct forces are called either tensile (A pull) or compressive (A push).

Tensile forces cause a bar to stretch and compressive forces cause a bar to contract.

Fig 1

Fig 1 Illustrates a bar acted upon by a tensile force at either end causing the bar to stretch.

F = The applied force
A = Cross sectional area of the bar
L = Original length of the bar
x = Change in length produced by the applied force, F

Stress

Forces on a component, e.g.  A bar, are often referred to as loads.
The load that a particular component can withstand will depend on the dimensions of the component as well as the material it is made from; in particular its cross sectional area when subjected to direct forces.

For example if the cross sectional area is doubled then the load that the component can withstand will be doubled.
Therefore what is important is the intensity of loading i.e.  The load per unit area, and this is given the name direct stress.  Often the word ‘direct’ is omitted.

Hence

Stress = Force
             Area

Or in symbols

σ = F
A

Example 1: A rectangular bar of dimensions 5mm breadth x 20 mm depth, supports a load of 4900 N. Determine the stress in the bar.

Known values

F = 4900 N and A = 5 x 20 = 100mm²

Then σ = 4900 = 49 N/mm²
100

Strain

In addition to controlling the stress produced in a component, the dimensional changes also require consideration.
In the case of a bar loaded in tension the extension of the bar depends upon its total length. The bar is said to be strained and the strain is defined as the fractional change in the length of the bar.

Hence

Strain = Change in length
Original length

Note strain does not have any units.

Example 2: A wire of length 2 m, extends by 0.25 mm when acted upon by a tensile force. Determine the strain in the wire.

From ε = x
l

ε = 0.25
2 x 10³

= 1.25 x 10¯⁴

Mechanical Vibration | Introduction To Machine Vibration | Causes of Machine Vibration

What is Machine Vibration?

Most of us are familiar with vibration; a vibrating object moves to and fro, back and forth. A vibrating object oscillates.

We experience many examples of vibration in our daily lives. A pendulum set in motion vibrates. A plucked guitar string vibrates. Vehicles driven on rough terrain vibrate, and geological activity can cause massive vibrations in the form of earthquakes.
There are various ways we can tell that something is vibrating. We can touch a vibrating object and feel the vibration. We may also see the back-and-forth movement of a vibrating object. Sometimes vibration creates sounds that we can hear or heat that we can sense. To observe how vibration can create sound and heat, rub your feet back and forth on a carpet.

In industrial plants there is the kind of vibration we are concerned about: machine vibration.

What is machine vibration? Machine vibration is simply the back and forth movement of machines or machine components. Any component that moves back and forth or oscillates is vibrating.

Machine vibration can take various forms. A machine component may vibrate over large or small distances, quickly or slowly, and with or without perceptible sound or heat. Machine vibration can often be intentionally designed and so have a functional purpose. (Not all kinds of machine vibration are undesirable. For example, vibratory feeders, conveyors, hoppers, sieves, surface finishers and compactors are often used in industry.)

At other times machine vibration can be unintended and lead to machine damage. Most times machine vibration is unintended and undesirable. This article is about the monitoring of undesirable machine vibration.

Shown below are some examples of undesirable machine vibration.

Machine Vibration | Causes of Vibration

What Causes Machine Vibration?

Almost all machine vibration is due to one or more of these causes:

(a) Repeating forces

(b) Looseness

(c) Resonance

(a) Repeating Forces

Imagine a boat anchored in a bay. Waves are slapping the sides of the boat, and as long as the waves continue to act on the boat we would expect the boat to rock.  The boat would be rocking because the waves would be exerting a repeating force on the boat – a force of a pattern repeated over and over again.

Most machine vibration is due to repeating forces similar to those causing the boat to rock. Repeating forces such as these act on machine components and cause the machine to vibrate. Where do the repeating forces that cause machine vibration come from?

Repeating forces in machines are mostly due to the rotation of imbalanced, misaligned, worn, or improperly driven machine components. Examples of these four types of repeating forces are shown below.

(b) Looseness

Looseness of machine parts causes a machine to vibrate. If parts become loose, vibration that is normally of tolerable levels may become unrestrained and excessive.

(c) Resonance

Imagine a child swinging freely on a swing, that is, without the child propelling himself or anyone pushing him. If we observe the motion closely we will see the child swinging at a particular rate. For example, we may see that it consistently takes him three seconds to complete one cycle of swinging.

The rate of the child’s free-swinging is in fact a physical property of the child-swing system – much as the weight of the child is a physical property of the child. It is the rate at which the child will tend to swing while seated on that particular swing. It is the child’s most natural swinging rate on the swing, and the only way he can change it is to interfere with the natural swinging by propelling himself with his feet, changing his posture, rubbing his feet on the ground and so on.

Machines also tend to vibrate at certain oscillation rates. The oscillation rate at which a machine tends to vibrate is called its natural oscillation rate. The natural oscillation rate of a machine is the vibration rate most natural to the machine, that is, the rate at which the machine ‘prefers’ to vibrate.

A machine left to vibrate freely will tend to vibrate at its natural oscillation rate. Most machines have more than one natural oscillation rate. For example, a machine comprising two substructures of different natural oscillation rates will exhibit at least two natural oscillation rates.

In general, the more complex the machine, the more natural oscillation rates it has.

Now consider again the child on the swing. If we aided the swinging motion by repeatedly pushing the child, we would expect the child to swing higher and higher over time.

We would however only cause the child to swing higher and higher if we pushed with the right rhythm. If our pushing rhythm is such that he is sometimes pushed down while he is ascending, we would not expect him to swing properly. To make him swing higher and higher, our pushing rhythm would in fact need to be in harmony with his natural oscillation rate.

For example, we could push him every time – or every alternate time – he reaches his highest point. Only by pushing the child at a rate which is in harmony with his natural or preferred oscillation rate can we cause him to quickly swing higher and higher.

What happens if a machine is ‘pushed’ by a repeating force with a rhythm matching the natural oscillation rate of the machine? A similar situation will arise – the machine will vibrate more and more strongly due to the repeating force encouraging the machine to vibrate at a rate it is most natural with. The machine will vibrate vigorously and excessively, not only because it is doing so at a rate it ‘prefers’ but also because it is receiving external aid to do so. A machine vibrating in such a manner is said to be experiencing resonance.

A repeating force causing resonance may be small and may originate from the motion of a good machine component. Such a mild repeating force would not be a problem until it begins to cause resonance. Resonance, however, should always be avoided as it causes rapid and severe damage. For example, whole bridges have collapsed due to their natural oscillation rates being excited by the mere rhythm of soldiers marching in unison across the bridges.

Introduction to Mechanical Vibrations

A body is said to vibrate if it has periodic motion. Mechanical vibration is the study of oscillatory motions of bodies. Vibrations are harmful for engineering systems. Some times vibrations can be useful. For example, vibratory compactors are used for compacting concrete during construction work. Excessive vibration causes discomfort to human beings, damage to machines and buildings and wear of machine parts such as bearings and gears. The study of vibrations is important to aeronautical, mechanical and civil engineers. It is necessary for a design engineer to have a sound knowledge of vibrations. The object of the sixth semester course on mechanical vibrations is to discuss the basic concepts of vibration with their applications. The syllabus covers fundamentals of vibration, un-damped and damped single degree of freedom systems, multi degrees of freedom systems and continuous systems.

Examples of vibration

1.Beating of heart
2. Lungs oscillate in the process of breathing
3. Walking- Oscillation of legs and hands
4. Shivering- Oscillation of body in extreme cold
5. Speaking – Ear receives Vibrations to transmit message to brain
6. Vibration of atoms
7. Mechanical Vibrations

Classification of vibrations

One method of classifying mechanical vibrations is based on degrees of freedom. The number of degrees of freedom for a system is the number of kinematically independent variables necessary to completely describe the motion of every particle in the system. Based on degrees of freedom, we can classify mechanical vibrations as follows:

1.Single Degree of freedom Systems
2.Two Degrees of freedom Systems
3.Multi degree of freedom Systems
4.Continuous Systems or systems with infinite degrees of freedom

Another broad classification of vibrations is:

1. Free and forced vibrations
2. Damped and un-damped vibrations.

Sometime vibration problems are classified as:

1.Linear vibrations
2. Non-linear vibrations
3. Random vibrations
4.Transient vibrations

A system is linear if its motion is governed by linear differential equations. A system is nonlinear if its motion is governed by nonlinear differential equations. If the excitation force is known at all times, the excitation is said to be deterministic. If the excitation force is unknown, but averages and standard derivations are known,the excitation is said to be random. In this case the resulting vibrations are also random. Some times systems are subjected to short duration non-periodic forces. The resulting vibrations are called transient vibrations. One example of a non-periodic short duration excitation is the ground motion in an earthquake

The main causes of vibrations are:

1. Bad design
2. Unbalanced inertia forces
3. Poor quality of manufacture
4. Improper bearings (Due to wear & tear or bad quality)
5. Worn out gear teeth
6. External excitation applied on the system

The effects of vibrations are as follows:

1. Unwanted noise
2. Early failure due to cyclical stress(fatigue failure)
3. Increased wear
4. Poor quality product
5. Difficult to sell a product
6. Vibrations in machine tools can lead to improper machining of parts