Chapter 1 introduction to mechanical vibration

Mechanical Vibration

Prof. Dr. Eng. Abdul Mannan Fareed

Faculty of Engineering

University of Aden

Nov 2016

Chapter (1): Introduction to

Mechanical Vibration

Example 1: Centrifugal pump on base plate

- Introduction to System Mechanical Vibration

Example 2: 4 Parallel shafts gear box

Example 3: 4-Cylinder in-line engine

Input Energy Output Energy

Machine

What is a Machine (Dynamic System):

� A number of rigid parts connected together in such a

form that if input energy is given to a particular

member, another member shall produce a prescribed

output energy with some losses.

+ Losses

Fig.1 Model of a Machine

� All dynamic systems are capable of vibration.

What is a Mechanical Vibration?

� Scientific Definition

Mechanical vibration is the oscillatory motion of dynamic systems.

Mechanical vibration deals with the relationship between forces

acting on the mechanical system and the oscillatory motion

of mechanical system about a point within the system.

� Engineering Definition

- Objectives of Vibration:

� A dynamic system is a combination of matter which possesses mass and whose parts are capable of relative motion. All bodies possessing mass and elasticity are capable of vibration. The mass is inherent of the body and the elasticity is due to the relative motion of the parts of the body.

� The system may be very simple or complex. It may in the form of a structure, a machine or its components or a group of machines.

� The oscillatory motion may be objectionable or necessary for performing a task.

� The objective of the designer is to control the vibration when it is objectionable and to enhance the vibration when it is useful.

� Objectionable vibration in the machine may cause loosening of the parts or its malfunctioning or even its failure.

� Shakers in foundries and vibrators in testing machines require vibration.

� Operation of many instruments depend upon the proper control of the vibration characteristics of the devices.

� The primary objective of our study is to analyze oscillatory motion of dynamic systems and the forces associated with the motion.

� The ultimate goal is to determine its effect on the performance and safety of the system under consideration.

- Examples of Vibration Motion:

Examples of Vibratory Motions

� To illustrate different types of vibratory motion let us

consider the spring-mass systems shown below.

Fig. Vibratory Motions at Different Frequencies

� When the motion is maintained by the restoring forces only, the vibration is described as free vibration. When a periodic force is applied to the system, the motion is described as forced vibration.

� When the frictional dissipation of energy is neglected, the motion is said to be undamped. Actually, all vibrations are damped to some degree.

Fig. Free and Forced Vibration Systems

Fig. Free Damped and Undamped Vibratory Motions

�Once the system is set into motion, it will tend to vibrate at its natural frequency as well as to follow the frequency of excitation.

� If the system possesses damping, the part of the motion not sustained by the sinusoidal excitation will eventually die out. This is transient motion, which is under free vibrations.

�The motion sustained by the sinusoidal function is called the steady-state-response. Hence this response must be at the excitation frequency regardless of the initial conditions.

Figure: Sinusoidal Vibratory Motions

Natural Frequency

� Fig. illustrates the undamped free vibration. Since the spring is initially deformed from equilibrium, the corresponding potential energy is stored in the spring.

� Through the exchange of the potential and kinetic energies between the spring and the mass, the system oscillates periodically at its natural frequency about its static equilibrium position.

� This motion is simple harmonic motion. Since the system is conservative, the amplitude of vibration will not diminish from cycle to cycle.

� Hence the natural frequency describes the rate of exchange between two types of energy storage elements, namely, the mass and the spring.

Damped Natural Frequency

� Fig. shows a mass-spring system with damping. In addition to the spring force, the mass is acted upon by the damping force, which opposes its motion.

1. If the damping is light, the system is said to be under-damped and the motion is oscillatory. Here the amplitude decreases with each subsequent cycle of oscillation.

2. If the damping is heavy, the motion is non-oscillatory and the system is said to be over-damped.

- Elements of Vibratory Systems:

� Three major elements comprise the vibratory systems; these are:

- Inertia element (Mass, kg )

- Elastic element (Spring, N/m) and

- Energy-dissipative element (Damper, Ns/m)

� The mass oscillates or vibrates while the spring stores energy temporarily during vibration and damper consumes or dissipates the energy.

The mass is assumed to be a rigid body. It executes the vibrations and can gain or lose kinetic energy.

The spring possesses elasticity. A spring force exists if the spring is deformed. The work done in deforming a spring is transformed into potential energy, that is, the strain energy stored in the spring.

The damper has neither mass nor elasticity. Damping force exists only if there is a relative motion between the two ends of the damper. The work or energy input to a damper is converted into heat.

- Examples of Vibration Systems:

� There are in general three types of vibration systems:

1. Axial or longitudinal

2. Lateral or bending

3. Torsional or rotational

� Next figures show the three types of vibration.

Fig. Types of Vibration Systems

Axial Vibration:

Lateral Vibration:

Lateral or Bending Vibration:

Torsional or Rotational Vibration:

Torsional Vibration Rotational Vibration

See this movie!!!!

Simple harmonic Motion, Vector

Representation of Harmonic

Motion:

Definition of Simple Harmonic Motion SHM:

� It is the simplest form of periodic motion.

� It is also the basis for more complex analysis using Fourier analysis.

� Steady-state analysis can be greatly simplified using vectors to represent harmonic motion.

� A simple harmonic motion is a reciprocating motion. It can be represented by circular functions, sine or cosine, or their combination in complex cases.

� Simple Harmonic Motion SHM

Fig. Periodic Motion and Time Period

� Time Period τ in seconds

� Frequency f in Hz

� Amplitude X in mm

� From previous figures, the oscillatory

motion has the following parameters:

Fig. Vibration at a Natural Frequency

� Time interval required for a system to complete a full cycle of the motion is the time period of the vibration.

� Number of cycles per unit time defines the frequency of the vibrations.

� Maximum displacement of the system from the equilibrium position is the amplitude of the vibration.

� Definition of the Parameters:

� The time period τ =��

�s/Cycles.

� The Frequency � =�

�=

�

��Cycles/s or Hz.

� ωis called the circular frequency in radians/s.

� If�( ) represents the displacement of a mass, then velocity and acceleration are the first and second time derivative of the displacement,

� Displacement: � = ��ω

� Velocity: �� = −ω��ω

� Acceleration: �� = −ω��ω

� A Sine or Cosine functions or their combinations can be used to represent simple harmonic motions.

� Examples:

� = �1��ω + �2��ω

= �(�1

��ω +

�2

��ω )

= �(��ω ��α + ��ω ��α )

= ��(ω + α)

Vector Representation of SHM:� A harmonic motion can be represented by a

rotating vector �of constant amplitude � at a constant angular velocity ω.

� The displacement of P from centre O along x-

axis is �� = � = ��ω .

Fig. Harmonic Motion represented by a Rotating Vector

� Similarly, the displacement of P from centre O

along y-axis is � = ! = ��ω .

� Naming the x-axis as real part and the y-axis as the

imaginary one, the rotating vector �is represented by the equation:

� = ��ω + "��ω = �#$�% .

� � represents the length of the vector and " = −1.

� The relations between the displacement, velocity

and acceleration are again as below:

� Displacement: � = ��ω

� Velocity: �� = −ω��ω

� Acceleration: �� = −ω��ω

Fig. Displacement, Velocity and Acceleration Vectors

What are the Causes of Vibrations in

Machines ?

Question ?

� Coupling Misalignment

� Mass Unbalance

� Eccentricity

� Other factors - These factors may be summarized as:

1. Friction between mating parts

2. Wearing and tearing of parts

3. Broken parts, for instance bearings

4. Etc.

What are the Effects of Vibration on

Dynamic Systems !!!

Question ?

Why is Vibration Important?

Fig. Failure of Tacoma Bridge in US, 7th November 1940

The Tacoma NarrowsBridge Disaster


Fig. Amplitude of vibration reached above 4 m!

See this Movie to realize Vibration

Effect:

What was the Cause of Vibrations in this Structure?

Answer is the physical phenomenon

Vortex Shedding.

Question ?

Vortex Shedding:

…Caused Wind-Induced High-amplitude Vibration (!!!!)


Example 1 : Tacoma Bridge

Example 2: Wing Flutter Failure of Planes

See this Movie to realise Vibration

Effect:

Example 3: Failure of Wind-energy Turbine

See this Movie to realize Vibration

Effect:

What are the Effects of Vibration on

Machines & Man?

Question ?

� Machines with repetitive disturbing forces such as engines, motors, turbines etc. often have vibration problems.

� Serious vibration problems may cause damage, malfunction or even failure of the structure or machine itself or machine parts their selves.

� Vibration causes interruption of production, reduction of working lives of machines, loss of power and energy.

� Vibration cause also uncomfortable feeling or noise, which can damage human ears permanently.

- Dynamic System Modelling:

� The vibration is very small so that sin θ ≈ θ.

� The structure is linear system whose dynamics may be represented by a set of linear, second order, differential equations.

� The structure obeys Maxwell’s theorem.

� The structure can be considered as time invariant; the coefficients in the linear, second order, differential equations are constants with respect to time.

Basic Assumptions:

Basic Concepts:

Every mechanical vibratory system has:

� Frequencies at which it “likes” to vibrate

� Characteristic geometries of vibration

Basic Concepts:

Every mechanical vibratory system has:

� Frequencies at which it “likes” to vibrate

� Characteristic geometries of vibration

Natural

Frequencies

1 24 34

Mode

Shapes of Vibrations

144424443

Modelling of Vibratory Systems:

The elements are:

1. Inertia (stores kinetic energy)

2. Elasticity (stores potential energy)

1

Realistic addition:

3. Energy Dissipation


The elements are:



1

Realistic addition:


2


The elements are:



1

Realistic addition:

3. Energy Dissipation3

2


The elements are:



2 3

1

Realistic addition:



The elements are:

1. Mass, m

2. Stiffness, k

3. Damping, ck c

m

x

How is this Model Useful?

k c

m

x

By building and solving equation of motion, we get the followings:

- Natural frequency and

- Mode shape

Basic Concepts:

� Resonance Condition

Basic Concepts:

� A vibration of large amplitude occurs.

� It occurs when a mechanical system is

forced to operate near its natural frequency.

� Under negligible damping, amplitude of

vibration may increase to infinity.

� Hence, mechanical system may fail to work

or break into pieces.

� Resonance Condition

Resonance Condition:

� A vibration of large amplitude.

� It occurs when a mechanical system is forced to operate near its natural frequency.

Dynamic System

Resonance Condition:

� A vibration of large amplitude

� It occurs when a mechanical system is forced to operate near its natural frequency

m

ck

x

tω

M

e

Dynamic System Model

Chapter 1 introduction to mechanical vibration

Engineering

Transcript of Chapter 1 introduction to mechanical vibration