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2103433

Introduction to MechanicalVibration

Nopdanai Ajavakom (NAV)

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Course Topics

• Introduction to Vibration

 – What is vibration? – Basic concepts of vibration

 – Modeling

 –Linearization

• Single-Degree-of-Freedom Systems

 – Free Vibration

•

Undamped• Damped

• Measurement and Design Considerations

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Course Topics – Forced vibration

• Harmonic

 – Applications

• Rotating Unbalance

• Base Excitation

• Measurement Devices

 – Forced vibration (more)

• Periodic

• Impact

• Arbitrary

• Multi-Degree-of-Freedom Systems

• Vibration Isolation and Suppression32103433 Intro to Mech Vibration, NAV

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Road Map

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What is Vibration?

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• Vibration is the study of repetitive motion of 

relative to the reference position or frame.• Examples:

 – Swinging pendulum

 – Spring mass system

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Where to find vibration?

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• Car

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Where to find vibration?

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• Machine

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Where to find vibration?

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• Structure

 – The collapse of Tacoma Bridge

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Where to find vibration?

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• Structure

 – Earthquake

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Elementary parts of vibrating systems

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A vibrating system is a model consisting of 

• 1. Elastic components

• 2. Inertia (mass) components

• 3. Damping components

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Elementary parts of vibrating systems

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• 1. Elastic components

 – store or release potential energy as itsdisplacement increases or decreases.

 – e.g. linear spring, helical spring, thin rod, elastic

torsion bar, cantilever beam etc.

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Elementary parts of vibrating systems

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• 1. Elastic components

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Elementary parts of vibrating systems

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• 1. Elastic components

 – Thin rod

 – Torsion bar

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Elementary parts of vibrating systems

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• 1. Elastic components

 – Cantilever beam

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Elementary parts of vibrating systems

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• 1. Elastic components

 – Combination of springsParallel Series

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Elementary parts of vibrating systems

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• 1. Elastic components

 –Proofs

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Elementary parts of vibrating systems

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• 2. Inertia components

 – store or release kinetic energy as velocities

increase or decrease.

 – e.g., mass (translation), mass moment of inertia

(rotation)

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Elementary parts of vibrating systems

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• 3. Damping components

 – Dissipate energy out of system into heat or sound

 – e.g. shock absorber, damper, material strain

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Elementary parts of vibrating systems

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• 3. Damping components

 – Viscous damper

• No damping

• With damping

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Elementary parts of vibrating systems

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• Summary

Linear Rotational

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Elementary parts of vibrating systems

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• Exercises

Find the equivalent single stiffness representation of 

the five-spring system shown in the figure.

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Modeling of Vibration Systems

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Modeling of Vibration Systems

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Modeling of Vibration Systems

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d l f b

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Modeling of Vibration Systems

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Wing flutters due to excitation e.g. from wind

Simplify the model of the wing as a beam

Continuous system with structural stiffness and damping

Physical model turns into a math model with a

governing partial differential equation

Simplify more and make the mass “lumped” together

d l f b

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Modeling of Vibration Systems

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A reciprocating engine is mounted on a foundation as shown. The

unbalanced forces developed in the engine are transmitted to the

frame and the foundation. An elastic pad is placed between the

engine and the foundation block to reduce the transmission of vibration. Develop the physical model.

f d ( O )

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Degree of Freedom (DOF)

2103433 Intro to Mech Vibration, NAV

Degree of freedom (DOF): The minimum number of 

independent coordinates required to determine all positionsof all parts of a system at any time.

• Single degree of freedom systems

D f F d (DOF)

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Degree of Freedom (DOF)

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• Two degrees of freedom systems

• Three degrees of freedom systems

D f F d (DOF)

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Degree of Freedom (DOF)

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• Infinite degree of freedom systems(continuous systems, distributed systems)

By increasing number of degrees of freedom

• More accurate result

• More complexity

M th ti l M d l

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Mathematical ModelEquation of Motion (EOM)

2103433 Intro to Mech Vibration, NAV

• Math modeling to find the equation that describe the

motion of our system. In our class, it is a linear second

order differential equations…called “Equation of Motion,”

EOM

•

Procedures(1) Define coordinates and their positive directions

Note the degrees of freedom (DOF)

Write geometric constraints

(2) Write necessary kinematic relations(3) Draw free-body diagram

(4) Apply Newton’s 2nd law on the free body(5) Combine all relations

M th ti l M d l

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Mathematical ModelEquation of Motion (EOM)

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• Example 1: Spring mass system

Find the EOM of the mass attached to a spring as shown.

i f i ( O )

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Equation of Motion (EOM)

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• Example 2: Hanging massFind EOM of the system

E i f M i (EOM)

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2103433 Intro to Mech Vibration, NAV

• Example 3: Pendulum

Find EOM of the system

Equation of Motion (EOM)

E i f M i (EOM)

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2103433 Intro to Mech Vibration, NAV

• Example 4: 2-DOF systemEquation of Motion (EOM)

E i f M i (EOM)

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2103433 Intro to Mech Vibration, NAV

• Example 4: 2-DOF systemEquation of Motion (EOM)

Ans

E ti f M ti (EOM)

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2103433 Intro to Mech Vibration, NAV

• Example 5: Pulley and mass systemEquation of Motion (EOM)

Li i ti

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2103433 Intro to Mech Vibration, NAV

Consider the EOM of a simple pendulum

It is non-linear, which is difficult to solve by hand for the

exact solution. To make it simpler to solve, we linearize it

into this form.

where

How to linearize?

Linearization

Li i ti

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2103433 Intro to Mech Vibration, NAV

Linearization

Li i ti

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Linearization

Li i ti

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2103433 Intro to Mech Vibration, NAV

Linearization• Example 6: Accelerator

Li i ti

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h ib i

Linearization• Example 7: Pendulum Mechanism