2103433 Introduction to Mechanical Vibrationpioneer.netserv.chula.ac.th/~anopdana/433/ch1.pdf ·...

2103433

Introduction to Mechanical Vibration

Nopdanai Ajavakom (NAV)

12103433 Intro to Mech Vibration, NAV

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


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

Road Map


What is Vibration?


• Vibration is the study of repetitive motion of relative to the reference position or frame.

• Examples:

– Swinging pendulum

– Spring mass system

Where to find vibration?


• Car



• Machine



• Structure

– The collapse of Tacoma Bridge



• Structure

– Earthquake

Elementary parts of vibrating systems


A vibrating system is a model consisting of

• 1. Elastic components

• 2. Inertia (mass) components

• 3. Damping components




– store or release potential energy as its displacement increases or decreases.

– e.g. linear spring, helical spring, thin rod, elastic torsion bar, cantilever beam etc.




– Thin rod

– Torsion bar




– Cantilever beam




– Combination of springs

Parallel Series




– Proofs



• 2. Inertia components

– store or release kinetic energy as velocities increase or decrease.

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




– Dissipate energy out of system into heat or sound

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




– Viscous damper

• No damping

• With damping



• Summary

Linear Rotational



• Exercises

Find the equivalent single stiffness representation of the five-spring system shown in the figure.

Modeling of Vibration Systems




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



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.

Degree of Freedom (DOF)


Degree of freedom (DOF): The minimum number of independent coordinates required to determine all positions of all parts of a system at any time.

• Single degree of freedom systems



• Two degrees of freedom systems

• Three degrees of freedom systems



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

By increasing number of degrees of freedom• More accurate result• More complexity

Mathematical ModelEquation of Motion (EOM)


• 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 directionsNote 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

Mathematical ModelEquation of Motion (EOM)


• Example 1: Spring mass systemFind the EOM of the mass attached to a spring as shown.

Equation of Motion (EOM)


• Example 2: Hanging massFind EOM of the system


• Example 3: PendulumFind EOM of the system

Equation of Motion (EOM)


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


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

Ans


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


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


Linearization


Linearization• Example 6: Accelerator


Linearization• Example 7: Pendulum Mechanism

2103433 Introduction to Mechanical Vibrationpioneer.netserv.chula.ac.th/~anopdana/433/ch1.pdf ·...

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