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Introduction to Mechanical Vibration

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

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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

Where to find vibration?

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• Car

Where to find vibration?

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• Machine

Where to find vibration?

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• Structure

– The collapse of Tacoma Bridge

Where to find vibration?

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• Structure

– Earthquake

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

Elementary parts of vibrating systems

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• 1. Elastic 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.

Elementary parts of vibrating systems

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• 1. Elastic components

Elementary parts of vibrating systems

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• 1. Elastic components

– Thin rod

– Torsion bar

Elementary parts of vibrating systems

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• 1. Elastic components

– Cantilever beam

Elementary parts of vibrating systems

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• 1. Elastic components

– Combination of springs

Parallel Series

Elementary parts of vibrating systems

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• 1. Elastic components

– Proofs

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)

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

Elementary parts of vibrating systems

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• 3. Damping components

– Viscous damper

• No damping

• With damping

Elementary parts of vibrating systems

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• Summary

Linear Rotational

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.

Modeling of Vibration Systems

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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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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

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.

Degree of Freedom (DOF)

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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

Degree of Freedom (DOF)

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• Two degrees of freedom systems

• Three degrees of freedom systems

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

Mathematical ModelEquation of Motion (EOM)

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• 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)

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

Equation of Motion (EOM)

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• Example 2: Hanging massFind EOM of the system

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• Example 3: PendulumFind EOM of the system

Equation of Motion (EOM)

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

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

Ans

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

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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

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Linearization

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Linearization

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Linearization• Example 6: Accelerator

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Linearization• Example 7: Pendulum Mechanism