2103433 introduction to mechanical vibration -...

2103433

Introduction to Mechanical Vibration

Nopdanai Ajavakom (NAV)

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Chapter 2: Free Vibration of SDOF Systems

• Introduction

• Undamped Free Vibration

• Damped Free Vibration

– Underdamped

– Overdamped

– Critically Damped

• Measurement

• Design Consideration

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2.1 Introduction• A system is said to undergo free vibration when it

oscillates only under an initial disturbance with no external forces after the initial disturbance.

Examples

• The oscillations of the pendulum of a clock

• The vertical oscillatory motion felt by a bicyclist after hitting a road bump

• The motion of a child on a swing under an initial push

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2.1 Introduction• EOM of a single degree of freedom system

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2.1 Introduction• Review of Linear Second Order Differential Equation

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2.1 Introduction• Three Solutions

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• Three Solutions

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2.1 Introduction

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2.2 Free Undamped Vibration

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2.2 Free Undamped Vibration

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2.2 Free Undamped Vibration

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2.2 Free Undamped Vibration• Relationship between displacement, velocity, and

acceleration

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2.3 Free Damped Vibration

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2.3 Free Damped Vibration

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2.3 Free Damped Vibration

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2.3.1 Underdamped Vibration

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2.3.2 Overdamped Vibration

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2.3.3 Critically Damped Vibration

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2.3 Free Damped Vibration• Exercises

1. A Spring-mass-damper system has mass of 100 kg, stiffness of 3000 N/m and damping coefficient of 300 kg/s. Calculate the (undamped) natural frequency, the damping ratio and the damped natural frequency. Does the solution oscillate? This system is given a zero initial velocity and an initial displacement of 0.1 m. Calculate the vibration response. [inman1.40, 1.42]

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2.3 Free Damped Vibration2. A Spring-mass-damper system has mass of 150 kg,

stiffness of 1500 N/m and damping coefficient of 200 kg/s. Calculate the undamped natural frequency, the damping ratio and the damped natural frequency. Is the system overdamped, underdamped or critically damped? Does the solution oscillate? This system is given an initial velocity of 10 mm/s and an initial displacement of -5 mm. Calculate the vibration response. [inman1.41, 1.43]

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2.3 MeasurementLogarithmic Decrement

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2.3 MeasurementLogarithmic Decrement

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2.3 MeasurementExample:

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2.3 MeasurementExample:

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2.4 Design Consideration

• Design in vibration: adjusting the physical parameters of a device to cause its vibration response to meet a specified shape or performance criterion.

Example

– Design a m-c-k system to have the desired response.

• underdamped, overdamped, critically damped

– Design a system that has a given natural frequency.

• Select connection of springs (series or parallel)

• Use elastic elements as springs

• Consider acceptable static deflection

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2.4 Design Consideration• Example:

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2.4 Design Consideration• Example:

Consider modeling the vertical suspension system of a small sports car, as a single-DOF system. The mass of the automobile is 1361 kg. The static deflection of the spring is 0.05 m. Calculate c and k of the suspension system of the car to be critically damped. If there are passengers and baggage of 290 kg in the car, how does this affect the damping ratio?