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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2.1 Introduction
• Three Solutions
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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 Vibration
2. 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 Measurement
Logarithmic Decrement
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2.3 Measurement
Logarithmic Decrement
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2.3 Measurement
Example:
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2.3 Measurement
Example:
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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?
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