2103433 introduction to mechanical vibrationpioneer.netserv.chula.ac.th/~anopdana/433/ch1.pdf ·...
TRANSCRIPT
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2103433
Introduction to Mechanical Vibration
Nopdanai Ajavakom (NAV)
12103433 Intro to Mech Vibration, 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
22103433 Intro to Mech Vibration, NAV
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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?
52103433 Intro to Mech Vibration, NAV
• 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
102103433 Intro to Mech Vibration, NAV
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 its displacement 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 springs
Parallel 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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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
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Modeling of Vibration Systems
2103433 Intro to Mech Vibration, NAV
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.
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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 positions of all parts of a system at any time.
• Single degree of freedom systems
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Degree of Freedom (DOF)
2103433 Intro to Mech Vibration, NAV
• Two degrees of freedom systems
• Three degrees of freedom systems
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Degree of Freedom (DOF)
2103433 Intro to Mech Vibration, NAV
• Infinite degree of freedom systems(continuous systems, distributed systems)
By increasing number of degrees of freedom• More accurate result• More complexity
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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 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
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Mathematical ModelEquation of Motion (EOM)
2103433 Intro to Mech Vibration, NAV
• Example 1: Spring mass systemFind the EOM of the mass attached to a spring as shown.
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Equation of Motion (EOM)
2103433 Intro to Mech Vibration, NAV
• Example 2: Hanging massFind EOM of the system
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2103433 Intro to Mech Vibration, NAV
• Example 3: PendulumFind EOM of the system
Equation of Motion (EOM)
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2103433 Intro to Mech Vibration, NAV
• Example 4: 2-DOF systemEquation of Motion (EOM)
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2103433 Intro to Mech Vibration, NAV
• Example 4: 2-DOF systemEquation of Motion (EOM)
Ans
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2103433 Intro to Mech Vibration, NAV
• Example 5: Pulley and mass systemEquation of Motion (EOM)
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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
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2103433 Intro to Mech Vibration, NAV
Linearization
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2103433 Intro to Mech Vibration, NAV
Linearization
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2103433 Intro to Mech Vibration, NAV
Linearization• Example 6: Accelerator
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2103433 Intro to Mech Vibration, NAV
Linearization• Example 7: Pendulum Mechanism