2103433
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
• Vibration Isolation and Suppression32103433 Intro to Mech Vibration, NAV
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