2103433 Introduction to Mechanical Vibration Nopdanai Ajavakom (NAV) 1
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 2
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 Suppression 3
Road Map 4
What is Vibration? Vibration is the study of repetitive motion of relative to the reference position or frame. Examples: Swinging pendulum Spring mass system 5
Where to find vibration? Car 6
Machine Where to find vibration? 7
Structure Where to find vibration? The collapse of Tacoma Bridge 8
Structure Where to find vibration? Earthquake 9
Elementary parts of vibrating systems A vibrating system is a model consisting of 1. Elastic components 2. Inertia (mass) components 3. Damping components 10
Elementary parts of vibrating systems 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. 11
Elementary parts of vibrating systems 1. Elastic components 12
Elementary parts of vibrating systems 1. Elastic components Thin rod Torsion bar 13
Elementary parts of vibrating systems 1. Elastic components Cantilever beam 14
Elementary parts of vibrating systems 1. Elastic components Combination of springs Parallel Series 15
Elementary parts of vibrating systems 1. Elastic components Proofs 16
Elementary parts of vibrating systems 2. Inertia components store or release kinetic energy as velocities increase or decrease. e.g., mass (translation), mass moment of inertia (rotation) 17
Elementary parts of vibrating systems 3. Damping components Dissipate energy out of system into heat or sound e.g. shock absorber, damper, material strain 18
Elementary parts of vibrating systems 3. Damping components Viscous damper No damping With damping 19
Elementary parts of vibrating systems Summary Linear Rotational 20
Elementary parts of vibrating systems Exercises Find the equivalent single stiffness representation of the five-spring system shown in the figure. 21
Modeling of Vibration Systems
Modeling of Vibration Systems
Modeling of Vibration Systems
Modeling of Vibration Systems 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 25
Modeling of Vibration Systems 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) 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) Two degrees of freedom systems Three degrees of freedom systems
Degree of Freedom (DOF) Infinite degree of freedom systems (continuous systems, distributed systems) By increasing number of degrees of freedom More accurate result More complexity
Mathematical Model Equation of Motion (EOM) 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 directions Note the degrees of freedom (DOF) Write geometric constraints (2) Write necessary kinematic relations (3) Draw free-body diagram (4) Apply Newton s 2 nd law on the free body (5) Combine all relations
Mathematical Model Equation of Motion (EOM) Example 1: Spring mass system Find the EOM of the mass attached to a spring as shown.
Equation of Motion (EOM) Example 2: Hanging mass Find EOM of the system
Equation of Motion (EOM) Example 3: Pendulum Find EOM of the system
Equation of Motion (EOM) Example 4: 2-DOF system
Equation of Motion (EOM) Example 4: 2-DOF system Ans
Equation of Motion (EOM) Example 5: Pulley and mass system
Linearization 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
Linearization
Linearization Example 6: Accelerator
Linearization Example 7: Pendulum Mechanism