41514 – Dynamics of Machinery– Theory, Experiment, Phenomenology and Industrial Applications –

Ilmar Ferreira Santos

1. Recapitulation

2. Simulation Models & Experimental Validation

3. Proportional Damping D = αM + βK

4. Theoretical Modal Analysis & Modal Parameters

5. Experimental Modal Analysis & Modal Parameters

6. Extracting Modal Parameters from Steady-State Responses (FRF)

7. Extracting Modal Parameters from Transient Vibrations

8. Extracting Mode Shapes from Steady-State Response (FRF)

* Mass Elements MParticle

Rigid Body

Distributed

* Spring Elements KElasticity Theory & Material

Magnetism

Fluid Mechanics

* Damping Elements D

Fluid Mechanics

Contact Mechanics (friction)

Mathematical

Model

Mechanical

Model

Physical

System

Assumptions

(simplifications)

Newton, Euler, D‘Alembert, Lagrange, Hamilton, Jourdain

(principles & axioms)

Static Equilibrium Position (Linearization)

(structure)

(machine)

Solution:

(eigenvalues)

(eigenvectors)

1. Recapitulation – Mathematical Modeling & Steps

* Mass Elements MParticle

Rigid Body

Distributed

* Spring Elements KElasticity Theory & Material

Magnetism

Fluid Mechanics

* Damping Elements D

Fluid Mechanics

Contact Mechanics (friction)

Mathematical

Model

Mechanical

Model

Physical

System

Assumptions

(simplifications)

Newton, Euler, D‘Alembert, Lagrange, Hamilton, Jourdain

(principles & axioms)

• Kinematics1. Reference Frames (systems of coordinates)

2. Transformation Matrices

3. Position Vectors

4. Velocity Vectors (linear and angular)

5. Acceleration Vectors (linear and angular)

• Dynamics 6. Mass Properties (mass center, moments of inertia)

7. Force and Moment Vectors

8. Dynamic Equilibrium:(Newton, Euler, Lagrange …)

9. Equilibrium Poition (Linearization, Vibration Analysis)

1. Recapitulation – Mathematical Modeling & Steps

2. Simulation Models & Experimental Validation

Experimental Validation

(Experimental Modal Analysis)

Undamped natural frequecies

Damped natural frequencies

Damping factors

Mode shapes

(Experimental Modal Analysis)

Undamped natural frequecies

Damped natural frequencies

Damping factors

Mode shapes

(Theoretical Modal Analysis)

Simulation Model

(Theory of Machinery Dynamics)

Examples of Industrial Application

3. Proportional Damping D = αM + βK

Simulation Models Fundamental Question: How to obtain α and β ?

damping factor

undamped natural frequency [rad/s]

• Link between damping factor and model parameters α and β

3. Proportional Damping D = αM + βK

Simulation Models Physical Meaning of Proportional Damping

• Link between damping factor and model parameters α and β

3. Proportional Damping D = αM + βK

damping factor

undamped natural frequency [rad/s]

• Link between damping factor and model parameters α and β

Steady-State Vibrations

Transient Vibrations

4. Theoretical Modal Analysis & Modal Parameters

5. Experimental Modal Analysis & Modal Parameters

Frequency Response Function & Modal Superposition

How to obtain Frequency Response Functions?

H1(ω), H2(ω) & Coherence Functions

5. Experimental Modal Analysis & Modal ParametersFrequency Response Functions – H1, H2 & Coherence

5. Experimental Modal Analysis & Modal ParametersFrequency Response Functions – H1, H2 & Coherence

6. Extracting Modal Parameters – Steady-State Responses (FRF)

Simulation Models

Experimental Validation

Simulation Models

From modal mass m, modal stiffness k and modal damping d to natural frequency ω & damping factor ξ

6. Extracting Modal Parameters – Steady-State Responses (FRF)

ω FRF(ω)

Experimental Frequency Response Function (frf-general.m)

Extracting Modal Parameters Using Least Square Method

6. Extracting Modal Parameters – Steady-State Responses (FRF)

Experimental Damping Factor (transient vibrations)

a) Least Square Method b) Half Power Points

c) Log Dec.

7. Extracting Modal Parameters – Transient Vibrations

Isolation of Mode Shapes and Resonance Testing (Transient Vibrations)

7. Extracting Modal Parameters – Transient Vibrations

7. Extracting Modal Parameters – Steady-State X Transient

8. Extracting Mode Shapes – Steady-State Response (FRF)

3 D.O.F – System

8. Extracting Mode Shapes – Steady-State Response (FRF)

8. Extracting Mode Shapes – Steady-State Response (FRF)

3 D.O.F

8. Extracting Mode Shapes – Steady-State Response (FRF)

8. Extracting Mode Shapes – Steady-State Response (FRF)

Summary & Overview

8. Extracting Mode Shapes – Steady-State Response (FRF)

• Numerical Example – Extracting the modes shapes from FRFs (text book)

8. Extracting Mode Shapes – Steady-State Response (FRF)

• Numerical Example – Extracting the modes shapes from FRFs (text book)

8. Extracting Mode Shapes – Steady-State Response (FRF)

• Numerical Example – Extracting the modes shapes from FRFs (text book)

8. Extracting Mode Shapes – Steady-State Response (FRF)

• Experimental Example – Modes shapes from FRFs H21(ω) H41(ω)