MECHANICAL VIBRATION MME4425/MME9510 Prof. Paul Kurowski

TEXT BOOKS REQUIRED RECOMMENDED

MME4425b web site http://www.eng.uwo.ca/MME4425b/2012/ Design Center web site http://www.eng.uwo.ca/designcentre/

Software used: SolidWorks Design and assembly of mechanisms and structures SolidWorks Simulation (add-in to SolidWorks) Structural analysis Motion Analysis (add-in to SolidWorks) Kinematic and dynamic analysis of mechanisms Excel

SolidWorks 2012 installation and activation instructions: Go to www.solidworks.com/SEK Use SEK-ID = XSEK12 Select release 2012-2013 When prompted enter serial number for activation

WHAT IS THE DIFFERENCE BETWEEN DYNAMIC ANALYSIS AND VIBRATION ANALYSIS?

DIFFERENCE BETWEEN A MECHANISM AND A STRUCTURE Structure is firmly supported, mechanism is not. Structure can only move by deforming under load. It may be one time deformation when the load is applied or a structure can vibrate about its neutral position (point of equilibrium). Generally a structure is designed to stand still. Mechanism moves without deforming it components. Mechanism components move as rigid bodies. Generally, a mechanism is designed to move.

DIFFERENCE BETWEEN A MECHANISM AND A STRUCTURE STRUCTURES MECHANISMS

RIGID BODY MOTION

How many rigid body motions?

DISCRETE SYSTEM VS. DISTRIBUTED SYSTEM Mass, stiffness and damping are separated Distributed system Mass, stiffness and damping are NOT separated

DISCRETE SYSTEM VS. DISTRIBUTED SYSTEM 1DOF.SLDASM 2DOF.SLDASM Discrete system Mass, stiffness and damping are separated Distributed system Mass, stiffness and damping are NOT separated

swing arm 01.SLDASM swing arm 02.SLDASM Discrete system Mass, stiffness and damping are separated Distributed system Mass, stiffness and damping are NOT separated

Discrete system Vibration of discrete systems can be analyzed by Motion Analysis tools such as Solid Works Motion or by Structural Analysis such as SolidWorks Simulation based on the Finite Element Analysis Distributed system Vibration of distributed systems must be analyzed by structural analysis tools such as SolidWorks Simulation based on the Finite Element Analysis.

SINGLE DEGREE OF FREEDOM SYSTEM LINEAR VIBRATIONS

SINGLE DEGREE OF FREEDOM SYSTEM, LINEAR VIBRATIONS Homogenous equation

FINDING GENERAL SOLUTION OF HOMOGENEOUS EQUATION By guessing solution How to solve this? We guess solution based on experience that the solution will be in the form: A – magnitude of amplitude Ф – initial value of sine function ωn – angular frequency

FINDING GENERAL SOLUTION OF HOMOGENEOUS EQUATION By guessing solution ωn – natural angular frequency found from system properties Where A and Ф are found from initial conditions

FINDING GENERAL SOLUTION OF HOMOGENEOUS EQUATION Using complex numbers method

FINDING GENERAL SOLUTION OF HOMOGENEOUS EQUATION Using complex numbers method We have found two solutions to equation and Since is linear, then the sum of two solutions is also a solution Using Euler’s relations: The equation can be re-written as: Where A and Ф are found from initial conditions

FINDING GENERAL SOLUTION OF HOMOGENEOUS EQUATION Using Laplace transformation Taking Laplace transform of both sides Using (5), (6)

Laplace transformation

Laplace transformation Inman p 619

QUANTITIES CHARACTERIZING VIBRATION Average value of amplitude is But average value of is zero. Therefore, average value of amplitude is not an informative way to characterize vibration. for this reason we use mean-square value (variance) of displacement: Square root of mean square value is root mean square (RMS). RMS values of are commonly used to characterize vibration quantities such as displacement, velocity and acceleration amplitudes.

QUANTITIES CHARACTERIZING VIBRATION Displacement Velocity Acceleration These quantities differ by the order of magnitude or more, hence it is convenient to use logarithmic scales. The decibel is used to quantify how far the measured signal x1 is above the reference signal x0

QUANTITIES CHARACTERIZING VIBRATION Lines of constant displacement For a device experiencing vibration in the frequency range 2-8Hz: The maximum acceleration is 10000mm/s^2 The maximum velocity is 400mm/s Therefore the maximum displacement is 30mm Lines of constant acceleration Nomogram for specifying acceptable limits of sinusoidal vibration (Inman p 18)

LINEAR SDOF

LINEAR SDOF 10kg mass Linear spring 400000N/m Base SDOF.SLDASM

Results of modal analysis LINEAR SDOF Results of modal analysis

Trigonometric relationship between the phase, natural frequency, and initial conditions. Note that the initial conditions determine the proper quadrant for the phase.

PENDULUM SDOF

PENDULUM SDOF Galileo Galilei lived from 1564 to 1642. Galileo entered the University of Pisa in 1581 to study medicine. According to legend, he observed a lamp swinging back and forth in the Pisa cathedral. He noticed that the period of time required for one oscillation was the same, regardless of the distance of travel. This distance is called amplitude. Later, Galileo performed experiments to verify his observation. He also suggested that this principle could be applied to the regulation of clocks.

PENDULUM SDOF pendulum 02.SLDPRT

PENDULUM SDOF Equations of motion method

The energy method is suitable for reasonably simple systems. The energy method may be inappropriate for complex systems, however. The reason is that the distribution of the vibration amplitude is required before the kinetic energy equation can be derived. Prior knowledge of the “mode shapes” is thus required.

PENDULUM SDOF Energy method

TORSIONAL SDOF

TORSIONAL SDOF polar moment of inertia of cross-section disk 01.SLDPRT

TORSIONAL SDOF

ROLER SDOF roler.SLDASM Inman p 32

ROLER SDOF

ROLER SDOF Inman p 32 k = k1 + k2 = 2000N/m m = 75.4kg r = 0.1m J = 0.3770kgm2

ROLER SDOF

MASS AT THE END OF BEAM rotation.SLDASM

MASS AT THE END OF BEAM mass 2.7kg cantilever.SLDPRT

RING Ring.SLDASM

HOMEWORK 1 Derive equation of motion of SDOF using energy method Find amplitude A and tanΦ for given x0, v0 Find natural frequency of cantilever, l=400mm, Φ=5mm, E=2e11Pa, m=2.7kg. Confirm with SW Simulation Work with exercises in chapter 19 – blue book

TORSONAL SDOF TRIFILAR 1060 alloy Model file trifilar.sldasm Configuration trifilar Model type solid Material as shown Supports as shown Objectives Find the natural frequency of trilifar Fixed support Custom material E = 10MPa ρ = 1kg/m3 very soft, very low density 1060 alloy Restraint in radial direction to force torsional mode trifilar.SLDASM

TORSIONAL SDOF BIFILAR

TORSIONAL SDOF TRIFILAR

TORSIONAL SDOF TRIFILAR Using energy method:

TORSIONAL SDOF TRIFILAR

TORSONAL SDOF TRIFILAR Trifilar can be used to find moments of inertia of objects placed on rotating platform

spur gear.SLDPRT