MULTI DEGREE OF FREEDOM SYSTEM Equation of Motion, Problem Statement &

Solution MethodsPertemuan 16

Matakuliah : Dinamika Struktur & Teknik GempaTahun : S0774

MDOF SystemsTopics:• Introduction to Multi DOF Systems• Close Coupled Systems• Far Coupled Systems• Orthogonality of Mode Shapes• Modal Analysis

– Undamped Analysis– Damped Analysis– Forced Vibration

Introduction Continuous Systems

• Any Mechanical System is Continuous in Mass and Stiffness Properties• Some Systems e.g. Turbine Blade Better Modeled as Distributed then Lumped• Partial Differential Equations• Solutions are Simpler and Accurate compared to MDOF System• Strings, Bars, Rods & Beams

Introduction Continuous Systems

Radial Drilling machine Modeled as MDOF System

MDOF SystemsTopics:• Introduction to Multi DOF Systems• Close Coupled Systems• Far Coupled Systems• Orthogonality of Mode Shapes• Modal Analysis

– Undamped Analysis– Damped Analysis– Forced Vibration

Close Coupled Systems

Mass Matrix

Close Coupled Systems

Free Vibrations

Close Coupled Systems

Eigen Value Problem

21

2

2n

pp

p

11 12

21

nn

X XX X

X

Natural Frequencies

Mode Shapes

Close Coupled Systems

MDOF SystemsTopics:• Introduction to Multi DOF Systems• Close Coupled Systems• Far Coupled Systems• Orthogonality of Mode Shapes• Modal Analysis

– Undamped Analysis– Damped Analysis– Forced Vibration

Far Coupled Systems

Influence Coefficient Method

Far Coupled Systems

MDOF SystemsTopics:• Introduction to Multi DOF Systems• Close Coupled Systems• Far Coupled Systems• Orthogonality of Mode Shapes• Modal Analysis

– Undamped Analysis– Damped Analysis– Forced Vibration

Orthogonality of Mode Shapes

Mode rMode s

2

T

T

U M U I

U K U

[U] is Orthonormal Modal Matrix

MDOF SystemsTopics:• Introduction to Multi DOF Systems• Close Coupled Systems• Far Coupled Systems• Orthogonality of Mode Shapes• Modal Analysis

– Undamped Analysis– Damped Analysis– Forced Vibration

Modal Analysis

Modal Analysis is a Powerful Tool to Determine the Free and Forced Vibrationsof MDOF systems

We can Consider Physical MDOF system to be replaced by several SDOFSystems, each SDOF system representing one Specific Natural Mode

This process of determining the modal masses and stiffness in each modeOf Vibration of a MDOF and determine the response in each of the modes toDetermine the Total Behavior is Modal Analysis

General Response can be written as:

Modal AnalysisUndamped Analysis

Modal AnalysisUndamped Analysis

Modal AnalysisDamped Analysis

Proportional Damping Decoupled Governing Equations

Modal AnalysisDamped Analysis

Rigid Body mode

For Non Rigid Body Modes

Modal AnalysisDamped Analysis

Modal AnalysisForced Vibration For Harmonic Excitation

Steady State Solution

Assignment

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Assignment

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Assignment

An automobile has an instrument of mass 200kg mounted on its chassis using an isolator of stiffness 580kN/m. The chassis has a mass of 1200kg and is sprung on the wheel axle through suspension of total stiffness 80kN/m. The axle has a mass of 220kg and tyre stiffness is 1000kN/m. Model the automobile as a three mass system with instrument, chassis and axle mass.

For the sake of quick assessment of vibratory response of the automobile, remodel the above system as a two mass system and then using a modal analysis approach, find the response of the instrument mounted on the chassis after the automobile encountered a step bump of 5cm.

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Assignment

Mc = 1200 kg Kc1 = 35kN/m

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Thank You