Matakuliah : Dinamika Struktur & Teknik Gempa Tahun : S0774 MULTI DEGREE OF FREEDOM SYSTEM Equation of Motion, Problem Statement & Solution Methods Pertemuan 16

MDOF Systems Topics: 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 Systems Topics: 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 Natural Frequencies Mode Shapes

Close Coupled Systems

MDOF Systems Topics: 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 Systems Topics: 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 r Mode s [U] is Orthonormal Modal Matrix

MDOF Systems Topics: 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 Vibrations of MDOF systems We can Consider Physical MDOF system to be replaced by several SDOF Systems, each SDOF system representing one Specific Natural Mode This process of determining the modal masses and stiffness in each mode Of Vibration of a MDOF and determine the response in each of the modes to Determine the Total Behavior is Modal Analysis General Response can be written as:

Modal Analysis Undamped Analysis

Modal Analysis Undamped Analysis

Modal Analysis Damped Analysis Proportional Damping Decoupled Governing Equations

Modal Analysis Damped Analysis For Non Rigid Body Modes

Modal Analysis Damped Analysis

Modal Analysis Forced Vibration For Harmonic Excitation Steady State Solution

Assignment 1

Assignment 2

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. 3

Assignment 4 Mc = 1200 kg Kc1 = 35kN/m

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