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MULTI DEGREE OF FREEDOM SYSTEM Equation of Motion, Problem Statement & Solution Methods Pertemuan 16. Matakuliah: Dinamika Struktur & Teknik Gempa Tahun: S0774. MDOF Systems. Topics: Introduction to Multi DOF Systems Close Coupled Systems Far Coupled Systems - PowerPoint PPT Presentation
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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
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
Mc = 1200 kg Kc1 = 35kN/m
4
Thank You