Matakuliah
Tahun
: Dinamika Struktur & Teknik Gempa
: 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





p2
2
  p1
 

 
 
 X 11

X

   X 21


X 12



pn2 



X nn 
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   M U    I 
T

T
U   K U   


2





[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
Rigid Body mode
For Non Rigid Body Modes
Modal Analysis
Damped Analysis
Modal Analysis
Forced Vibration
For Harmonic Excitation
Steady State Solution
Assignment
1
Assignment
2
Assignment
3 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.
Assignment
4
Mc = 1200 kg
Kc1 = 35kN/m
Thank You