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

3. 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

4. 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

5. Introduction Continuous Systems
Radial Drilling machine Modeled as MDOF System

6. 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

7. Close Coupled Systems
Mass Matrix

8. Close Coupled Systems
Free Vibrations

9. Close Coupled Systems Eigen Value Problem Natural Frequencies
Mode Shapes

10. Close Coupled Systems

11. 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

12. Far Coupled Systems
Influence Coefficient Method

13. Far Coupled Systems

14. 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

15. Orthogonality of Mode Shapes
Mode r
Mode s
[U] is Orthonormal Modal Matrix

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

17. 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:

18. Modal Analysis
Undamped Analysis

19. Modal Analysis
Undamped Analysis

20. Modal Analysis Damped Analysis Proportional Damping
Decoupled Governing Equations

21. Modal Analysis Damped Analysis For Non Rigid Body Modes

22. Modal Analysis
Damped Analysis

23. Modal Analysis Forced Vibration For Harmonic Excitation
Steady State Solution

24. Assignment 1

25. Assignment 2

26. 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

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

28. Thank You