Dynamic analysis

1. Free Vibration of SDOF Systems
Brief Introduction about why SDOF System & its importance , Basic concepts etc
Chap. 1:- Free Vibration of SDOF Un-damped Case.
Chap. 2:- Free Vibration of SDOF Damped Case.
Contents to be covered under Free Vibn under Part I of syllabus
• Part 1a: Brief Introduction about:- Why SDOF System & its importance etc
Some practical examples of SDOF sys
Physical & Idealized Model of Generalised SDOF System
FBD of idealized model & Generalized Governing Equation of Motion
with variable ‘x’ & its derivatives appear in 1st degree hence classified as linear but of 2nd order
Four Possible Cases with zero or non zero values for dyn. force & damping coef in Generalized Equation of Motion
• Part 1b: Stiffness of Dyn system & Concept of Equivalent Stiffness & Its application
• Chap-1: Un-damped - Free Vibration of SDOF
Physical Demonstration of the periodic motion of SDOF sys.
Solution of Governing Equation for Case-I:-Un-damped Free Vibration
Dynamic properties of Un-damped SDOF system
Natural Period/Frequency
Graphical plots of un-damped free vibration response
with varying initial conditions
Typical examples to help in clear understanding of above mentioned theoretical concepts
• Chap-2: Damped case - Free Vibration of SDOF

2. Brief Introduction about why SDOF System & its importance etc ....
Why SDOF System? 1) Majority of Civil Engg Buildings/Strs are simulated by MDOF Systems rather than by
SDOF. 2) Since SDOF being the simplest dynamic system, hence easy to understand its dynamic behaviour.
3) Solution of SDOF is also prerequisite for simplified dynamic analysis method i.e. Mode Superposition
Method for forced Vibn. of MDOF Sys, (wherein N deg. of MDOF sys is transformed into N SDOF systems).

3. Some practical examples of SDOF Systems
: 1. Tower with lumped mass at top like OHT 2. Single Storey Structure
• .

4. Before discussing Elements of Dynamic System & its Governing Equation (G E), let us first look at Elements of Static System & its G E.
Real Structure Idealized Model Free Body Diagram
Although static analysis is much simpler than dynamic analysis, still we need to simplify the
real structure in the form of simple idealized model for avoiding complexities in structural
analysis of both Static & Dynamic Analysis.
The above simplification is essential even for static analysis to express physical behavior of
static beam in terms of mathematical governing equation, Pmax=K . umax

5. Four Basic Elements of Dynamic System Required for its Analytical Modelling &
their important Dynamic Characteristics.

6. Physical & Idealized Model of Generalised SDOF System
Single Storey Physical Frame/Structure
Idealized Model for
SDOF systems

7. FBD of idealized model & Generalized Governing Equation of Motion Based on Newtons Law-NS
Mass
Dashpot
Spring
External load
External load
Dashpot force
Spring force
From Newton’s Law, F = mü
Q(t) - fD - fS = mü

8. Recall Newton’s 2nd Law of motion

9. In present course we consider only Linear Dynamic Analysis, hence the elements of dynamic model
must behave in a Linear manner as shown below, i.e . Both c & k have constant values
Elastic resistance
Viscous resistance
)
(t
Q
ku
u
c
u
m 

 



10. Role of D’ Alemberts Principle in Simplifying the formulation of Governing Equation of Complex Dynamic
Systems (for which use of Newton’s law not easy) in terms of established principles of Static equilibrium.
• .

11. For simple structural systems, Newton’s law can be used easily, but is not easy for complex structural systems
For such cases, D’ Alemberts Principle i.e.setting Dynamic Equilibrium into Static Equilibrium is very helpful.

12. 1. F. B. D. of Idealized Dynamic System of Fig (a) shown in Fig (b) with fictitious inertia force ( Fi )
= mass x accel acting in direction opposite to accel.
2. With inertia force Fi included in FBD, the principle of static equilibrium can be used to develop G E.
• .

13. Four Possible Cases with zero/non-zero values for Q(t) & C in Generalized Equation of Motion
Total Four cases
2 under Free vibration: i.e. Q(t) = 0
Un-damped : c = 0
Damped : c ≠ 0
2 under Forced vibration: i.e. Q(t) ≠ 0
Un-damped : c = 0
Damped : c ≠ 0
)
(t
Q
ku
u
c
u
m 

 



14. Repeated in different form:- Four Possible Cases with Corresponding values for P(t) & C

15. End of Part-1a: Brief Introduction about:- Why SDOF System &
its importance etc

16. Part 1b: Stiffness of Dynamic system & Concept of Equivalent Stiffness

17. Stiffness for commonly used structural elements:
a) bar element & (b to g) beam elements .
Ask to find these for few cases

18. Concept of Equivalent Stiffness & Its Application
case-1: Springs in parallel
Ask to solve these as static Analysis problem

19. Case-2 Springs in series

20. How to decide if springs in a complex system are in parallel or in series?
It is important to note that springs may have apparently complex
configuration, thus difficult to identify whether to consider them in
parallel or series. To sort out such an issue, adopt the strategy as under:
• In such situations, mass element of SDOF system may be imagined to be given a
displacement, if all the springs get displaced by same amount, then springs are
treated as parallel, otherwise they are in series.

21. End of Part 1b: Stiffness of Dynamic system, Concept of Equivalent
Stiffness & Its application in solving numerical problems
Asked to solve few cases of Keq determination in the class as home work.