Simple Dynamical System Response due to General Forces
BEHAVIOUR OF SYSTEMS
• System: collection of machine constituents
• Behavior of Physical Systems and Their Assumptions/Hypothesis
• Laws of mechanics and mathematical Model
• Resemble with physical behavior otherwise refinement in the model
• Types of Mathematical Models:
(a) Discrete Systems and (b) Continuous Systems
(a) Discrete Systems
• Example of a Spring Mass Damper System:
m – mass
k – stiffness
c – damping
F(t) – force
y(t) – response
t – time
Equations of motion can be written using Newton’s second law as:
- Representation of System Behavior
- Idealization in the Spring Mass Damper System
(b) Continuous Systems
• Physical Properties and System Behavior
• Example of a Bar
The partial differential equation describing the system:
• Response of a System
• Relation between the Excitation and Response is described by the following block
Where G(t) is in the form of the differential operator:
• Linear Systems: Dependent variable y has power 0 or 1. Superposition Principle holds.
• Non-Linear: Dependent variables have power (expect 0 or 1) or cross product.
A Simple way to test whether a system is Linear or Non-Linear
where y1(t)is the response of the system to an excitation of F1(t),
y2(t)is the response of the system to an excitation of F2(t) , and
G is the differential operator that reflects the property of the system only
Now if the system is excited by
with c1 and c2 as arbitrary constants
And we find :
then the system is Linear.
And if we find
then the system is Non-Linear.
Example of Harmonic Oscillator: Simple Pendulum
So the total energy of the conservative system is constant and its the statement of the
conservation of energy. It is seen that it is not restricted to small motion.