Mechanical Vibrations


Simple Dynamical System Response due to General Forces


Mechanical Vibrations
railway wagon

• 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:

equation (1.3)

• 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.


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