This document discusses forced harmonic motion modelling. It describes free vibration versus forced vibration, undamped versus damped systems, and common spring, mass, and damping elements. It provides the example of the 1940 collapse of the Tacoma Narrows Bridge due to resonant wind forces that matched the bridge's natural frequency. The document outlines modelling a spring-mass system to represent a real system and describes external forcing, base excitation, and rotor excitation models. It concludes with the equation of motion for an externally forced system using Newton's laws of motion.

1. FORCED HARMONIC MOTION MODELLING
OCTOBER 6,2015

2. Classification of Vibration
FREE VIBRATION VS FORCED VIBRATION

3. Classification Of Vibration
Undamped VS Damped

4. SPRING ELEMENTS
-A linear spring is a type of mechanical link thatis
generally assumed to have negligible mass and
damping.

5. Mass or Inertia Elements
- The mass of inertia element is assumed to be a rigid body.
It can gain or lose kinetic energy whenever the velocity of
the body changes.

6. Damping Elements
The mechanism which the vibrational energy is gradually
converted into heat or sound known as damping.

7. The Tacoma Narrows bridge collapsed due to wind induced
vibration on November 7th, 1940. Resonance is a process in
which an object's, in this case a bridge's, natural vibrating
frequency is amplified by an identical frequency.

8. Resonance
is a phenomenon that occurs when a given system
is driven by another vibrating system or external
force to oscillate with greater amplitude at a specific
preferential frequency.

9. In this case the identical frequency was caused by
strong wind gusts blowing across the bridge,
creating regions of high and low pressure above
and below the bridge (Bernouli's principle).
As the wind's intensity increased, the bridge rolled
in a corkscrewing motion faster and faster until it
was ripped apart.

10. TACOMA NARROWS BRIDGE COLLAPSE

11. In forced vibration there is an external
force acts on the system.
This external force which acts on the
system executes the vibration of the
system.

12. Examples of forced vibrations are air
compressors, I.C. engines, turbines,
machine tools etc,.

13. The external force may be harmonic and periodic, non-
harmonic and periodic or non periodic. In this report , only
external harmonic forces acting on the system are
considered.

15. Forced Harmonic Motion Modelling
-Used for analysis of forced vibrations system through a spring-
mass system as a model of a real engineering system.
-Is a simple method that involves replacing the distributed mass
or inertia of the system by a finite number of masses or rigid
bodies.

17. Modelling spring – mass system :
The masses are assumed to be connected by massless
elastic and damping members. Linear (or angular)
coordinates are used to describe the motion of the
masses(or rigid bodies).Such models are coordinates
necessary to describe the motion of the lumped masses
used in the model , the higher the accuracy of the resulting
analysis.

20. External Forcing models the behavior of a system which has a time
varying force acting on it. An example might be an offshore structure
subjected to wave loading.
Base Excitation models the behavior of a vibration isolation
system. The base of the spring is given a prescribed motion, causing
the mass to vibrate. This system can be used to model a vehicle
suspension system, or the earthquake response of a structure.
Rotor Excitation models the effect of a rotating machine mounted on
a flexible floor. The crank with small mass rotates at constant
angular velocity, causing the mass m to vibrate.

21. Equation of Motion for External Forcing

22. Newton’s law of motion gives: