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Peter's Index Physics Home Lecture 12 Course Index Lecture 14
Physics for Civil Engineering
This is an introduction to Electricity, Strength of Materials and Waves.
Lecture 13 (Oscillations and Waves)
In this lecture the following are introduced:
• Oscillations and Vibrations
• Simple Harmonic Motion
• Damped Simple Harmonic Motion
• Forced Oscillations
• Vibrations inside built structures
• Vibrations coming from outside built structures
• Positive and Negative Damping
Oscillations and Vibrations
If you pull a swing or pendulum to the side and release it then it will oscillate (Latin for "swing") back and forth. If you hang a weight on a spring, pull it down and release it, then the system will vibrate (Latin for "shake") up and down. Oscillations and vibrations are two words for one concept, i.e. repetitive motion.
When the time between repetitions is constant, the oscillation is called a harmonic motion and
the time between repetitions is called the period.
The number of repetitions per second is called the frequency and is the inverse of the period.
The period, T, is normally measured in seconds, and the frequency, ν, in Hertz.
When the oscillations (or vibrations) affect the material around them a wave is formed which transports energy away.
Simple Harmonic Motion
A body undergoes simple harmonic motion when it is displaced from a fixed point and there is a restoring force on it that is directly proportional to its displacement from that fixed point and always directed towards the fixed point.
Mass and spring on a smooth plane.
This is an example of free vibrations. Where: 

Integrating the acceleration gives speed as a function of displacement from the origin. Note the switching of the variable from time t to velocity v in the top line. 

A graph of velocity versus displacement is given on the right. 

Integrating further gives the displacement as a function of time. 

The result is: 

You always have the sine of an angle. 
α and φ are called phase angles,
because they relate particular displacements to the maximum displacement. 
From above, velocity and displacement are related by:
This can be written as: Displacement and time are related by:
These two results can be repesented graphically, as shown on the right. 

The velocity, as a function of time, can be found by differentiating displacement as a function of time . 
Displacement and velocity are seen to be 90° out of phase. 
Putting these relationships together as functions of both time and displacement.
Energy relationships in SHM
Representing this graphically: 
Example O1
A simple harmonic oscillator can be described by x = 0.3 sin (ωt + α) m.
It has a mass of 30 kg and a force constant of 480 N.m^{1}.
Also, x = 0.2 m at time t = 0.1 s.
>Find
(a) the maximum mechanical energy of the oscillator,
(b) the potential energy and kinetic energy when the displacement is 0.2 m,
(c) the angular frequency of the oscillation,
(d) the initial phase of the displacement.
Answer O1
Damped Simple Harmonic Motion
The "free" vibrations on a real physical system will decrease in amplitude with time because there is always some friction or viscous drag on the moving object.
Mass, spring and dashpot on a smooth plane.
The damping of the piston provides a typical resistive force, proportional to the velocity, i.e. The solution is probably a sine curve that decays exponentially,
so this solution is assumed and checked by substitution.


Trial solution: Force equation: 
Substituting the assumed solution for x into the force equation above gives: where ω_{0} is the frequency for a free vibration, i.e. with r=0. 
The damped frequency ω_{1} is smaller than the free vibration frequency, and three situations can result depending on the size of the damping parameter, r.
For small damping the damped frequency is only slightly smaller than the free vibration,
i.e. ω_{1} ≈ ω_{0} For large damping, the frequency is also zero Hz, but it takes a longer time to return to zero displacement. 

The total mechanical energy also decays exponentially with time. 
Example O2
A damped simple harmonic oscillator has a mass of 0.25kg attached to a spring with stiffness 85 N.m^{1} and a small damping constant 0.07 kg.s^{1}.
Find the number of periods it oscillates before the energy drops to half the initial value.
Answer O2
Since damping is small we can assume:
We find the time when:
From this i.e. 7.5 periods
Forced Oscillations
When a system is subjected to an external sinusoidal force it oscillates at the external frequency and not at its natural frequency. This is called forced oscillation.
For the springplusmass system, the natural frequency is: 
Writing the equation of motion, i.e. Newton's Second Law, we have:
Resultant force = spring force + friction force + harmonic driving force = mass × acceleration
The solution found (after much effort) will be a displacement x, that is simple harmonic (a sine function) at the forced frequency ω_{2}, but whose amplitude F_{m} from the harmonic driving force, is modulated by a term G, that depends on both the forced frequency and the natural frequency.
When the forcing frequency ω_{2} approaches the natural frequency ω_{0} in value, then G goes to a minimum . 

If G goes to a minimum, then the resultant amplitude will go to a maximum whose size is limited by the friction coefficient r. Larger friction causes a smaller maximum amplitude. 

As shown in the graph on the right, there is an increase in the amplitude of the oscillation when the forcing frequency approaches the natural frequency of the system. This kind of situation is called a resonance condition. 
Example O3
A cosine oscillator with amplitude 360mm drives a mass of 0.5kg attached to a spring with stiffness 162 N.m^{1} and a damping constant 0.1 kg.s^{1}. Find
(a) the resonant frequency, and
(b) an expression for the displacement at frequency 2.8 Hz.
Example O3
Vibrations and built structures
Vibrations in built structures can just be a nuisance, or cause illness, or even structural damage. Vibrations come from two main sources, internal or external.
Inside buildings, vibrations come from machines such as air conditioning units, pumps, elevators, fans, front end loaders, presses, etc, or from people walking, jumping, or running around.
External vibrations come from road and rail traffic, subways, construction, earthquakes and winds.
Winds apply intermittent and chaotic forces to buildings but they can set up oscillations by:
• aerodynamic instability (negative damping) producing selfinduced vibrations in the structure;
• periodic eddy formations (Strouhal vortices); and
• the random effects of fluctuations in wind velocity and direction. The turbulence generated by obstacles may last up to 100 times the height of the structure.
Tacoma Narrows Bridge
The famous failure of the Tacoma Narrows Bridge is an example of how winds can cause dangerous vibrations. It had shallow plate girders instead of the normal deep stiffening trusses.
When it twisted along its length Newton's law for angular motion applied i.e. 
where: 
The bridge failed because of torsional "flutter" where one mode at 0.2 Hz had negative damping from synchronised motioninduced pressures and deck motion. It is not properly "resonance" because there is no forcing frequency.
Damping
The damping capacity is the ability of the structure to absorb energy. Energy is naturally dissipated in the materials, joints and connections of the structure, or it can be increased by the addition of viscoelastic damping layers.
Electrohydraulic mass dampers, tuned massdampers (a large mass in some oil) and special friction joints can also be used.
Elastomeric pads can be placed at the basement level of every column to reduce the magnitude of vibrations in the remainder of the building.
External sources
When an external source sends vibrations through the soil into the foundations of a buiding, the vibrations move both sideways and upwards. Soft soils give larger amplitude vibrations than hard or rocklike materials, but the amplitude of those vibrations decreases more rapidly with distance. Extending the foundation down to hard layers will reduce these.
Barriers in the path to the structure can reduce the incidence of vibrations e.g. backfilled trenching or piles set in suitable locations
Another way is to reduce the production of the external vibrations.
• The allowed speed and weight of nearby vehicles can be reduced.
• Road surfaces can be smoothed and road beds can be improved.
• Isolation pads can be used between the rail and the subgrade for both surface and subway rails.
• The rolling stock can have rubber tires instead of steel wheels.
Internally:
• vibrating machines can be isolated from their mountings by springs or rubber pads.
• The use of rotating rather than reciprocating machines can be encouraged.
Summarising:
Simple Harmonic Motion. 

Damped Simple Harmonic Motion. 

Forced Simple Harmonic Motion. 
Vibrations in structures come from two main sources, internal or external.
Vibrations can be reduced by damping materials and design.
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