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.

	This is an example of free vibrations. When the mass is pulled to one side the spring provides a restoring force (Hooke's law) and there is no friction (smooth plane), so when it is released it oscillates indefinitely. Mathematically, from Hooke's law: Where: F is the restoring force on the moving body (in N); x is the body's displacement from the origin (in m); k is the force constant of the spring (in N.m-1); a is the body's acceleration (in m.s-2); m is the body's mass (in kg); and A gives the starting position i.e. maximum displacement of the body from the origin (in m).
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. The speed goes to zero at the extreme displacements, and to maximum at the origin.
Integrating further gives the displacement as a function of time. Note that the variables are separated to give only displacement on the LHS and only time on the RHS.
The result is:
You always have the sine of an angle. Here the angles, α and φ are not real angles in space.	α and φ are called phase angles, because they relate particular displacements to the maximum displacement. ωt has to have a unit of angle, and so ω must be in radian per second, rad.s-1. The phase of a point tells you how far through the cycle you are (like the phases of the Moon). The initial phase, α, tells you how far through the cycle a body is when it starts off.
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.

	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 ω0 is the frequency for a free vibration, i.e. with r=0.

The damped frequency ω₁ 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. ω1 ≈ ω0 For critical damping there is only a swing back to zero, i.e. the frequency is zero Hz. 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 spring-plus-mass 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 ω₂, 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 ω2 approaches the natural frequency ω0 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. As also indicated, the sharpness of the peak depends on on the friction coefficient r.

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 self-induced 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: τ is torque, θ is angle of twist, I is moment of inertia, r is damping ratio, and ω is the natural frequency.

The bridge failed because of torsional "flutter" where one mode at 0.2 Hz had negative damping from synchronised motion-induced 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 mass-dampers (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 rock-like 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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