1. VIBRATION BASICS

2. Introduction
Vibration is created by mechanical processes – Fans, motors, pumps, pile driving, trains, earthquakes, etc.
Human response to vibration varies from:-
just perceptible --- annoying --- unpleasant --- painful --- physical damage
Buildings may be affected if levels are very high.
Vibration can be transient (shock), steady, or random.
Simplest situation is a single frequency with simple harmonic motion (SHM) as experienced by a pendulum or a mass hanging on a spring.

3. VIBRATION BASICS
At rest
MASS, m
 direction
+ direction
Mass hanging from a spring at rest. Forces are balanced.
If the mass is displaced downwards, there will be a force, F acting upwards due to the extension of the spring
The force is given by:
F = -k.x
where k is spring stiffness (N/m) and x is the extension in (m)
Note:- The force is acting in the –ve direction. F x
Displaced before release
MASS, m
Displaced
by distance, x

4. What happens when the mass is released?
The mass will accelerate upwards due to the force applied by the spring.
It will accelerate until it reaches the original rest position (0).
The spring /mass will then be in equilibrium but the mass has momentum so it continues to move upwards.
Once it has passed (0) it will decelerate due to the force of gravity until it stops at the top of its motion.
It will then accelerate downwards due to gravity until it is finally arrested by the spring at the position from which it was released from.
The motion will then be repeated. This is simple harmonic motion, SHM.
MASS
 direction
+ direction

5. Simple Harmonic Motion
Graph of SHM showing displacement, velocity and acceleration during one complete oscillation (360°). Note that we could also plot it against time in seconds or radians.
The graph this is sinusoidal.

6. The natural frequency of oscillation is:
The equation of motion of the mass is can be found by equating the forces:
MASS
 direction
+ direction
The forces must balance at any time (t), therefore:
m.a(t) = - k.x(t)
Rearranging to get the equation of motion of the mass gives:
m.a(t) + k.x(t) = 0
This is a second order differential equation since a(t) = d2x/dt2 which has a general solution in the form of :
x(t) = Xcos(t) this is a sine wave
The natural frequency of oscillation is:

7. Example
Find the natural frequency of oscillation of a 2.0 kg mass supported by a spring of stiffness 2000 N/m
Increasing the mass reduces f0
Increasing the stiffness increases f0

8. HUMAN BODY VIBRATION
Parts of the body can resonate when subject to low frequency vibration as shown in the diagram.
In general, vibration at lower frequencies is more perceptible than at higher frequencies.
Human sensitivity to whole body vibration is also directional and therefore vibration must be measured in three planes x, y and z.

9. FREQUENCY WEIGHTINGS
BS 6841 Guide to measurement and evaluation of human exposure to whole-body mechanical vibration and repeated shock
For whole body vibration, filters are used to make the overall measured vibration level represent the body’s sensitivity to vibration in a similar manner to how the “A” weighting mimics the sensitivity of the ear.
BS 6841 gives some guidelines on motion sickness, discomfort and perception, effects on health and activities but no definitive limits.

10. VIBRATION INDICES ROOT MEAN SQUARED VIBRATION ACCELERATION , aRMS
PEAK PARTICLE VELOCITY (ppv)
A measurement of vibration in terms of velocity. Mostly used for transient vibration (blasts, piling, etc.).
ROOT MEAN SQUARED VIBRATION ACCELERATION , aRMS 
For steady vibration RMS works well at assessing human response.
ROOT MEAN QUAD (RMQ), aRMQ
If the vibration contains short duration peaks the RMQ works better.
VIBRATION DOSE VALUE (VDV)
The VDV is not an average but the cumulative vibration dose for a specified period and can be calculated from: 
VDV = (0T aw4(t) dt) m/s (note the interesting units)

11. ESTIMATED VIBRATION DOSE VALUE
Accurate measurement of RMQ and VDV is difficult requiring specialist equipment to calculate the 4th power time average.
In practice the estimated vibration dose value, eVDV is used.
eVDV is calculated from the rms acceleration & duration of the vibration.
eVDV = 1.4 × aw (rms) × t (ms-1.75)

12. Example
The student flat next to the communal laundry room experiences vibration from the washing machines when they are on the spin cycle.
The weighted rms acceleration of the floor is 0.02m.s-2 during each cycle and on an average day there are 36 spin cycles each lasting for 2 minutes.
The total exposure time, t is 32 x 120 seconds.
eVDV = 1.4 × aw (rms) × t 0.25
eVDV = 1.4 x 0.02 x (36 x120)0.25
eVDV = m.s-1.75

13. VIBRATION IN BUILDINGS
BS 6472 Guide to evaluation of human exposure to vibration in buildings
Based on assessing vibration levels in buildings to determine the possibility of adverse comments.
Part 2 is concerned with blast induced vibration;
Part 1 is all other vibration.
Part 1 (not blast induced) uses same coordinates and units as in BS 6841 but has different frequency weightings, b for vertical and d for horizontal.

14. The eVDV is calculated from the tri-axial, frequency weighted, rms acceleration, aw (rms)
This is calculated from 3-D Pythagoras
aw = sqrt(axw2 + ayw2 + azw2)
Example
axw = 0.1, ayw = 0.2, azw = 0.3
aw = sqrt( )
aw = 0.374

15. Evaluation (People’s Response)

17. Evaluation (Building Damage)

18. Assessment (Construction Sites)
BS Control of noise and vibration on construction sites.
Uses the Peak Particle Velocity

19. Vibration Sensitive Equipment

20. Vibrock tri-axial accelerometer and overpressure microphone.
Instrumentation
The accelerometer should be attached so that the vibration is recorded accurately.
Faithful coupling with the vibrating medium shall be achieved, e.g.
burying sensors
using 300 mm long steel spikes according to ground conditions;
fast setting epoxy resin for measurement on building foundations or walls;
double-sided adhesive tape on hard floor finishes;
heavy metal plates with spiked feet for vibration measurements on floors with resilient covering.
Vibrock tri-axial accelerometer and overpressure microphone.

21. HAND-ARM VIBRATION
Hand-arm vibration can lead to damage to the nerves and vascular system – vibration white finger.
Assessed by measuring the 8-h energy-equivalent frequency-weighted vibration total value, A(8)
Main damage at frequencies between 1 – 1000 Hz as represented by the weighting curve.

22. Measurement
BS EN ISO :2001 Measurement and evaluation of human exposure to hand-transmitted vibration.
Gives the three axes of vibration that need to be measured.
The total acceleration is the vector of the three weighted axial accelerations:-
The “Daily Dose” is given as:-

23. Potential for VWF

24. Effects of VWF

25. Example
Calculate the A(8) & determine the likelihood of VWF for a fettler whose vibration total values for exposure times of 1 h, 3 h and 0.5 h (within the same working day) are 2ms-2 3.5ms-2 and 10ms-2 respectively:
ms-2
From graph this would give a 10% chance of VWF after about 8 years of exposure to this daily dose.

26. Physical Agents Directive 2005
Exposure action value of 2.5 m/s2 A(8) at which level employers should introduce technical and organisational measures to reduce exposure.
Exposure limit value of 5.0 m/s2 A(8) which should not be exceeded.
Whole body exposure limit is eVDV of 21 m/s1.75

27. Links Health and Safety Executive
Dan Russell animations (Kettering University)
Noise Net