The Control of Vibration - and Damping

The control of vibration is important for two reasons: firstly, personal comfort, health and safety may be adversely affected if humans are subjected to excessive vibration. Secondly, equipment suffering from external vibration may not perform as required, or a piece of vibrating equipment may cause adjacent machinery to perform in an unsatisfactory way.

Careful control over manufacturing and assembly tolerances may make it possible to reduce vibration, but at some cost. It will not often be possible to totally eliminate dynamic forces from machinery. If after appropriate action has been taken vibration levels are still in excess of acceptable levels, then there are a number of methods of reducing or controlling vibration.

1. Modify the system so the natural frequencies do not lie near the operating speed or any of the frequencies excited by the operation of the system.

2. Prevent excessive response, even at resonance, by incorporating damping into the system.

3. Reduce the transmission of forces from one part of the system to another part by placing vibration isolators between them.

4. Reduce the response of the 'main' component by adding an auxiliary mass which acts as an absorber.

Implementing these options involves the following:

1. Either the mass, m or the stiffness k can be varied. It is often easier to vary k than m.

2. Damping is normally ignored to simplify analysis, but it is always present. In many engineering structures the damping is very light and the errors of ignoring it are very small except when operating close to resonant frequencies. In some situations significant damping is required, eg. automobile shock absorbers, where carefully computed damping is provided. In forced vibration, when operating close to resonances, response may be excessive unless damping is added.

Where a wide range of operating speeds are in use it will often not be possible to avoid a modal frequency and in these cases use of structural materials with high inherent damping, such as cast iron or laminated sandwich materials, should be considered.

Bolted and riveted joints introduce damping, but debris due to joint slip and fretting corrosion may be produced.

Viscoelastic materials can be used. A layer can be added to a plate prone to vibration (common in car construction) or a layer may be sandwiched between the elastic sheets. Tapes consisting of metal foil coated with viscoelastic material are available for applying to existing structures. A problem with analysing viscoelastic materials is that their properties vary according to temperature, strain and frequency.

The equation of motion of a single degree of freedom with internal damping system subject to harmonic excitation F(t)=Foei.w.t can be written as:

mx" + k(1 + i.eta)x = Foei.w.t where eta is the loss factor (or loss coefficient) and can be written:

eta = (energy dissipated during 1 cycle per radian)/(max. strain energy in the cycle)

and the amplitude of response of the system at resonance (w = wn) is given by:

Fo/k.eta = Fo/a.E.eta where a is a constant and E is Young's modulus.

It should be noted that the loss factor can not be uniquely defined for a non linear system. For a linear system subject to harmonic excitation the loss factor is twice the damping ratio, zeta. Some typical approximate values for loss factors and damping ratios are shown in the table below:

Material / Structure Loss factor, eta Viscous damping ratio, zeta
steel 0.0004
aluminium 0.0001
fiber mats 0.1
natural rubber 0.01 - 0.08
butyl rubber 0.05 - 0.5
polystyrene 2
welded structure 0.02
bolted structure 0.06

Vibration Isolation

The undesirable effects of vibration are reduced by placing an isolator (resilient body) between the vibrating component and the source of the vibration. It may be an active or passive isolator, the latter cosists of a resilient member (elastic) and a dissipating (damping) element. Passive isolators include rubber, metal and pneumatic springs, cork and fibrous materials. An active isolator consists of a servomechanism with a sensor, signal processor amplifier and actuator with generates forces / motion to keep the motion of the delicate body to a minimum.

There are two types of isolation needed:

i.) A delicate package may need protecting from motion of the foundation / environment.

ii.) Foundations / buildings may need protection from large cyclic forces generated by industrial equipment, such as a drop forge or high speed press.

Assuming the system can be treated as single dof, forces are transmitted throught the spring and damper and the effectiveness of the isolator is given by its transmissibility:

Tr=(transmitted force)/(exciting force)

Forces Transmitted From a Vibrating Body to the Foundations:

F(t)=kx(t) + cx'(t)

For the case of a delicate body to be protected from a vibrating base:

F(t)=mx"(t) = k[x(t)-y(t)]+c[x'(t)-y'(t)], where x-y and x'-y' are the relative displacement and velocity accross the spring and damper.

Reduction in Force Transmitted to Foundation

Assume the machine operation causes a harmonic force: F(t)=Focos.w.t then the equation of motion of the machine is given by:

mx"+cx'+kx = Focos.w.t

The steady state solution of this is given by: x(t)=Xcos(w.t-phi) where:

X=(Fo)/[(k-m.w2)2 + w2c2]0.5 and

phi = tan-1[(w.c)/(k-m.w2)]

The force transmitted to the ground is given by:

F(t)=kx(t)+cx'(t)=k.X.cos(w.t-phi)-c.w.X.sin(w.t -phi) and the magnitude of the force:

FT=[(kx)2+(cx')2]0.5=X[(k)2+(w.c)2]0.5

The transmissibility: Tr=FT/Fo

TT=[(k2+w2c2)/{(k-m.w2)2+w2c2}]0.5 which can be re-arranged as: Tr=[{1+(2.zeta.w/wn)2}/{(1-(w/wn)2)2+(2.zeta.w/wn)2}]0.5 [eqn a]

The value of w/wn = r is sometimes referred to as the frequency ratio.

The following should be noted:

1. From plots of curves of Tr against r, it can be seen that for the transmitted force to be less than the exciting force the forcing frequency has to be more than 20.5 times the natural frequency (wn) of the system.

2. To reduce the magnitude of the transmitted force, reduce the natural frequency of the system.

3. The force transmitted to the foundation can be reduced by reducing the damping ratio, but as vibration reduction requires operation at r>20.5 the machine will pass through wn on starting and stopping, so some damping is needed to prevent excessive displacements at resonance.

4. Damping reduces the amplitude of the body motion for all frequencies, but it only reduces the transmitted forces at r>20.5, above this value, adding damping increases the transmitted force.

5. Where the forcing frequency (speed of the machine) varies, some compromise will be needed in choosing the damping so as not to increase the transmitted force too much (r>20.5) while still restricting the motion at resonance.

Force Transmitted to a Delicate Mass from a Moving Base

The governing equation is given by: mz" + cz' + kz = -my" where z = x - y the displacement of the mass relative to the base. Where the base motion is harmonic, the motion of the mass will also be harmonic and the displacement transmissibility: Td =X/Y and this is given by equation [a] above. This ratio is also equal to the ratio of the maximum accelerations of the mass and base.

Force Transmitted to Base by Rotating Unbalance

Here the exciting force is given by: Ft = Fosin.w.t = m.e.w2sin.w.t and the transmissibility as:

Tr = FT/Fo = FT/m.e.w2

If the left hand side of eqn [a] is replaced with FT/m.e.wn2, then the right hand side of eqn [a] remains unaltered.

David Grieve, 31st October 2012.