Vibration Basics - Operating Principles
Operating Principles of an elastic suspension
Before studying complex assemblies, it is useful to consider how an elastic suspension operates with a single degree of freedom. Frequently, the following simple calculations establish the main criteria for solving a problem.
These calculations are reasonably accurate if the mountings are well positioned with respect to the center of gravity and if the adjacent structure is rigid and does not compromise the performance of the mounting system.
The technical departments of Hutchinson can produce comprehensive models of complex systems where the center of gravity is offset with respect to the mounting points for the resilient mountings. Six degree of freedom natural frequencies and mode shapes can be generated, as well as specific outputs for a variety of complex shock and vibration inputs, for systems with resilient mountings with linear or non-linear stiffness characteristics.
4.1 - Static operation
An elastic suspension provides better distribution of static loads and allows for manufacturing tolerances without excessive overstressing of the mounting points.
An elastic suspension takes up small displacements while avoiding damaging forces. These small displacements may be caused, for example, by expansion under heat or deformation of the chassis, casing, structure, etc...
4.2 - Dynamic operation
This is the main function of elastic mounting systems for vibration or shock. The following calculations are based on the mounting having linear or constant stiffness which is not always the case.
4.2.1 - Vibration with a single degree of freedom
The design of a mounting system is highly complex. A simple case is given as an illustration (fig.1)
A piece of equipment with mass M may only move parallel to the vertical axis Gz. It is fixed at its base by an elastic mounting system S whose stiffness along Gz is K.
(fig. 1)
Sinusoidal vibrations x = A sin ω t (fig. 2)
(fig. 2)
Frequency n = 1 ⁄ T = ω ⁄ 2π
- Maximum amplitude A
Instantaneous amplitude x = A sin ω t
- Maximum speed V = A ω
Instantaneous speed v = A ω cos ω t
- Maximum acceleration G= -A ω2
Instantaneous acceleration y = -A ω2 sin ω t
High frequency vibration (high ω) may therefore cause very high acceleration, even at low amplitude.
Free oscillation (natural)
a) Without damping (theoretical)
If the suspended system is moved from its equilibrium position by a distance A and released, it will oscillate following a sine wave.The position along the Z axis will be z = A sin w0
Natural frequency : ω0 = 
s-1 or f0 = ω0⁄2π Hz
The oscillation continues indefinitely with a maximum displacement of A (illustration fig. 2 with ω replaced by ω0).
b) With damping
In this case the machine will oscillate about its equilibrium position following a damped sine wave (see fig.3).
z = A.e ε' 0 ω'0 t.sin ω0
Natural frequency
ω'0 = (1 - ε '0 2 ) = ω'0 
ε'0 is the damping rate at frequency ω'0
As ε'0 is very close to ε0 the natural frequency can be expressed as :
ω'0 ≠ ω0
For natural rubber, ε'0 is much smaller than 1 (0.02 to 0.1). Therefore ω'0 is very close to ω0.
(fig. 3)
Forced vibration
Assuming that the suspended system is subjected to a vertical forced vibration which imposes a sinusoidal force at frequency w The exciting force will be : F = FM sin ω t
For rigidly mounted equipment the whole
disturbing force will be transmitted into the equipment.
For resilient mounted equipment with natural frequency
ω0 and damping coefficient ε0
At the start of the forced vibration at frequency ω, a vibration is induced at the natural frequency ω0. This is damped very quickly so that, within a very short time, only the vibration at the forcing frequency ω remains as a permanent vibration transmitting a sinusoidal force to the equipment.
Force transmitted : F' = F'M sin ω t.
The transmission coefficient λ can be defined. This is the ratio of the maximum force transmitted F'M to the maximum exciting force FM (or alternatively the force that would be transmitted if there were no elastic mounting system). For an elastic or elastomer mounting system this coefficient is :
λ = F'M/FM = 
In conclusion :
| Existing force | Force transmitted | Transmissibility | |
| Rigid mounting system | F=FM sin ω t | F=FM sin ω t | λ=1 |
| Elastic mounting system (ω0,ε0) |
F=FM sin ω t | F=F'M sin ω t | λ = F'M/FM = |
The variation in the transmission coefficient λ for the ratio ω/ω0 for various values of ε0 is shown in fig. 4.
Attenuation :
For anti vibration mountings the term 4 ε 02 is small compared to 1. The percentage attenuation is equal to 100 less the transmissibility coefficient. For a given excitation frequency ω, the attenuation depends on the natural frequency of the mounting system.
E% = 100 or 100  |
(fig. 4)
To obtain a good suspension :
high ω/ω0 → low ω0 → low λ
moderate ε0 → - limited amplification at resonance
- little effect below resonance
(fig. 5)
The diagram (fig. 5) gives a relationship between the static deflection of a mounting system and its frequency. It can only be used as a guide as it is strictly applicable only to perfect spring mounting with one degree of freedom. In practice, one must use a larger deflection to take into account the dynamic stiffening and the creep.
The diagram also gives the theoretical maximum coefficient of isolation, again corresponding to a perfect spring suspension with one degree of freedom and assumes also that the supporting structure is infinitely rigid. On a typical structure with low mass it is wise to use a larger deflection.
4.2.2 - Shocks - Protection against shock
There are two principal cases to be considered :
a) Limitation of the force transmitted to the equipment :
This case often appears in the following form : The equipment moving at a precise speed, meets an obstacle. The force that it can withstand without damage is limited to a known value. A system of rubber parts, which could be the flexible mounting system of the equipment, is placed between the equipment and the obstacle. These parts provide a constant stiffness Kz in the direction of the shock. If there is energy W to be absorbed, in the absence of damping :
W = ½ Kz Z2
The maximum force FM = KzZ = 2W/Z . The maximum force is inversely proportional to the travel.
The travel Z = 
. The travel is inversely proportional to the square root of the stiffness.
Note :
Some systems do not have a constant stiffness, but a stiffness which increases rapidly (e.g. compression system). It is clear that if the energy W is not absorbed before the stiffness increases the maximum force will be much higher than predicted by the formula.
b) Limiting the acceleration of particular parts of the equipment :
In this case the shock must be described in terms of its potential to destroy. The efficiency of the protection system is measured by its ability to reduce this potential.
A shock to the equipment can damage a component part if this part is induced to vibrate at an amplitude which is compatible with its mechanical characteristics thus causing it to break. A shock can be characterised by its action on a whole series of components. For the same shock, each component has its own specific response, which differs from one component to the next.
The shock spectrum is the graphical representation of the ratio of amplitude of vibration ( Γ ) of the components to the amplitude of the shock ( Γ0 ) as a function of the ratio of the duration of the t to the natural frequency T of the elements.
This is not a representation of the amplitude as the function of time, neither of the excitation nor the effects, but a convenient representation of the destructive power of shock.
. It is not possible to recover the form of the
shock from the spectrum.
. Two different shocks may well produce the
same spectrum.
Take, for example, the case of shock with a half sinusoidal acceleration. (fig. 6)
(fig. 6)
A piece of equipment must withstand a shock of Γ0 = 400 m/s2 maximum for a period τ = 8.75 ms.
| Component A of the equipment | Component B of the equipment | |
| Natural frequency mass |
40Hz 10 kg (22 lb) |
286Hz 1 kg (2.2 lb) |
| τ/T
Γ/Γ0 Load on mounting points |
8.75.10-3 x 40 = 0.35
1.25 400 x 1.25 x 10 = 5000 N |
8.75.10-3 x 286 = 2.5
1.1 400 x 1.1 x 1 = 440 N |
(fig. 7)
Study of the spectrum shows the performance of a mounting system is acceptable when it is possible to obtain a natural frequency T such as:
τ/T < in which case the ratio Γ/Γ0 is less than
1 and the component is protected.
If it is not possible, it is preferable to modify the suspension to prevent amplification in the range τ/T between 0.25 and 2.5
This simple case shows the role of a flexible mounting system and the importance of knowing the details (shock spectrum, amplitude as function of time) and above all the duration of the shock.
Damping
Damping may be useful to reduce rebound and the amplitude of successive oscillations. However, care must be taken to choose the right type of damping as damping can reduce the offered protection. The influence of the damping effect will be all the greater in the case of multi-frequency excitation where it is not always possible to select a natural frequency well away from the excitation frequencies. This is also true when searching for a compromise between shock attenuation (force transmission) and the limitation of displacement (fig.8).
(fig. 8)
Important
When designing the equipment it should be borne in mind that : Good protection requires great flexibility, which causes significant displacement between the equipment and its environment.
The equipment will oscillate and there must be space for the rebound in case of shock.
4.2.3 - Materials used for specific mounting systems. Selecting characteristics for different vibrational environments.
As already seen, the operating environment determines the type of mounting whether elastomeric or metallic or a combination of both to be used in the mounting system.
Hutchinson has available a comprehensive databank on materials which allows them to offer you the best compromise between the static or dynamic characteristics of each type of mounting, their ability to withstand certain environments and their life. Some of the characteristics of typical types of mountings are described below.
4.2.3.1 - Elastomeric mounting
Natural rubber
Natural rubber is known for its excellent vibrational behaviour (linearity) and its long life. However it may only be used in temperatures between -30°C and + 70°C.
Silicone rubber
Silicone rubbers are usually more damped than natural rubbers and can be used over a wider temperature range between -54°C to + 150°C.
Other elastomers
Other types of elastomer include butyl, neoprene and butadiene rubber which have specific properties. Please contact our technical services for additional information.
Changes in the Young's modulus and damping with temperature (fig.9) and amplitude for natural rubber and silicone rubber (fig.10) are illustrated through the following curves.
(fig. 9)
(fig. 10)
These variations are characteristic of the non linear behaviour of elastomers and should be taken into account when designing the suspension. The static stiffness of a mounting system is different from its dynamic stiffness. The stiffness to be considered when studying shock is also different from the dynamic stiffness of the suspension.
Important: the characteristics of products described in the data sheets are for typical ambient temperatures.
4.2.3.2 - Cable mounts
Made of a stainless steel cable, fixed to two light alloy bars, the cable has a high coefficient of internal damping given by the friction of the cables's strand against one another allowing it to have the job of spring damping. These isolators are particularly suitable for applications which at high shock where large deflection are required. Excellent temperature performance, resistance to a wide range of contaminants, chemical agents and ageing.
4.2.3.3 - Metallic mounts
A spring is an elastic material which allows large deflection and therefore it is possible to reach very low frequencies. The metallic cushion is made of a stainless wire 18/8 cold hammered, knit embossed and compressed. Its characteristics are excellent (high mechanic properties, wide temperature range, grease, oil, solvent, water, dust, chemical agents resistance).