The influence of the geometrical and material parameters of damping coatings, covering helical spring wires, on the resonant longitudinal vibration amplitudes is presented in the paper. It is shown that coatings with geometrical parameters that are possible to obtain in practice allow the efficient reduction of the maximal amplitudes of these vibrations, when the materials used for coatings have properties specific for macromolecular polymers. The considerations are illustrated with numerical examples

Krzysztof Michalczyk

English

Vibroengineering PROCEDIA

  © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533 37 
The influence of damping coating geometrical-material 
parameters on helical spring longitudinal vibration 
amplitudes 
Krzysztof Michalczyk 
AGH University of Science and Technology, Faculty of Mechanical Engineering and Robotics,  
30 Mickiewicza Avenue, 30-059 Krakow, Poland 
E-mail: kmichal@agh.edu.pl 
(Received 4 July 2016; accepted 6 July 2016) 
Abstract. The influence of the geometrical and material parameters of damping coatings, 
covering helical spring wires, on the resonant longitudinal vibration amplitudes is presented in the 
paper. It is shown that coatings with geometrical parameters that are possible to obtain in practice 
allow the efficient reduction of the maximal amplitudes of these vibrations, when the materials 
used for coatings have properties specific for macromolecular polymers. The considerations are 
illustrated with numerical examples. 
Keywords: helical springs, helical spring vibrations, vibration damping, damping coatings. 
1. Introduction 
Helical springs applied in general machine constructions are usually subjected to excitation, 
which consists of a static load component and a dynamic periodical load component. The static 
load component is mostly caused by a supported machine part weight or pre-tension, whereas the 
changing load component is a result of working machine excitations, foundation vibrations or both 
of these factors simultaneously. The natural frequencies of helical springs can be situated within 
the range of frequencies of the excitations mentioned above, both in steady and transient 
conditions, which may result in the generation of resonant spring vibrations, which are dangerous 
and detrimental from the spring durability point of view. The vast range of helical spring 
applications means that these vibrations constitute an important practical problem. Spring 
vibrations are a serious problem, among others, in vehicle suspension systems, fuel injection 
systems and timing systems in piston combustion engines [1]. Research focused on the reduction 
of combustion engine valve spring vibrations, which has been conducted for many years, indicates 
the importance of this problem [2-4]. There are many constructional solutions which aim to 
restrain this phenomenon [5-7]. The application of a damping coating covering the spring wire is 
one of methods that restrain the amplitudes of spring vibrations. This solution utilizes the internal 
material friction and, as opposed to the methods utilizing constructional friction, it also protects 
the spring wire against corrosion, which is a very important factor that negatively influences the 
fatigue strength of steel springs. 
2. Longitudinal vibrations 
In typical industrial applications of helical springs, like vibrating machine support or vibration 
isolation systems, one end of the spring is mounted to an element that can be treated as a fixed 
one, whilst the second end of the spring is subjected to vibrations, having an axial component of 
a certain amplitude ܽ and circular frequency ߱. In conditions of steady resonant vibrations, the 
maximal spring vibration amplitude ܣ௜  (a lower index ݅  denotes the number of the mode of 
longitudinal natural vibrations) can achieve such high values that it can be assumed with high 
accuracy that the mode of the spring vibrations corresponds to the mode characteristic for 
clamped-clamped boundary conditions.  
THE INFLUENCE OF DAMPING COATING GEOMETRICAL-MATERIAL PARAMETERS ON HELICAL SPRING LONGITUDINAL VIBRATION AMPLITUDES.  
KRZYSZTOF MICHALCZYK 
38 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533  
2.1. Equivalent rod model 
The equivalent rod model with suitable dynamic parameters can be utilised when considering 
the longitudinal vibrations of coated helical springs. The effects connected with damping can be 
initially omitted, as damping does not affect the normal mode shapes, and due to the relatively 
low damping obtained by elastomeric coatings, its influence on natural frequencies is small. Thus, 
the wave equation of undamped longitudinal vibrations of equivalent rod takes a conventional  
for ݉: 
߲ଶݕ(ݔ, ݐ)
߲ݐଶ =
ߛ
݉
߲ଶݕ(ݔ, ݐ)
߲ݔଶ , (1)
where ݉  denotes the linear density of equivalent rod, whereas ߛ  denotes an equivalent 
compression rigidity [8]: 
ߛ = (ܩ௪ܬ௪ + ܩ௖ܬ௖)ܮߨܴଷ݊ cos ߜ , (2)
where: ݊  – number of active coils, ܮ  – current active spring height, ܴ  – half of the nominal 
diameter of the spring, ܩ௪  and ܩ௖  – shear modulus of the spring wire and coating material 
respectively, ܬ௪  and ܬ௖  are area moments of inertia of the wire and coating cross-sections 
respectively. 
2.2. Amplitudes of resonant longitudinal spring vibrations 
The first four normal longitudinal modes of a clamped-clamped coated helical spring as well 
as plots of the displacement amplitudes of equivalent rod modelling for the spring are presented 
in Fig. 1. 
 
Fig. 1. The first four longitudinal natural modes of a clamped-clamped coated helical spring  
with corresponding plots of displacement amplitudes of equivalent rod 
Under the initial assumption that both ends of the spring are fixed, one can write the stationary 
solution of Eq. (1) and the time independent factor ௜ܺ(ݔ) of this solution, which describes the 
amplitudes of longitudinal vibrations of equivalent rod modelling for the considered spring: 
௜ܺ(ݔ) = ܣ௜sin ൬
݅ߨݔ
ܮ ൰, (3)
where ݅ = 1, 2, 3,... denotes the number of the natural longitudinal frequency ߱௜.  
Knowing the shape of the vibration mode, one can calculate the maximal elastic energy stored 
THE INFLUENCE OF DAMPING COATING GEOMETRICAL-MATERIAL PARAMETERS ON HELICAL SPRING LONGITUDINAL VIBRATION AMPLITUDES.  
KRZYSZTOF MICHALCZYK 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533 39 
in the spring in resonance conditions as a function of the maximal amplitude ܣ௜ and, subsequently 
using the definition of the specific damping capacity ߦ, the energy dissipated during one period of 
vibrations can be determined. According to this definition [9], energy dissipated during the cycle 
ܷ஽௝  is equal to the maximal potential energy ௝ܷ  of the vibratory system within one cycle 
multiplied by the specific damping capacityߦ. In the case of a spring with a damping coating on 
its wire, the equivalent relative energy dissipation ߦ can be written in the form [10]: 
ߦ = ߦ௪ܩ௪ܬ௪ + ߦ௖ܩ௖ܬ௖ܩ௪ܬ௪ + ܩ௖ܬ௖ , (4)
where: ߦ௪ is the specific damping capacity of the spring wire material, whereas ߦ௖ is the specific 
damping capacity of the coating material. Longitudinal deformations of the spring are connected 
mainly with the twisting of the wire and the coating, thus the energy dissipated by the spring ܷௗ௜ 
during one period of vibrations amounts to: 
ܷ஽௜ = ߦ න
൬d ௜ܺ(ݏ)dݔ ߛܴ cos ߜ൰
ଶ
2(ܩ௪ܬ௢௪ + ܩ௖ܬ௢௖) dݏ
ௌ
଴
, (5)
where: ܵ is the active length of the coating and the spring wire, ݏ – the coordinate along the spring 
wire axis, ߜ – the lead angle of the helix. 
Giving up the initial assumption that both ends of the spring are clamped and assuming in 
return that one end of the spring (at ݔ =  0) is moving in accordance with the equation  
ݑ(ݔ = 0, ݐ) = ܽsin(߱ݐ) and only the second end of the spring (at ݔ = ܮ) is clamped, the work ௜ܹ 
performed by the vibrating support on the spring can be determined. This work can be expressed 
in the following form: 
௜ܹ = න ቆ−ߛ
߲ݑ௜(ݔ, ݐ)
߲ݔ ∙
߲ݑ௜(ݔ, ݐ)
߲ݐ ቇቤ௫ୀ଴
ଶగ
ఠ೔
଴
dݐ. (6)
It should be mentioned that in resonance conditions, when ߱ = ߱௜ , there is a phase shift 
between the driving oscillation and vibrations equal to one quarter of the period, thus the function 
ݑ௜(ݔ, ݐ) appearing in Eq. (6) has the following form: 
ݑ௜(ݔ, ݐ) = ௜ܺ(ݔ) sin ቀ߱௜ݐ −
ߨ
2ቁ. (7)
Comparing the energy dissipated during one period of vibrations ܷ஽௜ (Eq. (5)) with the work 
௜ܹ performed on the spring during this period (Eq. (6)), after transformations we obtain the ratio 
of the maximal amplitude ܣ௜ to the amplitude of the spring end (at ݔ = 0) displacement ܽ: 
ܣ௜
ܽ =
4
݅ߦ. (8)
The relationship Eq. (8) therefore allows the maximal vibration amplitude of the spring with a 
damping coating on its wire to be determined, when the amplitude of driving oscillations is given.  
2.3. Numerical examples 
Fig. 2 presents the ratio between the maximal resonant vibration amplitude ܣ௜ of the spring 
with a damping coating to the maximal resonant vibration amplitude ܣ௡௖௜ of the same spring but 
THE INFLUENCE OF DAMPING COATING GEOMETRICAL-MATERIAL PARAMETERS ON HELICAL SPRING LONGITUDINAL VIBRATION AMPLITUDES.  
KRZYSZTOF MICHALCZYK 
40 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533  
without a damping coating, under the assumption that the excitation amplitude ܽ is the same in 
both cases as a function of the coating material shear modulus ܩ௖ and the ratio of coating and wire 
diameters. It was assumed that ߦ௪ = 0.01 and ߦ௖ =0.4.  
 
Fig. 2. The ratio of amplitudes ܣ௜ ܣ௡௖௜⁄  as a function of coating material shear modulus ܩ௖  
and the ratio of coating and wire diameters ݀௖ ݀௪⁄  
Fig. 3 presents the ratio of amplitudes ܣ௜ ܣ௡௖௜⁄  as a function of coating material shear modulus 
ܩ௖  and the coating material specific damping capacity ߦ௖  under the assumption that  
݀௖ ݀௪⁄ = 2 and ߦ௪ = 0.01.  
 
Fig. 3. The ratio of amplitudes ܣ௜ ܣ௡௖௜⁄  as a function of the coating material shear modulus ܩ௖  
and the coating material specific damping capacity ߦ௖ 
The assumed range of the shear modulus ܩ௖ and the specific damping capacity ߦ௖ of the 
damping coating material includes values of these parameters typical for both industrial rubber 
and plastics, such as polyurethane, polyethylene of high density or polyamide. Rubber-like 
materials usually have very good damping properties, but are accompanied by very low shear 
modulus values. Thus, although they are successfully applied in many different types of dampers 
[11], it can be seen in Fig. 3 that in the considered case such materials do not provide good results. 
Materials with good damping properties but higher shear modulus, such as typical polymers [12], 
provide much higher vibration damping efficiency. It is also visible in Fig. 2 that the relative 
thickness of the coating plays a major role in damping effectiveness – the increase in the ratio 
݀௖ ݀௪⁄  is accompanied by a slow increase in damping. 
3. Conclusions 
The presented model allows the maximum amplitude of coated helical spring vibrations to be 
THE INFLUENCE OF DAMPING COATING GEOMETRICAL-MATERIAL PARAMETERS ON HELICAL SPRING LONGITUDINAL VIBRATION AMPLITUDES.  
KRZYSZTOF MICHALCZYK 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533 41 
calculated when the amplitude of excitation is known. The considerations proved that the resonant 
longitudinal vibration amplitudes of helical springs can be effectively reduced using damping 
coatings applied on their wires. It was shown that the material used for the coating should have 
properties characteristic for typical macromolecular plastics rather than rubber.  
References 
[1] Berger C., Kaiser B. Results of very high cycle fatigue tests on helical compression springs. 
International Journal of Fatigue, Vol. 28, 2006, p. 1658-1663. 
[2] Yang O., Engel P., Shiva Prasad B. G., Woollatt, D. Dynamic response of compressor valve springs 
to impact loading. International Compressor Engineering Conference, 1996, p. 353-358. 
[3] Flenker Ch, Uphoff U. Efficient valve-spring modelling with MBS valve-train design. MTZ, Vol. 66, 
Issue 12, 2005, p. 946-950. 
[4] Liu H., Kim D. Effects of end coils on the natural frequency of automotive engine valve springs. 
International Journal of Automotive Technology, Vol. 10, Issue 4, 2009, p. 413-420. 
[5] Speckhart F. H., Mongold B. D., Chun H., Tyler V. M. Helical Coil Spring Damper Assembly. 
Patent US6789790 B2, 2004. 
[6] Kutzbach J. H., Menzel J. D. I. Anti-vibration Element. Patent DE10332637 A1, 2005. 
[7] Sai K. Valve Train of Internal Combustion Engine. Patent US8479696 B2, 2013. 
[8] Michalczyk K. Analysis of the influence of elastomeric layer on helical spring stresses in longitudinal 
resonance vibration conditions. Archives of Civil and Mechanical Engineering, Vol. 13, Issue 1, 2013, 
p. 21-26. 
[9] Rivin E. Passive Vibration Isolation. ASME Press, New York, 2003. 
[10] Michalczyk K. Influence of the elastomeric coating on parameters of steady state vibrations of coil 
springs in the resonance and outside it. Journal of Theoretical and Applied Mechanics, Vol. 52, Issue 2, 
2014, p. 507-518. 
[11] Dziurdź J., Zawisza M. The investigation of the impact of changes in the characteristics of the rubber 
element on the effectiveness of torsional vibration damper. Vibroengineering Procedia, Vol. 6, 2015, 
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[12] Chung D. D. L. B. Review materials for vibration damping. Journal of Materials Science, Vol. 36, 
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The influence of damping coating geometrical-material parameters on helical spring longitudinal vibration amplitudes

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The influence of damping coating geometrical-material parameters on helical spring longitudinal vibration amplitudes

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