Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

45, 2, pp. 363-377, Warsaw 2007

THEORETICAL AND EXPERIMENTAL STUDIES OF

ENERGY DISSIPATION IN A MODEL OF A RING SPRING

Zbigniew Skup

Institute of Machine Design Fundamentals, Warsaw University of Technology

e-mail: zskup@ipbm.simr.pw.edu.pl

The paper presents the problem of energy dissipation in ring spring jo-
ints with structural friction taken into account. A comparative analysis
has been conducted to compare theoretical results obtained bymeans of
numerical simulation with results of experimental research assisted by
an MTS testing machine. This contribution shows an outline of theore-
tical considerations, themethod of experimental tests aswell as selected
comparative results.

Key words: ring spring, energy dissipation, structural friction, hysteretic
loop, experimental testing

1. Introduction

The subject of analytical considerations is a friction pair consisting of co-
operating conical-shaped friction surfaces of internal and external rings of a
ring spring. That type of a spring is applied in ring buffers and friction dam-
pers used to dissipate energy. Contact deformation, friction and damping of
vibrations occurring in temporary fastenings andpermanent joints have essen-
tial influence on dynamical properties of machines and devices. Mathematical
description of structural friction phenomena is not easy due to complexity of
the friction process anddifficulties in describing the state of stresses anddefor-
mations occurring in joints of elements. Therefore, the description is based on
simplifying assumptions and fundamental mechanical laws that apply to pat-
terns of stress and deformations arising during tension, torsion, compression,
shearing (see References).

The following assumptions are made in order to analyze the ring spring:
the distribution of unit pressure between cooperating surfaces of the contact
joint is uniform; there is a constant friction coefficient of contacting elements
for an arbitrary value of the unit pressure; the friction force is subject to the



364 Z. Skup

Coulomb Law; and, consequently, the frictional resistance is proportional to
pressure, whilematerial properties are described by theHooke Law; friction is
fully developed in the sliding zone; inertia forces are neutral (due to very low
acceleration values); and flat sections are assumed (cross-sections remained
flat after the deformation of elements). Apart from a theoretical study of the
model shown inFig.2, experimental testshavebeenconductedonareal system
(Fig.1).

Fig. 1. A simplified model of a real ring spring

Fig. 2. A simplified model of the contact joint of the friction pair in the ring spring

2. Determination of displacements in particular stages during

loading the friction pair

Analytical investigations have been conducted on a ring spring model with
pure friction interaction between the elements. It is the closest approximation
of the results obtained from experiments. The simplifications taken in the pa-
per are a result of assumptions related to the theory of axially-symmetrical
thin-walled rings. Thus, a simplifiedmodel was assumed for analytical investi-
gations – a model of contact consisting of external and internal rings loaded



Theoretical and experimental studies of energy dissipation... 365

with an axial force αP, as presented in Fig.2, where: r is the average radius
of the conical surface of the internal ring, rz, rw – external and internal radii
of the conical surface of the ring, β – angle of tilt of the conical surface of the
ring.

Stage 1 – Loading the system with the axial force αP (0¬ α ¬ α1 =1)

By projecting the forces in the axial direction, we obtain the following
relationship

2πr(psinβ +µpcosβ)= αP (2.1)

while projection in the radial direction yields

2(pcosβ −µpsinβ)= qr (2.2)

where: p denotes the unit pressure per unit length, µ – friction coefficient,
αP – axial load of the system (0¬ α ¬ 1), qr – radial load per unit length.
Having determined the unit pressure p from equation (2.1), and substitu-

ted it to equation (2.2), we obtain

qr =
αP

πr

(cosβ −µsinβ

sinβ +µcosβ

)

=
αP

πr

(1−µtanβ

µ+tanβ

)

(2.3)

In accordance with the Hooke Law, given a uni-axial stress, the strain ε can
be described with the following dependence

ε =
N

EF
=
rqr

EF
(2.4)

where: E is Youngs elasticity modulus, F – cross-sectional area of the ring.
Due to deformation of the two co-operating rings, the axial displacement

can be determined from

u =2∆rcotβ =2εrcotβ (2.5)

The dependence between the displacement u of the ring and its external lo-
ad αP can be determined by substituting (2.3) and (2.4) to formula (2.5).
Therefore, after transformations, we can obtain the following

u =
2αPr

πEF

(cotβ −µ

tanβ +µ

)

(2.6)

Equation (2.6) shows interval 1 of the straight line OA1 in Fig.3. The ma-
ximum displacement umax for the final stage of loading with α = 1 can be
described with the following formula

umax =
2Pr

πEF

(cotβ −µ

tanβ +µ

)

(2.7)



366 Z. Skup

Fig. 3. Hysteretic loop for the friction pair of the ring spring

Stage 2 – Unloading of the system without sliding (α2 ¬ α ¬ α1)

The unloading decreases from the force value P down to α2P (interval 2
of the straight lineA1A2 – Fig.3), and it does not produce any changes in the
displacement umax. Therefore, the cooperating surfaces do not slide.

Stage 3 – Unloading of the system with sliding (α3 ¬ α ¬ α2)

When sliding occurs between the cooperating surfaces, this phenomenon is
accompanied by the change of sense of the friction forces. Therefore, equations
of equilibriumof forces projected on the axis of symmetry and radial direction
are the following

2πr(psinβ −µpcosβ)= αP
(2.8)

2(pcosβ −µpsinβ)= qr

Having determined the unit pressure from equation (2.8) and substituted it to
equation (2.9), we obtain

qr =
α2P

πr

(1+µtanβ

tanβ −µ

)

(2.9)

Tomake the ring springperform in the rightway, i.e. so that the considered
systemwouldnot lock itself, equation (2.10) has aphysical sensewhen tanβ >
µ – then the system will not be a self-locking one.
Following the steps taken at thefirst stage of loading, andapplying formula

(2.6) while taking account of (2.10), we obtain a relation between the axial
displacement of the rings and their loading

u =
2α2Pr

πEF

(cotβ +µ

tanβ −µ

)

(2.10)



Theoretical and experimental studies of energy dissipation... 367

From the equality of displacements for loading, see formula (2.7), and unlo-
ading, see (2.11), we find the parameter α2

α2 =
(cotβ −µ)(tanβ −µ)

(tanβ +µ)(cotβ +µ)
(2.11)

The minimum displacement can be determined from formula (2.11) with
α2 = α3, that is

umin =
2α3Pr

πEF

(cotβ +µ

tanβ −µ

)

(2.12)

The displacement umin corresponds to interval 4 of the straight line A3A4 of
the hysteretic loop (Fig.3).

Stage 4 – Repeated increase of loading in the system without sliding

(α4 ¬ α ¬ α3)

At this stage, when the loading is increased again from α3P to α4P, the
displacement does not change. The parameter α4 can be determined as shown
above from the comparison of the same displacements defined by (2.6) and
(2.13); thus

α4 =α3
(cotβ +µ)(tanβ +µ)

(tanβ −µ)(cotβ −µ)
=
α3

α2
(2.13)

3. Determination of energy dissipated during one cycle of loading

The dissipated energy in one cycle of loading equals is (Fig.3)

ψ =
(α4P −α3P +P −α2P

2

)

(umax−umin) (3.1)

Having used relationships (2.7), (2.11), (2.12) and (2.13), we obtained

ψ =
P2r

πEF

[d

a

(

1−
bd

ac

)

+
cα23
b

(

1−
ac

bd

)]

(3.2)

where

a =tanβ +µ b =tanβ −µ c =cotβ +µ

d =cotβ −µ tanβ > µ
(3.3)

Formula (3.2) has a physical sense in the case of tanβ > µ – the rings would
not jam then.



368 Z. Skup

4. Results of simulation testing

The simulation testing was conducted with the use of Mathematica 4.1 pro-
gram. Numerical calculations were carried out with the use of computer pro-
grams written in C language. These applications allowed for modifications
of the following: medium radius of cooperating surfaces of the friction pair
consisting of external and internal rings, friction coefficient, angle of tilt of
cooperating conical surfaces of the rings, Young’s modulus and the stress ran-
ge of rings of the spring (through the change of the parameters α1 and α3),
which made it possible to conduct a precise analysis of the joint. The nume-
rical calculations were made for the experimental model tested that had the
following parameters: F =132.94mm2, E =2.1 ·105N/mm2, r =37.25mm,
β =12◦, P =28kN, µ =0.15.

Table 1 shows a comparison of energy losses (the area of hysteretic loops)
obtained as a result of computational simulations.

Table 1. Energy losses obtained from computational simulations for dif-
ferent loadings

No. of diagram P [kN] umax [m] ψ [Nm]

1 15 0.000160066 0.000622207

4 20 0.000213422 0.001106150

3 25 0.000266777 0.001728350

4 28 0.000298791 0.002168040

The results of numerical calculations are shown inFig.4-Fig.12. Interesting
processes of energy losses in function of the ring average radius and in function
of the loading for different angles of tilt of the conical surface are show inFig.5
and Fig.6. Energy losses in function of the loading for different values of the
average radius are nonlinear (Fig.7).

Fig. 4. Hysteretic loops for different loadings P : 1 – 15kN, 2 – 20kN, 3 – 25kN,
4 – 28kN



Theoretical and experimental studies of energy dissipation... 369

Fig. 5. Energy losses in function of the average radius for different coning angles
(µ =0.15)

Fig. 6. Energy losses in function of the loading for different coning angles (µ =0.15,
r =37.25mm)

Fig. 7. Energy losses in function of the loading for different average radii (µ =0.15,
β =12◦)

The diagrams indicate that energy losses increase as the coning angle de-
creases. Figure 8 shows that in the case of complete unloading and given the
range of the coefficient of friction µ =0.14-0.22, the energy losses are optimal.
When the system is unloaded by 5-20%, the energy losses are optimal for the
coefficient of friction µ =0.21 (Fig.9). Theaxial displacement of rings in func-
tion of the coning angle or in function of the coefficient of friction is nonlinear
(Fig.10 and Fig.11). Figure 12 demonstrates that the axial displacement of



370 Z. Skup

the rings under the influence of loading and changes in the average radius of
the rings is linear.

Fig. 8. Energy losses in function of the coefficient of friction for completely unloaded
system (β =12◦)

Fig. 9. Energy losses in function of the coefficient of friction for different degrees of
unloading (β =12◦)

Fig. 10. The displacement u in function of the coning angle for µ =0.15



Theoretical and experimental studies of energy dissipation... 371

Fig. 11. The displacement u in function of the coefficient of friction (β =12◦)

Fig. 12. The displacement u in function of the average radius (β =12◦)

5. The experimental model

Thegoal of experimental testingwas to choose amathematicalmodel thatwo-
uld constitute the closest approximation to the real model, for which the area
of hysteretic loop is closest to the experimental area. In order to performexpe-
rimental tests, the model of the friction pair in the ring spring was designed
and built (Fig.13). Themodel wasmade of 45 steel, and the basic geometrical
parameters of the friction pair were the following: rz =77mm, rw =63mm,
β =12◦. The overall structural is shown in Figure 4. It was designed in such
amanner so that its position in themachine gripping jaws during the loading
process would not change (Fig.13 – elements 6 and 7). For measuring displa-
cements of the system, extensometers were used with a measurement base of
10mmand ameasurement nominal range ±1.2mm(sub-range of the nominal
range ±0.24mm), which were included in the standard machine equipment
(Fig.14 – elements 2 and 3).

Theway of attaching extensometers 2 and3 to elements 1, 2 and3 (Fig.13)
is presented inFig.14.On elements 3 and 5 (Fig.13), steady pins 6 and 7were
located in order to ensure a coaxialmeasurementbase for second extensometer
(Fig.14) in relation to the symmetry axis of the inner and outer part of the



372 Z. Skup

Fig. 13. Design of the experimental model of the friction pair of the ring spring
(decomposed system): 1 – outer grip, 2 – inner grip, 3 – shears to position the
coaxial friction pair and to fix an extensometer, 4 – grips to fix the second
extensometer, 5 – shears to position the inner and outer grip, 6 – steady pin to
position the friction pair, 7 – steady pin to position the inner and

outer grip

Fig. 14. Themode of fixing the model and an overview of its components: 1 – tested
model, 2 – first extensometer, 3 – second extensometer, 4 – gripping jaws of anMTS

testing machine

friction joint (elements 1, 2 and 3 in Fig.13). In order to provide the best
conditions for cooperating joined elements (maximum contact surface, surface
pressure, smoothness of motion), conical surfaces were subject to a surface
treatment by very precise grinding.



Theoretical and experimental studies of energy dissipation... 373

6. Results of experimental tests

The tests were conducted on anMTS testingmachine at the Institute of Ma-
chine Design Fundamentals of Warsaw University of Technology with the use
of TestWareSX software. Themode of fixing the tested model is presented in
Fig.14.Themethodology of the experiment consisted in loading the systemup
to themaximumvalue Pmax so that not to exceed a safe limit ofmaterial ela-
sticity. Next, the system was unloaded down to the pre-assumed value Pmin
and loaded up again. The loading process was controlled by a computer in
such away so that the torquemoment would equal zero while the experiment
was conducted. Before the measurement was started, both the extensometer
and the control-measurement system were subject to calibration. A Mathe-
matica 4.1 application was used to produce a graphical representation of the
testing results (Fig.15-Fig.20). Theprograms enabled us, among other things,
to calculate numerically the area of the hysteretic loop and to approximate
the obtained graphs. A series of experiments was conducted, and selected re-
sults were shown in Fig.15-Fig.18. By modifying the loading parameters, we
found singular (e.g. Fig.18) and cyclic hysteretic loops (e.g. Figs.15, 17, 18).
Moreover, graphs were arranged to show the change of the loading force and
the displacement of the joint in time (Figs. 16, 19). Figures 16 and 19 show
characteristics P(u), P(t), u(t) obtained for the model in four measurement
cycles. In that case, the system was loaded up to Pmax = 20kN and 28kN,
and then unloaded down to Pmin = 1kN. The whole process can be divided
into four stages (Fig.3).

Fig. 15. An experimental hysteretic loop for the loading force P
max
=20kN,

number of loading cycles: 4

The program enabled us to control the loading in the following manner:
preliminary loading (increase of the force up to Pmax and maintaining the
load for 10s (for one or four loading cycles).



374 Z. Skup

Fig. 16. Quadruple changes of the force and displacement in time for P
max
=20kN

Fig. 17. An experimental hysteretic loop for the loading force P
max
=28kN,

number of loading cycles: 4

Fig. 18. An experimental hysteretic loop for the loading force P
max
=28kN,

number of loading cycles: 4



Theoretical and experimental studies of energy dissipation... 375

Fig. 19. Quadruple changes of the force and displacement in time of P
max
=28kN

Fig. 20. Singular changes of the force (1) and displacement (2) in time

Table 2.Energy losses obtained from simulations and experimental tests

No.
Load Theoretical model Experimental model
[KN] ψ [Nm] ψ [Nm]

1 15 0.000622 0.000685

2 20 0.001106 0.001159

3 28 0.002168 0.002224

To sum up, it should be noted that the separation of the phenomenon
of structural friction in the experimental tests was very complicated and the
obtained results were influenced by simplified assumptions and internal fric-
tion, which was neglected in theoretical considerations. In order to present
the results of experiments in a graphical way, Microsoft Excel andMathema-
tica 4.1 programs were used. The programs enabled calculation of the area
of hysteretic loops and to approximate the obtained graphs. Table 2 shows a



376 Z. Skup

comparison of energy losses (the area of hysteretic loops) obtained as a result
of simulations and experimental tests.

The value of dissipated energy is proportional to the area contained inside
the hysteretic loop. The area was measured by a planimeter. The measure-
ment of the system was calibrated. In order to achieve identical measures for
the quantities, it was necessary to scale the measured systems and carry out
necessary calculations.

The comparisonwas compiled for the joint of internal and external rings of
the friction pair in the ring spring. Percentage differences between the results
(areas of hysteretic loops) obtained numerically and experimentally are shown
in Table 3.

Table 3.Differences (in %) between the average value (area of hysteretic
loops) of the results obtained from experimental tests and numerical simu-
lation

No. Loading [KN] Difference [%]

1 15 10.2

2 20 4.8

3 28 2.6

For better comparison, an average value of the hysteretic loop area obta-
ined from experiments on theMTS testing machine was calculated.

7. Conclusions

The paper presents amathematical model of a ring spring system and results
of experimental tests on a real model conducted on anMTS testing machine.
A comparative analysis allows one to formulate a conclusion that the pure
friction model is more similar to the real one. Characteristics shown in Figu-
res 15 and 17 are comparable – both quantitatively and quantitatively – in
graphs showing the results of theoretical considerations (Fig.13). The diver-
gence between theoretical and experimental results is caused by simplifying
assumptions made for the mathematical model. The following implifications
contributed to the discrepancy: constant friction coefficient, neglected internal
friction, unprecise constructionof the realmodel, problems related to thefixing
of themodel in the testingmachine (positioning in the gripping jaws) and dif-
ficulties occurring duringmeasurements (elimination of clearances), problems
with mounting of measurement elements.



Theoretical and experimental studies of energy dissipation... 377

References

1. Giergiel J., 1990,Tłumienie drgań mechanicznych, PWN,Warszawa

2. Grudziński K., Kostek R., 2005, Influence of normal micro-vibrations in
contact on sliding motion of solid body, Journal of Theoretical and Applied
Mechanics, 43, 37-49

3. Kaczmarek W., 2003, Analysis of a bolted joint with elastic and frictional
effects occurring between its elements, Machine Dynamics Problems, 27, 1,
21-40

4. KosiorA., SkupZ., 2000,Badanie rozpraszaniaenergiipoprzez tarcie suchew
połączeniu dwóch belek,Proceedings of the International Conference ”Friction
2000”,Modeling and Simulation of the FrictionPhenomena in the Physical and
Technical Systems, WarsawUniversity of Technology, 137-144

5. Osiński Z., 1998, Damping of Vibrations, A. A. BALKEMA, Rotterdam-
Brookfield

6. SkupZ., 1998,Wpływ tarcia konstrukcyjnegowwielotarczowymsprzęgle cier-
nym na drgania w układzie napędowym,Prace naukowe, Mechanika, 167, Ofi-
cynaWydawnicza PW

7. Skup Z., Kaczmarek W., 2005, Badania teoretyczne i doświadczalne roz-
praszania energii w połączeniu gwintowym z uwzględnieniem tarcia konstruk-
cyjnego, XV konferencja nt ”Metody i Środki Projektowania Wspomaganego
Komputerowo”, PolitechnikaWarszawska, 347-356

Badania teoretyczne i doświadczalne rozpraszania energii dla modelu

sprężyny pierścieniowej

Streszczenie

Wpracy przedstawiono zagadnienie rozpraszania energii wmodelu sprężynypier-
ścieniowej przy uwzględnieniu tarcia konstrukcyjnego. Przeprowadzono analizę po-
równawcząwynikówteoretycznychuzyskanychmetodą symulacji cyfrowej zwynikami
badań eksperymentalnych przeprowadzonych na maszynie wytrzymałościowej MTS.
W pracy omówione są rozważania teoretyczne, metoda przeprowadzenia badań oraz
porównanie wyników badań.

Manuscript received June 22, 2006; accepted for print December 20, 2006