Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

52, 1, pp. 189-197, Warsaw 2014

FORCED VIBRATIONS ANALYSIS OF A CONICAL SLEEVE-SHAFT

FRICTION JOINT

Andrzej Andrzejuk, Zbigniew Skup, Robert Zalewski

Warsaw University of Technology, Institute of Machine Design Fundamentals, Warszawa, Poland

e-mail: mang@ipbm.simr.pw.pl; zskup@ipbm.simr.pw.pl; robertzalewski@wp.pl

The paper presents a theoretical study of the damping process of non-linear vibrations in
a one-mass model of a mechanical system over a friction joint. The problem is considered
assuming a uniformunit pressure distribution between the contacting surfaces of the conical
sleeve-shaftneckof the friction joint.The steady-statemotionof the systemis subject tohar-
monic excitation. The analysis includes the following: the influence of geometric parameters
of the system, its external load amplitude, unit pressures and the friction coefficient upon
the amplitude-frequency characteristics and the phase-frequency characteristics. Structural
friction has been also taken into account. The equation ofmotion of the examinedmechani-
cal systemhas been solvedbymeans of the slowly varyingparameters (Van derPol)method
and a numerical simulationmethod.

Key words: conical joint, sleeve-shaft neck, energy dissipation, structural friction

1. Introduction

Forced vibration problems of mechanical systems with structural friction are widely discussed
in a number of domestic and foreign scientific papers, see Andrzejuk (2012), Gałkowski (1999),
Giergiel (1990), Grudziński and Kostek (2005), Kosior (2005), Mostowicz-Szulewski and Nizioł
(1992), Osiński (1998), Skup (2010), Zboiński and Ostachowicz (2001) or Badraghan (1994),
Meng (1989), Sanitruk et al. (1997), Sextro (2002), Wang and Chen (1993), Zahavi (1993).

Analytical considerations presented in this paper concern a real model of a friction joint
(Fig. 1). It consists of two cooperating conical friction pairs (Fig. 2). Such types of joints have
found an extensive application in different types of mechanical systems and devices. They are
often designed andmanufactured as natural energy dissipation elements.

Fig. 1. Components of the investigated sleeve-shaft neck friction joint: 1 – lower pressure plate grip of
the model, 2 – upper grip of the model, 3 – shaft neck, 4 – sleeve

Mathematical description of structural friction phenomena is not easy due to the complexity
of the friction process and difficulties in describing the state of stresses and deformations occur-
ring in cooperating elements. Therefore, the description is based on simplified assumptions and
fundamental mechanical laws that apply to the patterns of stress and deformations resulting
from tension, compression, torsion, shearing. A typical approach to such problems can be found



190 A. Andrzejuk et al.

Fig. 2. Physical model: (a) sleeve-shaft neck joint; (b) element of thickness ∆x at a distance x from the
larger end of the shaft; (c) element of thickness ∆x at a distance x from the larger end of the sleeve;

(d) displacement

in Gałkowski (1999), Kosior (2005), Osiński (1998), Skup (2010). The following assumptions
weremade in order to analyze the investigated model: the distribution of unit pressure between
cooperating surfaces of the joint contact elements is uniform; there is a constant friction coef-
ficient of the contacting elements for an arbitrary value of the unit pressure; friction force on
contact surfaces of the cooperating elements is subject to Coulomb’s law; and, consequently,
the frictional resistance is proportional to the pressure, while thematerial properties are descri-
bed by Hook’s law. The friction is fully developed in the sliding zone, the internal forces are
neutral (due to very low acceleration values) and, finally, the cross-sectional area of the coopera-
ting elements remains flat. Besides theoretical investigations of the model shown in Fig. 2, also
experimental tests on the real testing object (Fig. 1) have been carried out.

2. A mathematical model of the friction joint – analysis of forced vibration

In this Section, the solution of the problem concerning forced vibrations of the conical friction
joint is presented.Nonlinear vibrations of the examined elements under forced harmonic loading
(2.1) are examined. An additional assumption has been made that the considered friction joint
can be described as a single-mass system with a triangular hysteresis loop

P =P0cosωt (2.1)

Themathematical analysis is carried out considering the Van der Pol method.
The equation of motion of the system can be written as follows

mü+P(u, sgn u̇)=P0cosωt (2.2)

where m is the reduced mass, u – axial displacement; P(u, sgnu̇) – force represented by the
structural hysteretic loop (Fig. 3) dependent on the relative displacement, amplitude and sign
of velocity, P0 – excitation amplitude of the loading force, t – time; ω – angular velocity of the
excitation force.
Assuming the approximation of (2.2) in the form

u=Acos(ωt+φ) (2.3)

where φ denotes the initial forcing phase A, φ – slowly varying time functions.
Differentiating equation (2.3), we obtain

u̇= Ȧcos(ωt+φ)−Aω sin(ωt+φ)−Aφ̇sin(ωt+φ) (2.4)



Forced vibrations analysis of a conical sleeve-shaft friction joint 191

Fig. 3. Hysteresis loop for the investigated friction joint

By analogy to the Lagrange method of parameters variation, (2.4) may be written in the form

Ȧcos(ωt+φ)−Aφ̇sin(ωt+φ)= 0 (2.5)

Therefore

u̇=−Aωsin(ωt+φ) (2.6)

Thus differentiating (2.6) once again, gives

ü=−Ȧω sin(ωt+φ)−Aω2cos(ωt+φ)−Aωφ̇cos(ωt+φ) (2.7)

After introducing the denotation

z=ωt+φ (2.8)

and taking advantage of (2.7), differential equation (2.2) takes the form

−Ȧω sinz−Aω2cosz−Aωφ̇cosz+
P(u, sgnu̇)

m
=
P0
m
cos(z−φ) (2.9)

Multiplying equation (2.5) by −ωcosz and equation (2.9) by sinz, we obtain

Ȧωcos2z−Aωφ̇sinzcosz=0

Ȧω sin2z+Aω2 sinzcosz+Aωφ̇sinzcosz−
P(u, sgnu̇)

m
sinz=−

P0
m
sinzcos(z−φ)

(2.10)

Subtracting the system of equations (2.10), gives

Ȧω+Aω2 sinzcosz−
P(u, sgn u̇)

m
sinz=−

P0
m
sinzcos(z−φ) (2.11)

Since A and φ0 are slowly varying parameters in equation (2.2), equation (2.11) takes, after
integrating over the interval z∈ (0,2π), the following form

Ȧω

2π
∫

0

dz+Aω2
2π
∫

0

sinzcosz dz−
1

m

2π
∫

0

P(u, sgnu̇)sinz dz=−
P0
m

2π
∫

0

sinzcos(z−φ) dz (2.12)

Integrating both sides of equation (2.12), we get

2πȦω−
1

m

2π
∫

0

P(u, sgnu̇)sinz dz=−
P0π

m
sinφ (2.13)



192 A. Andrzejuk et al.

Multiplying equation (2.5) by ω sinz and equation (2.9) by cosz, one arrives at the following
system of equations

Ȧω sinzcosz−Aωφ̇sin2z=0

− Ȧω sinzcosz−Aω2cos2z−Aωφ̇cos2z+
P(u, sgn u̇)

m
cosz=

P0
m
coszcos(z−φ)

(2.14)

Adding both sides of equations (2.14) and averaging over one cycle of z∈ (0,2π), gives

−2πAωφ̇−πAω2+
1

m

2π
∫

0

P(u, sgnu̇)cosz dz=
P0π

m
cosφ (2.15)

Steady-state equations (2.13) and (2.15) can be obtained when Ȧ = φ̇ = 0, therefore these
equations are reduced to the form

sinφ=
1

P0π

2π
∫

0

P(u, sgnu̇)sinz dz

mω2+
P0
A
cosφ=

1

πA

2π
∫

0

P(u, sgn u̇)cosz dz

(2.16)

Integrating equations (2.16) produces a discontinuity of P(u, sgn u̇) for φ̇=0.To avoid this
problem, we confine our considerations to a single half-period (the motion between four stops).

Thus, the integration interval (from0 to2π) of the right-hand terms of the above equations is
divided into four sub-intervals. A similar procedure has been successfully adopted inBadraghan
(1994), Gałkowski (1999), Giergiel (1990), Kosior (2005), Osiński (1998) or Skup (2010). The
influence of elasto-frictional parameters k1 and k2, corresponding to tanξ1 and tanξ2, on the
investigated system is depicted in Fig. 3

k1 =tanξ1 =
P1
umax

k2 =tanξ2 =
P2
umax

umax =A (2.17)

where

P1 =α1P P2 =α2P α1 =1 (2.18)

Basing on the work by Skup (2010), the maximal axial displacement umax and dimensionless
parameter α2 are given by

u1(x=H)=umax =α1P(m17+m18) α2 =α1
m17+m18
m19+m18

(2.19)

where

m17 =
η3 tanβ(λ9R

λ10−λ9rλ9 −λ10r
λ10)

πr(rλ10 −Rλ10−λ9rλ9)
m18 =

η4
πr

m19 =
η9 tanβ(λ11R

λ12−λ11rλ11 −λ12r
λ12)

πr(rλ12 −Rλ12−λ11rλ11)



Forced vibrations analysis of a conical sleeve-shaft friction joint 193

and

η1 =
χ

E2

[

1−ν2+
r2z0
r2
(1+ν2)

]

+
1−ν1
E1

η2 =
cosβ(1−µtanβ)

2(tanβ+µ)tanβ

η3 = η1η2 η4 =
ν1
E1
tanβ

χ=
1

(rz0
r

)2

−1
η8 =

cosβ(1+µtanβ)

2tanβ(tanβ−µ)

η9 = η1η8 ∆41 =B
2
4 +4C12 > 0

λ9,10 =
B4∓

√
∆41
2

B4 =1+
η6

η3 tanβ

C12 =
η7

η3 tan
2β

η6 = η4−2η3 tanβ−η5

η7 = η4 tanβ+
1

E1
η5 =

ν1cosβ(1−µtanβ)

E1(tanβ+µ)

∆43 =B
2
5 +4C15 > 0 λ11,12 =

B5∓
√
∆43
2

B5 =1+
η10
η9 tanβ

C15 =
η11

η9 tan
2β

η10 = η4−2η9 tanβ− c1 η11 = η4 tanβ+
1

E1

c1 =
ν1cosβ(1+µtanβ)

E1(tanβ−µ)

I Stage of motion from 0 to π/2, P(u, sgnu̇)= k2u, u̇ < 0, u> 0.
II Stage of motion from π/2 to π, P(u, sgnu̇)= k1u, u̇ < 0, u< 0.
III Stage of motion from π to 3π/2, P(u, sgnu̇)= k2u, u̇ > 0, u< 0.
IV Stage of motion from 3π/2 to 2π, P(u, sgnu̇)= k1u, u̇ > 0, u> 0.
Therefore, substituting formulas (2.3), (2.8) and (2.18) into equations (2.16) and integrating,

gives

2π
∫

0

P(u, sgnu̇)sinz dz=

π

2
∫

0

k1usinz dz+

π
∫

π

2

k2usinz dz+

3π

2
∫

π

k1usinz dz

+

2π
∫

3π

2

k2usinz dz=A(k2−k1)

2π
∫

0

P(u, sgnu̇)cosz dz=

π

2
∫

0

k1ucosz dz+

π
∫

π

2

k2ucosz dz+

3π

2
∫

π

k1ucosz dz

+

2π
∫

3π

2

k2ucosz dz=
πA(k2+k1)

2

(2.20)

Finally, form (2.20), the expression for (2.16) is given by

sinφ=
1

P0π
A(k2−k1) mω

2+
P0
A
cosφ=

k1+k2
2

(2.21)

To introduce a dimensionless vibration amplitude to the system of equations (2.21), the follo-
wing notation was assumed: A – vibration amplitude, a – dimensionless vibration amplitude,



194 A. Andrzejuk et al.

xst – static axial displacement in form of the relative displacement of the friction joint elements,
ki – elasticity of the frictional parameters (i= 1,2), ∆k – dimensionless damping parameter,
ω0 – frequency of free vibrations of the system, γ – dimensionless frequency, kav – average ela-
sticity of the system, k – dimensionless elasticity parameter of the frictional joint. Additionally

kav =
k1+k2
2

∆k=
k1−k2
kr

=
2(1−k)

1+k
k=
k2
k1
=
P2
P1

ω0 =

√

kav
m

γ =
ω

ω0
kst =

P0
kav

a=
A

kst

ω2
ω1
=

√

k2
k1
=
√
k

Therefore

sinφ=−
a∆k

π
γ2+

1

a
cosφ=1 (2.22)

With the help of the above equations (2.22), the relation between the tangent of the phase
displacement angle φ and the dimensionless amplitude a can be calculated as

tanφ=−
∆k

π(1−γ2)
a=

1
√

(1−γ2)2+
(

∆k
π

)2
(2.23)

3. Numerical results

Numerical results for vibrations of the considered system have been obtained using theMathe-
matica environment. Typical results are depicted in Figs. 4-7. The basic geometrical parameters
andmaterial properties of the investigated frictional model are presented in Table 1.

Table 1.Parameters of the investigated model

No. Parameter [unit] Value

1 Loading force P1 =P [kN] 100

2 Dimensionless parameter α1 1

3 Dimensionless parameter α2 for β=12
◦ 0.200

4 Dimensionless parameter α2 for β=14
◦ 0.272

5 Dimensionless parameter α2 for β=16
◦ 0.298

6 Dimensionless parameter α2 for β=18
◦ 0.341

7 Poisson’s ratio ν 0.29

8 Young’s modulusE [N/mm2] 2.1 ·105

9 Friction coefficient µ 0.15

10 Coning angle of tilt β [◦] 12, 14, 16, 18

11 External radius of sleeve rz [mm] 42

12 Internal radius of shaft rw [mm] 28.03

14 Surface of cross-section field model [mm2] 2063.3

15 Reducedmassm [kg] 2.661



Forced vibrations analysis of a conical sleeve-shaft friction joint 195

Fig. 4. Typical amplitude-frequency characteristics for forced vibrations of the investigated system and
various angles β: (a) 12◦, (b) 14◦, (c) 16◦, (d) 18◦

Fig. 5. Global dimensionless amplitude-frequency characteristics for forced vibrations of the investigated
system and various values of angle β: 1 – 12◦, 2 – 14◦, 3 – 16◦, 4 – 18◦

Fig. 6. Relationship between the dimensionless damping parameter ∆k and the angle β [◦]



196 A. Andrzejuk et al.

Fig. 7. Graphs of the phase displacement angle φ of forced vibration of the friction joint as function of
the dimensionless frequency γ for various values of the angle β and dimensionless elasticity k;
1 – β=12◦, k=0.028, 2 – β=14◦, k=0.099, 3 – β=16◦, k=0.159, 4 – β=18◦, k=0.209

The numerical results for basic parameters of forced vibrations are presented in Table 2.

Table 2.Numerical data

Angle
β [◦]

Force
P1 [N]

Force
P2 [N]

Displacement
umax [mm]

Dimensionless Dimensionless
damping elasticity

parameter ∆k k

12 100000 20053.2 0.1055 1.3318 0.2005

14 100000 27182.1 0. 0900 1.1447 0.2718

16 100000 29766.6 0.0851 1.0825 0.2977

18 100000 34101.2 0.0755 0.9828 0.3410

4. Concluding remarks

Basing on detailed analysis of the acquired numerical data it was found that all resonance cha-
racteristics of dimensionless amplitudes start at 0.33-0.42 range (accordingly to ∆k parameter)
and tend asymptotically to zero in the post resonance range. In this range, the characteristics
exhibit a strong dynamical decrease in the amplitude values. Moreover, the increase in the re-
sonance amplitudes and the rightwards shift of the resonance can be observed for higher cone
angles (Figs. 4 and 5) while the other parameters remain unchanged.

Nonlinearities of investigated systems are observable for all considered amplitudes and vi-
bration frequencies. For the forced frequency ω, which is close to the natural frequency of
vibrations, non-dimensional amplitudes a assume higher values. Basing on the data depicted in
Figs. 4 and 5), the most dangerous range of frequencies for the investigated frictional joint is
0.85<γ< 1.15.

Values of the dimensionless damping parameter ∆k and dimensionless rigidity k (Table 2)
strongly depend on the angle β. These characteristics reveal a nonlinear character (Fig. 6). The
parameter ∆k can be treated as ameasure of damping of vibrations of the mechanical system.
For higher values of the parameter ∆k, the system reveal higher dissipative properties (higher
values of the resonance amplitude damping). This phenomenon is observable in Figs. 4 and 5.
For lower values of the angle β and parameter k, a decrease in the resonance amplitude values
is observed (Fig. 5). Such a phenomenon results from the increasing surface of themicro-sliding
zone of the cooperating elastic elements. The selection of the angle β should also take into
account the undesirable jamming phenomenon (where tanβ >µ).

Relationships of the phase shift φ and dimensionless frequency γ for various angles β are
depicted in Fig. 7. For higher angles β (lover sliding zone of the cooperating elements) the
angle φ nearby γ = 1 rapidly changes. For lower γ, φ angle remains small, thus vibrations



Forced vibrations analysis of a conical sleeve-shaft friction joint 197

are almost in phase with the excitation. For higher γ, an increase in φ is observable, tending
to 180◦ regardless of thedamping intensity.Thephasedisplacement angle reflects themagnitude
of damping in the system. Higher values of the angle φ results in an increase in the damping
properties of the system.
Thebest effect of dampingof vibrations is observable for a selected value of the friction force.

Then themicro-sliding zone of the cooperating parts of the conical joint is greater. Concluding,
thedampingof vibrations in the investigated systemdependson the followingparameters: forced
amplitude, the rigidity of the shaft and sleeve in the joint, unit pressure and friction coefficient.

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Manuscript received June 26, 2013; accepted for print September 2, 2013