Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

4, 39, 2001

TORSIONAL VIBRATION OF A SANDWICH SHAFT WITH

DAMPING INTERLAYER

Katarzyna Cabańska-Płaczkiewicz

Institute of Technology, Bydgoszcz Academy

Nataliya Pankratova

Institute of Applied System Analysis NTU ”KPI”, Kiev, Ukraine

This paper presents an analytical method of solving torsional vibration
problems concerning a sandwich circular shaft with a viscoelastic soft
and light interlayer. The elasticity and damping coefficients of the inter-
layer are assumed to be dependent on its geometrical characteristic and
viscoelastic properties of the interlayer material. Complex functions of
a real variable are applied in the solution to free and forced vibration
problems. Then, the property of orthogonality of complexmodes of the
free vibration, which is the basis for solving the free vibration problem
for arbitrary initial conditions, has been demonstrated. The solution to
the problemof real stationary forced vibration has been obtained on the
grounds of the complex stationarymodes of vibration.

Key words: sandwich shaft, damping, torsional vibration

Notations

ψ1,ψ2 – angles of torsion of shafts I and II, ψi =ψi(x,t), i =1,2
m2 – distributed load torque of the shaft II, m2 = m2(x,t)
µ – moment transfered through the interlayer from one shaft to

the other, µ =µ(x,t)
τ – tangential stress on the cylindrical surface of radius ρ,

τ = τ(x,ρ,t) and r1 ¬ ρ ¬ r2
r1,r2 – internal and external radius of the interlayer
r – external radius of the sandwich shaft
γ – shear strain on a surface of the interlayer, γ = γ(x,ρ,t)
G1,G2 – Kirchhoff’s moduli of shafts I and II



1002 K.Cabańska-Płaczkiewicz, N.Pankratova

G – Kirchhoff’s moduli of the interlayer
b – viscosity coefficient of the interlayer
E1,E2 – Young’s moduli of shafts for I and II
E – Young’s modulus of the interlayer
c – damping coefficient of the interlayer
k – elasticity coefficient of the interlayer
ρ1,ρ2 – massdensity of thematerial of shafts I and IIper unit length
I01,I02 – polar cross-section moments of inertia of shafts I and II
l – length of shafts I and II
x – longitudinal axis of shafts I and II
t – time.

1. Introduction

Complex torsional systems coupled together by viscoelastic constraints
play an important role in various engineering and building structures. Vi-
bration analysis of laminated layer elements such as plates, shells, beams and
shafts have been presented by Kurnik and Tylikowski (1997). Application of
piezoelectric vibration dampers in various elements have been discussed by
Tylikowski (1999), Przybyłowicz (1995).

Vibrationanalysis of complex structural systemswithdamping is adifficult
problem. In the above complex cases, especially where viscous and discrete
elements occur, it is recommended to adopt amethod of solving the dynamic
problemof the given system in the real domain of a variable complex function.

The property of orthogonality of free vibration complex modes in discrete
systems with damping was first presented by Tse et al. (1978), in discrete-
continuous systems with damping by Nizioł and Snamina (1990) and in con-
tinuous systems with damping by Cabańska-Płaczkiewicz (1998, 1999a,b),
Cabańska-Płaczkiewicz and Pankratova (1999).

Dynamic analysis of discrete-continuous complex torsional systems with
damping were also presented in the papers by Bogacz and Szolc (1993), Na-
dolski (1994), Pielorz (1995), Kasprzyk (1996).

In the papers by Cabańska-Płaczkiewicz (1998, 1999a,b), an analytical
method of solving the free vibration problem of continuous one- and two-
dimensional sandwich systems with damping, with manifold boundary condi-
tions and different initial conditions was presented.

The aim of this paper is to conduct a dynamic analysis of free and for-
ced vibration problems of a continuous torsional sandwich circular shaft with



Torsional vibration of a sandwich shaft... 1003

damping in the interlayer, in which the outer layers are made of an elastic
material, while the internal one possesses some viscoelastic properties and is
a soft and light structure.

2. Formulation of the problem

2.1. Physical model of the system

Fig. 1. Model of torsional vibration of the sandwich shaft with damping in the
interlayer

The sandwich system consists of an internal solid shaft I, and outer ring
roller II, coupled together by a viscoelastic ring-shaped interlayer (Fig.1). In-
ternal and outer layers I and II are made of a homogeneous, elastic material.
The viscoelastic interlayer is made of a light soft material with circumferen-
tial characteristic. A shearing, which is described by the Voigt-Kelvin model
(cf Nowacki, 1972; Osiński, 1979) is observed on the cylindrical surface of the
viscoelastic interlayer. It has been assumed that the interlayer does not trans-
fer torsional stresses in the transverse sections. Outer shaft II is subjected to
a torque acting at the point x0 = 0.5l, varying in time t, described by the



1004 K.Cabańska-Płaczkiewicz, N.Pankratova

function m2 = M2δ(x−x0)sin(ω0t). The load from shaft II on shaft I is trans-
ferred through tangential stresses on the cylindrical surface of the interlayer.
Deformation of separated segment of the interlayer is shown in Fig.2.

Fig. 2. Deformation of a separated segment of the interlayer

The transfered moment µ = 2πρ2τ occurring in the interlayer takes for
ρ = r1, τ = τ1 the form µ =2πr

2
1τ1, which implies the following relation

τ =
r21
ρ2
τ1 (2.1)

Making use of the constitutive equations of the Voigt-Kelvin model (cf
Nowacki, 1972; Osiński, 1979) into Eq. (2.1) we obtain a relationship for the
shear strain on the interlayer surface

γ =
r21
ρ2
γ1 (2.2)

In order to define the next geometrical relationships a segment of the in-
terlayer has been separated, as presented in Fig.2, and then a deformation of
this interlayer shown.
Having transformed the absolute shear strain ds = γdρ on the cylindrical

surface of the internal shaft we obtain

ds1 =
r31
ρ3
γ1dρ (2.3)

The arc length B∗B1 (Fig.2) has been denoted by ∆s1. Then, the geo-
metrical dependence has been determined

∆s1 =(ψ2−ψ1)r1 (2.4)



Torsional vibration of a sandwich shaft... 1005

Integrating Eq. (2.3) within the limits from r1 to r2, the following form has
been obtained

∆s1 =
r1γ1(r

2
1 −r

2
2)

2r22
(2.5)

After comparing Eq. (2.4) and Eq. (2.5), the shear strain of the interlayer
has been calculated

γ1 =
2(ψ2−ψ1)r

2
2

r21 −r
2
2

(2.6)

After substituting Eq. (2.6) in the constitutive equations of the Voigt-
Kelvin, the transfered moment can be rewritten in the following form

µ =
(

k+ c
∂

∂t

)

(ψ1−ψ2) (2.7)

where

k =4πG
r21r
2
2

r22 −r
2
1

c =4πb
r21r
2
2

r22 −r
2
1

(2.8)

2.2. Mathematical description of the model

The phenomenon of torsional vibration of the sandwich shaft with dam-
ping in the interlayer is described by the following heterogeneous system of
conjugate partial differential equations

R1
∂2ψ1

∂x2
−Γ1

∂2ψ1

∂t2
−
(

k+ c
∂

∂t

)

(ψ1−ψ2)= 0

(2.9)

R2
∂2ψ2

∂x2
−Γ2

∂2ψ2

∂t2
+
(

k+ c
∂

∂t

)

(ψ1−ψ2)= m2(x,t)

where

Ri = GiJ0i Γi = ρiI0i i =1,2

3. Solution to the boundary-value problem

By substituting (3.1) (cf Nowacki, 1972; Tse et al., 1978; Osiński, 1979;
Nizioł and Snamina, 1990) to the system of differential equations (2.9), on the
assumption that m2 =0

ψ1 = Ψ1(x)exp(iνt) ψ2 =Ψ2(x)exp(iνt) (3.1)



1006 K.Cabańska-Płaczkiewicz, N.Pankratova

the homogenous systemof conjugate ordinarydifferential equations describing
the complex modes of vibration of the shafts is obtained

d2Ψ1

dx2
+R−11

[

(Γ1ν
2−k− icν)Ψ1+(k+icν)Ψ2

]

=0

(3.2)

d2Ψ2

dx2
+R−22

[

(Γ2ν
2−k− icν)Ψ2+(k+icν)Ψ1

]

=0

where Ψ1(x), Ψ2(x) are the complex modes of the free vibration of shafts I
and II, and ν is the complex eigenfrequency of the sandwich shaft.
The general solution to the system of differential equations (3.2) has been

presented in the following form (cf Cabańska-Płaczkiewicz, 1998)

Ψ1(x)=
2
∑

υ=1

A∗υ sinλυx+A
∗∗

υ cosλυx

(3.3)

Ψ2(x)=
2
∑

υ=1

aυ(A
∗

υ sinλυx+A
∗∗

υ cosλυx)

where λυ are parameters describing the roots of the characteristic equation,
aυ are coefficients of amplitudes (cf Cabańska-Płaczkiewicz, 1998), and A

∗

υ,
A∗∗υ are integration constants.
In order to solve the boundary value problem, the following boundary

conditions are applied

Ψ1(0)= Ψ1(l)= Ψ2(0)= Ψ2(l)= 0 (3.4)

The following frequency equation of the free vibration has been obtained

ν4−
[

(R1λ
2
s +k+icν)Γ

−1
1 +(R2λ

2
s +k+icν)Γ

−1
2

]

ν2+

(3.5)

+λ2s

[

R1R2λ
2
s +(k+icν)(R1+R2)

]

(Γ1Γ2)
−1 =0

fromwhich a sequence of complex eigenfrequencies was determined

νn = iηn±ωn (3.6)

where

λs =
sπ

l
n =2s−δn,(2s−1) η =

c

2Γp
s =1,2, ...



Torsional vibration of a sandwich shaft... 1007

and δn,(2s−1) is the Kronecker number.
The coefficients of amplitudes have been found as

an =
R1λ

2
s −Γ1ν

2
n+k+icνn

k+icνn
=

k+icνn
R2λ

2
s−Γ2ν

2
n+k+icνn

(3.7)

By incorporating the sequences of λs and an to Eqs (3.3), the two following
sequences ofmodes of the free vibration for the two shafts have been obtained

Ψ1n(x)= sinλsx Ψ2n(x)= an sinλsx (3.8)

4. Solution to the initial value problem

The complex equation of motion, when ν = νn has the following form

Tn = Φnexp(iνnt) (4.1)

where Φn denote Fourier’s coefficients.
The free vibration of the shafts is presented in the form of Fourier’s series,

based on the complex eigenfunctions, i.e.

ψs(x,t)=
∞
∑

n=1

ΨsnΦnexp(iνnt) s =1,2 (4.2)

From the system of Eqs (3.2), after making some algebraic transformations,
adding the equations together, and then integrating both sides within the
limits from 0 to l, the property of orthogonality of the eigenfunctions is
obtained (cf Cabańska-Płaczkiewicz, 1998, 1999)

l
∫

0

i(νn+νm)(Γ1Ψ1nΨ1m+Γ2Ψ2nΨ2m)+ c(Ψ1n−Ψ2m) dx = Nnδnm (4.3)

where δnm is Kronecker’s delta and

Nn =2

l
∫

0

[

2iνn(Γ1Ψ
2
1n+Γ2Ψ

2
2n+ c(Ψ1n−Ψ2m)

2
]

dx (4.4)

The problem of the free vibration of the shafts is solved by application of the
following conditions

ψ1(x,0)= ψ01 ψ2(x,0)= ψ02

ψ̇1(x,0)= ψ̇01 ψ̇2(x,0)= ψ̇02
(4.5)



1008 K.Cabańska-Płaczkiewicz, N.Pankratova

Applying conditions (4.5) in series (4.2) and taking into consideration the
property of orthogonality (4.3), the following formula for Fourier’s coefficients
is obtained

Φn =
1

Nn

l
∫

0

[

Γ1(iνnΨ1nψ01+Ψ1nψ̇01)+

(4.6)

+ Γ2(iνnΨ2nψ02+Ψ2nψ̇02)+ c(Ψ1n−Ψ2n)(ψ01−ψ02)
]

dx

Substituting Eqs (3.8), (4.1) and Eq. (4.6) to Eqs (4.2) and performing trigo-
nometrical and algebraical transformations, the final form of free vibration of
the sandwich shaft with damping in the interlayer is obtained

ψs =
∞
∑

n=1

e−ηnt|Φn||Ψsn|cos(ωnt+ϕn+χsn) s =1,2 (4.7)

where
|Ψsn|=

√

X2sn+Y
2
sn χsn =argΨsn s =1,2

|Φn|=
√

C2n+D
2
n ϕn =argΦn

and
Xsn =re Ψsn Ysn = im Ψsn s =1,2

Cn =re Φn Dn = im Φn

5. Solution to the forced vibration problem

In the case when ν = ω0 (Eq. 3.3)

λ1 6= λ2 a1 6= a2 (5.1)

where ω0 is the frequency of the real stationary forced vibration.
After incorporating of Eqs (3.4), (5.1) to Eqs (3.3), the general solution

to the system of ordinary differential equations (3.2) in the following matrix
form is obtained

Ψ
∗(x)=

[

A1 sinλ1x+A2 sinλ2x

a1A1 sinλ1x+a2A2 sinλ2x

]

(5.2)



Torsional vibration of a sandwich shaft... 1009

The particular solutions, using the operator method (cf Osiowski, 1972),
are derived.
The system of equations (2.9) (after elimination of time) in amatrix form

is as follows

C2
d2ψ

dx2
+C0ψ =f(x) (5.3)

where

ψ(x)=

[

ψ1(x)

ψ2(x)

]

f(x)=

[

0

R−12 M2δ(x−x0)

]

C2 =

[

1 0

0 1

]

(5.4)

C0 =





R−11 (Γ1ω
2
0 −k− icω0) R

−1
1 (k+icω0)

R−12 (k+icω0) R
−1
2 (Γ2ω

2
0 −k− icω0)





where δ(x−x0) is the Dirac delta function.
The system of Eqs (5.3) is a normal system, because

detC2 6=0 (5.5)

Using Laplace’s transformations to Eqs (5.3), the operational equation is ob-
tained

G(s)×Y(s)=F(s)+L(s) (5.6)

where
G(s)=C2s

2+C0 (5.7)

is the characteristicmatrix, Y(s) and F(s) –matrix transforms,and ψ∗∗(s)=
Y(s) on the assumption that the volumematrix L(s)≡0.
Equations (5.6) can be written in the following form

Y(s)=G−1(s)[F(s)+L(s)] (5.8)

where

G(s)=





s2+R−11 (Γ1ω
2
0 −k− icω0) R

−1
1 (k+icω0)

R−12 (k+icω0) s
2+R−12 (Γ2ω

2
0 −k− icω0)



 (5.9)

and
K(s)=G−1(s) (5.10)



1010 K.Cabańska-Płaczkiewicz, N.Pankratova

Using inverse Laplace’s transformation L−1 of Eq. (5.6), the particular solu-
tions in the matrix form are obtained

y(x)= k(x)f(x) (5.11)

where

k(x)=L−1K(s) (5.12)

The elements of matrix (5.12) are described in the form of

k(x)=

[

k11 k12

k21 k22

]

(5.13)

After substituting Eq. (5.13) to Eq. (5.11), the particular solution of the
system of Eqs (5.3) in the matrix form is obtained

Ψ
∗∗(x)=











l
∫

0
M2k12(x− τ)δ(τ −x0) dτ

l
∫

0
M2k22(x− τ)δ(τ −x0) dτ











(5.14)

where

y(x)=Ψ∗∗(x) (5.15)

Themodes of the stationary forced vibrations of the two shafts can bewritten
in the form

Ψ
(x)=Ψ∗(x)+Ψ∗∗(x) (5.16)

The steady-state forced vibration of the sandwich shaft is

ψ(x,t)=Ψ(x)exp(iω0t) (5.17)

Substituting Eq. (5.16) to Eq. (5.17) andmaking trigonometric and algebraic
transformations, the forced vibration of the sandwich shaft with damping in
the interlayer is obtained

ψs = |Ψs|sin(ω0t+χs) s =1,2 (5.18)

where
|Ψs|=

√

X2s +Y
2
s χs =argΨs

Xs =re Ψs Ys = im Ψs s =1,2

and |Ψ1|, |Ψ2| – amplitudes of shafts I and II.



Torsional vibration of a sandwich shaft... 1011

6. Calculations

Calculations are carried out for the following data

E1 = E2 =2.1 ·10
11 Nm−2 E =107Nm−2 r1 =0.02m

ρ1 = ρ2 =7.8 ·10
3 Ns2m−4 r2 =0.05m r =0.06m

M2 =4000Nm ν0 =0.2 l =5m

c =2.5Ns k =2.5 ·104 N x0 =0.5l

Constants occurring in Eqs (5.2) are described in the following forms

A1 =
M2 sinλ1(l−x0)

R2λ1 sinλ1l
A2 =−

M2 sinλ2(l−x0)

R2λ2 sinλ2l
(6.1)

Tables 1-3 present values of the complex eigenfrequencies νn = iηn ±ωn
for s =1,2,3. The effects of the damping coefficients for c =2.5Ns (Table 1),
c =5Ns (Table 2) and c =7Ns (Table 3) on the system frequencies are shown.
The investigation of the complex eigenfrequencies of the sandwich shaft has
shown, that in the case, when s = 1 (n = 1), s = 2 (n = 3), s = 3
(n = 5), the real parts ωn of the complex eigenfrequencies for the damping
coefficients: c =2.5Ns (Table 1), c =5Ns (Table 2) and c =25Ns (Table 3)
donot change. In the case, when s =1 (n =2), s =2 (n =4), s =3 (n =6),
the real parts ωn of the complex eigenfrequencies for the damping coefficient
c =2.5Ns (Table 1)are correspondingly larger than for thedampingcoefficient
c = 5Ns (Table 2) and c = 25Ns (Table 3). In the case, when s = 1
(n = 2), s = 2 (n = 4), s = 3 (n = 6), the imaginary parts ηn of the
complex eigenfrequencies for the damping coefficient c =2.5Ns (Table 1) are
correspondingly smaller than for the damping coefficient c = 5Ns (Table 2)
and c =25Ns (Table 3).

Table 1.Complex eigenfrequencies νn for c =2.5Ns and s =1,2,3

s νn (n =1, ...,6)

1 ν1 =±2104.44 ν2 =±4126.35+651.102i

2 ν3 =±4208.88 ν4 =±5505.71+651.102i

3 ν5 =±6313.32 ν6 =±7242.66+651.102i

Table 2.Complex eigenfrequencies νn for c =5Ns and s =1,2,3

s νn (n =1, ...,6)

1 ν1 =±2104.44 ν2 =±3969.25+1302.2i

2 ν3 =±4208.88 ν4 =±5388.97+1302.2i

3 ν5 =±6313.32 ν6 =±7154.32+1302.2i



1012 K.Cabańska-Płaczkiewicz, N.Pankratova

Table 3.Complex eigenfrequencies νn for c =25Ns and s =1,2,3

s νn (n =1, ...,6)

1 ν1 =±2104.44 ν2 =±0+11505.3i

2 ν3 =±4208.88 ν4 =±0+9925.21i

3 ν5 =±6313.32 ν6 =±3238.31+6511.02i

After analysing the results shown in Tables 1-3 we state that:

• a decrease in the real parts ωn of the complex eigenfrequencies corre-
sponds to a larger period of damped vibration of shafts I and II

• an increase in the imaginary parts ηn of the complex eigenfrequen-
cies corresponds to smaller amplitudes (damping decrement) of shafts I
and II, Eq. (4.7).

Fig. 3. Complex modes of free vibrations of shafts I and II for s=1, (n =1,2) –
variants I, II



Torsional vibration of a sandwich shaft... 1013

Fig. 4. Complexmodes of free vibrations of shafts I and II for s =2, (n =3,4) –
variants I, II

The influence of the complex eigenfrequencies for small damping coefficient
c =2.5Ns (Table 1), and for large damping coefficient c =25Ns (Table 3) on
the complexmodes of free vibrations is illustrated in the Fig.3 and Fig.4 (va-
riants I, II). The complexmodes of the free vibrations for the eigenfrequencies
presented in Table 1 are given in variant I, for the eigenfrequencies presented
in Table 3 – given in variant II. The diagrams in Fig.3 and Fig.4 show the
complexmodes of the free vibrations for shafts I and II. The results are given
for s = 1 (n = 1,2) – Fig.3, s = 2 (n = 3,4) – Fig.4. The complex modes
for the real eigenfrequencies νn = ±ωn (n = 1,3) – Tables 1-3 have syn-
chronous character (Fig.3-4). In the case of higher complex eigenfrequencies
νn = iηn±ωn (n =2,4) – Tables 1-3, themodes have asynchronous character
(Fig.3-4).

Figure 5 shows the complexmodes of the stationary forced vibration of the
torsional sandwich circular shaft with damping in the interlayer for the set of



1014 K.Cabańska-Płaczkiewicz, N.Pankratova

Fig. 5. Complexmodes of stationary forced vibrations of shafts I and II



Torsional vibration of a sandwich shaft... 1015

Fig. 6. Amplitude-frequency characteristic of torsional vibration of the sandwich
shaft with damping in the interlayer at the point x0 =0.5l

real frequencies ω0 = {2000, 3200, 5200}. Figure 6 presents the amplitude-
frequency diagrams of the torsional sandwich shaft with damping in the in-
terlayer. The amplitude-frequency diagrams show changes of the amplitudes
|Ψ1| and |Ψ2| of shafts I and II for real stationary frequencies in the range of
0 < ω0 < 10000. After analysing the results presented in Fig.6 we state, that
internal shaft I can be a dynamic vibration damper for outer shaft II, which is
subjected to a torque acting at the point x0 =0.5l, varying in time, described
by the function m2 = M2δ(x−x0)sin(ω0t).

In this paper the system of two conjugate differential equations (2.9) of
torsional vibration of the system of two shafts coupled by a viscoelastic soft
light interlayer is solved. The viscoelastic interlayer is made of a light soft
material with a circumferential characteristic. The obtained solution can be
applicable to systems with interlayers of a small thickness, as well as small
damping for variant I and large damping for variant II.

7. Conclusions

• Complex modes of vibration and the property of orthogonality of these
modes have been presented in this paper. They a basis for solving the
free and forced vibration problems of the torsional sandwich circular
shaft with damping in the interlayer.

• Themethod presented in this paper is correct for small and large dam-
ping. In the case of small damping, the free vibrations have periodic



1016 K.Cabańska-Płaczkiewicz, N.Pankratova

character. For large (critical) damping, the lower components of the free
vibrations have non-periodic character.

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Drgania skrętne sandwiczowego wału z tłumieniem w przekładce

Streszczenie

Wpracyprzedstawionoanalitycznąmetodę rozwiązywaniaproblemudrgań skręt-
nych sandwiczowego okrągłego wału z lepko-sprężystą, miękką i lekką przekładką.
Współczynnik sztywności i tłumienia przekładki uzależniono od jej cech geometrycz-
nych oraz od lepko-sprężystych własności materiału przekładki. W rozwiązaniu za-
gadnienia drgań swobodnych i wymuszonych zastosowano funkcje zespolone zmiennej
rzeczywistej. Następnie wykazanowłasność ortogonalności zespolonychpostaci drgań
własnych, która jest podstawą rozwiązania zagadnienia drgań własnych przy dowol-
nych warunkach początkowych. Rozwiązanie zagadnienia rzeczywistych ustalonych
wymuszonych drgań otrzymano za pośrednictwem zespolonych ustalonych postaci
drgań.

Manuscript received November 9, 1999; accepted for print June 8, 2000