Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

44, 4, pp. 907-927, Warsaw 2006

VIBRATIONS AND STABILITY OF COLUMNS LOADED BY
FOUR-SIDE SURFACES OF CIRCULAR CYLINDERS

Lech Tomski
Iwona Podgórska-Brzdękiewicz

Institute of Mechanics and Machine Design Foundations, Technical University of Częstochowa

e-mail: iwonapb@imipkm.pcz.czest.pl

The paper presents a new type of specific load realised through compo-
nents making a side surface of four circular cylinders having pair-wise
equal radii. This is a conservative load representing a follower force di-
rected towards the positive pole. Theoretical considerations are made
related to the determination of boundary conditions with installation
of needle rolling bearings or rigid elements in the receiving head. The
course of the natural vibration frequency against the external load is
determined. The results of theoretical research are compared to results
of experimental examinations. The rigidity of the substitute rotational
spring is defined, considering ”the rigidity impact” of the free end of the
column.

Key words: elastic column, specific load, divergence instability, natural
frequency

1. Introduction

Only a conservative load is considered in the paper,without considering publi-
cations related to non-conservative systems. Systems subjected to conservative
loads (compare Gajewski and Życzkowski, 1969; Leipholtz, 1974; Timoshen-
ko and Gere, 1963; Ziegler, 1968) lose their stability by buckling. The system
change fromthe static into non-static condition occurswhen the value of natu-
ral vibration frequency reaches zero (ω=0) at the so called divergence critical
force. Solution to the problem of free vibration of slender systems exposed to
external compressive forces leads to a definite course of curves in the plane
load (P) – natural frequency (ω), called the characteristic curves. Leipholz
(1974) proved that all the characteristic curves of divergence system always
have a negative slope.



908 L. Tomski, I. Podgórska-Brzdękiewicz

In Tomski et al. (1994), the authors for the first time presented results
of theoretical, numerical and experimental research concerning a new type of
load, called the specific load.
The specific load of slender systems gave a new (not shown by other au-

thors) course of the characteristic curves. The function P(ω) of such systems
has the following course:
• if the load P ∈ 〈0,Pc) (Pc is the critical load), angle of the curve tangent
P(ω) may be a positive, equal to zero or negative value

• if P ≈Pc, the slope of the curve in the Pω-plane is always negative

• the change of the natural vibration form (fromthefirst to the second one
and inversely) takes place along the curve which determines the P(ω)
function for the basic frequency (M1,M2 – denote the first and second
form of vibration, respectively).

The system characterised by such a course of the characteristic curve in the
Pω-plane was named the divergence pseudoflutter system by Bogacz et al.
(1998).
The specific load is defined in the following way:
• general loading with a force directed to the pole – loading with a force
concentrated at the free end of the column directed to the fixed point
(pole) located on the undeformed axis of the column and causing a ben-
ding moment which is linearly dependent on this force (Tomski, 2004;
Tomski et al., 1994, 1995, 1996, 1999a; Tomski and Szmidla, 2004)

• loadingwith a follower force directed towards thepole – is realisedwith a
concentrated force the direction of which agrees with the tangent to the
deflection angle of the column and crosses the fixed point (pole) situated
on theundeformedaxis (Tomski, 2004;Tomski et al., 1998, 1999b, 2004c,
2006)

• loading through a stretched component of finite bending rigidity – lo-
ading with a force transferred to the compressed rods of the column by
an additional stretched componentwhich is rigidly connected to the end
of the system (Tomski et al., 1997, 2004b, 2005, 2006).

For each discussed type of load there is a design solution realised by struc-
tures consisting of linear components (Tomski et al., 1994, 1995, 1996, 1998,
2004) or circular components (Tomski, 2004; Tomski et al., 1995, 1999, 2004c,
2006; Tomski and Szmidla, 2004). The systems subjected to the specific load
depending on geometric parameters of the loading structures, may be of the
divergence pseudoflutter type, and also of the divergence type.
Through experimental examinations, it was found that selection of the

intermediate component in the receiving head (rolling bearing, slide bearing,
rigid component) may have considerable influence on the value of the basic



Vibrations and stability of columns... 909

experimental natural frequency of the system compared to the same value
obtained throughnumerical simulations (for the appliedmathematicalmodel),
seeTomski et al. (1999a, 2001, 2002, 2003, 2006).Theexaminations concerning
the influence of components used in the loading head on natural vibration
of slender systems with specific load and Euler’s load were included in the
following elaborations:
• with the generalised load with a force directed towards the positive pole
– Tomski et al. (1999a, 2003)
• loaded with a follower force directed towards the positive pole – Tomski
et al. (2002, 2003, 2006) and negative pole – Tomski et al. (2001)
• loaded with a force directed towards the positive pole – Tomski et al.
(1999a, 2003)
• with Euler’s load – Tomski et al. (1999a, 2001, 2002, 2003).

Making use of frequency values obtained from the approximation, the rigidity
of the substitute rotational spring (taken into account in the boundary condi-
tions) was determined against the external load (Tomski et al., 1999a, 2003,
2006) by comparing the obtained results with those obtained for Euler’s load.
It was demonstrated that the way of loading the column has important influ-
ence on the stiffening of the system free end. Changes in the flexural rigidity
for individual stages of the column gave similar results.

2. Formulation of the problem

This paper presents theoretical, numerical and experimental examinations of
the system shown in Fig.1. The examination of this system includes determi-
nation of:
• free vibration (characteristic curves)
• critical force
• influence of components used in the receiving head: needle rolling be-
arings (variant F1) or rigid elements (variant F2) on the system free
vibration.

The structure of the loading systemconsists of a forcinghead and receiving
head.The forcinghead ismadeof circular components of radii Randcentres in
points ORL and ORP (Fig.1). The receiving heads are composed of circular
components of the radii r and centres OrL and OrP . The segments of the
loading heads have two common points (points of tangency). The poles ORL
and ORP are located at the constant distance equal to (R− r) from centres
OrL and OrP .



910 L. Tomski, I. Podgórska-Brzdękiewicz

Fig. 1. The physical model of the system

In the structure of the load receiving heads, there is a rigid element (2)
installed orthogonally to the free end of the column (1), the length of which
2a is strictly connected with location of the centers of elements having the r
radius. Location of the centres of the forcing head circular components having
the R radii in relation to the neutral axis of the system is defined as c. There
is a c>a relation between the c and a values, and the analysis is carried out
in the range up to 2a.

2.1. Description of geometrical relations in the system

The location of the receiving system with the centres of circles of the r
radius in the points OrL and OrP is the initial configuration. The final con-
figuration, resulting from the location of the receiving element, is determined
by location of the OL and OP centres of these circles.



Vibrations and stability of columns... 911

The systemgeometry and components of the external load P aredescribed
by the following quantities:
a) related to the undeformed condition of the column:

S0 – components of the external load for the undeformed con-
dition of the column

S0x,S0y – horizontal and vertical components of the S0 forces
α – angle formed by the components of the P load with the

neutral axis of the system in the undeformed condition
a – location of the centres of the r circles in relation to the

neutral axis of the system
c – location of the centres of the R circles in relation to the

neutral axis of the system
b – quantity determining themutual location of the centres of

circular components (b= c−a), of the loading head

b) related to the deformed condition of the system:

S1,S2 – components of the external load for thedeformedcondition
of the column

S1x,S2x – vertical components of the S1 and S2 forces
S1y,S2y – horizontal components of the S1 and S2 forces
α1,α2 – angles of the S1 and S2 forces in relation to the neutral

axis of the system
β1,β2 – angles between the load components, for the initial (unde-

formed) and deformed condition of the column.

The geometrical parameters characterising the systemunder consideration
fulfil the following relation

sinα=
b

R−r
(2.1)

The S0 forces and their vertical S0x and horizontal components S0y equal to

S0 =
P

2
1
cosα

S0x =
P

2
S0y =

P

2
tanα (2.2)

For the deformed condition of the column, the external load P is distri-
buted into two components S1 and S2, and their angles in relation to the
undeformed axis of the system are

αi =α−βi(−1)
i for i=1,2 (2.3)

In order to determine the vertical and horizontal components of the S1 and
S2 forces and corresponding displacements, the following equations have been
determined (i=1,2)



912 L. Tomski, I. Podgórska-Brzdękiewicz

sinαi =
1
R−r

[

a+ b−acos
∂W(x,t)
∂x

∣

∣

∣

x=l
−W(l,t)(−1)i

]

(2.4)

cosαi =−
a

R−r
(−1)i sin

∂W(x,t)
∂x

∣

∣

∣

x=l
+cosα

[

1−
a2

2b2
sin2
∂W(x,t)
∂x

∣

∣

∣

x=l

]

The β angles are small, therefore, it has been assumed that sinβ1 =sinβ2 =
sinβ=β, and the following was determined

β=−
a

(R−r)sinα
sin
∂W(x,t)
∂x

∣

∣

∣

x=l
=−
a

b
sin
∂W(x,t)
∂x

∣

∣

∣

x=l
(2.5)

The components of the external load for the deformed condition are expressed
as follows: S1 = S2 = S0. The vertical and horizontal components of these
forces, after applying relations (2.4), and eliminating the non-linear elements,
amount to

Six =
1
2
P
[

1− (−1)i
a

b
tanαsin

∂W(x,t)
∂x

∣

∣

∣

x=l

]

(2.6)

Siy =
1
2
P

1
(R−r)cosα

[

a+ b− (−1)iW(l,t)+(−1)iacos
∂W(x,t)
∂x

∣

∣

∣

x=l

]

and the following relation is satisfied

S1x+S2x =P (2.7)

The displacements for the y1 and y2 horizontal components, after considering
relations (2.4), are determined as follows

yi =−a(−1)
i
[

1− cos
∂W(x,t)
∂x

∣

∣

∣

x=l

]

+W(x,t)
∣

∣

∣

x=l
+

(2.8)

−(−1)i
{

r sinα+
r

R−r

[

(−1)i(a+ b)−W(l,t)+acos
∂W(x,t)
∂x

∣

∣

∣

x=l

]}

The displacements x1 and x2 of appropriate vertical components are

xi =−a
( 1
R−r

+1
)

sin
∂W(x,t)
∂x

∣

∣

∣

x=l
−∆1(−1)

i+
(2.9)

−(−1)i
a2r

2b2
cosαsin2

∂W(x,t)
∂x

∣

∣

∣

x=l

where the ∆1 is a displacement resulting from shortening of the axis of the
column caused by bending, and amounts to

∆1 =
1
2

l
∫

0

[∂W(x,t)
∂x

]2
dx (2.10)



Vibrations and stability of columns... 913

The geometrical relations between the elements of the loading structures lead
to a relation between the transverse displacement and the deflection angle of
the column free end in the following way

W(l,t)=−
a(R−r)
b

cosαsin
∂W(x,t)
∂x

∣

∣

∣

x=l
(2.11)

3. Mechanical energy of the system. Boundary conditions

The kinetic energy T for the considered system in variant (F1) is as follows

T =T1+T2+T3 =
1
2
ρ0A

l
∫

0

[∂W(x,t)
∂t

]2
dx+

1
2
m
[∂W(x,t)
∂t

∣

∣

∣

x=l

]2
+

(3.1)

+
1
2
B
[∂2W(x,t)
∂x∂t

∣

∣

∣

x=l

]2

where B is themoment of inertia of themass placed at the end of the column
calculated with respect to the axis perpendicular to the plane of vibration.
The relations for the potential energy of the system are determined as

follows:
— energy of elastic strain

V1 =
1
2
EJ

l
∫

0

[∂2W(x,t)
∂x2

]2
dx (3.2)

— potential energy of the horizontal components of the P force

V2 =
1
2
[(S1y +S0y)y1− (S2y +S0y)y2] (3.3)

— potential energy of the vertical components of the P force

V3 =
1
2
[(S2x+S0x)x2− (S1x+S0x)x1] (3.4)

The formulation of the problem is carried out with the use of Hamilton’s
principle (Goldstein, 1950), i.e.

δ

t2
∫

t1

(

T −
3
∑

k=1

Vk

)

dt=0 (3.5)



914 L. Tomski, I. Podgórska-Brzdękiewicz

The commutation of integration (with respect to x and t) and variational
calculus have been used within Hamilton’s principle (3.5). The equation of
motion, after taking into account the commutation of variation and differen-
tiation operators and after integrating kinetic and potential energies of the
system, is obtained in the form

EJ
∂4W(x,t)
∂x4

+P
∂2W(x,t)
∂x2

+ρ0A
∂2W(x,t)
∂t2

=0 (3.6)

Equation (3.6) is a differential equation of column motion. Considering the
a priori known geometrical boundary conditions of the considered system in
relation (3.5) regarding the fixed point

W(0, t)=
∂W(x,t)
∂x

∣

∣

∣

x=0
=0 (3.7)

allows for determination of the missing boundary condition at the free end
of the column, which is necessary for the solution of the boundary problem
(Podgórska-Brzdękiewicz, 2004)

∂3W(x,t)
∂x3

∣

∣

∣

x=l
+

b

a(R−r)cosα

[∂2W(x,t)
∂x2

∣

∣

∣

x=l
+
B

EJ

∂3W(x,t)
∂t2∂x

∣

∣

∣

x=l

]

+

(3.8)

−
m

EJ

∂2W(l,t)
∂t2

+
P

EJ

[

1+
a

b
+
Rb(a+ b)

(R−r)3cos2α

]∂W(x,t)
∂x

∣

∣

∣

x=l
=0

For columns of variant (F2), boundary condition (3.8) should bemodified
by introducing the rigidity of the equivalent rotational spring C whichmodels
the stiffening of the system.Thedescribed value is determined in the following
way

C1 = g1+g2P (3.9)

where g1 and g2 coefficients define the stiffening of the system caused by:
g1 – ”preliminary tension” in the loading head, without participation

of the external loading,
g2 – influence of the external load.

Themodified boundary condition is described as follows

∂3W(x,t)
∂x3

∣

∣

∣

x=l
+

b

a(R−r)cosα

[∂2W(x,t)
∂x2

∣

∣

∣

x=l
+
B

EJ

∂3W(x,t)
∂t2∂x

∣

∣

∣

x=l

]

+

(3.10)

−
m

EJ

∂2W(l,t)
∂t2

+
P

EJ

[

1+
a

b
+
Rb(a+ b)

(R−r)3cos2α
+
C1

P

]∂W(x,t)
∂x

∣

∣

∣

x=l
=0



Vibrations and stability of columns... 915

4. Solution to the boundary problem

Considering a symmetric distribution of the flexural rigidity and themass per
unit length, the equations of motion for the reviewed system after distingu-
ishing the function variables Wi(x,t) in relation to time t and space x in the
form of

Wi(x,t) = yi(x)cos(ωt) i=1,2 (4.1)

are as follows

(EJ)iy
IV
i (x)+(S)iy

II
i (x)− (ρ0A)iω

2yi(x)= 0 (S)1+(S)2 =P (4.2)

where (S)i is the internal force in the ith rod of the system.
The boundary conditions in the fixing point and at the free end of the

column (in (F1) variant), after distinguishing the variables, have the following
form

y1(0)= y2(0)= 0 y
I
1(0)= y

I
2(0)= 0

y1(l)= y2(l) y
I
1(l)= y

I
2(l)

y1(l)=
a

b
(R−r)cosαyI1(l) (4.3)

yIII1 (l)+y
III
2 (l)−

b

a(R−r)cosα

[

yII1 (l)+y
II
2 (l)−

Bω2

(EJ)1
yI1(l)
]

+

+k2yI1(l)
[

1+
a

b
+
Rb(a+ b)

(R−r)3cos2α

]

+
mω2

(EJ)1
y1(l)= 0

General solutions to equations (4.2) are as follows

yi(x)=D1icosh(αix)+D2i sinh(αix)+D3icos(βix)+D4i sin(βix) (4.4)

where Dni are integration constants (n=1,2,3,4), and

α2i =−
1
2
k2i +

√

1
4
k4i +Ω

2
i β

2
i =
1
2
k2i +

√

1
4
k4i +Ω

2
i (4.5)

where

Ω2i =
(ρ0A)iω2

(EJ)i
ki =

√

(S)i
(EJ)i

(4.6)

Substituting solutions (4.4) into boundary conditions (4.3), yields a trans-
cendental equation for eigenvalues of the considered system.



916 L. Tomski, I. Podgórska-Brzdękiewicz

5. The experimental research

5.1. Experimental stand

The experimental research was carried out on a test stand designed and
built at the Institute of Mechanics and Machine Design Fundamentals of the
Technical University of Częstochowa. A constructional diagram of the stand
is shown in Fig.2 (Tomski et al., 2004a).

Fig. 2. The test stand for examination of natural frequencies of slender systems with
fixed columns (Tomski et al., 2004a)

The experimental stand consists of support frame 1 to which head 2 is
fixed.Theheadhas a screw system themovement ofwhich loads the examined
column. The loading force is measured by dynamometer 3. The requested
boundary conditions are realised by appropriate column supports fastened to
plates 4(1) and 4(2).
Plate 4.2 has a supportmounted to it realising appropriate boundary con-

ditions of the column fixing. Plate 4.1 has an element mounted to it, where
the loading headmoves in the longitudinal direction. Investigations of natural
frequencies are performed with the use of a two-channel vibration analyser
made by Brüel &Kjaer (Denmark). Vibration of the columns is induced by a
hammer. The acceleration of the individualmeasuring point wasmeasured by



Vibrations and stability of columns... 917

means of an accelerometric sensor. The signal from the sensor is transmitted
to the analyser.

5.2. Loading structure of the column

The constructional scheme of the column loading head is presented in
Fig.3. It is composedof part G forcing the loadandpart H receiving this load.
The forcinghead G consists of element (3)wherecubes (4) are characterisedby
circular outlines of the radius R. Element (3) is jointwithmovable system (5).
The receiving head H is composed of cube (2) in which pins (6.1) and (6.2)
are located on which circular outline elements (7.1) and (7.2) of the radius r
are mounted.
The circular outlines of elements (4) and (7) making a part of the forcing

and receiving head respectively, have the same symbol (R> 0 and r > 0).

Fig. 3. Structural scheme of the loading head of the column

The column consists of two rods (6.1 and 6.2) with the bending rigi-
dity (EJ)1 and (EJ)2 respectively, and the mass per unit length (ρ0A)1
and (ρ0A)2, and (EJ)1 = (EJ)2, (ρ0A)1 = (ρ0A)2, (EJ)1 +(EJ)2 = EJ,
(ρ0A)1+(ρ0A)2 = ρ0A, where: E – longitudinal modulus of elasticity of the
rodmaterial, J – central axial moment of inertia of the column rod, ρ0 –ma-
terial density, A – cross-section area.The lengths of the column rods are equal



918 L. Tomski, I. Podgórska-Brzdękiewicz

to l. The rods of the column have the same cross-sections and aremade of the
same material. The rods and their physical and geometrical parameters are
distinguished by 1, 2 indices, which are only needed to calculate symmetric
natural frequencies and to determine corresponding forms of vibration. Hen-
ce, we can assume a global bending rigidity EJ and elementary mass of the
column ρ0A in the following considerations.

5.3. Results of experimental examinations and numerical calculations

Based on the solution of the boundary condition (for the (F1) column),
numerical calculations of the natural frequency course against the external
load were performed. Then they were verified experimentally on a test stand
(Fig.2) for the considered design solutions of the receiving heads, i.e. using
needle bearings (F1) and for the rigid components (F2). The physical and
geometrical parameters of the systems as well as the values characterising the
forcing and receiving heads were distinguished by introduced designations A,
B and C as presented in Table 1.

Table 1.Physical and geometrical parameters of the considered systems

Column
EJ ρ0A l R b α m B

[Nm2] [kg/m] [m] [m] [m] [◦] [kg] [kgm2]

A(F1),A(F2) 152.68 0.631 0.7 0.05 0.01 19.47 0.615 5.5 ·10−4

B(F1),B(F2) 152.68 0.631 0.7 0.08 0.01 9.6 0.615 5.5 ·10−4

C(F1),C(F2) 152.68 0.631 0.7 0.14 0.02 9.6 0.615 5.5 ·10−4

a=0.04m, r=0.02

Theobtained results of the experimental testing andnumerical calculations
are shown inFig.4-Fig.6.The lines represent results of numerical calculations
(for the column F1). The points present the obtained results of the experi-
mental testingwith the needle bearings in the receiving head (columns A(F1),
B(F1) and C(F1)) or rigid components (A(F2), B(F2) and C(F2)). The re-
sults are limited to the first four basic natural frequencies (M1-M4) and
three additional frequencies (M2e,M3e,M4e) characterised by symmetry of
vibrations.

Installation of the needle bearings in the receiving head (columns A(F1),
B(F1) and C(F1)) gave a good conformity of the results of numerical calcu-
lations and experimental testing. It was found that the loading of the column
with a headwhere the rigid components take part in taking the load (columns
A(F2), B(F2) and C(F2)) cause a considerable increase in the basic natural
frequency of the system compared to its value obtained through numerical
simulations and experimental tests with the needle bearings.



Vibrations and stability of columns... 919

Fig. 4. Curves in the plane: load-natural frequency for columns A(F1),A(F2)

Fig. 5. Curves in the plane: load-natural frequency for columns B(F1),B(F2)

Considering the results of the experimental testing, the course of changes
of characteristic values against the external load is approximated in order
to obtain a theoretical characteristic curve with possibly highest correlation
coefficient and lowest standarddeviation.Appropriate resultsweredetermined
in dimensionless co-ordinates where λ∗ is a parameter of the external load,
and Ω∗ defines a dimensionless parameter of the system natural frequency

λ∗ =
Pl2

EJ
Ω∗ =

ρ0Aω
2l4

EJ
(5.1)



920 L. Tomski, I. Podgórska-Brzdękiewicz

Fig. 6. Curves in the plane: load-natural frequency for columns C(F1),C(F2)

Fig. 7. Curves in the plane: load-natural frequency for columns (a)A(F1),A(F2),
(b)B(F1),B(F2), (c)C(F1),C(F2)

The curves Ωa obtained from the approximation are presented in Fig.7,
and their equations including the determined correlation coefficients and stan-
dard deviations are shown in Table 2. The curves marked with indices b, c
determine the triple standarddeviations of the individual experimental results
from the expected value.



Vibrations and stability of columns... 921

Table 2. Equations of approximating functions, values of correlation co-
efficients and standard deviations

Column
Approximating
function

Correlation Standard
coefficient deviation

A(F2) Ω∗(λ∗)= 289.3−2.18λ∗−0.15(λ∗)2 0.982 11.64
B(F2) Ω∗(λ∗)= 295.3−2.38λ∗−0.14(λ∗)2 0.984 11.29
C(F2) Ω∗(λ∗)= 357.4−10.4λ∗−0.11(λ∗)2 0.985 10.25

Bymaking use of the results obtained from the linear regression and intro-
duced rigidity of the substitute rotational spring in the boundary conditions,
the external load for the systems under consideration is determined. The re-
sults of calculations are presented in Fig.8 within the scope of the external
load realised for the experimental measurements.
The rigidity of spring C1 is expressed in a dimensionless form

c∗1 =
C1l

EJ
(5.2)

Fig. 8. Change of the nondimensional parameter of the rotational spring rigidity c∗
1

against the external load parameter

The rigidity coefficient of the springmodelling the stiffening of the system
free end depends on the geometric parameters of the forcing and receiving
heads. The influence of the external force loading the system on the range
of change of the spring rigidity for a given geometry of the forcing head is
small. The discrepancies between some constant value of the c∗1 coefficient
(at constant geometry) observed in Fig.8 may result from the error of the
experimental tests. Themeasurement results are influenced by inaccuracies of
the system and simplifying assumptions resulting from:
• accuracy of physical and geometrical parameters assumed in the calcu-
lations and characterising the system

• assumption of infinite rigidity of the forcing and receiving head compo-
nents



922 L. Tomski, I. Podgórska-Brzdękiewicz

• assumption of infinite rigidity of the system fastening

• influence of properties of the testing stand on the examined system, etc.

For the applied mathematical model of the considered system, using the
needle bearings in the receiving head (type F1), numerical calculations were
made concerning the influence of selected parameters of loading structures on
the stability and free vibration of the system.Thevalue of the critical load and
parameters of the forcing and receiving heads are expressed in dimensionless
coordinates

λ∗c =
Pcl
2

EJ
R∗ =

R

l
r∗ =

r

l

∆r∗ =
R−r

l
a∗ =

a

l
c∗ =

b

a

(5.3)

Figure 9 presents the influence of change in the forcing head radius R∗ on
the critical load λ∗c for some selected values of the parameter c

∗.

Fig. 9. Change of the load critical parameter λ∗
c
against the column radius R∗ of

the forcing head for various values of the parameter c∗

The curves of the critical load changes are characterised by the presence
of the maximum value of the critical load parameter. The values related to
geometry of the loading heads are interdependent of geometrical relationships
(compare (2.1)), the radii R∗, r∗ have to satisfy the relation resulting from
the structure of elements realising the load (R∗ ­ r∗+ b).
If the radius R∗ of the forcing head increases to very high values

(R∗ →±∞), the critical force reaches the same value – point R(∞). Satisfying
∆r∗ = b∗ (point Rb), the critical parameter of load corresponds to the system
characterised by rigid fastening at x = l (compare Table 3). Values of the
natural frequency against the external load were numerically calculated.



Vibrations and stability of columns... 923

Table 3. Boundary conditions for columns having the limit values of the
parameter ∆r∗

∆r∗ →∞
∂W(x,t)
∂x

∣

∣

∣

x=l
=0

∂3W(x,t)
∂x3

∣

∣

∣

x=l
−
m

EJ

∂2W(l,t)
∂t2

=0

∆r∗ → b∗ W(l,t)= 0
∂W(x,t)
∂x

∣

∣

∣

x=l
=0

Determination of the nature of changes is limited to two basic natural
frequencies in a dimensionless form Ω∗t (t=1,2) and additional symmetrical
natural frequency Ω∗s2 (Tomski et al., 2001, 2004) against the dimensionless
parameter of load λ∗ (compare (5.1)) for selected parameters of the forcing
and receiving heads and a constant value of the concentrated mass m at the
free end of the system. The following is assumed

m∗ =
m

ρ0Al
(5.4)

For the columns loaded with heads having the limit values of the parame-
ter ∆r∗, Table 3 presents the boundary conditions at their free end.
Figures 10 and 11 present the influence of changes of the forcing head

geometry on the course of curves in the plane: loading parameter λ∗ -natural
frequency parameter Ω∗.

Fig. 10. Curves in the plane: load parameter λ∗ -natural frequency parameter Ω∗

for a column with r∗ =0.029 and c∗ =0.25

The discussed diagrams have been obtained for constant values of the ra-
dius parameter of the receiving head r∗, constant parameter a∗ characterising



924 L. Tomski, I. Podgórska-Brzdękiewicz

Fig. 11. Curves in the plane: load parameter λ∗ -natural frequency parameter Ω∗

for a column with α=20◦

the length of the rigid component of the receiving head aa well as a constant
value of the concentrated mass m∗. In Fig.10, the course of characteristic cu-
rves is limited by broken curves (1) and (7) made for the systems having the
limit values of the parameter ∆r∗.
The critical load is presented by points on the first natural frequency curve

of the system for Ω∗1 =0. The course of the additional natural frequency Ω
∗s

correspondedby the symmetrical formhaving a constant value independent of
geometry of the forcing and receiving heads is characteristic for the presented
diagrams.
The presented curves in the plane: loading parameter λ∗ -natural frequ-

ency parameter Ω∗ allow one to classify the considered columns among two
types of systems, i.e. divergent or divergent pseudoflutter ones.

6. Final remarks

Based on theoretical considerations, experimental tests and results of nume-
rical simulations related to stability and free vibration of considered columns
loaded by heads made of circular outline cylinders, the following conclusions
can bemade:

• geometric relations between the forcing and receiving structures lead to
determination of geometric relations between the transverse displace-
ment and the deflection angle of the free end of the column



Vibrations and stability of columns... 925

• selection of components used in the receiving head has influence on the
course of the basic experimental natural frequency giving a good confor-
mity of the results from numerical calculations and experimental tests
with the use of needle bearings referring to the same value obtained
through numerical simulations

• installation of rigid components in the loading head causes a considera-
ble increase of the basic natural frequency and requires modification of
the boundary conditions by introducing some rigidity of the substitute
rotational spring thatmodels the stiffening effect of the system free end

• depending on the course of curves shown in the loading-natural frequ-
ency plane, the considered systems can be classified as one of the two
possible types: i.e. as divergent or divergent pseudoflutter systems.

Acknowledgment

The authors would like to thankA. Kasprzycki formanufacturing the column lo-
ading structure andM.Gołębiowska-Rozanow for experimental examinations related
to rigid components in the receiving head.
This researchwas carried out within the task BW-1-101/204/03/Pand was sup-

ported by the StateCommittee for ScientificResearch (KBN),Warsaw,Polandunder
grant No. 4T07C04427.

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Drgania i stateczność kolumn obciążonych poprzez cztery pobocznice
walców kołowych

Streszczenie

Wpracyprezentuje się nowy typobciążenia swoistego zrealizowanegopoprzez ele-
menty tworzącepobocznicę czterechwalcówkołowychopromieniachparami równych.
Jest to obciążenie, które reprezentuje siłę śledzącą skierowanądo bieguna dodatniego.
Przeprowadza się rozważania teoretyczne dotyczące sformułowaniawarunkówbrzego-
wych przy zastosowaniu w głowicy przejmującej obciążenie łożysk tocznych igiełko-
wych lub elementów sztywnych.Określa się przebieg częstości drgańwłasnychw funk-
cji obciążenia zewnętrznego. Wyniki badań teoretycznych porównuje się z wynikami
badań eksperymentalnych. Wyznacza się sztywność zastępczej sprężyny rotacyjnej
uwzględniającej „usztywnienie” nieutwierdzonego końca kolumny.

Manuscript received April 20, 2006; accepted for print June 28, 2006