JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

42, 4, pp. 905-926, Warsaw 2004

VIBRATIONS AND STABILITY OF A TWO-ROD COLUMN

LOADED BY A SECTOR OF A ROLLING BEARING

Lech Tomski
Janusz Szmidla
Maria Gołębiowska-Rozanow

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

e-mail: szmidla@imipkm.pcz.pl

In thiswork, a new type of loading of slender systems,which is a follower
force directed towards a positive or a negative pole is presented. Con-
structional models of loading heads, which realize this type of loading,
are also presented. The variant of theoretical investigations concerning
formulation of boundary conditions is shown. It results from the ener-
getic formulation. Dependently on constructional variants of both the
loading and receiving heads, values of the critical force and courses of
the natural frequency as a function of the external loading for the ap-
plied geometry and physical constants of the column are determined.
Theoretical results are compared with those from an experiment.

Key words: elastic column, divergence instability, natural frequency

1. Introduction

1.1. Euler’s and Beck’s load. Plane load-natural frequency curves

The loading of a slender system, called Euler’s load, has been known since
the eighteenth century (compareEuler, 1774).This load is characterised by the
fact that a compressive force of a columnhas a constant point of its application
and a constant point of action, which are unchangeable during buckling.
A curve in the plane: load P -natural frequency ω (see Fig.1a) has always

a negative slope, which was proved by Leipholz (1974).
In 1952 Beck (compare Beck, 1953) reported the first solution for columns

with a non-conservative load (by follower force). This load is characterised by
a force that is tangential to the deflected axis of the column at the loaded end.



906 L.Tomski et al.

The curve in the plane: load-natural frequency is shown in Fig.1b (change
of vibration form takes place at O point). InFig.1 the critical force is denoted
as Pc, while M1 and M2 denote the first and second mode of vibrations,
respectively.

Fig. 1. Frequency curves in the plane: load P -natural frequency ω for
divergence (a), and flutter (b) systems

1.2. The generalised load and the condition of its potential

Let us consider a cantilever column shown in Fig.2. The flexural rigidity is
denotedas EJ,massdensityas ρA (where E isYoung’smodulus, J –moment
of inertia related to the neutral axis in the bendingplane, ρ –material density,
A – cross-section area). W(x,t) is a transverse displacement of the column,
m– concentratedmassat the free endof the column, C1 – rigidity of rotational
springmodelling elasticity of the fastened system. The column is loaded by a
longitudinal force P , shearing force H andbendingmoment M. According to
works byKordas (1963), Gajewski andŻyczkowski (1970, 1988), it is assumed
that the shearing force H and bending moment M linearly depend on the
displacement W(l, t) and deflection angle [∂W(x,t)/∂x]x=l of the free end of
the column in the following way

H =P
[

(1−µ)
∂W(x,t)
∂x

∣

∣

∣

x=l
+γW(l, t)

]

(1.1)

M =P
[

ρ
∂W(x,t)
∂x

∣

∣

∣

x=l
+νW(l, t)

]

where ρ, ν, µ, γ are determined coefficients.
The load is conservative (the external force has a potential) if rotation of

the gradient of its vector field is equal to zero, which leads to the relation
(Kordas, 1963; Gajewski and Życzkowski, 1970, 1988; Tomski et al., 1996)

∂H

∂W(x,t)

∣

∣

∣

x=l
=

∂M
∂W(x,t)
∂x
|x=l

(1.2)



Vibrations and stability of a two-rod column... 907

Fig. 2. A scheme of the cantilever column subjected to the generalised load

For the generalised load, relation (1.2) gives

ν+µ−1=0 (1.3)

Aphysical interpretation of this condition according to thefield theory is given
in the work by Tomski et al. (1996).
Apart from Euler’s, Beck’s and the generalised loads, the following loads

are referred to in the literature:

• the load developed by a force directed towards the positive pole – the
fixed point through which the direction of the force action passes is
placed below the free end of the column (Gajewski and Życzkowski,
1969; Timoshenko andGere, 1961)

• the load developed by a force directed towards the negative pole – the
fixed point through which the direction of the force action passes is
placed above the free end of the column (Gajewski and Życzkowski,
1969; Dąbrowski, 1984).



908 L.Tomski et al.

1.3. Authors’ research

1.3.1. Specific load. Load-natural frequency curves in the plane

In1994Tomski et al. describedanewsystemof a loaded columnandplanar
frame (Tomski et al., 1995). The vast results of theoretical and experimental
research on a column subjected to a specific load, are presented in a paper by
Tomski et al. (1996). A specific load can be called a generalised load with the
force directed towards the positive pole. Such a load can give an uncommon
course to the curve in the plane: load P -natural frequency ω (see Tomski et
al., 1996).
The function P(ω) (Fig.3) for this system has the following course:

• for P ∈< 0,Pc) (Pc is the critical load) the angle of the curve tangent
to P(ω) can take a positive, zero or negative value

• for P ≈ Pc load, the slope of the curve in the P(−ω) plane is always
negative

• change of the free vibration form (from the first to the second and in-
versely) takes place along the curvewhich determines the P(ω) function
for the basic frequency (M1, M2 denote the first and second forms of
vibrations, respectively).

Fig. 3. Frequency curves in the plane: load-natural frequency for the
divergence-pseudo-flutter type system

The system realising such a course was named the divergence-pseudo-
flutter type, Bogacz et al. (1998), as opposed to the already known separate
systems: divergence and flutter ones.
Further results of theoretical and experimental investigations, concerning

the specific load are presented in the following publications:

• a follower loadwith the force directed towards the positive pole, Tomski
et al. (1998)



Vibrations and stability of a two-rod column... 909

• a generalised load with the force directed towards the negative pole and
a follower loadwith the force directed towards the negative pole, Tomski
et al. (1999).

An attempt to optimise a column subjected to a generalised load by a
force directed towards the positive pole was made by Bogacz et al. (1998).
The results of experimental investigations, connected with changes of the free
vibration form along the load-natural frequency curve, were presented by Bo-
gacz et al. (1998). It was proved that the course of the curve was in good
agreement with that shown in Fig.3.

1.3.2. The extended condition for the load potential

In this paper, the condition for the potential of a generalised load has been
worked out according to the property that if the potential of external forces
exists (Levinson, 1966; Wallerstein, 2002), then

δV =−δL (1.4)

where δV is a variation of the potential energy and δL – variation of the
potential work

δV =
1
2

(

δM
∂W(x,t)
∂x

∣

∣

∣

x=l
+Mδ

∂W(x,t)
∂x

∣

∣

∣

x=l
+ δHW(l, t)+HδW(l, t)

)

(1.5)

δL=−
(

Mδ
∂W(x,t)
∂x

∣

∣

∣

x=l
+HδW(l, t)

)

while (Levinson, 1966)
δM = δH =0 (1.6)

The energy of the force P is neglected because it is a potential load.
Taking into account (1.1), (1.4)–(1.6), the extended condition for the exter-

nal load potential is

(µ+ν−1)
(∂W(x,t)
∂x

∣

∣

∣

x=l
δW(l, t)−W(l, t)δ

∂W(x,t)
∂x

∣

∣

∣

x=l

)

=0 (1.7)

If W(l, t) and [∂W(x,t)/∂x]x=l are linearly independent, the second factor
in relationship (1.7) is set to zero. The same condition for the external load
potential was obtained in the paper by Tomski et al. (1996), in which the
self-adjointness condition of differential operators describing free vibrations of
the columnwas borne in mind.



910 L.Tomski et al.

2. Statement of the problem

New heads loading and receiving the column are presented in this paper.
These heads are to realise a loadwith an a priori unknown force direction and
point of application. The value of the columnbendingmoment depends on the
point of force application P .
The boundary conditions are calculated according to the Hamilton princi-

ple with themanifestation of the load in two intersections arising from design
of the forcing and loading heads. The formulation of the load potential is used
here through the second factor of condition (1.7).
The results of numerical calculations, concerning the load-natural frequ-

ency curves, are verified by experimental investigations.
It must be noted that every specific load, i.e.:

• generalised load with a force directed towards positive or negative poles

• load by a follower force directed towards positive or negative poles, can
be realised with the use of loading and receiving heads in a few alterna-
tive designs, Tomski et al. (1996, 1998, 1999).

3. Structural schemes of the heads realising the load

The loading systems, which the column is subjected to, are presented in
Fig.4. These systems are composed of an enclosure (1) and end with rolling
guides (2). The outer race (3) (Figs4a,b) and internal rings (Fig.4c) of a
roll bearing (ball bearing) are mounted on the enclosure (1). The internal
ring (5) (Figs4a-b) and the outer race (Fig.4c) of the roll bearing are placed
in an element (4). The element (4) is connected to a block (7) by means of a
lock (6). Two rods (8) of the column are mounted in the block. It is assumed
that the elements of length l0 (lock (6), element (4), block (7)) are infinitely
rigid (this relates to constructional considerations). The elements (1, 3) make
up the loading head, while (4, 5, 6, 7) – the receiving head.
The column consists of two rods (8.1, 8.2) with the bending rigidity (EJ)1

and (EJ)2, respectively, and the mass per unit length (ρA)1 and (ρA)2 (and
(EJ)1 =(EJ)2, (ρA)1 =(ρA)2, (EJ)1+(EJ)2 =EJ, (ρA)1+(ρA)2 = ρA).
The rods of the column have the same cross-sections and are made of the
same material. The rods and their physical and geometrical parameters are
distinguished by 1, 2 indexes, which are only needed to calculate symmetrical



Vibrations and stability of a two-rod column... 911

Fig. 4. Structural schemes of heads realising the column load: (a) for R> 0,
R− l0 > 0; (b) for R> 0,R− l0 < 0; (c) for R< 0, l0 > 0

natural frequencies and to determine corresponding forms of vibration. Hence,
we can assume a global bending rigidity EJ and elementary mass of the
column ρA in the following considerations in this paper.



912 L.Tomski et al.

4. The physical model of the system

The physicalmodel of the considered system, in the constructional variant
shown in Fig.4a, is presented in Fig.5. The systems, shown in Figs4b,c, are a
specific case of the system presented in Fig.4a. The load is manifested in I-I
andII-II sections.Three-componentparts X,Y ,Z,which fulfil thedetermined
aims, can be distinguished by taking into account the system shown in Fig.5.
The system X creates external load 1 and internal forces 3a, which are

balanced by head 2 and rolling guides. The system Y consists of head 4 and
balanced forces 3a and 5a (Fig.5c). The systemZ is determined by column 6
and external forces 5b and columnmount 7.
The manifestation of the load in the Y and Z systems makes it possi-

ble to specify boundary conditions on the basis of mechanical energy balance
(vibration problem) or potential energy balance (statics criterion). The ma-
nifestation of the load in the Z system in section II-II makes it possible to
specify boundary conditions on the basis of mechanical energy or potential
energy, and also on the dependence of external load 5b on the internal forces
in column 6.
It should be underlined that if the generalised load is taken into account,

then the force direction and its point of application are a priori unknown for
the considered structure. As a result, the coefficient of the follower force η1
and coefficient of the bendingmoment η2 (Fig.5) are assumed in the following
considerations.
Geometrical dependences between elements of the structure and load pla-

cement (force P and bendingmoment M) lead to the following relationship

W(l, t)= (R− l0)
∂W(x,t)
∂x

∣

∣

∣

x=l
η2 =1−η1 (4.1)

4.1. Mechanical energyof the systemandHamilton’sprinciple.Boundary

conditions

Total potential energy of the system depicted in Fig.5 is examined with
respect to the place of its manifestation (Table 1):
— energy of the elastic strain

V1 =
EJ

2

l
∫

0

[∂2W(x,t)
∂x2

]2
dx (4.2)



Vibrations and stability of a two-rod column... 913

Fig. 5. The physical model of the object

—potential energy of elasticity of the fastening

V2 =
1
2
C1
[∂W(x,t)
∂x

∣

∣

∣

x=0

]2
(4.3)

— potential energy of the system Vkn (Table 1 – where k = 3,4,5,
n= ∗,∗∗,∗ ∗∗)



914 L.Tomski et al.

Table 1.Potential energy of the system

Manifestation of the load in section
I-I II-II I-I and II-II

Potential energy of the vertical component of the force P
V ∗2 =−P∆1−P∆2+P∆3 V

∗∗

2 =−P∆1 V
∗∗∗

2

Potential energy of the horizontal component of the force P
V ∗3 =PRη

2
1∆
2
l/2 V

∗∗

3 =Pη1W(l, t)∆l/2 V
∗∗∗

3

Potential energy of the bendingmoment
V ∗4 =0 V

∗∗

4 =−V
∗∗∗

4 V
∗∗∗

4

where
V ∗∗∗2 =−P∆2+P∆3

V ∗∗∗3 =
P

2
[Rη21∆

2
l −η1W(l, t)∆l]

V ∗∗∗4 =−
P

2
η2W(l, t)∆l

∆l =
∂W(x,t)
∂x

∣

∣

∣

x=l
∆1 =

1
2

l
∫

0

[∂W(x,t)
∂x

]2
dx

∆2 =
1
2
l0∆
2
l ∆3 =

1
2
R(1−η21)∆

2
l

Kinetic energy for the considered system is as follows

T =
1
2
ρA

l
∫

0

[∂W(x,t)
∂t

]2
dx+

m

2

[∂W(x,t)
∂t

∣

∣

∣

x=l

]2
(4.4)

In this paper, the formulation of the problem is carried out with the use of
Hamilton’s principle (Goldstein, 1950)

δ

t2
∫

t1

(T −V ) dt=0 (4.5)

The commutation of integration (with respect to x and t) andvariation calcu-
lation is usedwithin Hamilton’s principle (4.5). The equation ofmotion, after
taking into account the commutation of variation anddifferentiation 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

+ρA
∂2W(x,t)
∂t2

=0 (4.6)



Vibrations and stability of a two-rod column... 915

and after giving consideration to conditions (4.1), the following boundary con-
dition at the free end of the system is imposed

∂3W(x,t)
∂x3

∣

∣

∣

x=l
−
1
R− l0

∂2W(x,t)
∂x2

∣

∣

∣

x=l
−
m

EJ

∂2W(x,t)
∂t2

∣

∣

∣

x=l
=0 (4.7)

The conditions for the fastening are as follows

W(0, t)= 0 EJ
∂2W(x,t)
∂x2

∣

∣

∣

x=0
−C1
∂W(x,t)
∂x

∣

∣

∣

x=0
=0 (4.8)

Condition (4.7) is independent of the follower coefficients η1 and η2.
The loading of columns, for which the boundary conditions at the free end

(x= l) are definedby relations (4.1)1 and (4.7), depending on a constructional
variant of the loading head (sign of the radius of curvature R (Fig.4), assume
the name of:

• follower force directed towards the positive pole (R> 0 – Fig.4a,b)

• follower force directed towards the negative pole (R< 0 – Fig.4c)

independently of the length of the rigid element l0 of the head receiving the
load.

5. Solution to the boundary problem

The equations of motion for the considered column, with the function of
transverse vibration Wi(x,t) predicted in the form

Wi(x,t) = yi(x)cos(ωt) i=1,2 (5.1)

are as follows

(EJ)iy
IV
i (x)+Siy

′′

i (x)− (ρA)iω
2yi(x)= 0

2
∑

i=1

Si =P (5.2)

where a symmetrical distribution of the bending rigidity and mass per unit
length is assumed.



916 L.Tomski et al.

The boundary conditions at the fixed and free end of the column, with
regard to relationships (4.1)1 and (4.7), take the following form

y1(0)= y2(0)= 0 y′1(0)= y
′

2(0)

y1(l)= y2(l) y′1(l)= y
′

2(l)

y′′1(0)+y
′′

2(0)− c
∗

1y
′

1(0)= 0 y1(l)= (R− l0)y
′

1(l)

y′′′1 (l)+y
′′′

2 (l)−
1
R− l0

[y′′1(l)+y
′′

2(l)]+
mω2

(EJ)1
y1(l)= 0

(5.3)

where

c∗1 =
C1
(EJ)1

A general solution to Eqs. (5.2) is

yi(x)=C1i cosh(αix)+C2i sinh(αix)+C3i cos(βix)+C4i sin(βix) (5.4)

where Cji are integration constants (j =1,2,3,4), and

α2i =−
1
2
k2i +

√

1
4
k2i +Ω

∗2
i β

2
i =
1
2
k2i +

√

1
4
k2i +Ω

∗2
i

while

Ω∗2i =
(ρA)iω2

(EJ)i
ki =

√

Si
(EJ)i

Substitution of solutions (5.4) into boundary conditions (5.3) yields a trans-
cendental equation for eigenvalues of the considered system.

6. Experimental stand

The experimental stand for the examination of free vibrations of the con-
sidered columns is shown in Fig.6. It consists of head (1) which can be hori-
zontally shifted along guides (2). The load is applied to the tested column by
means of screw systems belonging to the head. The loading force is measu-
red by dynamometer (3). Column (5) is clamped to supports 4(1) and 4(2).
Support 4(1) enables fixing of loading head (6), see Fig.4. Tests of normal
frequencies were performed with the use of a two-channel vibration analyser
made by Brüel and Kjaer (Denmark).



Vibrations and stability of a two-rod column... 917

Fig. 6. The test rig for experimental research on the considered column

7. Numerical and experimental results

For the considered column numerical computations were accomplished on
the basis of the solution to the boundary value problem. Then, the course
of natural frequencies in relation to the external loads (for systems whose
physical and geometrical parameters are given inTable 2) was experimentally
verified on the stand (Fig.6).

The parameters of loading and receiving heads are also included in Ta-
ble 2. Systems K2, K7, K8 correspond with the head variant presented in
Fig.4c, systemsK5,K6 – Fig.4a, systemK1 – Fig.4b depending on the head
curvature and reciprocal relation between the rigid element l0 and radius R.

ColumnsK3,K4, for which R= l0, are the specific variant. The free end
of the system is at the non-deformed axis of the column for such a relation.



918 L.Tomski et al.

Table 2.Geometrical and physical parameters of the considered columns

Columns EJ [Nm2] ρA [kg/m] l [m] R [m] l0 [m] m [kg]

K1 152.68 0.631 0.7 0.0285 0.091 0.25
K2 152.68 0.631 0.7 −0.0285 0.091 0.4
K3,K4 152.68 0.631 0.71 0.058 0.058 0.335
K5,K6 152.68 0.631 0.71 0.058 0.025 0.25
K7,K8 152.68 0.631 0.71 −0.058 0.025 0.35

The boundary conditions for x= l can be stated as follows

y1(l)= y2(l)= 0 y
′

1(l)= y
′

2(l) y
′′

1(l)+y
′′

2(l)= 0 (7.1)

The results, obtained from experiments (points) and numerical computations
(lines), are presented in Fig.7-Fig.11, while columnsK4,K6,K8 are charac-
terised by c∗1 =0 joint attachment for x=0.The rigid attachment (1/c

∗

1 =0)
was applied in the remaining cases. The results are limited to the first three
basic natural frequencies (M1,M2,M3) and two additional frequencies (M2e,
M3e) characterised by symmetry of vibrations (compare Tomski et al., 1998).
Both numerical and experimental results are in good agreement.

Fig. 7. Frequency curves in the load-natural frequency fplane for columnK1

Additional results, connected with changes in the critical load and natural
frequency,were obtainedby taking into account the correctness of the assumed
mathematical model describing the variations. The rigid attachment of the
system 1/c∗1 =0was also taken into consideration.



Vibrations and stability of a two-rod column... 919

Fig. 8. Frequency curves in the load-natural frequency fplane for columnK2

Fig. 9. Frequency curves in the load-natural frequency fplane for columnsK3
andK4

Fig. 10. Frequency curves in the load-natural frequency fplane for columnsK5
andK6



920 L.Tomski et al.

Fig. 11. Frequency curves in the load-natural frequency fplane for columnsK7
andK8

The change in the critical load parameter is presented in Fig.12 in the full
range of the radius R of the loading head for three lengths A, B, C of the
rigid element carrying the load. The value of the critical force Pc is related to
the overall length of the system

λc =
Pcl
2
1

EJ
(7.2)

while R∗ =R/l1, l∗0 = l0/l1, l1 = l0+ l= const and l0 – length of the rigid
element, l – length of the column.
The curves A,B,C represent the value of the critical load parameter from

point A−∞, B−∞, C−∞ to point A0, B0, C0 for the column loaded by the
follower force towards the negative pole R∗− l∗0 < 0 and positive pole R

∗ > 0
with R∗− l∗0 < 0.
In the remaining range of the radius R∗ ((A0,A∞), (B0,B∞), (C0,C∞))

the system is loaded by the follower force towards the positive pole when
R∗− l∗0 > 0. For the considered values of the radius R

∗ of the loading head,
changes of every curve of the critical load are characterised by occurrence of
the maximum value of the critical load parameter λmax (Fig.13).
The extreme value for every R∗ and l∗0 fulfils the dependence

R∗− l∗0
1− l∗0

=
1
2

(7.3)

The value of the critical load parameter corresponding with R∗ → ±∞ is
specified by lines 1, 2, 3 (Fig.12). The points A0, B0, C0 describe the value



Vibrations and stability of a two-rod column... 921

Fig. 12. The change of the critical load parameter λ
c
as a function of radius R∗ of

the loading head

Fig. 13. The change of the critical load parameter λ
c
in relation to positive values of

the radius R∗ of the loading head

of the critical force for the column with the joint attachment at the free ends
(compare K3, K4 – Table 2).
The range R∗ ∈ (R∗

′

,R∗
′′

) of positive values of the radius R∗ of the loading
head exists for every length of the rigid element l∗0, see Fig.13. The considered
column is of divergence-pseudo-flutter type (A′,A′′ point) in the above range.
The considered system is of divergence type for the remaining positive and
negative values of R∗.
Thenumberingof the considered columnas one of the two types of systems

is associatedwith the courseof thenatural frequency in relation to the external
load, which is shown in Fig.14 and Fig.15.



922 L.Tomski et al.

Fig. 14. Frequency curves in the load-natural frequency plane for R∗− l∗
0
¬ 0

Fig. 15. Frequency curves in the load-natural frequency plane for R∗− l∗
0
­ 0

The presentationwas limited to the first two basic natural frequencies in a
dimensionless form Ωi and to an additional symmetric natural frequency Ωs2
in relation to the dimensionless loading parameter λ, while

λ=
Pl21
EJ

Ωi =
ρAω2l41
EJ

m∗ =
m

ρAl1
(7.4)



Vibrations and stability of a two-rod column... 923

The slope of the eigenvalue curves (Fig.14) is always negative for the co-
lumn loaded by a follower force directed towards the negative pole. That slope
canbepositive (curve 3 inFig.15), negative or zero (curves 2, 4) for the system
loaded by a follower force directed towards the positive pole.
The discussed curves were sketched for a constant length l∗0 of the rigid

element of the loading head and concentrated mass at the free end of the
column m∗. The value of the critical load Ω = 0 stays in accordance with
curve A in Fig.12.
The change of natural frequencies in relation to the dimensionless loading

parameter for a constant radius R∗ of the loading head (line 4 in Fig.12) is
shown in Fig.16.

Fig. 16. Frequency curves in the load-natural frequencies plane for R∗ = const

The considered column is of divergence-pseudo-flutter type independently
of the length l∗0 of the loading element for the chosen geometrical and physical
parameters of the system.

8. Conclusions

On the basis of experiments and carried out numerical simulations for the
presented variant of the specific loadby the follower force towards the negative
or positive pole, one can state that:



924 L.Tomski et al.

• correct boundary conditions for the considered structure canbedetermi-
ned on the basis of theminimumof potential energy (static problem) or
on the basis of theminimumofmechanical energy (Hamilton’s principle)

• the considered system can be of divergence or divergence-pseudo-flutter
type with regard to the design of loading and receiving heads, (value of
the radius R and length l0 of the rigid element of the head receiving the
load)

• there are such values of geometrical parameters of the loading and rece-
iving heads for which the maximum of the critical load is obtained

• the system is conservative according to the extended principle of poten-
tial, which was described by Tomski et al. (1996) and resulted from the
self-adjointness of differential operators, while in this paper the conse-
rvative system is determined from the relationship between the potential
energy and work of the potential system.

Acknowledgements

This researchhas been supported by the StateCommittee for ScientificResearch,
Warsaw, Poland, under grant No. 7T07C03218.

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926 L.Tomski et al.

Drgania i stateczność dwuprętowej kolumny obciążonej poprzez wycinek

łożyska tocznego

Streszczenie

Wpracyprezentuje sięnoweobciążenieukładówsmukłych,które jest obciążeniem
siłą śledzącą skierowanądobiegunadodatniego lubujemnego.Przedstawia się rozwią-
zania konstrukcyjne głowic obciążonych, które realizują to obciążenie. Prezentuje się
rozważania teoretycznedotyczące sformułowaniawarunkówbrzegowychnapodstawie
całkowitej energii układu. W zależności od rozwiązania konstrukcyjnego głowicy ob-
ciążającej i przejmującej obciążenie określa się wartość siły krytycznej oraz przebieg
częstości drgań własnych w funkcji obciążenia zewnętrznego dla zadanej geometrii
i stałych fizycznych kolumny.Wyniki badań teoretycznych porównuje się z wynikami
badań eksperymentalnych.

Manuscript received March 23, 2004; accepted for print April 24, 2004