Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

48, 2, pp. 345-365, Warsaw 2010

EXPERIMENTAL VERIFICATION OF FREE TRANSVERSE

VIBRATIONS OF COLUMNS LOADED BY HEADS

EQUIPPED WITH ROTARY NODES

Lech Tomski
Iwona Podgórska-Brzdękiewicz
Janusz Szmidla

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

Częstochowa, Poland; e-mail: iwonapb@imipkm.pcz.czest.pl

The main goal of this paper is to determine the influence of various
elements mounted in rotary nodes of a forcing head, i.e.: rolling needle
bearings or rigid elements on the natural transverse vibration frequen-
cy and value of the critical load of columns. The problem of stability
and course of characteristic values against the external force loading the
systems subjected to a specific load is discussed in the paper. A new
constructional scheme of the forcing and receiving heads serving for the
purposeof carryingout thegeneralised loadby the forcedirected towards
the positive pole and the load by a force directed towards the positive
pole is presented. Theoretical considerations related to determination of
boundary conditions using the method of mechanical energy variation
are shown. Verification of the assumed goal is realised by experimental
research.

Key words: elastic columns; divergence instability; natural transverse
vibrations

1. Introduction

Divergence systems (Gajewski and Życzkowski, 1969; Leipholz, 1974; Timo-
shenko and Gere, 1963; Ziegler, 1968) or divergence pseudoflutter systems
(Tomski et al., 1996, 1998, 1999, 2004;Tomski andSzmidla, 2004a,b, 2006) are
loaded by conservative forces. Only the conservative load is considered, witho-
ut considering publications related to non-conservative systems. The systems
subjected to conservative loads lose stability due to buckling, and the trans-
ition from the stable to unstable condition happens at frequencies of natural



346 L. Tomski et al.

vibrations equal zero for the so-called divergence critical force Pc1. By solving
the issue of free vibration of thementioned systems, a particular course of the
curves is obtained on the load (P) – natural frequency (ω) plane which is
presented in Fig.1.

Fig. 1. (a) Divergence system; (b) divergence pseudoflutter system

All curves characteristic for the divergence systems (Fig.1a) always have
a negative slope. In the case of divergence pseudoflutter systems, for the load
changing from 0 to the critical force (Pc1), the slope of the curves on the
load (P) – natural frequency (ω) plane may be positive, zero or negative.
For P ≈ Pc1, the natural frequency curve slope is always negative. For such
a system along the eigenvalue curves the natural vibration form change from
the first form to the second one and inversely (whereM1,M2 denote the first
and the second form of vibrations, respectively).
Considering conservative loads of slender systems and their properties, the

following cases can be distinguished:

• Euler’s load (Timoshenko and Gere, 1963)

• loading by the force directed towards the positive pole – Timoshenko
and Gere (1963), Gajewski and Życzkowski (1969) or the negative pole
Gajewski and Życzkowski (1969)

• specific loads:

– generalised load by the force directed towards the pole – Tomski et
al. (1996, 1999), Tomski and Szmidla (2004a,b)

– load by the follower force directed towards the pole – Tomski et al.
(1998, 2004), Tomski and Szmidla (2004a)

– load through a stretched element of finite bending rigidity –Tomski
and Szmidla (2006).

For each of the listed types of specific loads there is a constructional scheme
for the loading structure made of the forcing and receiving head.



Experimental verification of free transverse vibrations... 347

Researches concerning the influence of components used in the receiving
head (rolling bearing, plain bearing, rigid component) on natural vibrations of
slender systems are included in works byTomski andPodgórska-Brzdękiewicz
(2006a,b). It was found that the use of rigid components results in a conside-
rable increase in the basic experimental frequency of natural vibrations in the
systemwith respect to the samequantity obtained fromnumerical simulations
(for the appliedmathematic model). Using the obtained characteristic values,
the boundary conditions were modified by implementation of the equivalent
rigidity of rotational springmodelling the rigidity of the free endof the system.

2. Statement of the problem

Themain technical problemconsidered in this paper is the influence of various
elements mounted in the rotary nodes of the forcing head, i.e.: rolling needle
bearingsor rigid elements, on the stability and thenatural transversevibration
frequencyof columns.Anewconstructional schemeof the forcingandreceiving
head serving for the purpose of carrying out the generalised load is given.
Such a scheme can be implemented in the supporting systems for different
applications.
This paper presents theoretical considerations, numerical calculations and

experimental investigations concerning the stability and natural vibrations of
two types of cantilever columns (A and B) differing in the way of realising
the load. The schematic diagrams of the systems are shown in Fig.2, where
N is the value of external load necessary to obtain the compressive force of
the rods P (N = f(P)).
Depending on geometric parameters of the forcing and receiving heads, the

following cases can be distinguished (Fig.2):

(a) generalised load by the force directed towards the positive pole – sys-
tem A

(b) load by the force directed towards the positive pole – systemB.

Boundary conditions are formed for the distinguished systems using the
energetic method.
Examination of the systems concerns the influence that the elements used

in the rotary nodes of the forcing head (rolling needle bearings or rigid ele-
ments) and geometry of this head have on the course of curves of the charac-
teristic values and on the value of the critical force.



348 L. Tomski et al.

Fig. 2. Schematic diagrams of considered systems

2.1. Realisation of load

The structures loading the column (Fig.2) comprise: beam (2), linear ele-
ments (4) and (7) and a circular element (5). Linear elements (2), (4) and (7)
and structure (5) with radius R are characterised with infinite flexural rigidi-
ty. Nodes (3) and (6) aremade of rotary elements transferring the load to the
columnwhichmay comprise rolling needle bearings (F) or rigid elements (G)
(Fig.3). The radius of the element mounted in rotary node (6) has beenmar-
ked as r. The following designations were introduced – compare Fig.2:
• I – lower rotary node (3)

• II – upper rotary node (6).

Fig. 3. Elements used in the rotary elements of the forcing head

The basic features of the column loading structures are as follows:
• the forcing structure comprises: linear elements (2) and (4) connected
by a hinge joint and rotary nodes (3) and (6)



Experimental verification of free transverse vibrations... 349

• element (7) connected by a rigid joint with circular cylinder (5) make
the receiving head

• the centre of segment (5) having the circular cylinder outline is located
at thedistance lC fromthe free endof the system, andat thedistance lD
from node (3).

The direction of the loading force P is marked with a broken line (Fig.2).
It crosses the fixed point O (node (3)) located on the non-deflected axis of the
column below its free end, node (6) and the point determining the curvature
centre of a fragment of circular cylinder (5).
Values R and l0 resulting from the construction of the forcing head are

interrelated by the following relationship: lD =R−r+ l0.
The real system is made as a planar frame made of two rods (1.1 and

1.2) with the bending rigidity (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 denotes the longitu-
dinal modulus of elasticity of the rod material, J – central axial moment of
inertia of the column rod, ρ0 – material density, A – cross-section area. The
lengths of the column rods are equal to l. The rods of the column have the sa-
me cross-sections andaremade of the samematerial. Such arrangement allows
for forcing vibration in a privileged plane. There is symmetrical distribution
of bending rigidity in thismodel. The rods and their physical and geometrical
parameters are distinguished by (1.1), (1.2) indexes, which are only needed
to calculate symmetrical natural frequencies and to determine corresponding
forms of vibration (compare Tomski et al., 1999). Hence, we can assume the
global bending rigidity EJ and elementary mass of the column ρ0A in the
following considerations.

3. Column subjected to the generalised load by the force directed

towards the positive pole – system A

3.1. Physical model and analysis of the system geometry

The physical model of the system subjected to a generalised load by the
force directed towards the positive pole is shown in Fig.2a (A). The system
is loaded via beam (2) connected by a hinge joint with element (4) having the
length of l0. The column mounted rigidly (x= 0) at the free end (x= l) is
jointed with linear elements (7). Displacement of the column end in relation
to the axis x has the value of W(x,t), and the deflection angle of this end



350 L. Tomski et al.

compared to the non-deflected axis is ∂W(x,t)/∂x|x=l. Element (7) is rigidly
jointed with circular cylinder (5) on which the second end of the element is
moving (4). The totalmass m comprises concentratedmass m1 and corrected
mass mzr of elements (4), (5) and(7) in relation to thefixingpointofmass m1.
Figure 4 presents the physical model of the considered system, geometry

and three options of externalising the internal forces (sections I–I, II–II also
I–I and II–II).The formulas describing the total potential energy of the system
depend on the place of externalising the load. The boundary conditions of the
considered system can be obtained based on the total energy of the system
for externalising the load in sections I–I or II–II. By externalising the internal
forces in sections I–I and II–II, the potential energy of the system is equal to
zero.

Fig. 4. Deflected form of column typeA (see Fig.2a), geometry and externalisation
of internal forces

In order to unify the considerations presented in this work, the externali-
sation of the load in section II–II is examined (at the free end of the column).
Having examined geometry of the system (Fig.4a), angle β formed by the

direction of the force P with the non-deflected axis of the column is expressed
by the following relations

sinβ=
W0
lD

(3.1)

where W0 is the displacement of the point determining the centre of the cy-
linder element of radius R, is



Experimental verification of free transverse vibrations... 351

W0 =W(l,t)− lC sin
[∂W(x,t)
∂x

∣

∣

∣

∣

x=l
]

(3.2)

Based on the relationships resulting from Fig.4a, the condition of equality of
the angels at the free end of the column looks as follows

β+ϕ=
∂W(x,t)
∂x

∣

∣

∣

∣

x=l

(3.3)

where ϕ is the angle formed by the direction of the force P with the axis of
rigid element (7).
The theory of small displacement was used in the description – if there

are geometric relations between the elements of a given system, the following
holds

sinϕ=tanϕ=ϕ cosϕ=1−
ϕ2

2
(3.4)

Using relations (3.1), (3.3) and (3.4)1, the following is obtained after transfor-
mation

W0
lD
=
∂W(x,t)
∂x

∣

∣

∣

∣

x=l

−ϕ (3.5)

The final form of the expression determining the value of angle ϕ, determined
based on formulas (3.2)-(3.5), was described in the following way

ϕ=
(

1+
lC
lD

)∂W(x,t)
∂x

∣

∣

∣

∣

x=l

−
1
lD
W(l,t) (3.6)

The ∆1 value 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 (3.7)

3.2. Mechanical energy of the system. Boundary conditions

Kinetic energy T of the system is the sum of kinetic energy of the co-
lumn T1 and energy of the concencrated mass m−T2, and is expressed by
the following formulas

T =T1+T2 =
ρ0A

2

l
∫

0

[∂W(x,t)
∂t

]2

dx+
1
2
m
[∂W(l,t)
∂t

]2

(3.8)



352 L. Tomski et al.

where ρ0A is themass density per column length unit, m – concentratedmass
at the free end of the system (in point x= l).
Relations for potential energy of the system V =

∑4
k=1Vk (with externa-

lising the internal forces in section II–II) are determined as follows:
— energy of elastic strain

V1 =
1
2
EJ

l
∫

0

[∂2W(x,t)
∂x2

]2

dx (3.9)

— potential energy of the vertical component of the P force

V2 =−P∆1 =−P
1
2

l
∫

0

[∂W(x,t)
∂x

]2

dx (3.10)

— potential energy of the horizontal component of the P force

V3 =
1
2
PβW(l,t)=

1
2
P

lD

[

W(l,t)− lC
∂W(x,t)
∂x

∣

∣

∣

∣

x=l
]

W(l,t) (3.11)

— potential energy of the bendingmoment

V4 =
1
2
PlCϕ

∂W(x,t)
∂x

∣

∣

∣

∣

x=l

=

(3.12)

=
1
2
PlC
[(

1+
lC
lD

)∂W(x,t)
∂x

∣

∣

∣

∣

x=l

−
1
lD
W(l,t)

]∂W(x,t)
∂x

∣

∣

∣

∣

x=l

Formulation of theproblemconsisting in thedetermination of boundarycondi-
tions for the considered column ismade usingHamilton’s principle (Goldstein,
1950) that states

δ

t2
∫

t1

(T −V )dt=0 (3.13)

The commutation of integration (with respect to x and t) and variational
calculation is used within Hamilton’s principle (Eq. (3.13)). The equation of
motion, after taking into account the commutation of variation and differen-
tiation operators and after integrating the kinetic andpotential 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 (3.14)



Experimental verification of free transverse vibrations... 353

Using the a priori known geometric boundary conditions in the fixing point
of the column (x=0)

W(0, t)= 0
∂W(x,t)
∂x

∣

∣

∣

∣

x=0

=0 (3.15)

the remaining boundary conditions necessary to solve the boundary problem
are determined

EJ
∂2W(x,t)
∂x2

∣

∣

∣

∣

x=l

+P
[(

1+
lC
lD

)

lC
∂W(x,t)
∂x

∣

∣

∣

∣

x=l

−
lC
lD
W(l,t)

]

=0

(3.16)

EJ
∂3W(x,t)
∂x3

∣

∣

∣

∣

x=l

+P
[(

1+
lC
lD

)∂W(x,t)
∂x

∣

∣

∣

∣

x=l

−
1
lD
W(l,t)

]

−m
∂2W(l,t)
∂t2

=0

4. Column loaded by the force directed towards the positive pole

– system B

4.1. Physical model and analysis of the system geometry

The physical model of the system loaded by the force directed towards the
positivepole is presented inFig.5.Thecolumn is loadedby the force P applied
to its free end and is jointed with rigid element (7) bymeans of a structure of
mass m. The total mass m consists of the concentratedmass m1 and reduced
mass mzr of elements (4), (5) and (7) in relation to the fixing point of the
mass m1. The location of linear element (4) agrees with the direction of the
force P and makes an angle β with the axis x. Element (4) is connected by
a hinge joint with beam (2); its second end is moving on circular cylinder (5)
of radius R which is rigidly jointed with element (7).
Geometric relations between the elements and the loading structure lead to

a relation between the transverse displacement and the angle in the direction
of the force which, after considering some small displacement (3.4), is given in
the following form

β=
W(x,t)

∣

∣

x=l

lD
(4.1)

4.2. Total potential energy of the system. Boundary conditions

Potential energy of the system V , for externalisation of internal forces
presented in Fig.5b is the sum of elastic strain energy V1, potential energy



354 L. Tomski et al.

Fig. 5. Deflected form of columnB (see Fig.2b), geometry and externalisation of
internal forces

of the components of the force P: vertical – V2 and horizontal – V3, and is
expressed by the following formula

V =V1+V2+V3 =
1
2
EJ

l
∫

0

[∂2W(x,t)
∂x2

]2

dx−P∆1+
1
2
PβW(l,t) (4.2)

∆1 is the longitudinal displacement described in formulas (3.7).
Taking into account Hamilton’s principle (3.13) and considering relation

(4.2), kinetic energy variation (3.8) and known boundary conditions (formu-
las (3.15)), the equation of motion is obtained (compare Eq. (3.14)) and the
remaining boundary conditions necessary to solve the problem

∂2W(x,t)
∂x2

∣

∣

∣

∣

x=l

=0

(4.3)
∂3W(x,t)
∂x3

∣

∣

∣

∣

x=l

+
P

EJ

[∂W(x,t)
∂x

∣

∣

∣

∣

x=l

−
1
lD
W(l,t)

]

−
m

EJ

∂2W(l,t)
∂t2

=0

5. Solution to the boundary value problem

Considering the symmetrical distribution of flexural rigidity andmass per unit
of length of the considered columns, then distributing variables of the function



Experimental verification of free transverse vibrations... 355

Wi(x,t) versus time t and space x in the following form

Wi(x,t)= yi(x)cos(ωt) i=1,2 (5.1)

the equations of motion of the considered systems are obtained

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

II
i
(x)−(ρ0A)iω

2yi(x)= 0 (S)1+(S)2 =P (5.2)

where (S)i is the internal force in the i-th rod of the system.
The boundary conditions in the fixing point of systemsA and B (x=0)

were put in the following way

y1(0)= y2(0)= 0 yI1(0)= y
I
2(0)= 0

y1(l)= y2(l) yI1(l)= y
I
2(l)

(5.3)

and at the free end (after separation of variables), respectively for
— systems of typeA

yII1 (l)+y
II
2 (l)+

P

(EJ)1
lC
[(

1+
lC
lD

)

yI1(l)−
1
lD
y1(l)
]

=0

(5.4)

yIII1 (l)+y
III
2 (l)+

P

(EJ)1

[(

1+
lC
lD

)

yI1(l)−
1
lD
y1(l)
]

+
mω2

(EJ)1
y1(l)= 0

— systems of typeB

yII1 (l)+y
II
2 (l)= 0

(5.5)

yIII1 (l)+y
III
2 (l)+

P

(EJ)1

[

yI1(l)−
1
lD
y1(l)
]

+
mω2

(EJ)1
y1(l)= 0

General solutions to equations (5.2)1 are as follows

yi(x)=D1icosh(αix)+D2i sinh(αix)+D3icos(βix)+D4i sin(βix) (5.6)

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

α2
i
=−
1
2
k2
i
+

√

1
4
k4
i
+Ω2

i
β2
i
=
1
2
k2
i
+

√

1
4
k4
i
+Ω2

i
(5.7)

where

Ω2
i
=
(ρ0A)iω2

(EJ)i
ki =

√

(S)i
(EJ)i

(5.8)

Substitutionof solutions (5.6) intoboundaryconditions (5.3), and respecti-
vely forEqs. (5.4) orEqs. (5.5) yields a transcendental equation for eigenvalues
of the considered system.



356 L. Tomski et al.

6. Constructional scheme of the loading head

Figure 6 presents a constructional scheme of the head used for loading the
considered columns.
The structure consists of beam (1) fixed by a hinge joint to element (2)

and equippedwith rotary node (3), where the rolling element (FI) or rigid ele-
ment (GI) can bemounted. The load is transferred through stretched bolt (4)
mounted in sockets (4(1)) and ((4(2)), and pin (5) where a needle bearing
is mounted (FII) or a rolling element (GII) – (6). The compressive force is
carried by housing (7) jointed with element (9) which together with column
rods (10(1)) and (10(2)) is mounted in block (11) of mass m. It is assumed
that elements (1), (4), (4(1)), (4(2)), (7), (8), (9) and (11) are infinitely rigid,
which is justified due to constructional reasons.

Fig. 6. Constructional scheme of the head loading the columns

Elements (1), (3), (4), (4(1)) and (4(2)) constitute the forcing head, and
elements (7), (9), (11) – the receiving head. Intermediate element (8), and
rotary node (6) comprise the forcing head.

7. Experimental stand

The experimental research on natural frequencies of the considered systems
have been carried out on an experimental stand designed and built at the
Technical University of Czestochowa, Institute ofMechanics andMachine De-



Experimental verification of free transverse vibrations... 357

signFundamentals. Its constructional scheme is shown inFig.7 (Tomski et al.,
1998).

Fig. 7. Test stand for experimental research of the considered columns

The stand presented in Fig.7 has two loading heads 1(1) and 1(2) and
can be used to examine free vibration of the columns and frames located
vertically or horizontally. Head 1(1) was used for experimental research of
the considered systems, which can move on guides 2(1) and 2(2) both in the
longitudinal (horizontal) and transverse direction. Head 1(1) is equippedwith
bolt systems whose movement load the tested column. The measurement of
the loading force wasmade by dynamometer 3. The supports realising the set
boundary conditions of the columns were installed on plates 4(1-3).

Experimental research of the natural vibration frequencies were carried
out by the measuring system presented in Fig.7. The system is comprised of
accelerometer (5) (Brüel&Kjær – 4508B), two-channel vibration analyser (7)
by Brüel & Kjaer and computer (8). Vibration of the column fixed on the
experimental stand was induced by hammer (6) and then accelerations in
individual measuring points were measured by accelerometric sensor (5). The
signal from the sensor was transmitted to analyser (7), where it was processed
and sent to computer (8).



358 L. Tomski et al.

8. Experimental and numerical results

Based on the solution to the boundary value problem, numerical calculations
of the natural vibration frequency against the external load weremade. Then
they were verified on the experimental stand for natural vibration frequency
measurements of the columns and flat frames (Fig.7). The physical and geo-
metrical parameters of the systems as well as the values characterising the
forcing and receiving heads are presented in Table 1.

Table 1.Physical and geometrical parameters of the considered columns and
parameters the forcing and receiving heads

Column
(EJ)i (ρ0A)i l lC lD m R
[Nm2] [kg/m] [m] [m] [m] [kg] [m]

A 71.76 0.315 0.61 0.084 0.244 1.4 0.085
B 71.76 0.315 0.61 – 0.325 1.4 0.085

In order to verify themathematical model, experimental research was car-
ried out determining the characteristic values of the considered systems when
the rolling bearings (F) and/or slide bearings (G) were used in rotary nodes
(3), (6) of the loading heads (see Fig.3).
Themeasurementsweremade for the columns loaded in the followingway:

a) generalised load by the force directed towards positive poleA – Fig.8

b) load by the force directed towards the positive poleB – Fig.9.

Fig. 8. Curves in the plane: load – natural frequency for columnsA



Experimental verification of free transverse vibrations... 359

Fig. 9. Curves in the plane: load – natural frequency for columnsB

The results of numerical calculationsweremarkedwith lines.Theobtained
results of experimental results were shown with points, while rotary nodes I
and/or II of loading head were equipped with the rolling bearings (F) and
slide bearings (G). The results are limited to the first four basic natural fre-
quencies (M1-M4) and three additional frequencies (M2e-M4e) characterised
by symmetry of vibrations.
An arithmetic average of the three conducted measurements is presented

in Figs.8 and 9 by dots

fe =
1
3

3
∑

i=1

fe
i

(8.1)

where fe
i
denotes the results of consecutive measurements.

Analysis of the error of the proposed mathematical model is carried out.
The arithmetic average of experimental measurement (Eq. (8.1)) was treated
as an actual value of themeasured quantity that is the natural frequency. The
relative percentage error of the theoretical model is given by the expression

∆fe =
∣

∣

∣

ft−fe

ft

∣

∣

∣
·100% (8.2)

where ft is the value of the measured quantity, obtained theoretically.
There is very good consistence of the results of experimental investigations

withnumerical calculations in regards to thebasic natural vibration frequency.
In the case of remaining natural vibration frequencies, between the results
obtained in the experimental research and numerical computations the value
of the relative percentage error ∆f does not exceed 10% for over 250 points.
Only for the second natural vibration frequency for P = 0 (Fig.8) it differs



360 L. Tomski et al.

fromthe rest results.The lowest valueof ∆f of all accomplishedmeasurements
equals 0.24%, while the highest 31%.
Differences between the experimental and numerical resultsmay be caused

by the following factors:
• in the mathematical model – not taking into account the resistance to
motion – deformation of the contact surface,

• certain inaccuracy in the determination of Young’s modulus and the
material density,

• possibility of inaccurate mounting of the accelerometer in the plane of
vibrations,

• error in accurate determination of the loading force of the system when
using the dynamometer,

• assumption of the infinite rigidity of mounting of the system in thema-
thematical model,

• influence of the test stand on the tested column.

Itwas found that the loading of the columnby the headwithinwhich rigid
elements transfer the load (slide bearings) (GI and/or GII) does not influence
the course of characteristic values of the system against the external load.
The results of numerical simulations related to the determination of the

critical load Pc and the course of natural vibration frequency against the
external force loading the columnwere presented. The critical force, length lC
and distance lD as well as concentratedmass m located at the free end of the
column were expressed in non-dimensional coordinates

λ∗
c
=
Pcl
2

EJ
l∗
C
=
lC
l

l∗
D
=
lD
l

m∗ =
m

ρ0Al
(8.3)

The change of the critical parameter of load λ∗
c
against the parameter l∗

D

for some selected values of non-dimensional length l∗
C
was presented inFig.10.

The curves and points presented and listed in Fig.10 correspond to the
critical parameter of load or the course of its change for columns loaded re-
spectively by:
• generalised load by the force directed towards the positive pole (co-
lumnA) – curves 1-7,

• force directed towards the positive pole (column B, l∗
C
=0) – curve 8,

• follower force directed towards the positive pole (l∗
D
= 0) – points (◦)

marked on the axis of ordinates (e.g. Tomski et al., 2004; Tomski and
Szmidla, 2004a),



Experimental verification of free transverse vibrations... 361

Fig. 10. Change of the critical parameter of load λ∗
c
against the parameter l∗

D
for

various values of l∗
C

• Euler’s load for the system hinged at the free end (l∗
C
=0 and l∗

D
=0)

– point P ′,

• Euler’s load for the system hinged at both ends (l∗
D
=1) – point P ′′.

For the columns loaded with the considered types of load the courses of
natural vibration frequency against the external load were determined. Only
the character of change of the first two basic frequencies in non-dimensional
form were determined Ω∗

t
(t = 1,2) as well as an additional symmetrical

natural frequency Ω∗s2 against the non-dimensional parameter of load λ
∗,

where

λ∗ =
Pl2

EJ
Ω∗ =Ω2l4 =

ρ0Aω
2l4

EJ
(8.4)

Thecalculationsweremade for selected valuesofnon-dimensionally expres-
sed geometric parameters characterising the forcing and receiving heads and
for a constant value of the concentratedmass m located at the free end of the
system (Eqs. (8.3)). The influence of the change of parameter l∗

D
on the course

of natural vibration frequency against the external load parameter for selected
values of l∗

C
is shown in Fig.11 and Fig.12. The numerical simulations were

carried out for the systemA – Fig.11.
The course of curves on the plane P-ω presented in Fig.12 corresponds to

the system B. The range of changes of eigenvalues is limited with the curves
obtained for the systems loaded by the force directed towards the positive pole
(curve 1) and supported by hinge joints at both ends and exposed to the axial
compressive force (curve 7).
The influence of change of parameter l∗

C
on the course of natural vibra-

tion frequency against the external load parameter λ∗ for constant values l∗
D

and m∗ is shown in Fig.13.



362 L. Tomski et al.

Fig. 11. Curves on the plane: loading parameter – parameter of natural vibration
frequency of columnA

Fig. 12. Curves on the plane: loading parameter – parameter of natural vibration
frequency of columnB

Fig. 13. Curves on the plane: loading parameter – parameter of natural vibration
frequency of columns for various parameters l∗

C



Experimental verification of free transverse vibrations... 363

With the considered value of the parameter l∗
D
for the systemB (l∗

C
=0),

the course of natural vibration frequency against the external load allows
one to classify these systems among the divergence pseudoflutter ones. The
systemAmaybeof thedivergence pseudoflutter or divergence typedepending
on l∗

C
.

9. Final remarks

• Depending on the value of geometric parameters of the forcing heads,
characteristic cases were obtained for columns subjected to generalised
load by the force directed towards the positive pole and columns loaded
with the force directed to positive pole.

• Based on natural vibration frequency measurements and numerical si-
mulations, it was demonstrated that the type of elements used in the
rotary nodes of the forcing head does not influence the experimental
values of natural vibration frequency.

• For the columns subjected to the generalised load by the force directed
towards the positive pole, an increase in the critical force was obtained
compared to the system fixed with an articulated joint for x= l, where
the force maintains the constant direction (Euler’s load).

• The obtained results of numerical computations and experimental inve-
stigations regarding the course of natural frequencies in relation to the
external load showed good agreement.

• Appropriate selections of geometric parameters of the forcingheadsallow
obtaining an increase or decrease in the natural vibration frequency of
the system with an increase in the column loading force (divergence
pseudoflutter system).

Acknowledgment

The authorswould like to express their gratitude toDr.M.Gołębiowska-Rozanow
andA.Kasprzycki for partnership in their experimental research.The study has been
carried outwithin the framework ofWorkBS-1-101-302-99/PofTechnical University
of Czestochowa andResearchProject No. NN501117236 awarded by theMinistry of
Science and Higher Education.



364 L. Tomski et al.

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Experimental verification of free transverse vibrations... 365

Eksperymentalna weryfikacja swobodnych drgań poprzecznych kolumn

obciążonych poprzez głowice o węzłach obrotowych

Streszczenie

Głównym celem pracy jest określenie wpływu różnych elementów zamocowanych
w węzłach obrotowych głowicy obciążającej, tj. łożysk tocznych igiełkowych lub ele-
mentów sztywnych na częstość drgań własnych oraz wartość obciążenia krytycznego
kolumn.Wpracy rozważa się zagadnienia stateczności i przebieguwartości własnych
w funkcji zewnętrznej siły obciążającej układ, poddany obciążeniu swoistemu. Pre-
zentuje się nowe rozwiązaniekonstrukcyjne głowicyobciążającej, służącej do realizacji
obciążenia siłą uogólnioną skierowaną do bieguna dodatniego oraz siłą skierowaną do
biegunadodatniego.Przedstawia się rozważania teoretycznedotyczące sformułowania
warunkówbrzegowychmetodąwariacji energiimechanicznej poszczególnychukładów.
Weryfikacja założonego celu jest realizowana na drodze badań eksperymentalnych.

Manuscript received May 22, 2009; accepted for print September 24, 2009