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Journal of Physics: Conference Series
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Numerical calculation of the natural vibration frequencies of thermo-
elastic rods taking into account the effect of the temperature gradient
To cite this article: O G Kikvidze et al 2021 J. Phys.: Conf. Ser. 1901 012116
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MSTU 2021
Journal of Physics: Conference Series 1901 (2021) 012116
IOP Publishing
doi:10.1088/1742-6596/1901/1/012116
1
Numerical calculation of the natural vibration frequencies of
thermo-elastic rods taking into account the effect of the
temperature gradient
O G Kikvidze1, G G Sakhvadze2and G Zh Sakhvadze2
1Akaki Tsereteli State University, Kutaisi, Georgia
2 Mechanical Engineering Research Institute of Russian Academy of Sciences
(IMASH RAN), Moscow, Russia
Abstract. Elastic rod elements are considered, the axial line of which in the natural state is a
flat curve and remains a flat curve after loading. The following assumptions are used: the
normal cross-sections, which are flat before deformation, remain flat and normal to the
deformed thermo-elastic axial line after deformation; the temperature field is stationary and
non-uniform. Nonlinear equations of motion of the rod are obtained, taking into account the
rotational inertia of cross-section and the extensibility of the axial line.
Keywords: thermo-elastic rod, dynamic loading, temperature gradient, vibration frequency
1. Introduction
The use of new light metallic materials in building structures makes it relevant to study the effect of
the temperature factor on the strength and stiffness within the limits of elasticity, both under static and
dynamic loads. The definition of natural vibration frequencies of an elastic rod, taking into account the
axial line extension, is considered in the papers [1-3]. Non-uniform temperature distribution at small
temperature change can have a significant impact on the frequency of free vibrations of the rod. The
use of numerical calculation methods makes it possible to solve the nonlinear problem and automates
calculations for various anchoring conditions and complex rod geometry.
2. Problem statement
The aim of the presented article is to develop a method of numerical calculation of the dynamics of
rod elements of structures under thermo-mechanical loading. Such calculations are necessary when
designing light metallic materials in building structures and other related industries, and today such a
method of calculation is not found in publications.
3. Theory
Let us consider the rod vibrations in the plane in which the axial line is located. We shall investigate
the rod motion relative to the natural (unloaded) state. The temperature field is stationary and varies
only along the height of the cross-section in the bending plane. In this formulation the static equations
at large displacements are obtained in the paper [4].
Let us consider the rod element that has a forward velocity relative to the у and z axes and an
angular velocity relative to the x axis. In its original state we denote the radius of curvature of the
MSTU 2021
Journal of Physics: Conference Series 1901 (2021) 012116
IOP Publishing
doi:10.1088/1742-6596/1901/1/012116
2
axial line by r0, and the angle of inclination of the tangent to the z axis by θ0. In general, the rod
element can be affected by distributed and concentrated forces and moments that are time-variant. In
the Cartesian reference system, the forces and moments that affect the rod element when it moves in
the bending plane are shown in Fig. 1.
Figure 1. Illustration to the derivation of the equation of the rod motion.
Using the d’Alambert principle and the static equations [4] and taking into account the forces and
moments of inertia, the differential equations of motion will take form
()
=
+,
()
=
+, (1)
( )
=
++cossin,
where ρ is the density of the material; А is the cross-sectional area; v, w are the displacements in the
direction of the y and z axes, respectively; Ix0 is the physical moment of inertia of the unit length rod
element. For the principal axes of the cross-section Ix0=Ixρ, Ix is the geometrical moment of inertia of
the cross-section, M is the bending moment; R, H are the components of the internal force vector; qy,
qz are the components of the distributed external force vector; m is the intensity of the external
bending moment; l is the arc length of the deformed thermo-elastic line.
For the kinematics of deformation, the relations obtained in the paper [4] are valid. By replacing
the usual derivatives in them with partial derivatives, we obtain
=(1 + )sinsin,
=(1 + )coscos, (2)
=
+R
x,
where ɛ0 is the deformation of the thermo-elastic line; l0 is the arc length of the non-deformed thermo-
elastic line; kx characterizes the curvature change.
The ɛ0 and kx values are determined by the formulas [4]
=
+ [d]/, =
+ [d]/
, (3)
where A*= is the generalized area; Ix*= is the generalized moment of inertia; E=E(T)
is the material elasticity modulus ; T=T(y) is the temperature; ɛT is the temperature deformation.
MSTU 2021
Journal of Physics: Conference Series 1901 (2021) 012116
IOP Publishing
doi:10.1088/1742-6596/1901/1/012116
3
The normal force N in the cross-section is [4]
N=Hcosθ + Rsinθ
According to the formulas (3), we shall represent the deformation ɛ0 and curvature kx as a sum of
components from the corresponding force factors and temperature
=
+
, =
+
,
=
,
=
,
=
d ,
=
d .
For straight rods r0→∞, θ0=0 and equations (1) and (2) take the following form:
()
=
+,
()
=
+,
()
=
++cos, (4)
=sin,
=cos
,
=
.
3.1. Small free vibrations of a straight rod
When designing elastic rod elements operating in dynamic modes, it is necessary to determine the
spectrum of frequencies (more specifically, the first few frequencies) depending on the anchoring
conditions and the static stress-strain state. The frequencies are determined from the equations of small
free vibrations of the rod relative to its natural state or relative to the state of equilibrium.
We shall obtain the equations of small free vibrations of the rod, assuming that the inner forces and
displacements arising during the vibrations are small. The components of the displacement and force
vectors (,,,,,) are values of the first order of smallness, therefore their products are
neglected in the derivation of the equations of motion. Let us set the external loads equal to zero
== 0, = 0.
In the case of small displacements sin,cos 1 ,
1 and we disregard the
summand of the second order of smallness . Consequently, the nonlinear equations (4) are
simplified and take the form [4]: ()
=
,
= ,
()
=
,
=, (5)
()
=
+,
= .
The equations for an elastic rod without taking into account the axis deformation and temperature
changes are listed in [5].
3.2. Numerical calculation method
MSTU 2021
Journal of Physics: Conference Series 1901 (2021) 012116
IOP Publishing
doi:10.1088/1742-6596/1901/1/012116
4
To determine the frequencies of free vibrations, we present the force and kinematic factors in the
form:
(,)=(), (,)=
(),
(,)=(,), (,)=(,), (6)
(,)=(), (,)=() ,
(,)=
(), (,)=() .
From the system of equations (5) taking into account formulas (6), we obtain ordinary differential
equations:
=() ,
=(),
=(),
=
(), (7)
= ,
=
.
To integrate the system of differential equations, (7) boundary conditions are necessary that reflect
the anchoring of the rod ends. For the numerical solution of the problem we introduce dimensionless
quantities [4]:
=
, =
, = , =
, =
, =
, =
, =
,
λ1=
, λ2=
, λ3=
, λ4=
,
=
,
=
,
Ω2=/() ,
where:
00
,ET
is the room temperature and the corresponding modulus of the elasticity of the material,
is the coefficient of thermal expansion,
L
is the beam length, is the area and the moment
of inertia of the cross-section at the origin of the coordinates.
In dimensionless values, the system of equations (7) has the form:
=vR
,
= ,
= ,
=+
, (8)
= ,
=+
.
The system of ordinary differential equations (8) can be written in vector-matrix form. To do this,
let us introduce designations: =, =, =,=, =, = . (9)
Consequently, the system of equations (8) will take the form:
=+ , (10)
where:
=
, =
0
0
0
0
, =
0 0 1 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
.
00
,
x
IA
MSTU 2021
Journal of Physics: Conference Series 1901 (2021) 012116
IOP Publishing
doi:10.1088/1742-6596/1901/1/012116
5
There are six boundary conditions on both ends of the vibrating rod, from which relations between
the constants of the general solution and the frequency equation (the eigenvalue problem) can be
obtained. Thus, the forms of free vibrations and their frequencies will be established.
On the computational side, the eigenvalue problems are very similar to boundary problems, to
solve which we use the shooting method. The difference lies in the shooting not only on the missing
left boundary conditions, but also on the required eigenvalues.
The mathematical editor Mathcad uses the sbval and bvalfit functions to solve the eigenvalue
problems. The system of equations (8) is supplemented by a differential equation with a boundary
condition:
= 0 , (0)= ,
where: = , is the shooting parameter.
4. Obtained simulation results and discussion
Calculations have been performed for a free cantilever beam of rectangular cross-section. The
temperature in the cross-section changes according to the square function =+()(
),
where is the height of the cross-section
The boundary conditions have the form (0)= 0; (0)= 0; (0)= 0; (1)= 0; (1)=
0; (1)= 0. Figures 2 and 3 show the results of the calculation. For strength calculations, the first
natural frequency is essential [5]. According to the calculation, it equals to 6.2, which is about 11%
less than for an elastic rod without taking into account the axial deformation [6].
Figure 2. The dependence of the dimensionless
displacement components on the
dimensionless coordinate .
Designations: 1 - 10 ,
2 -
10
, 3 -
10
.
Figure 3. The dependence of the dimensionless
components of the internal force vector on the
dimensionless coordinate Z .
Designations: 1 - 10 ,
2 -
10
.
The cantilever has been calculated when a kinematic perturbation, varying according to the cosine
law, acts in the cross-section of the anchorage. The temperature deformations are not taken into
account in the calculations. The dimensionless peak value of the kinematic perturbation is assumed to
be 0.001. The boundary conditions have the form: (0)= 0.001; (0)= 0. ; (0)=
0. ; (1)= 0. ; (1)= 0. ; (1)= 0.
MSTU 2021
Journal of Physics: Conference Series 1901 (2021) 012116
IOP Publishing
doi:10.1088/1742-6596/1901/1/012116
6
Figure 4 shows the calculation results.
Figure 4. The dependence of dimensionless
displacement components on the coordinate.
Designations: 1 - 10 , 2 -
10 .
Figure 5. The dependence of
dimensionless vibration frequency on the
temperature (
0
C).
Designations: 1 - = 10, 2 - =
10 are the vibration frequencies at non-
uniform and uniform temperature
distributions, respectively
By numerical calculation, the value of eigenvalue Ω=0.339 is obtained; the cross-sectional force
and the bending value are actually equal to zero.
Figure 5 shows the graphs of change in the dimensionless vibration frequency when the rod is non-
uniformly heated in the cross-section, when the temperature on the one side of the rod is constant and
equal to 200С, and on the other side it changes and when the rod is uniformly heated. The boundary
conditions are: (0)= 0; (0)= 0; (0)= 0; (1)= 0; (1)= 0; (1)= 0.
5. Conclusion
The most general nonlinear equations of the rod motion in the plane under thermo-mechanical loading
have been obtained, taking into account the rotational inertia of cross-section and the deformation of
the thermo-elastic axial line. The numerical calculation method of natural vibration frequencies is
developed in the mathematical editor Mathcad. The results of calculations show that if the rod is
uniformly heated, the first natural vibration frequency decreases with increasing temperature. It has
been found that with non-uniform heating in the cross-section, the temperature gradient significantly
affects the value of the vibration frequency. It has also been found that the vibration frequency in
general is three times greater than with a uniform temperature distribution. Calculations have shown
that the difference between the vibration frequencies increases with an increase in the temperature
gradient in the cross-section. With the help of the developed method we can also determine the
resonance zones during thermo-metrical loading of rod elements of structures or buildings, which are
usually calculated according to the rod model.
6. References
[1] Valle J, Fernandez D, Madrenas J 2019 Int J Mech Sci 153-154 pp 380–390
[2] Bokaian A J 1990 Sound Vib 142 (3) pp 481–498
[3] Li X, Tang A, Xi L J 2013 Constr Steel Res 80 pp 15–22
[4] Kikvidze O 2003 Problems of mechanical engineering and machine reliability 1 pp 49–53
[5 Kiselev V A 1980 Structural mechanics. Dynamics and stability of structures (Moscow: Stroyizdat)
[6] Timoshenko S P, Yang D H, Weaver W 1985 Fluctuations in engineering (Moscow:
Mashinostroenie) (in Russian)
