An essentially new method for non-destructive testing of elastic electrically conductive rods using non-vortex electromagnetic induction is proved theoretically. An experimental technique for defining a location of a cross crack is offered.
Developing efficient and highly precise methods for diagnosing products is a relevant science task: non-destructive testing allows defining timely defects of elements of important constructions. There are several methods of non-destructive testing: magnetic-particle, eddy-current, liquid penetrant, acoustic, optical, radiation [1,2]. The Shock Pulse Method is widely applied [
The authors from Kazan [
Akhtyamov A.M. and Karimov A. R. [
In this work, an essentially new method is offered, which lets define experimentally frequencies of the normal mode of a rod’s longitudinal oscillation and determine the location of a crack.
Parts of a rod shape are often used in engineering: different shafts, axes of wheel sets, stocks, etc. The main type of defects for them is cross-cracks. A rod with a cross-crack is modeled by the system shown in
The natural oscillation in the system shown in
Let us write down boundary condition for the left part
of the rod:
where
We consider quasisolid motion of a rod (without any internal deformations). The differential equation looks like this if you do not consider the mechanical resistance:
The quasisolid motion of a body occurs with a frequency:
Since
However, it is impossible to define
Let us consider now elastic oscillation initiated in a rod taking into account the linear internal resistance, which is characterized by the coefficient
where
We apply the Fourier method to the Equation (5):
Here
where
Taking into account the terms of orthogonality:
Let us define the set of damped frequencies:
Using the boundary condition on the right end of the part 1, we obtain the equation of frequencies for damping oscillation:
Let us consider the Equations (4) and (10) together with
The values of frequencies
In case of absence of a crack, quasisolid motion of any of its parts is excluded. Then, the first frequency of damping elastic oscillation can be defined by the wellknown equation:
Thus, if the first frequency measured in an experiment coincides with the calculated value (12), we can conclude that the defect as a longitudinal crack is absent.
Note that the linear theory does not always take into account internal friction properly. Besides, the precise value of a coefficient
Let us state the concept of the electromagnetic way of defining natural frequencies of longitudinal oscillation of an elastic rod. We use the recently discovered potential component of the magnetic field [6-9]. It is described by the scalar functions
Conditions, under which a SMF can be created, and experiments with it are described in details in the monograph [
In the monograph [
where
It can be seen from the Equation (14) that transverse electromagnetic force is formed due to interaction of an intrinsic vector magnetic field of a conductor with an external vector magnetic field, whereas longitudinal force is formed due to interaction of an intrinsic scalar magnetic field of a conductor with an external SMF. Such an approach corresponds well to the common field theory and the Helmholtz’s theorem, because the superposition principle of solenoidal and potential components holds.
The certain similarity between phenomena of vortex and irrotational nature is observed. Along with the Ampere’s force, which is perpendicular to the current, the magnetic force was discovered, which is directed along the current or against it depending on the sign of an external SMF. The phenomenon of irrotational electromagnetic induction [
where
This phenomenon can be used for experimental studying of the process of longitudinal oscillation in a resilient rod. Let us look at the task on longitudinal oscillation of an electrically conductive rod in a SMF. We assume that an external scalar magnetic field is created on some (active) section of a rod
Let us write the differential equation for longitudinal oscillation of a resilient electrically conductive rod, based on the D’Alembert’s principle:
(16)
The first term in the equation represents the fictitious force, the second term characterizes the elastic force, and the third one corresponds to the internal dissipation. The last term in the equation represents the electromagnetic force according to the law (14).
In the works [6-8], it is shown that the SMF
Consequently:
The beginning of the crosshatched coordinate coincides with the left boundary of the active section. And
Taking into account the Equation (15), we have:
(17)
Note that this force has infinite values at the ends of the active section. This is because we used the model of a linear conductor, i.e. we omitted transverse sizes when calculating the SMF. Moreover, it was assumed that the SMF declines unevenly from some value to zero at the boundaries of the active section. Later, we will analyze a case close to real, when the SMF is heterogeneous and its strength equals zero at the ends of the active section. In this case, there is no uncertainty when calculating the longitudinal magnetic force.
Taking into account the Equation (17), the Equation (16) looks like this:
Let us apply the Fourier method using decomposition (6). We get:
Using amplitude functions (7) and the terms of orthogonality, we get the system of ordinary differential equations:
We introduce some notation:
We get:
Let us factorize the system (20) and rewrite it, so the first equation contains the first term of the sum, and the second one has two terms, etc.:
We can extract the equations for damped frequencies from here:
where
is a damping factor of kth partial oscillation.
It can be seen that the process is polyharmonic, i.e. there is oscillation of many frequencies. Every oscillation corresponds to a certain electrical signal appeared in a closed circuit. Using a frequency analyzer spectrum, we can extract and measure every frequency that is presented in an experiment. Thus, we can use (11) to calculate the length Lz, which defines the location of a crack.
It can be seen from the Equations (20) and (21) that there is no electromagnetic effect on such oscillation at
Thus, it was shown that it is possible to define the presence and location of a transverse crack in an electrically conductive rod. However, the case considered is idealized: in reality, it is impossible to create a homogeneous SMF with sharply defined boundaries.
Let us analyze a case, which is more common, when the SMF is heterogeneous
If we apply the Fourier method, we get:
We use eigen amplitude functions (7). We multiply the Equation (24) by
Let us introduce some notation:
Taking into account the notation introduced, the Equations (25) will look like this:
We factorize the system of Equations (28):
We write the equation for the set of natural damped frequencies:
where
is a damping factor of kth partial oscillation.
Let us analyze the particular case. Let the external SMF be distributed within the active section according to the following function:
Here
It can be seen on a diagram that the external SMF
We can see from the Equations (28) and (29), that at
(32)
electromagnetic effect on a kth oscillation is absent. Therefore, at a given location and width of an active sec-
tion, electrical signals, which correspond to some shapes of oscillation, are not present. In order to record effectively all natural frequencies, we need to change the location of the active section or its width several times.
The differential Equations (28) are more accurate comparing with (20), because they take into account the real distribution of a SMF.
We describe the experimental installation and the methods of the experiment. The part analyzed has to be made out of electrically conductive material and have a shape similar to a rod. The geometry of a transverse section can be different if cross section dimension is significantly smaller than the length of the part. In addition, the cross sectional dimensions have to meet another requirement: an external SMF has to be as homogeneous as possible within the range of transverse coordinates.
The part is hung horizontally on elastic ropes. A vertical stop clamp fixes one of the part’s ends. It is necessary to eliminate electric contact between the part and the clamp. Conductors with small impedance are attached to the ends of the rod, which are enclosed in a frequency spectrum analyzer. The instrument used has to register small current impulses (µA) in a range of low frequencies (kHz).
The conditions of SMF initiation, its characteristics and topology are described in the monograph [
alent current (shown as the arrow (
In order to create a quite strong SMF, an inductor as a toroidal coil can be used. A vortex magnetic field in this case is concentrated inside of the coil, and a SMF of different signs is formed on the edges of the toroid.
Cross sectional dimensions of the studying part cannot exceed sizes of the external SMF created by the inductor. The principal device of the experimental installation is shown in
A rod hanging horizontally sets against a heavy vertical wall. The end of the rod is fixed in a stop, so the rod cannot move away from the wall while testing. Conductors with small resistance are attached to the ends of the conductors, and are connected to a frequency analyzer. In the left side of the figure, there is a striker. Its transverse section has to exceed cross sectional dimensions of a part tested. This provides plane shift of sections of the rod at elastic oscillation. An inductor of a SMF is situated so the rod’s transverse section is entirely inside of the SMF in some areas.
Let us describe the experiment technique (method) stage by stage.
Define the location and the width of an active section for an inductor of SMF that is used.
Create elastic oscillation by hitting horizontally the free end of the rod several times.
Measure first several frequencies starting with the frequency of quasisolid motion. It is necessary to change the location of the inductor several times in order to record oscillation of several first shapes without missing any.
Check if the first frequency coincides with the value calculated by the Equation (13). If yes, we conclude that there is no crack.
Calculate
Let us give a numerical example. The following parameters are given: the maximum value of induction of an external SMF is
the elastic modulus of steel is
length of the rod studied is
We calculate the first frequency of oscillation of the defect free rod using the Equation (13):
We assume that the following frequencies were defined during the experiment:
and
We solve the Equation (13) numerically and get:
Physical phenomena connected with a potential component of a magnetic field are discovered quite recently. The knowledge on new electromagnetic phenomena allows us to find innovative technical and technological solutions of applied problems.
The method of non-destructive testing, which we offered, uses the phenomenon of non-vortex electromagnetic induction. That makes it fundamentally new and differs from other known methods. It allows carrying out non-destructive testing of a rod using a quite simple experimental device, and defining the location of a crack.
The work is carried out within the frameworks of the RFFR grant No. 13-01-90904.