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In this paper we consider of natural oscillations cylindrical bodies with external friction. Complex rates changes from friction parameters are shown. Rate equations are solved numerically—by method of Muller.

Simulation of vibrations of bodies located in the deformable medium is studied with many scientists and by various methods [_{0} and R at deformable surroundings are modeled (

Study fluctuations pipeline located in an elastic medium are considered different methods [

In this paper, fluctuations pipelines are modeled as a cylindrical body with a radius

tion. The main goal of the work is to study the natural oscillations a cylinder with external friction. In the study mentioned above, the optimal values of damping coefficients, in which the oscillations are, damped pipelines as possible.

Consider the problem of the oscillations of an infinite elastic cylinder with external friction at the interface (

where

Consider the problem in cylindrical coordinates

With boundary conditions:

at

where R-radius of the cylinder;

Then we call the boundary value problem (2a)-ant plane and (2b)-flat or planar problem of oscillations of a cylinder.

We call the natural oscillations of an elastic cylinder solution of (2a) and (2d) (f = 0) for ant plane case types:

problem (2b and 2d) for the planar case types:

where

Ant planar

It can be shown that the conditions for the finiteness of all the unknowns in the center for a solid cylinder equivalent to the following boundary conditions-for flat cylinder oscillation:

At n > 1 can be set equal to zero any set of two unknowns. Indeed, if the conditions of the limb in the center of the cylinder, then

where

Then substituting (5a) to (4a) we obtain a system of four equations. Equating them coefficients of like powers of r we obtain the recurrence relation:

At

If ant planar cylinder oscillation: the n = 0

where

Substituting (5b) in (4b), we obtain a system of two equations. Equating them to obtain coefficients of the same recurrence relation:

Then for

The general solution of the system of Equations (2a, 2, c) can be expressed in terms of Bessel and Neman functions n―the order:

where A, B―arbitrary constants;

When substituting (6) into (2c and 2d), we obtain the characteristic equation for w:

Theorem. Let the Eigen values

Simple. Then the system

Proof. To prove the theorem using the definition from [

where

Definition 1 is the number of

Definition 2 will be called the boundary conditions (9) normalized if any n boundary conditions, they are equivalent, i.e., received (9) are linear combinations of not less than the total order. Given the total order of the boundary conditions (9) is called a total order of the boundary conditions, we obtain from (9) after normalization.

Rewrite Equation (8a) and boundary conditions (8b) in the form (9). To make the replacement is necessary, so that the new unknown changed from zero to unity.

We introduce

We introduce the coefficients

Boundary conditions:

c = 1 ? total order.

The characteristic equation for (9) has the form:

Its roots are denoted

If the roots of the characteristic Equation (12) is simple, and the coefficients

fundamental system of solutions

where

of sector and the sequence of the system of recurrence equations.

We represent the solution of our problem in the form (14):

The eigenvalues of the problem (9), (10) are determined by the zeros of the characteristic determinant

Expanding the determinant, we obtain:

where

Then

We introduce the definition of [

Definition 3. Boundary value problem (9), (10) is said to be regular if all the coefficients

Let M denote the smallest convex polygon containing the point

Definition 4. Regular boundary value problem is said to be strongly regular if the zeros of the characteristic determinant

And obtained Equation (9), the coefficients

We assume without loss of generality that

The equation becomes:

At

Hence the zeros of the characteristic determinant

The results of calculations are given in dimensionless system of units in which the value of the shear modulus m, density cylinder

Then, after the stand (17) (7) have:

where

With the growing number of fashion maximum value increases, the value of a, to which he achieved increases, while remaining less than one. Meaning a, which peaks are shown in

Model number | The rooms of harmonics | ||||
---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | |

1 | 0.90 | 0.86 | |||

2 | 0.94 | 0.94 | 0.93 | 0.90 | 0.95 |

3 | 0.98 | 0.98 | 0.98 | 0.92 | 0.95 |

4 | 0.99 | 0.99 | 0.98 | 0.97 | 0.96 |

Mode number | The rooms of harmonics | ||||
---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | |

1 | 2.795 | 2.506 | |||

2 | 1.319 | 1.561 | 1.871 | 2.371 | 2.246 |

3 | 1.635 | 1.760 | 1.903 | 2.039 | 2.012 |

4 | 1.796 | 1.906 | 2.021 | 2.142 | 2.200 |

with one branch tend to zero

For n = 3 also exists a critical value

A similar case is obtained for n = 4. At the critical value

Thus for all n there is considered imaginary branch of the natural frequencies of the parameter

Solution of the resulting task was carried out by separation of variables and note to the solution of the transcendental equation. All results of the calculations are given in dimensionless system of units in which the value of the shear modulusm, density cylinder

For n = 0 the problem is divided into two independent tasks. In this case, as in the case ant planar cylinder oscillation, depending on the real parts of the eigen values have the form of smooth decreasing steps with a maximum angle of inclination of the tangent in the case of radial oscillations in the interval

As in the case anticline, with increasing numbers of harmonics increases the maximum value, except for the first radial root (see

For all the above cases, for n > 1, there is a clear separation of the roots into two types. The differences between these types of roots appear as a character of the dependence of the eigen values of the parameters of the external friction, and in the value of the form. For example, for the first root of the first harmonic-“twist”, the maximum value of the real part of the radial component of the voltage waveform is about three times smaller than the value of the real part component forms torsion stresses. For the second root of the first harmonic-“radial” ―the maximum value of the real part of the radial component of its own form of stress three times more than the maximum torsion component (see

Mode number | ||||
---|---|---|---|---|

1 | 10.3 | 1.7 | 0.9 | 0.95 |

2 | 4.08 | 1.7 | 1.09 | 0.95 |

3 | 4.1 | 1.7 | 1.23 | 0.95 |

4 | 4.12 | 1.7 | 1.46 | 0.98 |

ues-lose to a linear form (see

Values | Considering the first mode | Considering the second mode | |||
---|---|---|---|---|---|

1 | 3.287 | 1.7 | 8.218 | 0.905 | |

2 | 12.415 | 1.717 | 16.272 | 0.952 | |

3 | 25.48 | 1.727 | 27.39 | 0.978 | |

vibrations of the cylinder there is no expression of the growth of the maximum values with increasing mode and shift it to the axis

For all the cases considered flat fluctuations atn > 1, there is a clear separation of the roots into two types. The differences between these types of roots appear as a character of the dependence of the eigen values of the parameters of the external friction, and in the value of the form. For example, for the first root of the first harmonic-“twist”, the maximum value of the real part of the radial component of the voltage waveform is about three times smaller than the value of the real part component forms tensional stresses.

The imaginary parts of their own forms of valid order of magnitude smaller and do not have such a pronounced difference. When changing

For all cases considered anti plane oscillations n there imaginary branch of the natural frequencies of the parameter