Determination of the dimensions of the cylindrical weight at the end of the rod. Multiphase Systems. 14 (2019) 3. 214–217 (in Russian).
Determination of the dimensions of the cylindrical weight at the end of the rod
Mavlyutov Institute of Mechanics UFRC RAS, Ufa
This article discusses free transverse vibrations of a homogeneous rod. The left end of the rod is clamped, and a
cylindrical weight is concentrated at the right end. The eigenfrequencies of the rod vibration are known. The purpose
of this work is to determine the parameters of the end cylindrical weight of the rod (mass, moment of inertia, length
and radius) by the natural frequencies of the rod vibrations. We use a partial differential equation derivative of the
fourth order to solve this problem. This equation and boundary conditions are reduced to a spectral problem. To
find the mass and moment of inertia of the weight, the «Method of an additional unknown» was applied. In the
characteristic determinant of the spectral problem, there are terms that contain products of unknown coefficients.
The essence of the «Method of an additional unknown» is that some of these products are proposed to be considered
new additional unknowns, through which the rest can be expressed. It is shown that the mass and moment of inertia
of the weight can be found using the three natural frequencies of the rod vibrations. Formulas for finding the length
and radius of a cylindrical weight are obtained, and corresponding examples of finding unknown parameters are
Problem: To determine the parameters of the end cylindrical weight using the natural
frequencies of the rod’s vibrations. The parameters are the mass and moment of inertia, length and radius of the cylindrical end weight.
Methods: This article discusses the free bending vibrations of a homogeneous rod. The left end of the rod is sealed,
and a cylindrical weight of mass m1 and moment of inertia I1 is concentrated on the right end.
Length h and radius r2 are the dimensions of the cylindrical weight. These parameters are considered unknown.
The eigenfrequencies of vibrations of the rod is used as known acoustic data.
To solve this problem, we use a fourth-order partial differential equation that describes the bending vibrations of the rod that
has a constant bending stiffness. The equation and boundary conditions are reduced to the problem with the spectral parameter λ
by replacement and variable separation method. Next, we find the characteristic determinant of the spectral problem. In the characteristic
determinant containing unknown coefficients of the boundary conditions, there are also products of unknown coefficients. To find the mass
and moment of inertia of the weight, the method of additional unknown quantity is used. The essence of the method is that some of these
products of unknown coefficients are proposed to be considered new additional unknowns, through which the rest can be expressed.
Thus, new coefficients are introduced: a1 = m/(ρFL), a2 = I1/(ρFL),
a3 = a1a2. Substituting three known eigenvalues into the characteristic
determinant we obtain a system of three equations from three unknowns. To solve the system, we use Kramer's method and find the
dyenhrtcoefficients a1 and a2. Using the known formulas for determining the mass and moment of
inertia of the cylinder, as well as knowing the coefficients a1 and a2, we find the length and radius of the weight:
h = a12r12ρL/(2a2ρ1L2 -
r2 = √((2a2L2 -
where ρ1 is the density of material of weight, ρ is the density of the material of the rod,
Lis the length of the rod, r1is the radius of the rod (inner radius of the cylinder).
In a study was determined:
The mass and moment of inertia of the cylindrical weight attached to the right end of the rod are found by three natural frequencies of bending vibrations of the rod.
Formulas for finding the length and radius of the end cylindrical weight are obtained.
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