The work focuses on mechanical systems with lumped parameters. Using Lagrange’s formalism, the governing equations of motion are developed.
The generalized external force acting at each coordinate may either arise from imposed motion or be an externally applied dynamic load. The system of equations is expressed in compact matrix form:
where and denote respectively the inertia, damping, and stiffness matrices of order The perturbation is described by the matrix of the column {f}. The physical meaning of the matrix coefficients is as follows: is the component of the amount of motion along at unit velocity along the damping force along j at unit velocity along the elastic force along due to unit displacement along .
For oscillations defined by acceleration, this expression looks like this:
where is the angular frequency, is the corresponding frequency. If all excitation frequencies are integer multiples of a base frequency, the oscillations are strictly periodic; otherwise, the motion becomes quasi-periodic.
Accelerations can be determined from velocities by multiplying them by , or, in complex form, by , where the imaginary Eq. (3) introduces a phase shift of .
Steady-state oscillations are considered an approximate mathematical model. If the amplitudes and phase parameters or vary slowly, the resulting motion may be regarded as quasi-periodic. For strictly periodic oscillations, typically only amplitude values or their Root Mean Square (RMS) equivalents are taken into account, while phase angles are often omitted.
Instantaneous accelerations and the magnitudes of harmonics at the same frequency are summed vectorially. The phase difference plays a key role in resonance analysis, particularly when experimentally determining resonant frequencies. In the vicinity of resonance, the phase changes more rapidly than the frequency itself. Vector of external forces Eq. (2) describes the system’s free vibrations; when it corresponds to forced oscillations.
Free oscillations of dissipative systems. Let us consider a linear dissipative system whose motion is described by the matrices [M], [C], and [K]. The solution of Eq. (3) can be represented as where is a complex number, is a complex matrix of numbers - column. The numbers λ are called characteristic indicators, and the numbers (or - ) are called complex frequencies. Characteristic indicators must be the roots of the characteristic equation.
The defining expression in its expanded form can be presented as follows:
A mechanical system with n degrees of freedom is characterized by a total of 2n characteristic indicators
If these characteristic indicators form the simple roots of Eq. (5), then the general solution of Eq. (2) is represented as a superposition of the particular solutions given in Eq. (5).
Here are arbitrary complex constants, are columns of the numerical matrix.
Let's express the characteristic levels as follows:
Here and represent the damping coefficients and the natural frequencies of the dissipative system, respectively. If and satisfy Eq. (5), then their complex conjugate values are also solutions for this equation. In the absence of damping, all roots are located on the imaginary axis, and in the presence of damping, their real parts shift slightly to the negative side, i.e., to the left half-plane.
The corresponding eigenvectors fulfill the orthogonality relations:
where the superscript denotes transposition.
When orthogonality can also be ensured for multiple roots by appropriately choosing eigenvectors.
The damping coefficient has a unit of measurement To represent the degree of damping, you can introduce a convenient dimensionless parameter, such as:
with the help of the logarithmic decrement, it is possible to draw accurate conclusions about the degree of attenuation, dynamic stability, and changes in amplitude around resonance.
The attenuation decrement is one of the main parameters characterizing the intensity of oscillations around: dynamic stability and resonance states.
If is positive, then and represent the degree of decrease in the amplitude of oscillations, respectively. The relationship between the attenuation coefficient and the amplitude ratio is given by the following expression.
For systems with symmetric mass, damping and stiffness matrices the task reduces to finding the complex eigenvalues:
and the corresponding complex eigenvectors:
which satisfy Eq. (5). In comparison with the undamped case, the determination of complex eigenvalues is significantly more complicated and has been studied to a lesser extent. In a conservative system (
When dissipation is incomplete, some eigenvalues remain on the imaginary axis while others shift to the left, so the system includes periodic modes corresponding to undamped degrees of freedom. If dissipation is negative, certain eigenvalues acquire positive real parts (
Position of characteristic indicators for conservative, dissipative, and negatively damped mechanical systems; a. conservative, b. with complete dissipation, c. with incomplete dissipation, and d. with negative dissipation.
In a stationary problem, the vector function changes according to the harmonic law with a given frequency and amplitude Initial conditions are not set. Instead, the solution must satisfy the periodicity condition at the excitation frequency This requirement leads to an algebraic system of equations formulated with respect to the complex components of the unknown vector
The System (15) with complex coefficients can be solved, for example, by the Gauss method. A linear oscillatory system is considered, the position of which in space is determined by generalized coordinates It is assumed that generalized forces of two types act on the system: 1) these are forces linearly dependent on the generalized coordinates, and this dependence is hereditary:
Here generalized forces of the hereditary type; are such known constants that the quadratic
form is positively determined, and are known influence functions, and 2) these are generalized
forces, obviously dependent on time, and this dependence is harmonic. Due to the linearity of the problem, it is sufficient to consider the case when all apparently time-dependent generalized forces have equal periods and phases:
The Lagrange Eq. (17) for the system under consideration have the form:
In them - coefficients of a positively defined quadratic form The problem consists of finding the
periodic solution of the System (18). We reduce the System (18) to the normal coordinates of the elastic system obtained from Eq. (18) at . For this purpose, we introduce the transformation of generalized
coordinates. where are the normal coordinates, are the transformation matrix
coefficients, and The System (18) in the coordinates takes the form:
where are the natural frequencies of the elastic system, and are the amplitudes of the generalized
forces corresponding to the normal coordinates, each of which is determined by the relation
where is the influence function: The periodic solution of the System (19) is sought in the
form:
where are constants determined from the system of transcendental equations:
there
The System (21) was obtained by substituting Eqs. (19) and (20) and comparing the coefficients for sinpt and cospt. In this case, identity is used.
where The System (21) can be considered as a linear algebraic system with respect to the unknowns After determining , we will find the sought constants from the relations:
The problem is significantly simplified in the case when the influence functions are proportional, and the proportionality coefficients are the generalized stiffnesses , i.e., in the case when
An example of such a system can be an absolutely rigid body installed on shock absorbers with identical hereditary-elastic parameters. In the considered case, the System (19) takes the form:
The solution of this system is expressed by Formula (14), in which
Consider the forced oscillations of a system with n degrees of freedom described by Eq. (11). The application of the Fourier transform to differential Eq. (16) leads to a system of linear algebraic equations.
where and are matrices - image columns corresponding to the matrices and and the vectors and are defined by the initial conditions. The resulting system is solved using the Gauss method with the isolation of the main element. Applying the inverse Fourier transform gives the desired solution. Apply to one-dimensional equations of motion written in the form:
Fourier transform. For this, we multiply the equation by and integrate within Then:
or after integration by parts, from where:
To determine from the found function one can use the subtraction method or decomposition into elementary fractions, as well as the inverse Fourier transform.