, ( 1)
Where a is the maximum value of displacement x, called amplitude, which represents the intensity of vibration; ωn represents the amplitude-angle increment of vibration per second, which is called angular frequency, also called circular frequency; It is called the initial stage. F=ωn/2π represents the number of cycles of vibration per second, which is called frequency; Its reciprocal, T= 1/f, represents the time required for a period of vibration, which is called a period. The amplitude a, frequency f (or angular frequency ωn) and initial phase are called simple harmonic vibration.
Figure 1 simple harmonic dynamic curve
As shown in fig. 2, a simple harmonic oscillator consists of a lumped mass m connected by a linear spring. When calculating the vibration displacement from the equilibrium position, the vibration equation is:
Where is the stiffness of the spring? The general solution of the above formula is (1). When t=0, a sum can be determined by initial position x0 and initial speed:
But ωn is only determined by the characteristics m and k of the system itself, and has nothing to do with the external initial conditions, so ωn is also called natural frequency.
Fig. 2 Single degree of freedom system
For a simple harmonic oscillator, the sum of kinetic energy and potential energy is a constant, that is, the total mechanical energy of the system is conserved. In the process of vibration, kinetic energy and potential energy are constantly transformed into each other. Vibration with decreasing amplitude due to friction, medium resistance or other energy consumption. For micro-vibration, the velocity is generally not very large, and the medium resistance is directly proportional to the first power of the velocity, so it can be written that c is the damping coefficient. Therefore, the single-degree-of-freedom vibration equation with linear damping can be written as:
, (2)
Where β=c/2m is called damping parameter variable. The general solution of formula (2) can be written as follows:
. (3)
According to the numerical relationship between ωn and β, it can be divided into the following three situations:
①ωn & gt； β (small damping case) particle generates damping vibration, and its vibration equation is:
Its amplitude decreases with time according to the exponential law shown in the equation, as shown by the dotted line in Figure 3. Strictly speaking, this kind of vibration is aperiodic, but the frequency of its peak can be defined as:
It is called damping rate, where is the vibration period. The natural logarithm δ of amplitude reduction rate is called logarithmic reduction rate; Obviously, δ = β, where =2π/ω 1. The sum of δ is determined directly by experiments, and C can be obtained by using the above formula.
② (Critical damping case) At this point, the solution of formula (2) can be written as:
In the direction of initial velocity, it can be divided into three vibration-free situations, as shown in Figure 4.
③ωn & lt； The solution of β (large damping case) formula (2) is shown in formula (3). At this point, the system no longer vibrates. Vibration of the system under constant excitation. Vibration analysis is mainly to investigate the response of the system to excitation. Periodic excitation is a typical repetitive excitation. Because periodic excitation can always be decomposed into the sum of several harmonic excitations, according to the superposition principle, the total response of the system to periodic excitation can be obtained as long as the response of the system to each harmonic excitation is obtained and then superimposed. Under harmonic excitation, the differential equation of motion of a single-degree-of-freedom damping system can be written as:
Its response is the sum of two parts, one part is the response of damping vibration, which decays rapidly with time; Another part of the forced vibration response can be written as:
Fig. 3 Damping vibration curve
Fig. 4 Three initial condition curves of critical damping
formula
H/F0=H (), which is the ratio of steady-state response amplitude to excitation amplitude, and represents amplitude-frequency characteristic or gain function; ψ is the phase difference between steady-state response and excitation, which represents the phase-frequency characteristic. Their relationship with the excitation frequency is shown in Figures 5 and 6.
It can be seen from the amplitude-frequency curve (Figure 5) that the amplitude-frequency curve has a single peak under the condition of small damping; The smaller the damping, the steeper the peak value; The frequency corresponding to the peak value is called the * * * vibration frequency of the system. In the case of small damping, the vibration frequency of * * * is not much different from the natural frequency. When the excitation frequency is close to the natural frequency, the amplitude increases sharply, which is called * * * vibration (resonance). * * * When vibrating, the system gain takes the maximum value, that is, the forced vibration is the most intense. So in general, we always try to avoid * * * vibration, unless some instruments and equipment use * * * vibration to obtain a wide range of vibration.
Fig. 5 Amplitude-frequency curve
It can be seen from the phase-frequency curve (Figure 6) that the phase difference ψ=π/2 at ω0 can be effectively used for * * * vibration measurement without considering damping.
In addition to steady-state excitation, the system sometimes encounters unsteady excitation. It can be roughly divided into two categories: one is sudden impact. The second is the lasting influence of arbitrariness. Under unsteady excitation, the response of the system is also unstable.
A powerful tool for analyzing unsteady vibration is the impulse response method. It describes the dynamic characteristics of the system through the transient response of the system unit pulse input. The unit pulse can be expressed by δ function. In engineering, the δ function is usually defined as:
Where 0- represents the point on the T axis that tends to zero from the left; 0+ indicates a point from the right towards 0.
Fig. 6 Phase-frequency curve
Any input of Figure 7 can be regarded as the infinitesimal sum of a series of pulses.
The system response h(t) corresponding to the unit pulse acting at t=0 is called the impulse response function. Assuming that the t< system is static before the pulse action, when T
According to the superposition principle, the total response of the system corresponding to x(t) is:
This kind of integral is called convolution integral or superposition integral.