Nonlinear characteristics of nonlinear vibration, linear vibration Nonlinear vibration vibration

Answer] :d first step to determine the logical relationship of the question stem.

"Linear vibration" and "nonlinear vibration" are both "vibration", and "linear vibration" and "nonlinear vibration" are contradictory relationships in the juxtaposition.

The second step is to identify the options. Bu Zhi.

Item A: "petals" and "stamens" are opposing relationships in the juxtaposition relationship, which are inconsistent with the logical relationship of the subject stem and are excluded;

Item B: "Herbivores" and "Carnivores" are both animals, but "herbivores" and "carnivores" are opposing relationships in the juxtaposition relationship, which are inconsistent with the logical relationship of the subject stem and are excluded;

Item C: "Market entity" includes "laborers", "investors" and "operators", so "investors" and "operators" are the opposing relationship types in the juxtaposition relationship, which are inconsistent with the logical relationship of the topic stem and are excluded;

Item D: "Main contradiction" and "secondary contradiction" are both "contradictions", and the "main contradiction" and "secondary contradiction" are contradictory relationships that are consistent with the logical relationship of the topic stem.

So, choose option D.

Check-in task: 1: What are the nonlinear phenomena in life?

Life presents unfairness in a non-linear way, where a small advantage in life can bring amazing rewards, and people who are good for nothing are rewarded by randomness.

The most common method in science is called the sand pile effect. Also, the wisdom of the predecessors is like a straw that crushes a camel.

The sand pile effect, where a grain of sand destroys the entire structure of the castle, what we see here is that a linear force creates a non-linear effect on an object.

On the Internet, it is often seen that the sudden death at work at a young age is obviously the result of the usual overwork.

Why actors and writers are not particularly suitable for most professions (think in terms of randomness and probability).

Actor - the formation of fame has its own dynamic process, the actor is known by another group of people because one group of people knows him, this fame is like a propeller and the starting point is in the audition room. He will be chosen with a lot of randomness, probably for a variety of reasons, such as a ridiculous detail that happens to match the examiner's taste, or because of his looks - or because the examiner is in a good mood, and many other reasons.

Thinking from the perspective of probability, the unsuccessful actors are the vast majority, and the selected ones are very few.

Writers are also similar to actors, which are not suitable for most people, and writers also need to be famous before there will be a market. However, the world is full of polarization, and sometimes everyone rushes to publish your book, and sometimes no one cares. The one who wins is all-en-all, and the loser is difficult to survive!

What inspires you?

In my own field, the condition for success is strength + luck (randomness), which is very different from the investment field, in order to increase the probability of success, then it is necessary to spend time on the technology to polish, with this it is possible to compete on a rebate platform, just like the actor who goes to audition, you must meet the conditions to enter the door waiting for the audition, and only with strength can you have the possibility of being chosen by others. There is not such a song titled "Love Will Win", there is such a sentence in the lyrics: Three points are destined, and seven points depend on hard work.

A sentence that impressed?

The arrangement of the letters on the typewriter is an example of the least competent winning the letters, and the letters on the typewriter keyboard are not arranged in the most ideal order. The current order slows down typing, rather than making it easier for us to type, because the ribbon of mechanical typewriters was prone to jamming in the first place.

1.Natural frequency characteristics.

The natural frequencies of a linear system do not depend on the initial conditions of the motion, but only on the parameters (mass and stiffness) of the system. This is not the case with nonlinear vibrating systems. Since the stiffness varies with the magnitude of the deformation, the natural frequency of the system also varies with the magnitude of the motion.

A spring whose stiffness increases with deformation is called a hardening spring; Otherwise, it is called a softening spring. The natural frequency of a progressively hardened nonlinear system increases with the increase of amplitude. The opposite is true for progressively softening nonlinear systems.

2.Self-excited vibration.

When a nonlinear autonomous system has equivalent negative damping, adjust the steady-periodic vibration that exists when the equivalent damping is reduced to zero. An autonomous system is a system in which time is not evident in the differential equations of motion.

The divergent type corresponds to the negative damping situation, the conservative type corresponds to the undamped situation, and the attenuation type corresponds to the positive damping situation. It is only in the conservative case that the motion of the system is harmonious, forming a family of periodic motions with a continuous amplitude distribution (i.e., not isolated) according to the magnitude of the energy.

In a nonlinear autonomous system, in addition to the non-isolated periodic motion in the conservative case, there may also be an isolated periodic motion in the non-conservative case. When the damping is nonlinear, the damping coefficient varies with the motion, so it is possible that the equivalent damping is negative at small amplitudes. At large amplitudes, the equivalent damping is positive; At an intermediate amplitude, the corresponding equivalent damping is zero, and correspondingly, there is a constant periodic vibration, which is called self-excited vibration, or natural vibration for short. This vibration is isolated, and its amplitude change and period depend only on the system parameters and are independent of the initial state within certain ranges.

The natural vibration of a weakly nonlinear system is close to harmonic; The natural vibration of a strong nonlinear system is far from harmony. In the latter, the process of slowly accumulating energy and the process of releasing energy almost instantaneously alternates, so it is figuratively called relaxation vibration.

3.Jumping phenomenon.

The amplitude of a nonlinear system.

a) The curve of the harmonic and external disturbance frequency ( ) can have several branches, slowly changing the disturbance frequency, and the amplitude can change abruptly at some frequencies. Unlike linear systems, differential equations that describe nonlinear systems may have multiple periodic solutions under the same set of parameters. Only those solutions that satisfy the stability condition correspond to physically achievable motions. In nonlinear systems, the diversity and stability of motion cannot be ignored.

The phase plane method is commonly used. The differential equation of motion of the second-order autonomous system is written: where p(x,y) and q(x,y) are real analytic functions.

By subtracting the variable t from the equation, we get that x, y are regarded as the Cartesian coordinates of a point in the plane, which is called the phase plane, and the points (x, y) are called the phase points. The phase point describes the state of motion of the system at a certain moment.

Corresponding to any particular motion of the system x=x(t), y=y(t), there is a definite curve on the phase plane, called the phase rail. Phase-orbit describes the entire state of motion of a system. In the phase plane, all points where p(x,y) and q(x,y) are zero at the same time are called singularities.

In dynamics problems, the singularity corresponds to the equilibrium state of the system. A singularity, if the integral curves from its neighborhood all converge to it, or stay in its neighborhood, it is called a stable singularity; Otherwise, it is called an unstable singularity.

Of particular importance is one of the isolated orbits of second-order autonomous systems, which is called limit loops. Phases that depart from any point in the neighborhood on one side of it either approach it or leave it. A limit ring is stable if the phase rails in the inner and outer neighborhoods are close to it; Otherwise, it is unstable.

The stable limit loop corresponds to the natural vibration in the physical system. The fundamental difference between the limit ring and the free orbit closure of a conservative system is that the limit loop is isolated, i.e., there are no other orbits in its vicinity; The periodic vibrations corresponding to the limit ring do not depend on the initial conditions of the system.

The most commonly used quantitative method is the average method. Investigate single-degree-of-freedom nonlinear autonomous systems:

Derivative of Eq. (2) to t yields:

Eq. (4) compared to Eq. (3) yields:

Eq. (3) is derived for t, then there is:

Substituting Eq. (6) into Eq. (1) yields:

From Eq. (5> and Eq. (7), it can be solved:

As an example, examine the Rayleigh equation:

The stationary solution that can be solved from Eq. (10):

The former corresponds to:

an unstable singularity in the phase plane, i.e., an equilibrium state corresponding to the instability of the system; The latter corresponds to a stable limit ring in the phase plane, i.e. to a stable self-excited vibration of the system.

Random vibration simulation analysis is used to determine the response of a structure to random loads. At present, random vibration analysis is widely used in the design of vehicles, civil structures, and airborne electronic equipment. Unlike deterministic vibration, random vibration obeys the laws of probability and statistics, which are described by probability and statistical methods, and only the statistical values of physical quantities (mean, root mean square, standard deviation) can be known.

The research object of nonlinear system theory is nonlinear phenomena, which reflect the nature of the motion of nonlinear systems, and cannot be explained by the theory of linear systems. The main nonlinear phenomena include frequency dependence on amplitude, multi-value response and jump resonance, split-harmonic oscillation, self-excited oscillation, frequency capture, asynchronous suppression, bifurcation and chaos. This nonlinear phenomenon occurs only in the free oscillation of a class of nonlinear systems.

A well-known example is the Doufen equation m + f + kx + k'x3=0

The free oscillation of a class of mechanical systems described. where m is the mass of the weight, x is the displacement of the weight, the first and second derivatives of x, and f is the viscous friction coefficient of the damper, kx+k'x3 represents the nonlinear spring force. The parameters m, f, and k are all positive constants.

Parameter k'is called a hard spring, k'When it is negative, it is called a soft spring. After an initial displacement of the weight, the system oscillates freely. From the experiments, it can be observed:

In k'is positive, and the frequency value increases with the decrease of the free oscillation amplitude. In k'When negative, the amplitude decreases with the free oscillation, and the frequency value decreases. Figure 2: k'The waveform at =0 has 7 peaks with equal spacing, indicating that the frequency does not change with the decrease of amplitude, k'At >0, the time to reach the 7th peak was k'Short at =0; It is shown that the frequency increases with the decrease of amplitude; k'At <0, there are only 6 waves in the same amount of time, indicating that the frequency decreases with decreasing amplitude.

In 1963, the meteorologist Lorenz first discovered a nonlinear phenomenon in numerical experiments on atmospheric convection models to study weather forecasting problems. It is characterized by the fact that some nonlinear systems become very sensitive to the initial conditions within certain parameters, resulting in non-periodic, chaotic-looking outputs. Later, chaotic phenomena were also found in studies such as ecosystems.

Since the 80s, the study of chaos has become a very active field, with some rigorous mathematical results, but more computer experiments, and real physical experiments.

Nonlinear system: A system whose output is not proportional to the input.