The elasticity of the components in the system obeys Hooke's law, and the damping force generated when moving is proportional to the first power of the generalized speed
Linear system is usually an abstract model of micro amplitude vibration of actual system.Linear vibration system is applicablesuperposition principle That is, if the system response is y1 under the action of input x1 and y2 under the action of input x2, the system response under the combined action of input x1 and x2 is y1+y2.On the basis of superposition principle, an arbitrary input can be decomposed into a series of micro elementsimpulseAnd then get the total response of the system;A periodic excitation can also be Fourier transformed into the sum of a series of harmonic components, and the effect of each harmonic component on the system can be investigated separately. Then, the total response of the system can be obtained by adding them together.Therefore, the response characteristics of constant parameter linear systems are availableimpulse response Or frequency response description.Impulse response refers to the response of the system to the unit impulse, representing theresponse characteristic。Frequency response refers to the response characteristics of the system to the unit harmonic input, which represents the system'sfrequency domainResponse characteristics within.The corresponding relationship between them is determined by Fourier transform.
classification
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Linear vibration is divided into linear vibration of single degree of freedom system and linear vibration of multi degree of freedom system.
(1) The linear vibration of a single degree of freedom system can beGeneralized coordinateTo determine the linear vibration of the system position.It is the simplest vibration, and many basic concepts and characteristics of vibration can be derived from it.It includesSimple harmonic vibration、free vibration 、Attenuate vibrationAnd forced vibration.
Simple harmonic vibration: under the restoring force which is proportional to the displacementBalance positionThe neighborhood moves in a sinusoidal manner.
Damped vibration: friction andmediumResistance or other energy consumption, which makes the vibration amplitude continuously attenuate.
Forced vibration: the vibration of the system under the action of regular excitation.
(2) The linear vibration of a multi degree of freedom system isfreedomThe vibration of a linear system with n ≥ 2.A system of n degrees of freedom has nnatural frequencyAnd n mainMode shape。Any vibration form of the system can be expressed as a linear combination of each main vibration mode.Therefore, in the dynamic response analysis of multi degree of freedom systems, the mainMode superposition method。In this way, the systemnatural vibrationThe test and analysis of characteristics has become a routine step of system dynamic design.The dynamic characteristics of multi degree of freedom system can also be described by frequency characteristics.Since there is a frequency characteristic function between each input and output, it forms a frequency characteristic matrix.There is a definite relationship between the frequency characteristic and the main mode shape.Different from single degree of freedom system, amplitude frequency characteristic curve of multi degree of freedom system has multiple resonance peaks.
Linear vibration of single degree of freedom system
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It refers to the linear vibration where a generalized coordinate can be used to determine the position of the system.It is the simplest and most basic vibration, and many basic concepts and characteristics of vibration can be derived from it.It includes simple harmonic vibration, damped vibration and forced vibration.
Simple harmonic vibration
Under the action of restoring force proportional to the displacement, the object moves back and forth near its equilibrium position according to the sine law (Figure 1).withxRepresents displacement,tRepresents time, and the mathematical expression of this vibration is:
， （1）
WhereAIs displacementxThe maximum value of is called amplitude, which represents the intensity of vibration;ωnRepresents the angular increment of vibration per second, called angular frequency, also called circular frequency;It is called the initial phase.withf=ωn/2 π represents the number of cycles of vibration per second, called frequency;Its reciprocal,T=1/f, represents the time required for one week of vibration, called cycle.amplitudeA, Frequencyf(or angular frequencyωn）The initial phase is called the three elements of simple harmonic vibration.
Fig. 1 Simple harmonic vibration curve
As shown in Figure 2, the concentrated mass connected by the linear springmForm a simple harmonic oscillator.When the vibration displacement is calculated from the balance position, its vibration equation is:
，
Where is the stiffness of the spring.The general solution of the above equation is (1).AAnd can bet=Initial position at 0xzeroAnd initial speed:
，
butωnOnly by the characteristics of the system itselfmandkIt is determined that it is independent of the imposed initial conditions, soωnAlso called natural frequency.
Figure 2 Single Degree of Freedom System
For a simple harmonic oscillator, the sum of its 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.
Damped vibration
Vibration with friction, medium resistance or other energy consumption, which makes the amplitude continuously attenuate.For micro vibration, the speed is generally not very large, and the medium resistance is proportional to the first power of the speed, which can be written,cIs the damping coefficient.Therefore, the single degree of freedom vibration equation with linear damping can be written as follows:
， （2）
Whereβ=c/2mIt is called the damping parameter,.The general understanding of formula (2) can be written as follows:
。 （3）
basisωnandβThe numerical relationship between them can be divided into the following three cases:
①ωn>β(In case of small damping) particle produces attenuation vibration, and its vibration equation is:
，
Its amplitude decreases with the passage of time according to the exponential law shown in the equation, as shown by the dotted line in Figure 3.Strictly speaking, this vibration is non periodic, but the frequency of its peak value can be defined as:
。
Is called amplitude reduction rate, where is the vibration period.Natural logarithm of amplitude reduction rateδIt is called logarithmic decrement (amplitude) rate;obviously,δ=β, where=2 π/ωone。Determined directly by experimentδAnd can be obtained by using the above formulac。
② (Critical damping case) The solution of equation (2) can be written as follows:
It can be divided into three non vibration cases as shown in Figure 4 with the direction of initial velocity.
③ωn<β(For the case of large damping), the solution of equation (2) is shown in equation (3).At this time, the system is no longer vibrating.
Forced vibration
The vibration of the system under constant excitation.Vibration analysis is mainly to investigate the response of the system to excitation.Periodic incentive is a typical regular incentive.Since the periodic excitation can always be decomposed into the sum of several harmonic excitations, according to the superposition principle, as long as the response of the system to each harmonic excitation is calculated, and then they are superimposed, the total response of the system to the periodic excitation can be obtained.Under the action of harmonic excitation, the differential equation of motion of a single degree of freedom system with damping can be written as follows:
。
Its response is the sum of two parts. One part is the response of damped vibration, which decays rapidly with time;The response of another part of forced vibration can be written as:
，
Fig. 3 Damping vibration curve
Fig. 4 Curve of three initial conditions of critical damping
Where
h/Fzero=H() is the ratio of the constant response amplitude to the excitation amplitude, representing the amplitude frequency characteristics, or called the gain function;ψIt is the phase difference between the constant response and the excitation to characterize the phase frequency characteristics.The relationship between them and the excitation frequency is shown in Figure 5 and Figure 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;The frequency corresponding to the peak is called the resonance frequency of the system.In the case of small damping, there is little difference between the resonance frequency and the natural frequency.When the excitation frequency is close to the natural frequency, the amplitude increases sharply, which is called resonance (resonance).At resonance, the gain of the system takes the maximum value, that is, the forced vibration is the most intense.Therefore, in general, resonance is always avoided, unless some instruments and equipment need to use resonance to obtain large amplitude vibration.
Figure 5 Amplitude Frequency Curve
It can be seen from the phase frequency curve (Figure 6) that, regardless of the dampingωzeroPosition, phase differenceψ=π/2, which can be effectively used for resonance measurement.
In addition to steady excitation, the system sometimes encounters unsteady excitation.It can be roughly divided into two categories: one is the sudden impact.The second is the lasting effect of arbitrariness.Under unsteady excitation, the response of the system is also unsteady.
A powerful tool for analyzing unsteady vibration is the impulse response method.It describes the dynamic characteristics of the system by the transient response of the unit pulse input of the system.The unit pulse can be represented by δ function.In engineering,The δ function is often defined as:
，
，
Where 0-expresstThe point on the axis that tends to zero from the left;zero+Represents a point tending to 0 from the right.
Fig. 6 Phase frequency curve
Fig. 7 Any input can be regarded as the sum of a series of pulse elements
The system corresponds tot=Response generated by the unit pulse applied at 0h(t）, called impulse response function.Assuming that the system is static before the pulse action, then whentWhen<0, yesh(t)=0。Knowing the impulse response function of the system, the system can be evaluated for any inputx(t）Response from.At this time, you canx(t）See the sum of a series of pulse elements (Fig. 7).It is equivalent to a pulse acting at time, and the response of the system is:
。
Based on the superposition principle, the system corresponds tox(t）The total response of is:
。
This integral is called convolution integral or superposition integral.
Linear vibration of multi degree of freedom system
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freedomnVibration of linear system ≥ 2.
Figure 8 shows two simple harmonic oscillator systems connected by coupling springs.Because it is a two degree of freedom system, its position can be determined by two independent coordinates.This system has two natural frequencies:
Each frequency corresponds to a vibration pattern.Each simple harmonic oscillator carries out harmonic vibration of the same frequency, passes through the balance position synchronously, and then reaches the extreme position synchronously. This vibration is called the main vibration.On corresponds toωoneAmong the main vibrations ofxone=xtwo；On corresponds toωtwoIn the main vibration of.In the main vibration, the displacement ratio of each mass maintains a certain relationship and forms a certain vibration mode, which is called the main vibration mode or natural vibration mode.There is orthogonality about mass and stiffness between the main vibration modes, which reflects the mutual independence of the main vibration modes.The natural frequency and main mode shape characterize the inherent vibration characteristics of a multi degree of freedom system.
Figure 8 Multi degree of freedom system
OnenThe degree of freedom system hasnNatural frequencies andnThree main modes.Any vibration form of the system can be expressed as a linear combination of each main vibration mode.Therefore, the main mode superposition method is widely used in the dynamic response analysis of multi degree of freedom systems.In this way, the test and analysis of the natural vibration characteristics of the system will become a routine step in the dynamic design of the system.
The dynamic characteristics of multi degree of freedom system can also be described by frequency characteristics.Since there is a frequency characteristic function between each input and output, it forms a frequency characteristic matrix, and there is a definite relationship between the frequency characteristic and the main mode.Different from single degree of freedom system, amplitude frequency characteristic curve of multi degree of freedom system has multiple resonance peaks.
Elastomer vibration
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The above multi degree of freedom system is an approximate mechanical model of the elastic body.Elastomers have an infinite number of degrees of freedom.There are quantitative differences but no essential differences between the two.Any elastic body has an infinite number of natural frequencies and corresponding main modes, and there is orthogonality about mass and stiffness between these main modes.Any vibration form of the elastic body can also be expressed as the linear superposition of the main vibration modes.Therefore, the main mode superposition method is still applicable to the dynamic response analysis of elastic bodies (see linear vibration of elastic bodies).
Take the vibration of a string as an example.Set the mass per unit length asmThin string of, longl, both ends are tensioned, and the tension isT。At this time, the natural frequency of the string is determined by the following formula:
f=na/2l（n=1，2，3，…），
Where, is the propagation velocity of the shear wave along the chord line.The natural frequencies of each order of the string happen to be the fundamental frequency α/2lInteger multiple of.This integer multiple relationship leads to a pleasant homophonic structure.In general, there is no such integer multiple relationship between the natural frequencies of each order of the elastomer.
The first three modes of tension string are shown in Figure 9, and there are some nodes on the main mode curve.In the main vibration, there is no vibration at each node.Figure 10 shows several typical vibration modes of a circular plate with peripheral fixed supports. There are some pitch lines composed of circles and diameters on the figure.
The exact formulation of elastic body vibration problem can be reduced to the boundary value problem of partial differential equation.But only in some of the simplest cases can we find the exact solution, so we have to resort to the approximate solution for the complex vibration problems of elastic bodies.The essence of various approximate solutions is to change infinity into finiteness, that is, to discretize a multi degree of freedom system (continuous system) without limbs into a finite multi degree of freedom system (discrete system).There are two kinds of discretization methods widely used in engineering analysis: finite element method and modal synthesis method.
Figure 9 Vibration Mode of Chord
Figure 10 Vibration Mode of Circular Plate
The finite element method is a composite structure that abstracts a complex structure into finite elements and connects them at finite nodes.Each element is an elastomer element;The distributed displacement of the element is expressed by the interpolation function of node displacement;Then the distributed parameters of each element are concentrated on each node according to a certain format, and the mechanical model of the discrete system is obtained.
The modal synthesis method is to decompose a complex structure into several simpler substructures.On the basis of clarifying the vibration characteristics of each sub structure, these sub structures are combined into an overall structure according to the coordination conditions on the interface, and then the vibration morphology of the overall structure is obtained by using the vibration morphology of each sub structure.
These two methods have both differences and connections, which can be referred to.The modal synthesis method can also be effectively combined with the experimental measurement to form an analysis method for vibration problems of large systems that combines theory with experiment.
