**Linear vibration**: the elasticity of components in the system is subject to hooke’s law, and the damping force generated during the motion is proportional to the first equation of the generalized velocity (time derivative of the generalized coordinates).

## concept

Linear system is usually an abstract model of the vibration of real system.The linear vibration system applies the superposition principle, that is, if the response of the system is y1 under the action of input x1, and y2 under the action of input x2, then the response of the system under the action of input x1 and x2 is y1+y2.

On the basis of superposition principle, an arbitrary input can be decomposed into the sum of a series of infinitesimal impulses, and then the total response of the system can be obtained.The sum of the harmonic components of a periodic excitation can be expanded into a series of harmonic components by Fourier transform, and the effect of each harmonic component on the system can be investigated separately.Therefore, the response characteristics of linear systems with constant parameters can be described by impulse response or frequency response.

Impulse response refers to the response of the system to the unit impulse, which characterizes the response characteristics of the system in the time domain.Frequency response refers to the response characteristic of the system to the unit harmonic input.The correspondence between the two is determined by the Fourier transform.

## classification

Linear vibration can be divided into linear vibration of single-degree-of-freedom system and linear vibration of multi-degree-of-freedom system.

(1) linear vibration of a single-degree-of-freedom system is a linear vibration whose position can be determined by a generalized coordinate.It is the simplest vibration from which many basic concepts and characteristics of vibration can be derived.It includes simple harmonic vibration, free vibration, attenuation vibration and forced vibration.

Simple harmonic vibration: the reciprocating motion of an object in the vicinity of its equilibrium position according to a sinusoidal law under the action of a restoring force proportional to its displacement.

Damped vibration: vibration whose amplitude is continually attenuated by the presence of friction and dielectric resistance or other energy consumption.

Forced vibration: vibration of a system under constant excitation.

(2) the linear vibration of the multi-degree-of-freedom system is the vibration of the linear system with n≥2 degrees of freedom.A system of n degrees of freedom has n natural frequencies and n main modes.Any vibration configuration of the system can be represented as a linear combination of the major modes.Therefore, the main mode superposition method is widely used in dynamic response analysis of multi-dof systems.In this way, the measurement and analysis of the natural vibration characteristics of the system becomes a routine step in the dynamic design of the system.The dynamic characteristics of multi-dof systems can also be described by frequency characteristics.Since there is a frequency characteristic function between each input and output, a frequency characteristic matrix is constructed.There is a definite relation between the frequency characteristic and the main mode.The amplitude-frequency characteristic curve of the multi-freedom system is different from that of the single-freedom system.

## Linear vibration of a single degree of freedom system

A linear vibration in which the position of a system can be determined by a generalized coordinate.It is the simplest and most fundamental vibration from which many basic concepts and characteristics of vibration can be derived.It includes simple harmonic vibration, damped vibration and forced vibration.

Harmonic vibration

Under the action of restoring force proportional to the displacement, the object reciprocates in a sinusoidal manner near its equilibrium position (FIG. 1).X represents the displacement and t represents the time. The mathematical expression of this vibration is:

(1)Where A is the maximum value of displacement x, which is called the amplitude, and represents the intensity of the vibration;Omega n is the amplitude Angle increment of the vibration per second, which is called the angular frequency, or the circular frequency;This is called the initial phase.In terms of f= n/2, the number of oscillations per second is called the frequency;The inverse of this, T=1/f, is the time it takes to oscillate one cycle, and that’s called the period.Amplitude A, frequency f (or angular frequency n), the initial phase, known as simple harmonic vibration three elements.

FIG. 1 simple harmonic vibration curve

As shown in FIG. 2, a simple harmonic oscillator is formed by the concentrated mass m connected by a linear spring.When the vibration displacement is calculated from the equilibrium position, the vibration equation is:

Where is the stiffness of the spring.The general solution to the above equation is (1).A and can be determined by the initial position x0 and initial velocity at t=0:

But omega n is only determined by the characteristics of the system itself m and k, independent of the additional initial conditions, so omega n is also known as the natural frequency.

FIG. 2 single degree of freedom system

For a simple harmonic oscillator, the sum of its kinetic energy and potential energy is 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.

The damping vibration

A vibration whose amplitude is continually attenuated by friction and dielectric resistance or other energy consumption.For micro vibration, the velocity is generally not very large, and the medium resistance is proportional to the velocity to the first power, which can be written as c is the damping coefficient.Therefore, the vibration equation of one degree of freedom with linear damping can be written as:

(2)Where, m =c/2m is called the damping parameter, and.The general solution of formula (2) can be written:

(3)The numerical relationship between omega n and PI can be divided into the following three cases:

N > (in the case of small damping) particle produced attenuation vibration, the vibration equation is:

Its amplitude decreases with time according to the exponential law shown in the equation, as shown in the dotted line in FIG. 3.Strictly speaking, this vibration is aperiodic, but the frequency of its peak can be defined as:

Is called the amplitude reduction rate, where is the period of vibration.The natural logarithm of the amplitude reduction rate is called the logarithm minus (amplitude) rate.Obviously, =, in this case, is equal to 2/1.Directly through the experimental test delta and, using the above formula can be calculated c.

At this time, the solution of equation (2) can be written:

Along with the direction of initial velocity, it can be divided into three non-vibration cases as shown in FIG. 4.

N < (in the case of large damping), the solution to equation (2) is shown in equation (3).At this point, the system is no longer vibrating.

Forced vibration

Vibration of a system under constant excitation.Vibration analysis mainly investigates the response of the system to excitation.Periodic excitation is a typical regular excitation.Since periodic excitation can always be decomposed into the sum of several harmonic excitation, according to the superposition principle, only the response of the system to each harmonic excitation is required.Under the action of harmonic excitation, the differential equation of motion of a single degree of freedom damped system can be written:

The 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:

FIG. 3 damped vibration curve

FIG. 4 curves of three initial conditions with critical damping

Type in the

H /F0= h (), is the ratio of steady response amplitude to excitation amplitude, characterizing amplitude-frequency characteristics, or gain function;Bits for steady state response and incentive of phase, characterization of phase frequency characteristics.The relation between them and excitation frequency is shown in FIG. 5 and FIG. 6.

As can be seen from the amplitude-frequency curve (FIG. 5), in the case of small damping, the amplitude-frequency curve has a single peak.The smaller the damping, the steeper the peak;The frequency corresponding to the peak is called the resonant frequency of the system.In the case of small damping, the resonance frequency is not much different from the natural frequency.When the excitation frequency is close to the natural frequency, the amplitude increases sharply. This phenomenon is called resonance.At resonance, the gain of the system is maximized, that is, the forced vibration is the most intense.Therefore, in general, always strive to avoid resonance, unless some instruments and equipment to use resonance to achieve large vibration.

FIG. 5 amplitude frequency curve

Can be seen from the phase frequency curve (figure 6), regardless of size of damping, in omega zero phase difference bits = PI / 2, this characteristic can be effectively used in measuring resonance.

In addition to steady excitation, systems sometimes encounter unsteady excitation.It can be roughly divided into two types: 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 with the transient response of the unit impulse input of the system.The unit impulse can be expressed as a delta function.In engineering, the delta function is often defined as:

Where 0- represents the point on the t-axis that approaches zero from the left;0 plus is the point that goes to 0 from the right.

FIG. 6 phase frequency curve

FIG. 7 any input can be considered as the sum of a series of impulse elements

The system corresponds to the response h(t) generated by the unit impulse at t=0, which is called the impulse response function.Assuming that the system is stationary before the pulse, h(t)=0 for t<0.Knowing the impulse response function of the system, we can find the response of the system to any input x(t).At this point, you can think of x(t) as the sum of a series of impulse elements (FIG. 7).The response of the system is:

Based on the superposition principle, the total response of the system corresponding to x(t) is:

This integral is called a convolution integral or a superposition integral.

## Linear vibration of a multi-degree-of-freedom system

Vibration of a linear system with n≥2 degrees of freedom.

Figure 8 shows two simple resonant subsystems connected by a coupling spring.Because it is a two-degree-of-freedom system, two independent coordinates are needed to determine its position.There are two natural frequencies in this system:

Each frequency corresponds to a mode of vibration.The harmonic oscillators carry out harmonic oscillations of the same frequency, synchronously passing through the equilibrium position and synchronously reaching the extreme position.In the main vibration corresponding to omega one, x1 is equal to x2;In the main vibration corresponding to omega omega two, omega omega one.In the main vibration, the displacement ratio of each mass keeps a certain relation and forms a certain mode, which is called the main mode or the natural mode.The orthogonality of mass and stiffness exists among the main modes, which reflects the independence of each vibration.The natural frequency and main mode represent the inherent vibration characteristics of the multi-degree of freedom system.

FIG. 8 system with multiple degrees of freedom

A system of n degrees of freedom has n natural frequencies and n main modes.Any vibration configuration of the system can be represented as a linear combination of the major modes.Therefore, the main mode superposition method is widely used in dynamic response analysis of multi-dof systems.In this way, the measurement and analysis of the natural vibration characteristics of the system becomes a routine step in the dynamic design of the system.

The dynamic characteristics of multi-dof systems can also be described by frequency characteristics.Since there is a frequency characteristic function between each input and output, a frequency characteristic matrix is constructed.The amplitude-frequency characteristic curve of the multi-freedom system is different from that of the single-freedom system.

## The elastomer vibrates

The above multi – degree of freedom system is an approximate mechanical model of elastomer.An elastomer has an infinite number of degrees of freedom.There is a quantitative difference but no essential difference between the two.Any elastomer has an infinite number of natural frequencies and an infinite number of corresponding modes, and there is orthogonality between the modes of mass and stiffness.Any vibrational configuration of the elastomer can also be represented as a linear superposition of the major modes.Therefore, for dynamic response analysis of elastomer, the superposition method of main mode is still applicable (see linear vibration of elastomer).

Take the vibration of a string.Let’s say that a thin string of mass m per unit length, long l, is tensioned at both ends, and the tension is T.At this time, the natural frequency of the string is determined by the following equation:

F =na/2l (n= 1,2,3…).

Where, is the propagation velocity of the transverse wave along the direction of the string.The natural frequencies of the strings happen to be multiples of the fundamental frequency over 2l.This integer multiplicity leads to a pleasant harmonic structure.In general, there is no such integer multiple relation among the natural frequencies of the elastomer.

The first three modes of the tensioned string are shown in FIG. 9. There are some nodes on the main mode curve.In the main vibration, the nodes do not vibrate.FIG. 10 shows several typical modes of the circumferentially supported circular plate with some nodal lines composed of circles and diameters.

The exact formulation of the elastomer vibration problem can be concluded as the boundary value problem of partial differential equations.However, the exact solution can only be found in some of the simplest cases, so we have to resort to the approximate solution for the complex elastomer vibration problem.The essence of various approximate solutions is to change the infinite to the finite, that is, to discretize the limb-less multi-degree of freedom system (continuous system) 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.

FIG. 9 mode of string

FIG. 10 mode of circular plate

Finite element method is a composite structure which abstracts a complex structure into a finite number of elements and connects them at a finite number of nodes.Each unit is an elastomer;The distribution displacement of element is expressed by interpolation function of node displacement.Then the distribution parameters of each element are concentrated to each node in a certain format, and the mechanical model of the discrete system is obtained.

Modal synthesis is the decomposition of a complex structure into several simpler substructures.On the basis of understanding the vibration characteristics of each substructure, the substructure is synthesized into a general structure according to the coordination conditions on the interface, and the vibration morphology of the general structure is obtained by using the vibration morphology of each substructure.

The two methods are different and related, and can be used as reference.The modal synthesis method can also be effectively combined with the experimental measurement to form a theoretical and experimental analysis method for the vibration of large systems.

Post time: Apr-03-2020