Modal-Modal is an inherent vibration characteristic of vibration system (mechanical structure), which generally includes frequency, vibration mode and damping. ......

When an object vibrates at a certain natural frequency, the displacement of each point on the object from the equilibrium position satisfies a certain proportional relationship, which can be expressed by a vector, which is called mode.

Modal parameters-Modal parameters refer to natural frequency (modal frequency), modal shape, damping ratio (modal damping), modal mass, modal stiffness, etc.

Main mode, main space and main coordinates-each mode of undamped system is called main mode, the space expanded by each mode vector is called main space, and its corresponding modal coordinates are called main coordinates.

Modal order-Modal order refers to the order of modal shape (mode shape). The order corresponds to the vibration mode, and there are as many vibration modes as there are. The vibration mode of general shape can be regarded as the combination of many shapes with different orders. The vibration mode corresponding to the basic period is called the first-order vibration mode, and the vibration mode corresponding to the first period (the second period) is called the second order ... the nth order, and so on.

Modal truncation-ideally, we want to get a complete modal set of a structure, which is neither possible nor necessary in practical application.

Different modes have different contributions to the response, for example, for low-frequency response, higher-order modes have less influence.

For the actual structure, we are often interested in its first few or more modes, and the higher-order modes are often discarded. Although this will cause a little error, the matrix order of the frequency response function will be greatly reduced, thus greatly reducing the workload. This processing method is called modal truncation.

Modal leakage (I don't know if there is such a concept)—

Modal analysis-the classical definition is to transform the physical coordinates in the vibration differential equation of linear time-invariant system into modal coordinates, so that the equations are decoupled into a set of independent equations described by modal coordinates and modal parameters, and then the modal parameters of the system are obtained. The transformation matrix of coordinate transformation is modal matrix, and each column is modal shape.

Modal analysis refers to the process of finding modal parameters, which is divided into analytical (theoretical) modal analysis, experimental modal analysis and working modal analysis.

The essence of finite element modal analysis is to solve the eigenvalue problem of matrix, so "order" refers to the number of eigenvalues. Arrange the eigenvalues in order from small to large. The actual analysis object is infinite, so its mode has infinite order. But only the first few modes play a leading role in the motion, so it is necessary to calculate the first few modes in the calculation.

Second, the use of modal analysis

The ultimate goal of modal analysis is to identify the modal parameters of the system, which provides a basis for the analysis of vibration characteristics of structural systems, the diagnosis and prediction of vibration faults and the optimal design of structural dynamic characteristics.

The application of modal analysis technology can be summarized as follows:

1. Evaluate the dynamic characteristics (natural vibration period, natural vibration frequency, vibration mode and damping) of the existing structural system;

2. In the design of new products, predict and optimize the structural dynamic characteristics; Diagnose and predict the faults of structural systems;

Through modal analysis, we can clearly understand the main modal characteristics of the structure in a certain susceptible frequency range, and it is possible to predict the actual vibration response of the structure under the action of various external or internal vibration sources in this frequency band. Therefore, modal analysis is an important method for structural dynamic design and equipment fault diagnosis.

3. Control the radiation noise of the structure;

4. Identify the load of the structural system.

Thirdly, modal analysis &; finite element analysis

1. How to combine finite element analysis to carry out modal analysis on the structure;

A. The finite element analysis model is used to determine the measuring point, excitation point and supporting point (suspension point) of modal test, and the modal parameters of the test are identified and named with reference to the calculated vibration modes, which is particularly important for complex structures.

B, using the test results to modify the finite element model to meet the requirements of industry standards or national standards.

C. The finite element model is used to simulate and analyze the errors caused by boundary condition simulation, additional mass and additional stiffness and their elimination.

D. spectral consistency and mode correlation analysis of the two models.

E. using finite element model to simulate and analyze and solve the problems in the test.

2. Modification of finite element results

Fourth, modal analysis method

Modal analysis methods include time domain method and frequency domain method.

Time domain method.

The time domain method directly obtains the modal parameters from the time domain free response of the structure. Typical methods include random decrement method and time series method;

Random decrement method

Time series method

2. Frequency domain method

The frequency domain method first converts the test data into frequency domain data, and then identifies the modal parameters. It mainly includes main mode method and transfer function method. Experimental modal analysis is to determine modal parameters through experimental data, which belongs to frequency domain method.

The main mode method uses multi-point sinusoidal excitation to make the system vibrate in pure mode and obtain modal parameters.

The transfer function method generally adopts single point excitation. Firstly, the transfer function of the structure is obtained, and then the modal parameters are determined.

Verb (abbreviation of verb) analysis (theoretical) modal analysis

Experimental modal analysis of intransitive verbs

Apply exciting force to each point on the measured part and measure its response at the same time; Then, the transfer function between the excitation point and the response point is obtained by signal analysis equipment. If the vibration mode is needed, the transfer function of each point on the specimen needs to be obtained repeatedly. Then the parameters such as natural frequency, modal stiffness, modal damping, modal mass and modal shape are identified by curve fitting. Finally, according to the obtained modal parameters, the dynamic process of vibration mode is displayed on the display screen.

The process of experimental modal analysis: simulate the tested system and measure its response at the same time; The data acquisition and processing subsystem obtains the transfer function between the excitation point and the response point, and then obtains the natural frequency, modal damping, modal shape and other parameters of the tested system through curve fitting.

Excitation subsystem

It mainly includes signal source, power amplifier and vibration exciter, which are divided into fixed and non-fixed types. At present, the most widely used fixed vibration excitation system mainly includes electric vibration exciter and electro-hydraulic vibration exciter, and the most common example of non-fixed vibration excitation system is hammer vibration excitation.

Most vibration testing systems need a device to make the test object produce some kind of vibration. According to whether it is connected with the structure or not, the device can be divided into connected type and unconnected type. In connection excitation, the most typical device is composed of one or several vibrators placed on the ground (or fixed on the bracket) and connected to the test object, or the vibrators are only connected to the structure. In these cases, the vibration exciter has certain influence on the dynamic characteristics of the structure. In other cases, unconnected excitation is used: the excitation device is not connected with the test object, and hammer excitation is the most familiar example. Sometimes the static load can be preloaded on the structure, and the sudden release of this preload will produce a step input force. In addition, acoustic excitation and magnetic excitation also belong to connection excitation.

At present, the common vibration exciter in fixed excitation system is mainly electric vibration exciter and electro-hydraulic vibration exciter. Electric vibration exciter is the most popular vibration exciter. The input signal passes through a coil placed in a magnetic field. When the signal current alternates, the coil moves due to the alternating force. The test structure is driven by the connecting device of the moving coil, thus generating vibration. The electrical impedance of this device varies with the motion amplitude of the follower. This kind of vibration exciter can work normally in the range of 30 Hz-50 kHz. Electro-hydraulic vibration exciter uses hydraulic principle to amplify power and produce huge exciting force. Moreover, it can add both static load and dynamic load, and the whole mechanism is complex and expensive, so it is generally used in the case of low-frequency excitation and large excitation force.

The added mass of the vibration exciter to the test object will always have a certain impact on the vibration characteristics of the structure. Usually, the connection between the vibration exciter and the structure is realized by a unidirectional force sensor. In order to effectively measure the excitation force, it is necessary to ensure that the structure is excited in the direction of force measurement (for example, when using a tension-compression dynamometer, do not apply a bending moment to the structure). Therefore, the connection between the vibration exciter and the test object should be rigid in the measurement direction and flexible in all other directions. In addition, the vibration exciter may add some mass, damping and stiffness to the structure.

The most important advantage of non-fixed excitation system is that it does not add any mass to the structure, so it will not affect the dynamic characteristics of the test object. The most common example is hammer excitation, as well as pre-tightening-releasing excitation, acoustic excitation and magnetic excitation. The purpose of exciting the test object is to generate a certain magnitude of force within a specified frequency range. For example, when a hammer inputs a pulse, it will produce a force that extends smoothly to a specified frequency. Hammer and force sensor are combined into an instrument, which is a force hammer. The energy and frequency broadening of the excitation force depend on the operator's effort, the weight of the hammer, the hardness of the hammer and the plasticity of the hit point in the structure. The closer the input force is to the pulse (the duration is zero, the force amplitude is infinite, and the impulse is one unit), the wider the baseband frequency spread. If the hammer head is hard, the weight of the hammer head is small, and the surface of the test object is hard, the contact time between the hammer head and the test object is short, which makes the excitation signal close to the pulse, and the baseband frequency of the excitation will reach a very high frequency (for example, 10KHz). If the hammer is heavy and soft, the contact time will be longer and the lower frequency can be excited. In extreme cases, heavy structures with low vibration frequency, such as buildings, trains, ships and foundations, can be excited by hammering.

Measurement subsystem

It is mainly composed of force sensor and motion sensor. The commonly used sensors in modal analysis and testing are force sensors and acceleration sensors with piezoelectric crystals as sensitive elements.

The measurement subsystem mainly includes sensors, adaptive amplifiers and related connecting components. The most commonly used sensor is piezoelectric sensor. The function of the adaptive amplifier is to adjust the small signal generated by the sensor so that it can be sent to the analyzer for measurement.

The measurement subsystem is mainly composed of force sensor and motion sensor.

When the structure vibrates under the excitation of vibration exciter or force hammer, the signal input to the mechanical system and the signal output from the system must be measured. The input of the system is generally force, which is measured by the force sensor. The output of the system is usually the displacement, velocity or acceleration of some points of interest on the structure, and these outputs are measured by motion sensors.

The commonly used motion sensor in modal analysis test is an acceleration sensor with piezoelectric crystal as sensitive element. When the crystal is deformed, its two polar faces will produce charges proportional to its deformation, and the deformation is proportional to the force on the crystal.

In most modal analysis and measurement, piezoelectric sensors use strain gauges instead of traditional dynamometer. The main characteristic indexes of piezoelectric sensor are maximum force, minimum frequency and maximum frequency (related to load) and sensitivity. For very low frequency measurement, strain dynamic dynamometer is still in use. Generally speaking, force sensors have less influence on modal analysis and measurement than accelerometers.

In the modal analysis test of mechanical structures, the response is usually the movement of structural objects, which is expressed by displacement, velocity or acceleration. Theoretically, it doesn't matter which of the three motion parameters is measured. Measuring displacement is more important for low frequency situations, while measuring acceleration is more important for high frequency situations. The root mean square value of velocity is called "vibration intensity" because there is a simple relationship between vibration velocity and vibration energy. This may be an important reason for measuring speed.

However, displacement sensors and speed sensors are generally heavy. Most motion sensors are mass spring systems with * * * vibration frequency. The output signal of the displacement sensor is proportional to the displacement in the frequency band above its own vibration frequency. This inevitably requires that the vibration frequency of * * * is very low, which requires a large mass, and the accelerometer is the opposite. The smaller the mass, the smaller the influence of sticking to the structure and the more accurate the measurement.

Another advantage of accelerometer is that it can integrate the acceleration signal correctly through the integration circuit when doing conventional vibration analysis, so as to get the velocity and displacement. Speed sensors and displacement sensors are not suitable for use with differential circuits because they will amplify high-frequency noise. Based on the above considerations, accelerometer has become the most widely used motion sensor in modal analysis and testing.

Data acquisition subsystem

Record and process test data, such as the determination of frequency response function;

Record and process the signal data obtained from the test of force sensor and motion sensor, such as determining the frequency response function.

Data processing subsystem

Modal parameters (natural frequency, damping ratio, vibration mode, etc.). ) determined by curve fitting of test transfer function;

The modal parameters (modal frequency, modal damping ratio, modal shape vector, etc.) are derived and determined. ) the frequency response function obtained from the test;

In mechanics, the vibration mode is the ratio of the amplitude of each point and the characteristic vector corresponding to the characteristic equation.

boundary conditions

(1) Constrained brace mode

Install the test object on the foundation. The ideal situation is that the foundation is absolutely rigid, that is, when the test object is excited, the foundation is absolutely motionless, that is, the function value of the displacement frequency response of the excitation force to the foundation is zero. In fact, this is impossible. Generally speaking, if the frequency response function value of the foundation is much smaller than that of the structure of the test object in the whole test frequency band, it can be approximately considered to meet the requirements of constrained support. Therefore, the foundation quality is usually required to be at least 10 times that of the test object, so that the influence of the foundation on the dynamic characteristics of the test object can generally be ignored.

(2) Free support mode

The ideal free state is that the test object is suspended. At this time, the structure of the test object has six rigid modes with zero natural frequency, three of which are translation modes. Three are rotation modes. In fact, it is difficult to realize the real free state in the test room, and the free state can only be approximately simulated by supporting the test object in some appropriate way (such as air spring and pneumatic and magnetic suspension devices). At this time, the modal frequency of the test object is no longer zero, and its value is related to the mass characteristics of the test object and the stiffness characteristics of the support device. In order to reduce the influence of suspension system (the test object is a system composed of rigid body and elastic support device) on the elastic mode of the test object structure, it is required that the suspension system has low stiffness, small added mass and zero friction. The natural frequency of suspension system and the arrangement of suspension points shall generally meet the following requirements:

1) The natural frequency of the suspension system is lower than the basic natural frequency of the elastic mode of the tested structure110-1/5. Otherwise, the influence of suspension system on the elastic modal characteristics of the test object should be considered;

2) The suspension point should be selected as far as possible near the node with high structural stiffness of the test object, so as to avoid the change of structural stiffness caused by static stress of structural suspension and ensure the stability of the suspension system;

3) reduce the influence of additional damping caused by suspension system on the structural test object;

4) The hanging direction of the test object is preferably perpendicular to the main vibration direction of the structure.

3. In modal test, the design and verification of test fixture and supporting system are very important. When it is found that the dynamic characteristics of fixture and supporting system have obvious influence on the tested structure, the test object and fixture should be dynamically analyzed as a whole. With the increasing structure of the test object, it is becoming more and more difficult and expensive to design a fixture with ideal interface or small coupling with the test object. Some of them need to solve these contradictions from the test methods (such as inertial mass interface and residual flexibility).

Layout of measuring points

Finally, the modal diagram of the mode will be represented by the vibration of the measuring point, so the selection of the position and distribution density of the measuring point is very important. Too dense arrangement of measuring points will increase unnecessary workload, and too thin arrangement may make the expression of test vibration mode unclear. Therefore, the principle of layout is to simplify as much as possible without missing modes. If it is difficult to predict the vibration mode of the structure, modal analysis can be carried out by finite element software, and the modal characteristics of the measured structure can be roughly predicted, and then the layout of the measuring points can be determined.

First, the best suspension position

When doing modal test, it is generally hoped that the suspension point of the test object will be selected at a position with small amplitude. Therefore, it is necessary to determine the optimal suspension position in advance.

2. Optimal excitation position

In order to ensure the identifiability (controllability and observability) of the system, it is generally required that the excitation point should not be too close to the node or node line. This requires ODP (optional

The displacement response value of the optimal excitation point is not equal to zero. The excitation point should avoid choosing the place where the value of ODP optimal excitation point is equal to zero, and some modes will not be excited at this point.

When hammering method is used, the selection of optimal excitation position should not only satisfy that the value of ODP optimal excitation point is not equal to zero, but also avoid selecting those points with high average driving freedom speed, because double-clicking phenomenon is easy to occur at those points with high average driving freedom speed.

When the vibration exciter is used for vibration excitation, the selection of the optimal excitation position should not only ensure that the value of the ODP optimal excitation point is not equal to zero, but also avoid selecting those points with large average driving freedom acceleration, because the additional mass of the vibration exciter has great influence on those points with large average driving freedom acceleration.

Three. Accuracy requirements of the best test point

The information measured at the test point needs as high a signal-to-noise ratio as possible, so the test point should not be close to the node. Note that in practice, acceleration sensors are usually used. In fact, all the measured acceleration signals are acceleration signals, so the average driving freedom acceleration value should be large at the best test point. The method of determining the best test point usually uses EI (effective independence) method [22].

Related parameter setting

Sensor sensitivity, sampling frequency, test band selection, average calculation, trigger mode, signal recording length, force signal plus square window, acceleration signal plus exponential window.

1) Sensor sensitivity setting

The signal in signal analysis often appears in the form of voltage, and the analysis result is also a quantity related to voltage, which has a certain conversion relationship with the actual physical quantity in engineering. In order to reduce the analysis error, it is best to send the standard known physical signals to the analysis equipment in order to establish a direct relationship between the numerical values on the analysis equipment and the actual physical quantities. That is, set the sensitivity of the sensor and establish the conversion relationship between voltage unit and physical unit.

2) Sampling frequency

If the signal is analyzed in time domain, the higher the sampling frequency, the better the signal elasticity. The sampling frequency is 10 times of the highest frequency of the signal. For some signal analysis equipment, the number of sampling points is limited. If the sampling frequency is high, the recorded length of the collected signal will be very short, which will affect the integrity of the signal.

In frequency domain analysis, in order to avoid aliasing, the minimum sampling frequency must be greater than or equal to twice the highest frequency in the signal, that is, the sampling theorem. In practical analysis, the general sampling frequency is 3 ~ 4 times of the highest frequency in the signal. If only some frequency components in the signal are of interest, the highest frequency in the analysis can be taken as the highest frequency of interest. It is worth noting that when some signal analysis devices do frequency domain analysis, the number of sampling points is fixed and increased, the bandwidth of the combined analysis band increases and the frequency resolution becomes worse.

3) Number of sampling points

In time domain analysis, the more sampling points, the closer to the original signal. In frequency domain analysis, for the convenience of FFT calculation, the number of sampling points is generally a power of 2, such as: 32,64,128,256, etc.

4) Length of recording signal

When the number of sampling points n is determined, the recording length of the analysis signal is determined. The length of each sample is. In order to reduce the amplitude error of the analysis, the average processing is often used in the analysis, and the recording length of the signal is also related to the average number of segments Q. The recording length of the signal is, that is, the segment signal length.

5) Average calculation

In order to improve the accuracy of spectrum estimation, it is necessary to average the sampled data. The signal is sampled several times and then averaged. Generally, there are two processing methods: one is linear average; The other is exponential average.

6) Selection of test frequency band

The selection of test frequency band should consider the frequency range of excitation force of machinery or structure under normal working conditions. It is generally believed that the modes far away from the vibration source frequency band have little contribution to the actual vibration response of the structure, and even the response excited by low-frequency excitation does not include the contribution of higher-order modes. In fact, the contribution of high-frequency modes is not only related to the excitation frequency band, but also related to the distribution of excitation force. Therefore, the test frequency band should be appropriately higher than the vibration source frequency band. In addition, if it is a component test, the test results will be used for assembly comprehensive analysis together with other components to obtain the modal of the whole structure. Then, in order to make the whole mode have higher accuracy, the test frequency band of component modes should be relaxed appropriately to obtain more modes. When there are too few modes of parts and too many connection points between parts during assembly, it may make the overall comprehensive analysis impossible.

6) Trigger mode

The trigger mode determines the starting point of each sample when sampling. Its reasonable selection is of great significance for capturing transient signals or requiring the same operation for collecting signals. There are generally the following methods to solve the problem: free trigger, signal trigger, pre-trigger, external trigger and so on. For pulse signals, it is generally difficult to capture. The early sampled signal didn't arrive, and the late sampled signal passed. In this case, it can be triggered by the level of the signal itself. The trigger level can be adjusted to be slightly higher than the noise level, so that when there is no pulse signal, the noise cannot trigger the sampling system and sample; When the pulse signal appears and reaches the preset trigger level, the sampling system samples immediately. Using this trigger method, the pulse signal to be analyzed can be ensured. If there is no signal, the sampling system will not work until the next pulse signal appears. This not only ensures that the desired pulse signal is collected without omission every time, but also eliminates a lot of unnecessary noise.

Prepare for testing-reciprocal analysis

(1) linear assumption, that is, assume that the structure and its dynamic characteristics are linear. In other words, the output caused by any combination of inputs is equal to the combination of their respective outputs.

(2) Time-invariant (instant invariant) assumption, that is, it is assumed that the model of the structure and its dynamic characteristics do not change with time, so the coefficient matrix of the differential equation is a constant matrix independent of time. When the added mass of the system is generated by testing additional sensors, it remains unchanged at that time.

(3) Observability hypothesis, that is, it is assumed that all the data needed to determine the dynamic characteristics of the system we care about can be measured. In order to avoid the problem of observability, the degree of freedom of response should be reasonably selected.

(4) Reciprocity assumption, that is, assuming that the structure follows Maxwell's reciprocity principle, that is, input the quotation at Q point.

During the experiment, due to the influence of many practical factors, the original data obtained by the experiment often contain interference factors. An important premise of using experimental modal analysis technology to study the dynamic characteristics of machine tools is that the structure of machine tools should meet various hypothetical conditions and ranges. Especially for complex machine tool structural systems with various connections, in order to ensure the reliability and effectiveness of the test, the following preliminary tests should be carried out before the modal test:

Reciprocity test: the theoretical basis of modal analysis is based on linear system. This requires that the nonlinear error of the machine tool structure before testing is relatively small. In the pulse excitation test, the method of reciprocal measuring point and tapping point can be used for the test, which not only satisfies the reciprocity theorem:

Pre-test-coherence analysis

During the experiment, due to the influence of many practical factors, the original data obtained by the experiment often contain interference factors. Using experimental modal analysis technology to study the dynamic characteristics of structures has an important premise, that is, the structure should meet the conditions and scope of various assumptions. Especially for the research system of joint surface, in order to ensure the reliability and effectiveness of the test, the following preparatory tests should be carried out before testing the data: the coherence function can be calculated by using the frequency spectrum of excitation force and acceleration. The coherence function is between 0 and 1, which represents the reliability of experimental results and evaluates the reliability of transfer function estimation. Generally speaking, the closer to 1, the less interference the experiment receives and the more reliable the experimental results are. Generally, the coherence function should be greater than 0.8, preferably greater than 0.9.

A) test the linear hypothesis of the measured structure. Reciprocity test is generally used, that is, the position of response and excitation is interchanged, and its transfer function changes little in the corresponding direction.

B) Reliability analysis of response signal. That is, the coherence function can be calculated according to the response signal spectrum and the excitation signal spectrum. Coherence function

In the range of 0 ~ 1, it represents the reliability of experimental results and evaluates the reliability of transfer function estimation. Generally speaking, the closer to 1, the less interference the experiment receives and the more reliable the experimental results are. Generally, the coherence function should be greater than 0.8, preferably greater than 0.9.

Transfer function test

According to the modal testing theory, all modal information can be obtained only by transferring one row or one column in the function matrix. Therefore, there are two methods to measure the transfer function: one is to fix the excitation and pick up the vibration point by point; The other is fixed response and point-by-point excitation. In order to eliminate the interference signal as much as possible, it is usually necessary to measure several times and then take the average.

Modal parameter identification

Finally, the measured transfer function is imported into modal identification software, and modal parameters of each order are obtained by curve fitting identification, mainly including modal frequency, modal shape and modal damping ratio.

Modal analysis of intransitive verbs &; finite element analysis

Combining modal analysis with finite element analysis;

1. The finite element analysis model is used to determine the measuring point, excitation point and supporting point (suspension point) of modal experiment, and modal parameters are identified and named with reference to the calculated modal parameters, especially for complex structures.