JP2003287461A - Vibration analytical method for object to be measured symmetric with respect to central axis, program for executing the same and computer-readable recording medium with recorded program - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To model a dynamic specification of an object to be measured in the object to be measured which is symmetric with respect to a central axis. <P>SOLUTION: An excitation point in one place of the object to be measured symmetric with respect to the central axis is excited, and a transfer function between the excitation point and an observation point on the object to be measured is measured. The transfer function between the excitation point and the observation point is spatially Fourier-transformed at each frequency, and the transfer function to a mode shape from a point is found. A curve is adapted from the point to the mode phase, and a mode characteristic of the object to be measured is found. By using an identified specification, the dynamic specification of the object to be measured is modeled by modeling the object to be measured in a form of a characteristic matrix. <P>COPYRIGHT: (C)2004,JPO

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention is intended for an object to be measured which is symmetrical with respect to a central axis, such as a vehicle tire, and is subjected to signal processing from the measurement results of the exciting force and response obtained in a vibration experiment. The present invention relates to a vibration analysis method for obtaining a transfer function and identifying dynamic characteristics in the form of a characteristic matrix by modal analysis. Further, the present invention relates to a program for executing the method and a computer-readable recording medium recording the program.

[0002]

2. Description of the Related Art The characteristics of vehicle tires have a great influence on the behavior, vibration and noise of the vehicle. Methods for analyzing tire characteristics are generally classified into theoretical identification methods and experimental identification methods. In the theoretical identification method, a method in which the measured object is discretized by using the finite element method and the characteristic matrix is determined is the mainstream. On the other hand, in the experimental identification method, a method in which the transfer function is obtained by signal processing from the measurement results of the excitation force and the response obtained in the excitation experiment to identify the mode characteristic is the mainstream.

[0003]

By the way, a vehicle tire is composed of rubber, fibers, wires and the like, and it is difficult to obtain an accurate model by a theoretical identification method such as the finite element method. Even if the eigenmode and the natural frequency are obtained with satisfactory accuracy, the modal damping ratio is not theoretically obtained.Therefore, for the DUT with large damping such as a vehicle tire, the theoretical identification method is used. Is not suitable.

On the other hand, even in the experimental identification method based on the vibration experiment, it is difficult to perform the mode analysis on the object to be measured which is symmetrical with respect to the central axis. That is, in a vehicle tire, it is difficult to perform modal analysis due to a heavy root due to the symmetry of the shape, a large damping, or the like. In addition, there is also a method in which the vehicle tire is set as a spring-mass model with relatively few degrees of freedom, and the rigidity, the mass, and the damping coefficient are set so that the natural frequencies from the first order to the several orders coincide with each other. Although proposed, this method is not suitable for noise / vibration analysis in the frequency range of about 300 Hz or higher that is of practical interest because the frequency range that can be analyzed is not high. There is also the problem of overlooking some of the modes.

An object of the present invention is to obtain a point-to-point transfer function by signal processing from a measurement result of an exciting force and a response obtained in a vibration experiment on an object to be measured which is symmetric with respect to the central axis.
An object of the present invention is to provide a method for performing vibration analysis of an object to be measured by converting the transfer function into a transfer function for each mode shape of a stationary wave generated in the circumferential direction of the tread surface. Another object of the present invention is to provide a program for executing the method and a computer-readable recording medium recording the program.

[0006]

SUMMARY OF THE INVENTION Therefore, according to the present invention, a step of arranging an exciter at an exciter point at one location of an object to be measured, which is symmetrical with respect to a central axis; Disposing sensors at a plurality of observation points above; measuring a transfer function between the excitation point and the observation point by the sensor; and a transfer function between the excitation point and the observation point Is spatially Fourier-transformed for each frequency to obtain a transfer function from a point to a mode shape. A vibration analysis method for an object to be measured symmetrical with respect to a central axis is provided.

The present invention is particularly useful for vibration analysis of a tire for a vehicle, which is symmetrical with respect to the central axis and has viscoelasticity, as an object to be measured, and a tread surface of the tire. A step of arranging an exciter at an oscillating point at one of the locations; a step of arranging a sensor at an observing point on the wheel; a step of arranging a sensor at an observing point on a tread surface of a tire attached to the wheel; Measuring the transfer function between the excitation point and the observation point by a sensor; determining the transfer function between the excitation point and the observation point of the wheel by curve fitting, and the mode characteristic of the wheel and the Step of obtaining a mode characteristic of a wheel; Step of spatially Fourier transforming a transfer function between the excitation point and an observation point of the tire for each frequency to obtain a transfer function from a point to a mode shape; Obtaining a mode of movement of the wheel on the tread surface from 0th and 1st order transfer functions of the transformed transfer function and the mode characteristic of the wheel; from the mode characteristic of the wheel and the mode of the tread surface Calculating the six modes with movement of the wheel and the vibration mode of air to obtain the mode with movement of the wheel; curve-fitting a transfer function of the second or higher order among transfer functions after Fourier transformation 2 of the tread surface
Determining more than the following modes; and the tire dimensions,
A step of calculating a rigid body mode of the tire with 6 degrees of freedom from the mass, the moment of inertia and the center of gravity to obtain the rigid body mode of the tire is provided.

In an object to be measured, which is symmetrical with respect to the central axis, such as a tire, it is difficult to identify the mode due to the presence of a heavy root. In the step of finding the second or higher order mode of the tread surface by curve fitting, the phase of the mode is spatially shifted by π / 2 to generate the transfer function of the root of the mode of the tread surface. There is.

Further, in the present invention, as a result of the vibration analysis, the dynamic characteristics are expressed in the form of a characteristic matrix by the mode in which the wheel is moving, the second or higher order mode of the tread surface, and the rigid body mode of the tire. It is characterized by outputting.

[0010]

BEST MODE FOR CARRYING OUT THE INVENTION An embodiment of the present invention will be described below with reference to the drawings.

FIG. 1 is a block diagram of this embodiment. In FIG. 1, the DUT 10 is a tire. A vibrator 12 and a plurality of sensors 14 are attached to the tire 10. The control unit 16 controls the vibrator 12 to the tire 10
A dynamic action, that is, vibration is applied to. Shaker 12
Is, for example, a hammer. The control unit 16 causes the vibrator 12 to perform a dynamic action by the signal generator 18. The dynamic action exerted by the exciter 12 travels through the tire 10 and is measured by the sensor 14. Signal processing means 2
0 applies signal processing to the vibration input of the vibration exciter 12 and the response measured by the sensor 14. The storage unit 22 stores the signal processing method. In addition, the storage means 22 stores FD and C
It may be D. The input means 24 is an input device such as a keyboard. The output means 26 is an output device such as a display. The control unit 16 performs a Fourier transform on the response measured by the sensor 14 based on the program stored in the storage unit 22, identifies the mode characteristic by the experimental mode identification method, and finally expresses the characteristic matrix. Generate a model of the measured object.

The object to be measured of the present invention is typically a vehicle tire. Such a tire is symmetrical with respect to the central axis and is composed of rubber, fibers, wires, and the like. The present invention obtains a transfer function from a point to a mode by spatially performing Fourier transform for each frequency by utilizing the symmetry of the tire. Then, what kind of vibration the vibration will be in such an object to be measured will be examined. FIG. 2 is a schematic view of a side surface of a tire. For an object to be measured that is symmetrical with respect to the central axis, such as a tire, an integral multiple of the wavelength must be the circumference of the tire in order for the wave to be a standing wave. There is. Assuming that the natural angular velocity of the standing wave of wave number n generated on the tread surface of the tire is Ω _n , the displacement y at the position θ of the standing wave is

[Equation 1] Can be expressed as Equation 1 is the eigenmode itself, and includes a mode in which the entire observation point uniformly spreads in the radial direction and a mode in which the observation point is twisted when n = 0.

Since the rigid body motion of the object to be measured has 6 degrees of freedom of translation and rotation, the rigid body motion observed on the circumference is expressed by polar coordinates as follows:

[Equation 2] Is expressed as Here, C ₀ , C ₁ and C ₂ are constants determined by the speed and direction of the rigid body motion.

2N observation points are arranged at equal intervals on the circumference of the tire, and the kth position of the observation point is θ = π (k-1).
When expressed as / N, the number 1 is

[Equation 3] Can be transformed into The standing waves that can be represented by Equation 3 are in the range of n ≦ N.

Next, consider the response on the circumference when the observation point i is excited. Since this response is expressed in a form in which the eigenmodes of Equation 1 are superimposed, the transfer function (compliance) at the response position θ is

[Equation 4] It is represented by. Here, ζ _n is a modal damping ratio. From equations 4 and 3, the transfer function (compliance) between the observation point i and the observation point k is

[Equation 5] It is represented by. Here, A _{i, j} , C _{i, l} , d _{i, k} are given as complex numbers, and A _{i, j} , C _{i, l} are respectively A
It is a conjugate complex number of _{i, j} and C _{i, l} . Since the mode shape can be expressed only up to the Nth order, the residual rigidity d _{i, k}
Has been introduced.

Mathematically transforming equation (5),

[Equation 6] Becomes As a result, the transfer function of each point when the observation point i is excited is

[Equation 7] Expressed as The above Equation 7 is the Fourier transform of the transfer function of the point-to-point response between the excitation point and the observation point measured by the sensor 14 in the real space for each angular velocity. In the present invention, the transfer function of the response from the excitation point to the observation point obtained by the experiment is Fourier-transformed, so that the transfer function is converted from the point to the mode shape as shown in Formula 7, and is expressed in the form of the characteristic matrix. .

In the above expression 7, in the transfer function H _{i, n (ω)} after the Fourier transform, A _{i , k} , C _{i, k} , D
By identifying the values of _{i and k} , the vibration analysis of the measured object can be modeled. In this modal analysis,
Because the modes are separated, the analysis can be done with a one degree of freedom curve fit. However, in reality,
Since the generated waves often have bending, twisting, compression, etc. at the same time, they may not always have one degree of freedom, but the analysis method of the present invention can significantly reduce the degree of freedom during mode analysis. Becomes

Next, an example in which the analysis method of the present invention is applied to a vehicle tire as an object to be measured and modeling is performed will be described. FIG. 3 shows the experimental setup. The tire 10, which is the object to be measured, was placed on the tube to realize a state in which it can be regarded as a substantially free state around the periphery, and a vibration test was conducted. First, wheel 1
The sensor 14 is arranged at the observation point 1 and the tire 1
The sensor 14 is arranged at the observation point on the 0 tread surface. Next, the vibration exciter 12 is arranged at a vibration point at one place on the tread surface of the tire 10. The observation points of the wheel 11 are one at the center of the wheel and four at every π / 2 on the circumference of the wheel. The tread surface has three observation points in the axial direction of the tread surface, and 16 points at every π / 8 in the circumferential direction of the tread surface, for a total of 48 points. There is one excitation point between adjacent observation points in one row on the tread surface.

In the present embodiment in which 16 sensors 14 are arranged in the circumferential direction, modes up to the 8th order (N / 2) can be represented, but the more sensors 14 on the circumference can be identified. The frequency band expands. The mode order is related to the width of the frequency band, and in the case of this embodiment in which 16 sensors 14 are arranged, a transfer function can be obtained up to a frequency band of about 300 Hz, and 24 sensors 14 are arranged. In that case, a transfer function can be obtained up to a frequency band of about 400 Hz. Thus, according to the present invention, the transfer function can be obtained up to a high frequency band.

When a dynamic action is applied to the tire 10 at the excitation point by the exciter 12 shown in FIG. 1, the vibration is transmitted from the excitation point to each observation point, and the vibration is measured by the sensor 14 as a response. To be done. The response of the sensor 14 is processed by the signal processor 20 as a transfer function.

FIG. 4 is a Bode diagram showing the phase and transfer function (acceleration) between the excitation point 10000 and the observation point 1001. The phase and transfer function (acceleration) at other observation points. Is measured similarly.
The transfer function between the excitation point and each observation point obtained by the experiment is Fourier-transformed “spatial for each frequency” based on Equation 7. This Fourier transform is performed at observation points 1001 to 1016, 2001 to 2016, and 300 on the tread surface.
Responses in the circumferential direction, the axle direction, and the radial direction are separately divided for the three rows of observation points 1 to 3016. Further, the Fourier transform is similarly performed for the observation points 1 to 5 of the wheel 11. FIG. 5 shows the excitation point 100
Observation points 1000 to 10 when No. 00 is excited in the radial direction
The absolute value of the transfer function for each mode is shown for the 16th radial response. It can be understood from the figure that the modes are separated. In this experiment, 300
There are two vibration modes of the tread surface existing in a frequency band of Hz or less, that is, bending in the radial direction and bending in the axle direction accompanied by torsion in the radial direction. The large peaks appearing in each order in FIG. Bending mode, and a smaller peak is a bending mode in the axial direction with radial twist. The modal characteristics are identified by performing a curve fit on these peaks.

By the way, in an object to be measured which is symmetrical with respect to the central axis, such as a tire, there is a root of eigenvalue. That is, in the case of a cylindrical structure such as a tire, there is a geometrical feature that it is symmetrical with respect to a plane including an axis. Therefore, if the structure is symmetric with respect to the plane and the mode shape that the structure has is not symmetric with respect to the plane, there are double root modes having the same eigenvalue and symmetric mode with respect to the plane. However, the modes identified by the vibration experiment are
Although it is only a mode shape that the excitation point of the multiple solution is the belly,
In the present invention, since the mode is identified as the standing wave generated on the tread surface, the phase of the standing wave is spatially shifted by π / 2 to generate the double root mode. FIG. 6 shows a third bending mode in the radial direction, and shows a double root mode in which the phase is shifted by π / 2. By doing so, even in an object to be measured having a geometrical characteristic such as a tire, it is possible to reliably capture the mode by Fourier-transforming the response measured by the sensor 14.

Now consider the wheel response. Figure 7
[Fig. 4] is a Bode diagram showing the phase and transfer function (acceleration) of the axle-direction self-response of observation point No. 1 of wheel 11 obtained in a vibration experiment. In comparison with FIG. 4, the wheel 11 does not show the second or higher order modes on the tread surface, which are present in a plurality of frequency bands of 120 Hz or higher. From this fact, the tire is sufficiently symmetrical with respect to the axis, the momentum of vibration due to the standing wave of the second order or higher generated on the tread surface is balanced by the symmetry, and the movement at the wheel center does not occur. Therefore, it is sufficient to carry out the modal analysis at the wheel observation points only for the few modes in which the wheel is moving, namely the 0th and 1st order modes.

The analysis method of the present invention based on the above theory and experiment will be described with reference to the flowchart of FIG. First, the analyst determines that the tire 10 and the wheel 11 are in the peripheral free state as shown in FIG.
The sensors 14 are arranged at the observation points 1 to 5 of the tire 10 and observation points 1001 to 1016 of the tread surface of the tire 10
Sensors 1 to 001 to 2016 and 3001 to 3016
Place 4 Then, the vibration exciter 12 is arranged at a vibration point at one place on the tread surface of the tire 10.

When the analyst operates the input means 26 to activate the analysis device, the control means 16 reads and executes the program stored in the storage means 22. The analyzer is
Through the output means 26, the analyst is prompted to input the dimensions and the like of the object to be measured. The analyst measures various dimensions, masses, moments of inertia, and center of gravity of the tires to be measured, and inputs the respective numerical values from the input means 24.

When the control section 16 applies a dynamic action from the exciter 12 to the excitation point 10000 of the tire 10 by the signal generator 18, the sensor 14 at each observation point measures the response of the dynamic action. (Step S10).

When the sensor 14 measures the response from the excitation point to each observation point, the signal processing device 20 obtains a transfer function from the response based on the response. Then, the signal processing device 20
Calculates the rigid body mode of the tire with six degrees of freedom from the tire size, mass, moment of inertia and center of gravity to obtain the rigid body mode of the tire (step S20). Further, the signal processing device 20 curve-fits the transfer function on the wheel to determine the mode characteristic of the wheel, and also determines the natural frequency and modal damping ratio of the wheel (step S30). Further, the signal processing device 20 Fourier-transforms the transfer function at the observation point on the tread surface in the circumferential direction for each frequency (step S40).

The transfer function after the Fourier transform is the sensor 14
Up to the Nth order is obtained according to the number of In this embodiment, 1
Since the six sensors 14 are used, the transfer function up to the eighth order, which is a half thereof, can be obtained. Of these, the 0th-order and 1st-order transfer functions are modes in which the wheel moves. Step S
Among the transfer functions after the Fourier transform obtained in 40, the signal processing device 20 is obtained by the transfer functions of which the tread surface response shape is 0th order and 1st order, the natural frequency of the wheel and the modal damping ratio.
Calculates the mode shape of the tread surface by the method of least squares (step S41).

Further, the signal processing device 20 has a step S
Specifying the wheel mode obtained in step 30, and step S41
Based on the mode shape of the tread surface obtained in step 6, the six modes in which the wheel is moving and the vibration mode of the air are calculated to obtain the mode in which the wheel is moving (step S3).
1).

Next, the signal processor 20 curve-fits a transfer function of the second or higher order out of the transfer functions after the Fourier transform as a mode in which the wheel moves, and obtains a mode of the second or higher order of the tread surface ( Step S42). By curve-fitting the transfer function of the second or higher order on the tread surface,
The mode can be surely obtained from the peak of the root.

Then, the signal processing device 20 calculates the rigid body mode of the tire, the mode in which the wheel is moving, and the modes of the second or higher order of the tread surface in the form of a characteristic matrix. This characteristic matrix is represented in the form of Equation 7 and displayed on the output means 26. Then, the obtained characteristic matrix is stored in the storage means 22, and the program ends.

A transfer function calculated from the characteristic matrix,
By comparing with the transfer function obtained by the experiment, it was verified whether the modeling was done in the desired frequency band. Figure 9
[Fig. 4] is a diagram comparing a transfer function obtained from an experimental value in a free state with a periphery and a transfer function obtained as a mode form from a characteristic matrix. In FIG. 9, the experimental value indicated by the dotted line is the transfer function of the 3009 radial response at the time of No. 1000 radial excitation,
The transfer function calculated from the characteristic matrix shown by the solid line is 1001.
The transfer function of the 3009 radial response at the time of vibration excitation is shown. Since the excitation point 10000 and the observation point 1001 are only slightly (5 cm) apart in the circumferential direction, they were compared in this verification.

In FIG. 9, the experimental values and the calculated values of the characteristic matrix are well matched up to around 270 Hz where the 8th-order radial bending mode is present. Further, although not shown in FIG. 9, other observation points and other transfer functions also showed high agreement. From this experimental result, it is shown that a viscoelastic object to be measured symmetrical with respect to the central axis can be accurately modeled. Further, since the model is obtained in the form of a characteristic matrix, it is possible to obtain the natural frequency and modal damping ratio obtained by performing the eigenvalue analysis.

[0034]

As described above, according to the present invention, in the object to be measured which is symmetrical with respect to the central axis, the transfer function from point to point is obtained by signal processing from the measurement results of the excitation force and the response obtained in the vibration experiment. "The transfer function from the point to the mode shape is obtained by spatially Fourier transforming the transfer function for each frequency." Even for an object to be measured made of an elastic material, there are no omissions of vibration modes, and many resonance modes and frequencies can be identified. Moreover, since the model can be calculated in the form of the characteristic matrix, the boundary conditions can be changed easily, and thus the modeling that is ideally easy to use in design analysis can be realized. Furthermore, the problem of the heavy root peculiar to the DUT that is symmetric with respect to the central axis can be surely caught in the mode of the heavy root, and even in the high frequency band (about 300 Hz or more) of interest in the vehicle, Experimental modeling is possible.

[Brief description of drawings]

FIG. 1 is a block diagram illustrating a vibration analysis method according to the present invention.

FIG. 2 is a schematic view of a side surface of a tire.

FIG. 3 is a schematic diagram of an experimental device for carrying out the analysis method of the present invention.

FIG. 4 is a Bode diagram showing the phase and transfer function (acceleration) between the excitation point 10000 and the observation point 1001.

FIG. 5 is a diagram showing the absolute value of a transfer function for each mode with respect to the radial response of observation points 1000 to 1016 when the excitation point No. 10000 is excited in the radial direction.

FIG. 6 is a diagram showing a third bending mode in a radial direction and showing a cubic root mode with a phase shifted by π / 2 in a three-dimensional manner.

FIG. 7 is a Bode diagram showing the phase and the transfer function (acceleration) of the self-response in the axle direction of the observation point No. 1 of the wheel 11 obtained in the vibration experiment.

FIG. 8 is a flowchart for executing the vibration analysis method according to the present invention.

FIG. 9 is a transfer function obtained from experimental values in a free state in the periphery,
The Bode diagram comparing the transfer functions obtained as the mode form from the characteristic matrix.

[Explanation of symbols]

10 DUT (tire) 11 wheels 12 shaker 14 sensors 16 Control unit 18 signal generator 20 Signal processor 22 storage means 24 Input means 26 Output means

Claims (12)

[Claims] 1. A step of arranging a vibration exciter at a vibration point at one location of an object to be measured which is symmetrical with respect to a central axis; sensors are arranged at a plurality of observation points on a circumferential surface of the object to be measured. Measuring the transfer function between the excitation point and the observation point by the sensor; and spatially Fourier transforming the transfer function between the excitation point and the observation point for each frequency. And a step of obtaining a transfer function from a point to a mode shape. A method of analyzing vibration of an object to be measured symmetrical with respect to a central axis. 2. A step of arranging a vibration exciter at a vibration point at one place on a tread surface of a tire; a step of arranging a sensor at an observation point on a wheel; a plurality of tread surfaces of a tire attached to the wheel. Arranging a sensor at the observation point of: a step of measuring a transfer function between the excitation point and the observation point by the sensor; a curve of a transfer function between the excitation point and the observation point of the wheel Determining the modal characteristics of the wheel and the modal characteristics of the wheel by fitting; spatially Fourier transforming the transfer function between the excitation point and the observation point of the tire for each frequency to form the mode A transfer function to the tread surface of the tread surface by the 0th and 1st order transfer functions of the transfer function after Fourier transform and the mode characteristics of the wheel. Step; calculating six modes with movement of the wheel and vibration modes of air from the mode characteristics of the wheel and modes of the tread surface to obtain a mode with movement of the wheel; after Fourier transform Curve-fitting a transfer function of the second or higher order among the transfer functions of, to obtain a second or higher mode of the tread surface;
And a step of calculating a rigid body mode of the tire with six degrees of freedom from the size, mass, moment of inertia and center of gravity of the tire to obtain the rigid body mode of the tire, which is symmetric with respect to the central axis. Vibration analysis method for DUT. 3. In the step of curve-fitting a transfer function of the second or higher order among transfer functions after Fourier transform to obtain a mode of the second or higher order of the tread surface, the phase of the mode is spatially shifted by π / 2. 3. The vibration analysis method according to claim 2, wherein the root of the mode of the tread surface is generated. 4. The vibration analysis method according to any one of claims 1 to 3, wherein a characteristic matrix is formed by a mode in which the wheel is moving, a second or higher mode of the tread surface, and a rigid body mode of the tire. A vibration analysis method including a step of calculating. 5. A process of obtaining a transfer function between sensors at a plurality of observation points on the circumferential surface of the measured object from an excitation point at one location of the measured object that is symmetrical with respect to the central axis; And a process of spatially Fourier transforming a transfer function between the excitation point and the observation point for each frequency to obtain a transfer function from a point to a mode shape. A computer-readable program for analyzing the vibration of a symmetrical object to be measured. 6. A process for obtaining a transfer function between an excitation point at one location on a tread surface of a tire and an observation point on the wheel; a process from an excitation point at one location on the tread surface of the tire to the wheel. A process for obtaining a transfer function between an observation point on the tread surface of an attached tire; a transfer function between the excitation point and the observation point for the wheel is obtained by curve fitting, and a mode characteristic of the wheel and the wheel. Processing for obtaining the mode characteristic of the above; processing for spatially Fourier transforming the transfer function between the excitation point and the observation point of the tire for each frequency to obtain a transfer function from the point to the mode shape; A process of obtaining a mode in which the wheel moves on the tread surface based on the 0th and 1st order transfer functions of the transfer function and the mode characteristics of the wheel; A process of calculating the six modes in which the wheel is moving and the vibration mode of air from the modes of the tread surface to obtain the mode in which the wheel is moving;
A process of curve-fitting a transfer function of the second or higher order out of the transfer functions after Fourier transform to obtain a second or higher order mode of the tread surface; and six degrees of freedom of the tire from the size, mass, moment of inertia and center of gravity of the tire. A computer-readable program for analyzing the vibration of an object to be measured symmetrical with respect to a central axis, the process comprising: calculating a rigid body mode of a degree to obtain the rigid body mode of the tire. 7. In a process of curve-fitting a transfer function of a second order or higher among transfer functions after Fourier transform to obtain a second order or higher mode of the tread surface, the phase of the mode is spatially shifted by π / 2. The computer-readable program according to claim 6, further comprising a process of generating a root of a mode of a tread surface. 8. The computer-readable program according to any one of claims 5 to 7, characterized by a mode in which the wheel moves, a second or higher mode of the tread surface, and a rigid body mode of the tire. A computer-readable program comprising a process of calculating a matrix. 9. A process of obtaining a transfer function between a sensor at a plurality of observation points on a circumferential surface of the measured object from a vibration point at one location of the measured object that is symmetrical with respect to a central axis; And a process of Fourier-transforming a transfer function between the excitation point and the observation point to obtain a transfer function from a point to a mode shape. An object to be measured symmetric with respect to a central axis. A computer-readable recording medium for recording a program for causing a computer to execute a program for analyzing the vibration of the. 10. A process of obtaining a transfer function between an excitation point at one location on a tread surface of a tire and an observation point on the wheel; a process from an excitation point at one location on the tread surface of the tire to the wheel. A process for obtaining a transfer function between an observation point on the tread surface of an attached tire; a transfer function between the excitation point and the observation point for the wheel is obtained by curve fitting, and a mode characteristic of the wheel and the wheel. Processing for obtaining the mode characteristic of the above; processing for spatially Fourier transforming the transfer function between the excitation point and the observation point of the tire for each frequency to obtain a transfer function from the point to the mode shape; A process of obtaining a mode in which the wheel is moving on the tread surface based on 0th and 1st order transfer functions of the transfer function and the mode characteristics of the wheel; A process of calculating the six modes in which the wheel is moving and the vibration mode of air from the modes of the tread surface to obtain the mode in which the wheel is moving; transfer of the second or higher order of transfer functions after Fourier transform Rigid body mode of the tire by calculating a rigid body mode of 6 degrees of freedom of the tire from the size, mass, moment of inertia and center of gravity of the tire by curve fitting a function to obtain a second or higher mode of the tread surface. And a computer-readable recording medium for causing a computer to execute a program for analyzing vibration of an object to be measured that is symmetrical about a central axis. 11. In a process of curve-fitting a transfer function of a second order or higher among transfer functions after Fourier transform to obtain a second order or higher mode of the tread surface, the phase of the mode is spatially shifted by π / 2. The method further includes a process of generating a multiple root of a mode of the tread surface.
0 Computer readable recording medium. 12. The computer-readable recording medium according to any one of claims 9 to 11, wherein a mode in which the wheel moves, a secondary or higher mode of the tread surface, and a rigid body mode of the tire are used. A computer-readable recording medium including a process of calculating a characteristic matrix.