Automobile ride comfort simulation method under non-stationary and non-linear conditions

Abstract

The invention provides an automobile ride comfort simulation method under non-stationary and non-linear conditions, and relates to the technical field of automobile engineering. For the nonlinear problem of an automobile vibration system, a statistical linearization method is adopted to approximate a nonlinear system into a linear system. In order to improve the solving efficiency, a non-stationary virtual excitation method is adopted to construct a modulation function of the road surface. In order to improve the solving precision and efficiency, the exponential function is processed by adopting fine integration. According to the method, time-frequency domain solving is carried out on the vibration system, the response quantity statistical characteristic of the time domain can be obtained, the response quantity statistical characteristic of the frequency domain can also be obtained, the smoothness of the automobile can be comprehensively evaluated, the precision and efficiency problems of automobile vibration solving under the non-stationary and non-linear conditions are solved, and the method can be used for evaluating the smoothness of the automobile under the complex conditions of the automobile.

Description

Technical Field

[0001] The present invention relates to the field of automobile engineering technology, and in particular to a method for simulating automobile ride comfort under non-stationary and nonlinear conditions. Background Art

[0002] With the development of society and advancements in technology, people's demands for automotive performance are becoming increasingly stringent. Vehicle ride comfort, one of the six key performance characteristics of modern vehicles, is crucial for maintaining passenger comfort within certain limits due to vibration and impact generated during driving. For trucks, this also includes ensuring the integrity of cargo. As a key tool for vehicle evaluation, vehicle ride comfort simulation offers advantages such as high efficiency, low cost, and shortened R&D cycles, making it a crucial tool in automotive development.

[0003] Traditional vehicle ride comfort simulations often simplify the vehicle system into a linear model, ignoring the nonlinear vertical force-deformation relationship of the tire, as well as the nonlinear spring stiffness and damping characteristics of the suspension. This makes it difficult for the simulation to truly reflect the actual situation of the vehicle during actual driving. The complex vibration response caused by these nonlinear factors leads to a large deviation between the simulation results and the actual situation.

[0004] Furthermore, traditional vehicle ride comfort simulations are often based on the assumption of smooth driving conditions, using standard road surface roughness models and constant speed. However, in reality, vehicle driving conditions are complex and varied, involving non-steady maneuvers such as acceleration, deceleration, and switching between different road surfaces, and road conditions also vary widely. Traditional methods are unable to accurately simulate the actual vehicle motion and vibration under these non-steady conditions, and therefore cannot provide effective support for vehicle ride comfort optimization design.

[0005] Because they fail to fully account for the nonlinearities and non-stationary driving conditions of vehicle systems, traditional simulation results often deviate significantly from actual conditions, rendering them inaccurate. This can lead to vehicles designed based on traditional simulation experiencing poor ride comfort and prone to component fatigue damage during actual driving, making them unable to meet the engineering application requirements for improving vehicle quality and market competitiveness. Summary of the Invention

[0006] The technical problem to be solved by the present invention is to address the deficiencies of the above-mentioned existing technologies and provide a method for simulating automobile smoothness under non-stationary and nonlinear conditions, so as to solve the problems of accuracy and efficiency in solving automobile vibration under non-stationary and nonlinear conditions, and can be used to evaluate automobile smoothness under complex conditions.

[0007] In order to solve the above technical problems, the technical solution adopted by the present invention is:

[0008] A method for simulating vehicle ride comfort under non-stationary nonlinear conditions, the specific steps are as follows:

[0009] Step 1: Input basic vehicle parameters: body mass m2, unsprung mass m1, suspension stiffness k, suspension damping c, tire stiffness k t , stiffness nonlinear coefficient a k , damping nonlinear coefficient a c ;

[0010] Step 2: Calculate the exponential matrix using precise integration , and then we get the matrix ;

[0011] Step 3: Loop with frequency w and calculate the non-stationary virtual excitation at each frequency. Calculate the virtual response based on the acceleration, dynamic deflection and power spectrum density of relative dynamic load of the wheel;

[0012] Step 4: After the cycle ends at frequency w, calculate the dynamic deflection y i and relative speed The equivalent stiffness k is calculated by using the statistical linearization method. e and equivalent damping c e ;

[0013] Step 5: Judgement Is it satisfied? If so, return to step 2 and recalculate the exponential matrix; if not, proceed to the next step;

[0014] Step 6: Determine whether time t reaches running time t u ; If not, increase the time t+dt according to the time gradient and return to step 2; if the running time t is reached u , then the loop ends and the result graph is output.

[0015] Furthermore, the specific method for calculating the non-stationary road surface excitation in step 3 is:

[0016] The uniformly modulated evolving random excitation has the following form:

[0017] (16);

[0018] in, For time, is a uniform modulation function, is a stationary random process, and its power spectral density is ; Then the virtual incentive is:

[0019] (17);

[0020] in, is the time frequency, is an imaginary number;

[0021] When the car changes speed, the power spectrum density of the road surface roughness is for:

[0022] (18);

[0023] (19);

[0024] in, is the road roughness coefficient; is the spatial frequency, , unit m -1 ; v is the speed, is the initial velocity, is the acceleration;

[0025] Assume that the uniform modulation function of the road surface excitation is:

[0026] (20);

[0027] The virtual incentive is:

[0028] (twenty one);

[0029] in,

[0030] (twenty two).

[0031] Furthermore, the specific content of the fine integration method in step 2 is as follows:

[0032] The motion equation under virtual excitation is:

[0033] (1);

[0034] in, is the mass matrix; is the virtual displacement matrix, is the equivalent damping matrix, is the equivalent stiffness matrix; is the virtual incentive matrix;

[0035] Write equation (1) in the form of state equation:

[0036] (2);

[0037] in, , I represents the identity matrix, , ;

[0038] The solution of formula (2) is the sum of the homogeneous solution and the special solution, that is:

[0039] (3);

[0040] In t , homogeneous solution for:

[0041] (4);

[0042] in, is an undetermined constant vector, which is determined by the initial state t=t k decided, and represents the kth and k+1th moments in the time loop; represents the matrix exponential within the integration step τ; , then we can get ; and then we can conclude:

[0043] (5);

[0044] According to formula (4), we can get:

[0045] (6);

[0046] In an integration step , The internal subdivision is:

[0047] (7);

[0048] Among them, N=20; , which means that the integration step size Divide into m very small interval segments;

[0049] According to the addition theorem of exponential functions:

[0050] (8);

[0051] In a very small interval Perform Taylor series expansion on the exponential matrix:

[0052] (9);

[0053] make

[0054] (10);

[0055] but

[0056] (11);

[0057] The actual iteration process is carried out as follows:

[0058] (12);

[0059] According to the characteristics of the virtual excitation formula (21), and at t In a very small range, It is regarded as a constant value, so the simple harmonic external load is used:

[0060] (13);

[0061] in, represents the excitation frequency;

[0062] Substituting formula (13) into formula (2), we get:

[0063] (14);

[0064] in, , ;

[0065] Substituting formula (14) into formula (5), we get the final iterative format:

[0066] (15).

[0067] Furthermore, in step 4, the statistical linearization method is used to calculate the equivalent stiffness k e and equivalent damping c e The specific methods are as follows:

[0068] For a nonlinear vibration system of an automobile with n degrees of freedom, its dynamic equation is:

[0069] (twenty three);

[0070] Where y is the relative displacement; M is the mass matrix; is the sum of the nonlinear spring force and the damping force, ; F(t) is the road surface excitation;

[0071] Assume that the differential equation of the statistical equivalent linear system of the original nonlinear system is:

[0072] (twenty four);

[0073] in, is the equivalent damping matrix, is the equivalent stiffness matrix;

[0074] Subtracting equation (24) from equation (23), we get the error:

[0075] (25);

[0076] The essence of statistical linearization is to minimize the mean square value of the error ε:

[0077] (26);

[0078] The necessary condition for realizing Equation (26) is to and the equivalent damping matrix The partial derivative of is 0, so:

[0079] (27);

[0080] Where y={y1,y2,…y n}, y1, y2, ..., y n represents n relative displacements; , ;

[0081] Then we have:

[0082] (28);

[0083] in, ; 、 They are the equivalent stiffness matrices and the equivalent damping matrix The element in row i and column j of ;

[0084] set up and are the elastic force and damping force of elastic element i, respectively. Then the statistical equivalent stiffness and damping coefficient are obtained as follows:

[0085] (29);

[0086] because and are mutually independent stationary Gaussian processes with zero mean, so we have:

[0087] (30);

[0088] in, and They are and The root mean square values ​​of are calculated by the following formula:

[0089] (31);

[0090] in, and They are and The autopower spectral density function of .

[0091] The beneficial effects of adopting the above technical solution are: the method for simulating automobile smoothness under non-stationary nonlinear conditions provided by the present invention adopts a statistical linearization method to approximate the nonlinear system into a linear system for the nonlinear problem of the automobile vibration system. In order to improve the solution efficiency, a non-stationary virtual excitation method is adopted to construct the modulation function of the road surface. In order to improve the solution accuracy and efficiency, a precise integration is used to process the exponential function. The present invention solves the vibration system in the time-frequency domain, which can obtain the statistical characteristics of the response quantity in the time domain and the statistical characteristics of the response quantity in the frequency domain, and can comprehensively evaluate the smoothness of the automobile. The method of the present invention provides a new idea and reliable means for the smoothness analysis under complex working conditions in the field of automobile engineering. It has important theoretical significance and practical application value for improving the level of automobile design, optimizing product performance, and enhancing market competitiveness. It is expected to be widely promoted and applied in automobile research and development, manufacturing and other links. BRIEF DESCRIPTION OF THE DRAWINGS

[0092] Figure 1 A flow chart of a method for simulating vehicle ride comfort under non-stationary and nonlinear conditions provided by an embodiment of the present invention;

[0093] Figure 2 A schematic diagram of a vehicle vibration model provided by an embodiment of the present invention;

[0094] Figure 3 A comparison diagram of the RMS value curve simulation results of the vehicle body acceleration of the two methods provided in the embodiments of the present invention;

[0095] Figure 4 A comparison diagram of the RMS curve simulation results of dynamic deflection of the two methods provided in the embodiment of the present invention;

[0096] Figure 5 A comparison chart of the simulation results of the relative dynamic load root mean square value curves of the two methods provided in the embodiments of the present invention;

[0097] Figure 6 A diagram showing the power spectrum density of vehicle acceleration obtained by the method of this embodiment of the present invention;

[0098] Figure 7 A diagram showing the power spectrum density of dynamic deflection obtained by the method of this embodiment provided in an embodiment of the present invention;

[0099] Figure 8 This is a diagram of the wheel relative dynamic load results obtained by the method of this embodiment provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0100] The following embodiments of the present invention are described in further detail with reference to the accompanying drawings and examples. The following examples are used to illustrate the present invention but are not intended to limit the scope of the present invention.

[0101] like Figure 1 As shown, the method of this embodiment is as follows.

[0102] A method for simulating vehicle ride comfort under non-stationary nonlinear conditions, the specific steps are as follows:

[0103] Step 1: Input basic vehicle parameters: vehicle body mass m2, unsprung mass m1, suspension stiffness k, suspension damping tire stiffness c, tire stiffness k t , nonlinear stiffness coefficient a k , nonlinear stiffness coefficient a c .

[0104] Step 2: Calculate the exponential matrix using precise integration , and then we get the matrix The specific content of the refined integration is as follows:

[0105] The precise integration method is a numerical method based on the principle of numerical integration that solves ordinary differential equations and partial differential equations by finely processing the integral terms. It has the advantages of high accuracy, good stability, and high efficiency. The equation of motion under virtual excitation is:

[0106] (1);

[0107] in, is the mass matrix; is the virtual displacement matrix, is the equivalent damping matrix, is the equivalent stiffness matrix; is the virtual incentive matrix.

[0108] In order to facilitate the solution, formula (1) is written in the form of a state equation:

[0109] (2);

[0110] in, , I represents the identity matrix, , .

[0111] The solution of formula (2) is a homogeneous solution and special solution The sum is:

[0112] (3);

[0113] In t , homogeneous solution for:

[0114] (4);

[0115] in, is an undetermined constant vector, which is determined by the initial state t=t k decided, and represents the kth and k+1th moments in the time loop; , then we can get . We can further conclude that:

[0116] (5);

[0117] Represents the matrix index within the integration step τ, numerical integration solution The method can be conventional Duhamel integral, Newmark and Wilson-θ methods. This embodiment proposes to use the precise integration method for solution, which has the advantages of high solution accuracy and high efficiency. The core problem of precise integration is the calculation of matrix exponents. According to formula (4), we can get:

[0118] (6);

[0119] In an integration step , The internal subdivision is:

[0120] (7);

[0121] Where N=20, , which means that the integration step size Divide into m very small interval segments.

[0122] According to the addition theorem of exponential functions:

[0123] (8);

[0124] In a very small interval Perform Taylor series expansion on the exponential matrix:

[0125] (9);

[0126] make

[0127] (10);

[0128] but

[0129] (11).

[0130] In the actual iteration process, the following formula is used instead of formula (11), which can avoid the impact of rounding error on the accuracy, thus ensuring high calculation accuracy. Only one calculation is required, which is highly efficient. In the actual iteration process, the following formula is used:

[0131] (12).

[0132] Next, we derive the special solution According to the characteristics of the virtual excitation formula (21), and at t In a very small range, It is regarded as a constant value, so the simple harmonic external load is used:

[0133] (13);

[0134] in, represents the excitation frequency;

[0135] Substituting formula (13) into formula (2), we get:

[0136] (14);

[0137] in, , .

[0138] Substituting formula (14) into formula (5), we get the final iterative format:

[0139] (15).

[0140] Step 3: Calculate the non-stationary road excitation by looping at frequency w, and calculate the power spectrum density of the virtual excitation, acceleration, dynamic deflection, and relative dynamic load of the wheel at each frequency. The non-stationary virtual excitation method converts the non-stationary excitation into the product of a modulation function and a stationary excitation. The specific method is:

[0141] The uniformly modulated evolving random excitation has the following form:

[0142] (16);

[0143] in, For time, is a uniform modulation function, is a stationary random process, and its power spectral density is . Then the virtual incentive is:

[0144] (17);

[0145] in, is the time frequency, Is an imaginary number.

[0146] When the car changes speed, the power spectrum density of the road surface roughness is for:

[0147] (18);

[0148] (19);

[0149] in, is the road roughness coefficient; is the spatial frequency, , unit m -1 ; v is the speed, is the initial velocity, is the acceleration.

[0150] Assume that the uniform modulation function of the road surface excitation is:

[0151] (20);

[0152] The virtual incentive is:

[0153] (twenty one);

[0154] in,

[0155] (twenty two).

[0156] Step 4: After the loop ends with frequency w, calculate y i The root mean square root of the dynamic deflection is used to calculate the equivalent stiffness and equivalent damping using the statistical linearization method.

[0157] Statistical linearization approximates a nonlinear system as an equivalent linear system. By using statistical averaging (e.g., minimizing the mean square error), the nonlinear terms are converted to linear form, allowing the random vibration response to be solved using linear system theory. Its advantages include its ability to handle strongly nonlinear systems and its high computational efficiency.

[0158] For a nonlinear vibration system of an automobile with n degrees of freedom, its dynamic equation is:

[0159] (twenty three);

[0160] Where y is the relative displacement; M is the mass matrix; is the sum of the nonlinear spring force and the damping force, ; F(t) is the road surface excitation.

[0161] Assume that the differential equation of the statistical equivalent linear system of the original nonlinear system is:

[0162] (twenty four);

[0163] in, is the equivalent damping matrix, is the equivalent stiffness matrix.

[0164] Subtracting equation (24) from equation (23), we get the error:

[0165] (25);

[0166] The essence of statistical linearization is to minimize the mean square value of the error ε:

[0167] (26);

[0168] The necessary condition for realizing Equation (26) is to and the equivalent damping matrix The partial derivative of is 0, so:

[0169] (27);

[0170] Where y={y1,y2,…y n}, y1, y2, ..., y n represents n relative displacements; , .

[0171] Then we have:

[0172] (28);

[0173] in, ; 、 They are the equivalent stiffness matrices and the equivalent damping matrix The element at row i and column j of .

[0174] set up and are the elastic force and damping force of elastic element i, respectively. Then the statistical equivalent stiffness and damping coefficient are obtained as follows:

[0175] (29);

[0176] because and are mutually independent stationary Gaussian processes with zero mean, so we have:

[0177] (30);

[0178] in, and They are and The root mean square values ​​of are calculated by the following formula:

[0179] (31);

[0180] in, and They are and The autopower spectral density function of .

[0181] Step 5: Determine |(k ei -k ei-1 ) / k ei |>0.0001&&|(c ei -c ei-1 ) / c ei Is |>0.0001 satisfied? If so, return to step 2 and recalculate the exponential matrix; if not, proceed to the next step.

[0182] Step 6: Determine whether time t reaches running time t u ; If not, increase the time t+dt according to the time gradient and return to step 2; if the running time t is reached u , then the loop ends and the result graph is output.

[0183] Taking the 1 / 4 model of a car with 2 degrees of freedom as an example, the vibration dynamics model is as follows Figure 2 As shown, the nonlinearity of the suspension stiffness and the non-stationary condition of the car's acceleration are considered. The car parameters are shown in Table 1. The suspension stiffness has nonlinear characteristics, and its stiffness is ky+a k ky 3 , the damping is c +a c csign( ) . Where sign( ) is a sign function, when >0, sign( )=+1, when 0 o'clock, sign( )=0, <0, sign( ) = -1. The simulation condition is a Class C road, and the car acceleration is 1 m / s 2 , the acceleration time is 30 s, and the initial speed is 0 m / s.

[0184] Table 1 Vehicle parameters

[0185] Symbol / Unit significance Numerical <![CDATA[m2 / kg]]> Body quality 264.3 <![CDATA[m1 / kg]]> Unsprung mass 25.78 <![CDATA[k / Nm -1 ]]> Suspension stiffness 14984.6 <![CDATA[c / Nsm -1 ]]> Suspension damping 1081.6 <![CDATA[k t / Nm -1 ]]> Tire stiffness 116918.8 <![CDATA[a k ]]> Stiffness nonlinear coefficient 5000 <![CDATA[a c ]]> Damping nonlinear coefficient 1

[0186] Taking the acceleration of the vehicle body center of mass, the suspension dynamic deflection and the relative dynamic load of the wheel as the response quantities, the virtual response of each response quantity is:

[0187] (32);

[0188] (33);

[0189] (34);

[0190] According to equations (32) to (34), the root mean square value of the response at different times is calculated. The new method used in this embodiment is compared with the traditional method of Monte Carlo time domain solution, such as Figure 3-Figure 5 shown. Figure 3 The RMS curves of the vehicle body acceleration of the two methods have the best consistency ratio, with the maximum relative error being 2.26%. Figure 4 The RMS curves of dynamic deflection of the two methods are relatively different, with the maximum relative error being 4.75%. Figure 5 The relative dynamic load RMS curves of the two methods show good consistency, with a maximum relative error of 4.49%. Therefore, the correctness of the new method adopted in this embodiment can be verified. The runtime of the new method in this embodiment is 925 seconds, while the runtime of the Monte Carlo time-domain solution is 44,740 seconds. This new method in this embodiment saves 97.9% of the runtime, significantly improving operational efficiency.

[0191] At the same time, the method of this embodiment can obtain the time-frequency domain power spectrum density of the response quantity, such as Figure 6-Figure 8 As shown, Figure 6 is the vehicle body acceleration power spectrum density result diagram, Figure 7 is the dynamic deflection power spectrum density result diagram, Figure 8 Figure 2 is the wheel relative dynamic load result diagram. It can be seen from the figure that under non-stationary nonlinear conditions, the power spectral density of the response quantity is not only related to time, but also to frequency. As time increases, that is, the vehicle speed increases, the power spectral density of the response quantity increases. Within the frequency range, when w = 12 rad / s, the power spectral density of acceleration, dynamic deflection and wheel relative dynamic load reaches its maximum value.

[0192] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope defined by the present invention.

Claims

1. A method for simulating vehicle ride comfort under non-stationary and nonlinear conditions, characterized by: The method comprises the following steps: Step 1: Input basic vehicle parameters: body mass m2, unsprung mass m1, suspension stiffness k, suspension damping c, tire stiffness k t , stiffness nonlinear coefficient a k , damping nonlinear coefficient a c ; Step 2: Calculate the exponential matrix using precise integration , and then we get the matrix ; Step 3: Loop with frequency w and calculate the non-stationary virtual excitation at each frequency. Calculate the virtual response based on the acceleration, dynamic deflection and power spectrum density of relative dynamic load of the wheel; Step 4: After the cycle ends at frequency w, calculate the dynamic deflection y i and relative speed The equivalent stiffness k is calculated by using the statistical linearization method. e and equivalent damping c e ; Step 5: Judgement Is it satisfied? If so, return to step 2 and recalculate the exponential matrix; if not, proceed to the next step; Step 6: Determine whether time t reaches running time t u ; If not, increase the time t+dt according to the time gradient and return to step 2; if the running time t is reached u , then the loop ends and the result graph is output.

2. The method for simulating vehicle ride comfort under non-stationary and nonlinear conditions according to claim 1, characterized in that: The specific method for calculating the non-stationary road surface excitation in step 3 is: The uniformly modulated evolving random excitation has the following form: (16)； in, For time, is a uniform modulation function, is a stationary random process, and its power spectral density is ; Then the virtual incentive is: (17)； in, is the time frequency, is an imaginary number; When the car changes speed, the power spectrum density of the road surface roughness is for: (18)； (19)； in, is the road roughness coefficient; is the spatial frequency, , unit m -1 ; v is the speed, is the initial velocity, is the acceleration; Assume that the uniform modulation function of the road surface excitation is: (20)； The virtual incentive is: (21)； in, (22)。 3. The method for simulating vehicle ride comfort under non-stationary and nonlinear conditions according to claim 2, characterized in that: The specific content of the fine integration method in step 2 is as follows: The motion equation under virtual excitation is: (1)； in, is the mass matrix; is the virtual displacement matrix, is the equivalent damping matrix, is the equivalent stiffness matrix; is the virtual incentive matrix; Write equation (1) in the form of state equation: (2)； in, , I represents the identity matrix, , ; The solution of formula (2) is the sum of the homogeneous solution and the special solution, that is: (3)； In t , homogeneous solution for: (4)； in, is an undetermined constant vector, which is determined by the initial state t=t k decided, and represents the kth and k+1th moments in the time loop; represents the matrix exponential within the integration step τ; , then we can get ; and then we can conclude: (5)； According to formula (4), we can get: (6)； In an integration step , The internal subdivision is: (7)； Among them, N=20; , which means that the integration step size Divide into m very small interval segments; According to the addition theorem of exponential functions: (8)； In a very small interval Perform Taylor series expansion on the exponential matrix: (9)； make (10)； but (11)； The actual iteration process is carried out as follows: (12)； According to the characteristics of the virtual excitation formula (21), and at t In a very small range, It is regarded as a constant value, so the simple harmonic external load is used: (13)； in, represents the excitation frequency; Substituting formula (13) into formula (2), we get: (14)； in, , ; Substituting formula (14) into formula (5), we get the final iterative format: (15)。 4. The method for simulating vehicle ride comfort under non-stationary and nonlinear conditions according to claim 3, characterized in that: In step 4, the statistical linearization method is used to calculate the equivalent stiffness k e and equivalent damping c e The specific methods are as follows: For a nonlinear vibration system of an automobile with n degrees of freedom, its dynamic equation is: (23)； Where y is the relative displacement; M is the mass matrix; is the sum of the nonlinear spring force and the damping force, ; F(t) is the road surface excitation; Assume that the differential equation of the statistical equivalent linear system of the original nonlinear system is: (24)； in, is the equivalent damping matrix, is the equivalent stiffness matrix; Subtracting equation (24) from equation (23), we get the error: (25)； The essence of statistical linearization is to minimize the mean square value of the error ε: (26)； The necessary condition for realizing Equation (26) is to and the equivalent damping matrix The partial derivative of is 0, so: (27)； Where y={y1,y2,…y n }, y1, y2, ..., y n represents n relative displacements; , ； Then we have: (28)； in, ; 、 They are the equivalent stiffness matrices and the equivalent damping matrix The element in row i and column j of ; set up and are the elastic force and damping force of elastic element i, respectively. Then the statistical equivalent stiffness and damping coefficient are obtained as follows: （29）； because and are mutually independent stationary Gaussian processes with zero mean, so we have: (30)； in, and They are and The root mean square values ​​of are calculated by the following formula: (31)； in, and They are and The autopower spectral density function of .

Images