CN109918714B - Non-stationary random road surface unevenness overrun probability determination method based on Wiener process - Google Patents
Non-stationary random road surface unevenness overrun probability determination method based on Wiener process Download PDFInfo
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Abstract
The invention discloses a method for determining the out-of-limit probability of non-stationary random road surface unevenness based on a Wiener process, which specifically comprises the following steps: the road surface unevenness is simulated by a random process, wherein the random process B (T) is set as a standard Wiener process, and the safety limit is B and T 1 Time to first exceed safety margin, T, of B (T) 2 For the time B (T) crosses the safety margin for the second time, calculate T 1 Probability distribution function of
And T 2 Of the probability density function
Calculating T 1 And T 2 Of a joint probability density function
Calculating the time interval Δ T 1 Has a probability distribution function of
And a time interval Δ T 1 Probability density function of
Description
Technical Field
The invention belongs to the technical field of random process probability analysis and automobile mechanical engineering, and particularly relates to a method for determining out-of-limit probability of non-stationary random road surface unevenness based on a Wiener process.
Background
The excitation of the automobile during running mainly comes from the road surface, and the simulation and analysis of the unevenness of the road surface are the basis for analyzing the running smoothness of the automobile. The road surface unevenness is the height of the road surface with respect to a certain reference plane. Macroscopically, the road pavement is substantially flat. In practice, however, the road surface unevenness is often simulated by a random process because the road surface height varies randomly with its position in the road advancing direction at a fine observation scale (a 1cm difference in height causes a large vibration of a vehicle traveling at high speed). If the road surface unevenness is large and exceeds a certain limit, the vehicle is subjected to unacceptable vibration, and the riding comfort is seriously affected. However, due to the limit value of the construction process level, the pavement construction is difficult to be controlled accurately, the height of the road surface exceeding a certain limit value is difficult to avoid, and the height exceeding once occurs at a certain distance. Therefore, the distance between the two adjacent vehicle bodies is the overrun height (i.e. the distance between the two adjacent vehicle bodies is the impact), which is one of the key parameters for designing the anti-vibration system of the vehicle.
Since the road surface unevenness is a random process, the problem of the overrun height can be encountered by the distance between the road surfaces, namely the time interval between two times of overrun limits in the road surface random process. Obviously, the time interval is a random variable, and the probability information of the random variable is the road impact information required to be known by the anti-vibration design of the automobile.
The problem of the time interval between two times of road height overrun impacts in the vehicle driving process is not related to analysis means, and quantitative calculation results related to the problem are not available. But the problem can provide important basis for the anti-vibration design of the vehicle, and effectively improve the driving comfort of the vehicle.
Disclosure of Invention
The invention aims to provide a method for determining the out-of-limit probability of the unevenness of a non-stationary random road surface based on a Wiener process, and solves the problems that in the prior art, the length of a time interval between two times of road height out-of-limit impact in the driving process of a vehicle is not analyzed and calculated quantitatively, and further parameters cannot be provided for anti-vibration design of the vehicle.
The technical scheme adopted by the invention is that the method for determining the out-of-limit probability of the non-stationary random road surface unevenness based on the Wiener process is specifically carried out according to the following steps:
s1, simulating the road surface unevenness by using a random process, and setting a random process B (t) as a standard Wiener processRange, safety margin b, T 1 Time to first exceed safety margin of B (T), T 2 Time B (T) to the second time of exceeding the safety margin, T 1 、T 2 Is a random variable, T 1 ＜T 2 ，T 2 Has a probability distribution function of
Due to T 1 ＜T 2 Therefore, it is possible to
Also means that the random process B (T) is in the time period (T) 1 T) probability of occurrence of a safety margin b;
step S2, calculating T 1 Probability distribution function of
And T 2 Probability density function of
Step S3, calculating T 1 And T 2 Of a joint probability density function
Step S4, time interval delta T 1 ＝T 2 -T 1 Calculating the time interval DeltaT 1 Has a probability distribution function of
Step S5, calculating a time interval delta T 1 Of the probability density function
Step S6, utilizing the time interval delta T 1 Probability density function of
And carrying out vibration damping design on the automobile in the road driving process.
Further, the step S2 is specifically performed according to the following steps:
step S21, a new random process B '(T), B' (T) = B (T) -B (T) is established 1 )(t≥T 1 ) (ii) a According to the translational invariance of the standard Wiener process, B '(T) is still the Wiener process, and B' (T) 1 )＝B(T 1 )-B(T 1 )＝0；
Step S22, setting the random event as A (T) 1 ,t)，A(T 1 T) is the new random process B' (T) over a time period (T) 1 T) zero crossing occurs, and a random event A (T) is obtained according to the law of inverse cosine 1 T) has a probability of
"random Process B (T) over a time period (T) 1 T) occurrence of the safety margin B "is equivalent to" a new random process B' (T) during the time period (T) 1 T) zero crossings occur, so that the random process B (T) takes place over a time period (T) 1 T) probability of occurrence of safety margin b
Equals random event A (T) 1 T) probability P { A (T) 1 T) }, as shown below:
step S23, T 1 Is a probability distribution function of
To obtain T 1 Has a probability density function of
T 2 Has a probability density function of
Further, the formula in step S22
In the above formula, T 2 Obeying a composite profile; t can be obtained according to the theorem of composite distribution 2 Of the probability density function
Wherein,
represents T 1 Is determined by the probability density function of (a),
and
the meaning of (A) is consistent with that of (B),
the independent variable t is changed into the independent variable x
Will T 1 Probability density function of (1) and T 2 Into a probability density function
x<t, obtaining
Wherein,
according to the specific value of the safety limit b
Explicitly parsing the expression.
Further, the step S3 is specifically performed according to the following steps:
time T of B (T) to first exceed safety limit 1 And B (T) time T for the second exceeding of the safety limit 2 Has a correlation, according to the conditional probability theory, at T 1 ＝t 1 Under the condition of (1) 2 Has a probability density function of
T 1 And T 2 Is a joint probability density function
Wherein,
is at t = t 1 Time T 1 Probability density function of (a), t 1 、t 2 Is that
An independent variable of (d);
is composed of
So as to obtain the composite material,
comprising T 1 When T is 1 When determined, then
Can also be determined, therefore, formula
Substantially at T 1 ＝t 1 Under the condition of (1) 2 Probability density of, i.e.
T 1 And T 2 Is a joint probability density function
Further, the step S4 is specifically performed according to the following steps:
time interval Δ T 1 Has a probability distribution function of
Wherein,
further, the step S5 is specifically performed according to the following steps:
to pair
Derived on both sides by a time interval Δ T 1 Of the probability density function
Wherein,
representing a non-perfect gamma function, i.e.
The method has the advantages that the probability density function of the time (random variable) required by exceeding the safety threshold value twice in the road surface random process is established, namely the probability distribution of the time interval between the vehicle and the larger impact again after encountering the larger impact once is provided, the information provides the road impact extreme condition required to be known by the automobile anti-vibration design, and the accurate and complete design basis is provided for further improving the vibration reduction effect in the vehicle driving process according to the road surface condition.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
According to the research of the prior literature, the power spectral density is generated by the Wiener process
Road surface chart (Yushisheng-automobile theory [ M)]Beijing, mechanical industry Press, 2000) have the same form and connotation, and therefore, the vibration simulation analysis of automobiles can be performed using Wiener process as a non-stationary random road surface excitation signal (the variance σ of Wiener process represents the unevenness condition of the road surface, and ω represents the frequency). Because the road surface can be simulated by the Wiener process, the time interval of two times of road height overrun impacts in the vehicle running process is the time interval of two adjacent times of the Wiener process exceeding the safety limit, and the time interval is a continuous random variable with the value range of (0, ∞). To accurately predict the road twice suffered by a vehicle in the driving processThe length of the time interval of the high overrun impact is to obtain the probability distribution of the time interval of two adjacent overrun safety limit values of the Wiener process.
According to the definition of the standard Wiener process, the standard Wiener process is an independent incremental process, so the standard Wiener process has Markov property, and the standard Wiener process is also a Markov process.
According to the time interval Δ T 1 Probability density function of
And giving a safety limit b to obtain an accurate probability density function of a time interval between two adjacent overtaking safety limits, namely waiting time for the vehicle to be subjected to the overrun impact again after the vehicle is subjected to the overrun impact once, and providing quantitative basis for the targeted vibration resistance design of the vehicle. In the field of random process probability analysis, a probability density function of any two adjacent crossing time intervals of a Wiener process and a safety boundary is obtained for the first time, and the probability density function is an explicit analytical expression and is very convenient to calculate and apply. The probability distribution of the interval between any two adjacent crossing times of the Wiener process and the safety boundary can be directly applied to the problem that the distance between road surfaces is too long, the problem of the overrun height is met, namely, the distance that an automobile runs is subjected to overrun impact once, and key parameters are provided for the vibration reduction design of the automobile in the road running process.
Examples
According to the method for determining the out-of-limit probability of the unevenness of the non-stationary random road surface based on the Wiener process, a formula is used
Namely, the probability prediction can be carried out on the time interval which is subjected to the two times of road height overrun impacts in the vehicle running process, and when the safety limit b =1, the probability density function of the time interval which is subjected to the two times of road height overrun impacts in the vehicle running process is
Can be based on
The provided probability information of the time interval between two times of road height overrun impacts during the running process of the vehicle is subjected to anti-vibration design in a targeted mode.
All the embodiments in the present specification are described in a related manner, and the same and similar parts among the embodiments may be referred to each other, and each embodiment focuses on the differences from the other embodiments. In particular, for the system embodiment, since it is substantially similar to the method embodiment, the description is simple, and for the relevant points, reference may be made to the partial description of the method embodiment.
The above description is only for the preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention shall fall within the protection scope of the present invention.
Claims (6)
1. The method for determining the out-of-limit probability of the unevenness of the non-stationary random road surface based on the Wiener process is characterized by comprising the following steps:
s1, simulating the road surface unevenness by using a random process, wherein the random process B (T) is set as a standard Wiener process, the safety limit is B, and T is set as 1 Time to first exceed safety margin, T, of B (T) 2 Time B (T) to the second time of exceeding the safety margin, T 1 、T 2 Is a random variable, T 1 ＜T 2 ，T 2 Has a probability distribution function of
Due to T 1 ＜T 2 Therefore, it is possible to
Also means that the random process B (T) is in the time period (T) 1 T) probability of occurrence of a safety margin b;
step S2, calculating T 1 Probability distribution function of
And T 2 Probability density function of
Step S3, calculating T 1 And T 2 Is a joint probability density function
Step S4, time interval delta T 1 ＝T 2 -T 1 Calculating the time interval DeltaT 1 Is a probability distribution function of
Step S5, calculating a time interval delta T 1 Probability density function of
Step S6, utilizing the time interval delta T 1 Of the probability density function
And carrying out vibration damping design on the automobile in the road driving process.
2. The Wiener process-based method for determining the out-of-limit probability of non-stationary random road surface unevenness according to claim 1, wherein the step S2 is specifically performed according to the following steps:
step S21, a new random process B '(T) is established, B' (T) = B (T) -B (T) 1 ) (t≥T 1 ) (ii) a According to the translational invariance of the standard Wiener process, B '(T) is still the Wiener process, and B' (T) 1 )＝B(T 1 )-B(T 1 )＝0；
Step S22, setting the random event as A (T) 1 ,t)，A(T 1 T) is the new random process B' (T) over a time period (T) 1 T) zero crossing occurs, and a random event A (T) is obtained according to the law of inverse cosine 1 T) has a probability of
"random Process B (T) over a time period (T) 1 T) occurrence of the safety margin B "is equivalent to" a new random process B' (T) during the time period (T) 1 T) zero crossings occur, so that the random process B (T) occurs over a time period (T) 1 T) probability of occurrence of the safety margin b
Equals random event A (T) 1 T) probability P { A (T) 1 T) }, as shown below:
step S23, T 1 Has a probability distribution function of
To obtain T 1 Has a probability density function of
T 2 Has a probability density function of
3. The Wiener process-based method for determining out-of-limit probability of unevenness of non-stationary random road surface according to claim 2, wherein the formula in the step S22
In the above formula, T 2 Obeying a composite profile; t can be obtained according to the theorem of composite distribution 2 Of the probability density function
Wherein,
represents T 1 Is determined by the probability density function of (a),
and
the meaning of (A) is consistent with that of (B),
the independent variable t is changed into the independent variable x
Will T 1 Probability density function of (1) and T 2 By a probability density function
To obtain
Wherein,
according to the specific value of the safety limit b
Explicit analytic expressions ofFormula (II) is shown.
4. The Wiener process-based method for determining the out-of-limit probability of non-stationary random road surface irregularity according to any of claims 2 to 3, wherein the step S3 is specifically performed according to the following steps:
time T of first exceeding of safety limit of B (T) 1 And B (T) time T for the second exceeding of the safety limit 2 Having a correlation, according to the conditional probability theory, at T 1 ＝t 1 Condition (2) T 2 Has a probability density function of
T 1 And T 2 Of a joint probability density function
Wherein,
is at t = t 1 Time T 1 Probability density function of t 1 、t 2 Is that
An independent variable of (d);
is composed of
So as to obtain the compound with the characteristics of,
comprising T 1 When T is 1 When determined, then
Can also be determined, therefore, formula
Substantially at T 1 ＝t 1 Under the condition of (1) 2 Probability density of, i.e.
T 1 And T 2 Is a joint probability density function
5. The Wiener process-based method for determining the out-of-limit probability of non-stationary random road surface irregularity according to claim 3, wherein the step S4 is specifically performed according to the following steps:
time interval Δ T 1 Has a probability distribution function of
Wherein,
6. the Wiener process-based method for determining the out-of-limit probability of non-stationary random road surface unevenness according to claim 5, wherein the step S5 is specifically performed according to the following steps:
for is to
Derived on both sides by a time interval Δ T 1 Probability density function of
Wherein,
representing a non-perfect gamma function, i.e.
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