Real-time hybrid test time delay compensation method and bridge vibration simulation device

Abstract

The application discloses a real-time hybrid test time delay compensation method and a bridge vibration simulation device, and the method comprises the following steps: a corresponding mathematical model is established for each shaking table subsystem, and the mathematical model is discretized; a model prediction control module is established for each shaking table subsystem, and a shaking table subsystem control command sequence and a cost function are constructed; at each control time step, each model prediction control module solves an optimization problem based on a system input reference motion state, a predicted control command sequence, a predicted motion state, a measured motion state, an assumed motion state and an assumed motion state of a neighbor shaking table subsystem, obtains an optimal control command sequence, and issues the first group of elements of the optimal control command sequence to the corresponding shaking table subsystem as a control command of the current time step. The application provides a station array cooperative control algorithm for a mixed test of a running vehicle on a bridge under an earthquake, so that the influence of time delay on the result in the test is reduced, and the precision and stability of loading control are ensured.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of magnetic suspension test, in particular to a real-time hybrid test time lag compensation method and a bridge vibration simulation device. BACKGROUND

[0002] The construction of the maglev line is also inevitable to cross the earthquake zone or along the earthquake zone, and to cross the river, valley, ecological protection area, etc., and the maglev line is often constructed on the viaduct, so that the probability of the high-speed maglev train running on the bridge during the earthquake is greatly increased, and therefore it is necessary to study the safety of the high-speed maglev train running on the bridge under the action of the earthquake.

[0003] However, in the actual situation, due to the unpredictability of the earthquake and the difficulty of the test, it is difficult to measure the data of the train running on the bridge under the action of the earthquake in the actual earthquake damage, and the importance of the hybrid test in the laboratory is highlighted. The hybrid test is a new type of test method combining numerical simulation and physical test, that is, the structure to be tested is divided into numerical structure and physical structure, the numerical structure is simulated in the computer, and the physical structure is loaded in the laboratory, and the coupling of the numerical and physical parts is realized through sensors, actuators and the like. The real-time hybrid test of the maglev and high-speed train running on the bridge under the earthquake has many difficulties, among which in the multi-array hybrid test, the shaking table has a time lag phenomenon in the loading control, and due to the uneven distribution of the mass and stiffness of the loading structure or equipment on each shaking table, the time lags of the shaking tables are inconsistent. In the hybrid test of the maglev train running on the bridge, the inconsistent time lag phenomenon may cause the system to be unstable, the numerical algorithm cannot converge, and the inconsistent time lag phenomenon of the loading control reduces the influence on the test results, and it is of great significance to use a high-precision multi-array time lag compensation control algorithm to ensure the stability of the real-time hybrid test loading control. SUMMARY

[0004] In view of the problems existing in the prior art, the present application provides a real-time hybrid test time lag compensation method and a bridge vibration simulation device, which is based on a distributed model predictive control and considers the collaborative control strategy under the condition of inconsistent time lag of each shaking table subsystem, and can provide high-precision multi-array time lag compensation for the test research related to the safety of the maglev train running on the bridge under the action of the earthquake to ensure the stability of the real-time hybrid test loading control.

[0005] In the first aspect, a real-time hybrid test time lag compensation method is provided, comprising:

[0006] S1: establishing a corresponding mathematical model for each shaking table subsystem and discretizing it;

[0007] S2: establish a model predictive control module for each shaker subsystem based on model predictive control theory, and construct a control command sequence of the shaker subsystem and a cost function considering tracking error of the shaker subsystem motion and control command limit value;

[0008] S3: at each control time step, the model predictive control module corresponding to each shaker subsystem solves an optimization problem based on the cost function based on the system input reference motion state of itself, the predicted control command sequence, the predicted motion state, the measured motion state, the assumed motion state, and the assumed motion state of the neighbor shaker subsystem, obtains the optimal control command sequence, and issues the first group of elements of the optimal control command sequence to the corresponding shaker subsystem as the control command of the current time step.

[0009] Further, in step S1, the mathematical model of each shaker subsystem is expressed by a state equation and discretely expressed as follows:

[0010] X i [t+1]=A i X i [t]+B i u i [t]+C i ω i [t]

[0011] Y i [t]=D i X i [t]

[0012] In the formula, X i [t] represents the state of the i-th shaker subsystem at the t-th time step; Y i [t] represents the measured pose of the i-th shaker subsystem at the t-th time step; u i [t] represents the control command input to the i-th shaker subsystem at the t-th time step; ω i [t] represents the influence of the neighbor shaker subsystem on the i-th shaker subsystem at the t-th time step; A i , B i , C i , D i are the state matrix, control coefficient matrix, coefficient matrix, and output state control matrix of the i-th shaker subsystem, respectively.

[0013] Further, the state matrix, control coefficient matrix, coefficient matrix, and output state control matrix of each shaker subsystem are obtained by the following method:

[0014] Preliminary establishment of the dynamic model of each shaker subsystem;

[0015] A set of control commands are input to each shaker subsystem, and the position and posture of each shaker subsystem are measured and recorded by sensors;

[0016] Based on the preliminary established dynamic model of each shaker subsystem, the control commands input to each shaker subsystem and the corresponding measured position and posture, the state matrix, the control coefficient matrix, the coefficient matrix and the output state control matrix in the state equation are identified with high precision.

[0017] Further, in step S2, the cost function is expressed as follows:

[0018]

[0019] In the formula, J i represents the cost function of the i-th shaker subsystem in the prediction interval [t, t+T]; respectively represent the tracking error and the control command limit value of the motion of the i-th shaker subsystem; respectively represent the system input reference motion state, the measured motion state, the predicted motion state and the assumed motion state of the i-th shaker subsystem and the assumed motion state of the neighbor shaker subsystem of the i-th shaker subsystem; respectively represent the optimal control command sequence and the predicted control command sequence, respectively represent the optimal control command and the predicted control command of the i-th shaker subsystem at the time t in the prediction interval τ. j represents the neighbor shaker subsystem of the i-th shaker subsystem, is the set of neighbor shaker subsystems of the i-th shaker subsystem; [τ|t] represents the i-th shaker subsystem at the time t in the prediction interval τ, and τ is the prediction interval step τ=t, t+1,..., t+T-1.

[0020] Further, the cost function satisfies the following constraints:

[0021]

[0022] In the formula, A i , B i , C i respectively represent the state matrix, the control coefficient matrix and the coefficient matrix of the i-th shaker subsystem; ω i [τ|t] represents the influence of the neighbor shaker subsystem on the i-th shaker subsystem at the time t in the prediction interval τ.

[0023] Further, at the initial control time, the system input reference motion state of each shaker subsystem and the optimal control command sequence and the assumed motion state of the current time prediction interval are given.

[0024] Further, at each control time step, the measured motion state of the shaker subsystem is obtained based on a state estimation of a mathematical model of the shaker subsystem and the measured pose at the current time step.

[0025] Further, an optimization problem L i is represented as follows:

[0026] L i = min J i

[0027] wherein J i denotes a cost function of the i-th shaker subsystem over the prediction interval [t, t+T];

[0028] Solving the optimization problem yields an optimal control command sequence and an optimal predicted state sequence for the prediction interval [t, t+T].

[0029] Further, the first element of the optimal control command sequence is issued as the control command for the current time step to the corresponding shaker subsystem, and the remaining elements are the predicted control command sequence for the next time step; the first element of the optimal predicted state sequence is the predicted motion state for the next time step, and the remaining elements are the assumed motion state for the next time step.

[0030] In a second aspect, a bridge vibration simulation device is provided, comprising:

[0031] a shaker array;

[0032] a plurality of model predictive control modules connected one-to-one with a plurality of shaker subsystems formed by the shaker array and a loading structure thereon, and adjacent model predictive control modules communicate with each other to exchange assumed motion states at each control time step;

[0033] Each model predictive control module is configured to perform the following steps:

[0034] establishing a mathematical model for the corresponding shaker subsystem and discretizing it;

[0035] constructing a control command sequence for the shaker subsystem and a cost function considering tracking errors of the shaker subsystem motion and control command limits;

[0036] At each control time step, based on the system input reference motion state, the predicted control command, the predicted motion state, the measured motion state, the assumed motion state of the corresponding shaker subsystem, and the assumed motion states of the neighboring shaker subsystems, an optimization problem based on the cost function is solved to obtain an optimal control command sequence, and the first element of the optimal control command sequence is issued as the control command for the current time step to the corresponding shaker subsystem.

[0037] The application provides a real-time hybrid test time delay compensation method and a bridge vibration simulation device, and has the following beneficial effects.

[0038] 1. Time delay is reduced. In real-time hybrid test on a bridge, due to high bridge structure frequency, long wave and short wave irregularities of the bridge, high-frequency vibration is excited by coupling effect when a maglev train passes through the bridge at high speed, which puts forward higher requirements for loading control of the real-time hybrid test. Time delay of the loading control is equivalent to a negative damping effect in the real-time hybrid test, if the time delay is too large, the system may be unstable, numerical algorithm cannot converge, the real-time hybrid test result has a large error, and the loading signal may be divergent to cause test safety problems. The model prediction control algorithm is adopted, a mathematical model of a control object is established, tracking error and control command change are considered for constraint optimization, and time delay can be effectively compensated.

[0039] 2. Cooperative control. Due to different mass and stiffness distribution of loading structures or devices on each vibration table, time delays of the vibration tables in the loading control process are inconsistent, cooperative control of each vibration table subsystem can be realized, stability and safety of the real-time hybrid test are ensured, and the accuracy of the real-time hybrid test is improved.

[0040] 3. Safety limit. Control command limit value is considered in the cost function, change of the control command is limited, the control command changes stably, input to the vibration table is relatively gentle, extreme condition test can be limited in advance, one more safety consideration is provided, and maximum performance of the system is exerted in the safety range. DETAILED DESCRIPTION

[0041] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings needed to be used in the embodiments or the prior art description. Obviously, the drawings in the following description only show some embodiments of the present application, and for those skilled in the art, other drawings can also be obtained from these drawings without creative labor.

[0042] Figure 1 FIG. 1 is a schematic diagram of application of the multi-array time delay compensation method in real-time hybrid test provided by the embodiment of the present application;

[0043] Figure 2 FIG. 2 is a schematic diagram of the multi-array time delay compensation method provided by the embodiment of the present application;

[0044] Figure 3 FIG. 3 is a schematic diagram of the model prediction control module architecture provided by the embodiment of the present application. DETAILED DESCRIPTION

[0045] In order to make the objects, technical solutions and advantages of the present application clearer, the technical solutions of the present application will be described in detail below. Obviously, the described embodiments are only some of the embodiments of the present application, but not all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative work belong to the scope of protection of the present application.

[0046] The embodiment of the present application provides a real-time hybrid test time delay compensation method suitable for multiple arrays, which is applied to the specific application link of the magnetic levitation train bridge on-track hybrid test, such as shown in the figure. Figure 1 Figure 2 The figure shows a schematic diagram of a real-time hybrid test time delay compensation method suitable for multiple arrays. A model predictive control module (MPC control module) is established for each vibration table subsystem based on model predictive control theory. The model predictive control modules of the vibration table subsystems communicate and exchange the assumed motion state in the test process, and calculate the compensated command and issue it to the vibration table subsystem. In order to further understand the technical solutions of the present application, the real-time hybrid test time delay compensation method provided by the embodiment of the present application will be described in detail below.

[0047] A real-time hybrid test time delay compensation method, comprising:

[0048] S1: a corresponding mathematical model is established for each vibration table subsystem, and is discretized.

[0049] In the real-time hybrid test of the magnetic levitation train bridge on-track under the earthquake, the magnetic levitation train vehicle is physically loaded on the vibration table array as a physical substructure, and the rest of the track and the bridge part is numerically modeled and calculated in the computer. The numerical substructure and the physical substructure satisfy the boundary coordination condition, the interaction between the numerical substructure and the vibration table array is realized through the hybrid test platform, the numerical substructure receives the boundary signal, returns to the vibration table array after calculation, and loads the physical substructure, and the new boundary signal is transmitted to the numerical substructure through the hybrid test platform, and the cycle is repeated to realize the closed loop. Each vibration table and the structure loaded thereon is regarded as a vibration table subsystem. The mathematical model of each vibration table subsystem is expressed by a state equation and discretized as follows:

[0050] X i [t+1]=A i X i [t]+B i u i [t]+C i ω i [t]

[0051] Y i [t]=D i X i [t]​

[0052] wherein X i [t] represents the state of the i-th shaker subsystem at the t-th time step; Y i [t] represents the measured pose of the i-th shaker subsystem at the t-th time step; u i [t] represents the control command input to the i-th shaker subsystem at the t-th time step; ω i [t] represents the influence of the neighboring shaker subsystems on the i-th shaker subsystem at the t-th time step; A i , B i , C i , D i are the state matrix, the control coefficient matrix, the coefficient matrix, and the output state control matrix of the i-th shaker subsystem, respectively.

[0053] In implementation, the state matrix, the control coefficient matrix, the coefficient matrix, and the output state control matrix of each shaker subsystem are obtained by the following method:

[0054] Preliminary dynamic models of the shaker subsystems are established;

[0055] A set of control commands are input to each shaker subsystem, and the poses of the shaker subsystems are measured and recorded by sensors;

[0056] Based on the preliminary dynamic models of the shaker subsystems, the control commands input to the shaker subsystems, and the corresponding measured poses, the state matrix, the control coefficient matrix, the coefficient matrix, and the output state control matrix in the state equation are identified with high precision.

[0057] S2: A model predictive control module is established for each shaker subsystem based on the model predictive control theory, and a control command sequence of the shaker subsystem and a cost function considering the tracking error of the shaker subsystem motion and the control command limit value are constructed.

[0058] In this embodiment, the control command is considered in the vertical, pitch, and nodding directions of the shaker, and the control command is u i = [u i-z , u i-roll , u i-pitch ], wherein u i-z , u i-roll , and u i-pitch correspond to the commands in the vertical, pitch, and nodding directions of the shaker, respectively, and the control command sequence includes the commands at each time step in the prediction interval.

[0059] The cost function is represented as follows:

[0060]

[0061] wherein Ji Ji(t,τ) represents the cost function of the ith shaker subsystem in the prediction horizon [t, t+T], T is the time step number corresponding to the prediction horizon; respectively represent the tracking error and the control command limit value of the ith shaker subsystem motion, considering the control command limit value The change of the control command can be limited, so that the control command changes smoothly; respectively represent the system input reference motion state (i.e. the motion state that the shaker is expected to reproduce), the measured motion state, the predicted motion state, the assumed motion state of the ith shaker subsystem, and the assumed motion state of the neighboring shaker subsystems; respectively represent the optimal control command sequence and the predicted control command sequence, respectively represent the optimal control command and the prediction of the ith shaker subsystem at time t in the prediction horizon τ at time t; j represents the neighboring shaker subsystem of the ith shaker subsystem, is the set of the neighboring shaker subsystems of the ith shaker subsystem; [τ|t] represents the ith shaker subsystem at time t in the prediction horizon τ at time t, τ is the prediction horizon step, τ = t, t+1,..., t+T-1.

[0062] In addition, the cost function needs to satisfy the following constraints:

[0063]

[0064] wherein A i , B i , C i are respectively the state matrix, the control coefficient matrix, and the coefficient matrix of the ith shaker subsystem; ω i [τ|t] represents the influence of the neighboring shaker subsystems on the ith shaker subsystem at time t in the prediction horizon τ at time t.

[0065] S3: At each control time step, the model predictive control module corresponding to each shaker subsystem solves an optimization problem based on the cost function, based on the system input reference motion state, the predicted control command sequence, the predicted motion state, the measured motion state, the assumed motion state of the corresponding shaker subsystem, and the assumed motion state of the neighboring shaker subsystems, to obtain the optimal control command sequence, and the first group of elements of the optimal control command sequence is issued as the control command of the current time step to the corresponding shaker subsystem.

[0066] The optimization problem L i based on the cost function is as follows:

[0067] L i = min J i

[0068] wherein Ji represents the cost function of the i-th shaker subsystem in the prediction horizon [t, t+T] ;

[0069] Solving the optimization problem obtains the optimal control command sequence and the optimal predicted state sequence in the prediction horizon [t, t+T]. The first group of elements of the optimal control command sequence is issued as the control command of the current time step to the corresponding shaker subsystem, and the remaining elements are the predicted control command sequence of the next time step; the first group of elements of the optimal predicted state sequence is the predicted motion state of the next time step, and the remaining elements are the assumed motion state of the next time step.

[0070] Wherein, at each time step, the measured motion state of the shaker subsystem is obtained by state estimation based on the mathematical model of the shaker subsystem and the measured pose of the current time step through the unscented Kalman filter.

[0071] It should be noted that at the initial control time, the system input reference motion state of each shaker subsystem, the optimal control command sequence and the assumed motion state of the initial prediction horizon need to be given, and then step S3 is executed in real-time hybrid test process to achieve the purpose of multi-shaker loading control time delay compensation.

[0072] In summary, the architecture of the model predictive control module can be represented as Figure 3 As shown, at each control time step of the hybrid test process, each model predictive control module solves the cost function based on the system input reference motion state, the predicted control command sequence, the measured motion state, the predicted motion state, the assumed motion state and the assumed motion state of the neighboring shaker subsystem to obtain the optimal control command sequence and the optimal predicted state sequence, wherein the first group of elements in the optimal control command sequence is issued as the control command of the current time step after time delay compensation to the shaker subsystem, and the remaining commands are the predicted control command sequence of the next time step; the first group of elements of the optimal predicted state sequence is the predicted state of the next time step, and the remaining is the assumed motion state of the prediction horizon of the next time step. It should be noted that the model predictive control module can be a single physical unit module, or two or more unit modules integrated in one unit module, which can be realized in the form of hardware or software.

[0073] The embodiment of the present application also provides a bridge vibration simulation device, comprising:

[0074] A shaker array;

[0075] A plurality of model predictive control modules are connected one by one corresponding to a plurality of shaker subsystems formed by the shaker array and the loading structure thereon, and adjacent model predictive control modules communicate with each other to exchange assumed motion states at each control time step;

[0076] Each model predictive control module is configured to perform the following steps:

[0077] establishing a mathematical model for the corresponding shaker subsystem and discretizing it;

[0078] constructing a control command for the shaker subsystem and a cost function that takes into account a tracking error of the shaker subsystem motion and a control command limit;

[0079] at each control time step, solving an optimization problem based on the cost function, based on the own system input reference motion state, the predicted control command, the predicted motion state, the measured motion state, the assumed motion state, and the assumed motion states of the neighbor shaker subsystems, to obtain an optimal control command sequence, and issuing the first element of the optimal control command sequence as the control command for the corresponding shaker subsystem at the current time step.

[0080] It can be understood that the same or similar parts in the above embodiments can be mutually referenced, and the contents not described in detail in some embodiments can be referred to the same or similar contents in other embodiments.

[0081] Although the embodiments of the present application have been shown and described above, it can be understood that the above embodiments are exemplary and cannot be understood as limiting the present application, and the ordinary skilled in the art can make changes, modifications, replacements and variations to the above embodiments within the scope of the present application.

Claims

1. A real-time hybrid experimental time delay compensation method, characterized in that, include: S1: Establish a corresponding mathematical model for each shaking table subsystem and discretize it; S2: Based on model predictive control theory, a model predictive control module is established for each shaking table subsystem, and a control command sequence for the shaking table subsystem and a cost function considering the tracking error and control command limit of the shaking table subsystem motion are constructed. S3: At each control time step, the model predictive control module corresponding to each vibration table subsystem solves the optimization problem based on the cost function based on its own system input reference motion state, predictive control command sequence, predicted motion state, measured motion state, assumed motion state and the assumed motion state of the neighboring vibration table subsystems, obtains the optimal control command sequence, and sends the first set of elements of the optimal control command sequence as the control command for the current time step to the corresponding vibration table system. In step S2, the cost function is expressed as follows: ； In the formula, Indicates the first i The shaking table system is within the prediction range. The cost function; , They represent the first i Tracking error and control command limits of the motion of a vibration table subsystem; , , , , They represent the first i The system inputs of each shaking table subsystem are: reference motion state, measured motion state, predicted motion state, assumed motion state, and assumed motion state of its neighboring shaking table subsystems. , These represent the optimal control command sequence and the predictive control command sequence, respectively. , They represent in t Prediction interval of time Time of the first i Optimal control commands and predictive control commands for a vibration table subsystem; middle Indicates the first i The neighboring vibration table systems of a vibration table subsystem, For the first i The set of neighboring vibration table subsystems of a vibration table subsystem; To predict the interval step, .

2. The real-time hybrid test time delay compensation method according to claim 1, characterized in that, In step S1, the mathematical model of each shaking table subsystem is represented by state equations and discretized as follows: ； ； In the formula, Indicates the first i The vibration table system at the first t The state of time step; Indicates the first i The vibration table system at the first t The measurement pose of the time step; Indicates the first t Step input number i Control commands for a vibration table subsystem; Indicates the effect of the neighboring vibration table subsystem on the first i The vibration table system at the first t The influence of time step; , , , The first i The state matrix, control coefficient matrix, coefficient matrix, and output state control matrix of each vibration table subsystem.

3. The real-time hybrid test time delay compensation method according to claim 2, characterized in that, The state matrix, control coefficient matrix, coefficient matrix, and output state control matrix of each vibration table subsystem are obtained through the following method: Preliminary dynamic models of each vibration table subsystem were established; Input a set of control commands to each vibration table subsystem, and use sensors to measure and record the pose of each vibration table subsystem; Based on the preliminary established dynamic models of each shaking table subsystem and the input control commands and corresponding measurement poses of each shaking table subsystem, the state matrix, control coefficient matrix, coefficient matrix and output state control matrix in the state equation are identified with high precision.

4. The real-time hybrid test time delay compensation method according to claim 1, characterized in that, At the initial control moment, the system input reference motion state of each vibration table subsystem is given, along with the initial optimal control command sequence and assumed motion state for the current prediction interval.

5. The real-time hybrid test time delay compensation method according to claim 1, characterized in that, At each control time step, the measured motion state of the vibration table subsystem is estimated based on the mathematical model of the vibration table subsystem and the measured pose at the current time step.

6. The real-time hybrid test time delay compensation method according to claim 1, characterized in that, The cost function satisfies the following constraints: ； In the formula, , , The first i State matrix, control coefficient matrix, and coefficient matrix of a vibration table subsystem; Indicates the effect of the neighboring vibration table subsystem on the first i A vibration table system in t Prediction interval of time The impact of time.

7. The real-time hybrid test time delay compensation method according to claim 1, characterized in that, Optimization problems based on cost functions It is expressed as follows: ； In the formula, Indicates the first i The shaking table system is within the prediction range. The cost function; Solving the optimization problem yields the prediction interval. The optimal control command sequence and the optimal predicted state sequence.

8. The real-time hybrid test time delay compensation method according to claim 7, characterized in that, The first set of elements of the optimal control command sequence is sent to the corresponding vibration table subsystem as the control command for the current time step, and the remaining elements are used as the predictive control command sequence for the next time step; the first set of elements of the optimal predictive state sequence is used as the predicted motion state for the next time step, and the remaining elements are used as the assumed motion state for the next time step.

9. A bridge vibration simulation device, characterized in that, include: Vibration table array; Multiple model predictive control modules are connected one-to-one with multiple shaking table subsystems consisting of a shaking table array and its loading structure, and adjacent model predictive control modules communicate with each other to exchange assumed motion states at each control time step; each model predictive control module is configured to perform the following steps: A mathematical model is established for the corresponding shaking table subsystem, and then discretized. Construct the control command sequence for the shaking table subsystem and the cost function that considers the tracking error of the shaking table subsystem motion and the control command limit; At each control time step, based on its own system input reference motion state, predictive control command, predicted motion state, measured motion state, assumed motion state, and assumed motion state of neighboring shaking table subsystems, the optimization problem based on the cost function is solved to obtain the optimal control command sequence, and the first set of elements of the optimal control command sequence is sent to the corresponding shaking table system as the control command for the current time step. The cost function is expressed as follows: ； In the formula, Indicates the first i The shaking table system is within the prediction range. The cost function; , They represent the first i Tracking error and control command limits of the motion of a vibration table subsystem; , , , , They represent the first i The system inputs of each shaking table subsystem are: reference motion state, measured motion state, predicted motion state, assumed motion state, and assumed motion state of its neighboring shaking table subsystems. , These represent the optimal control command sequence and the predictive control command sequence, respectively. , They represent in t Prediction interval of time Time of the first i Optimal control commands and predictive control commands for a vibration table subsystem; middle Indicates the first i The neighboring vibration table systems of a vibration table subsystem, For the first i The set of neighboring vibration table subsystems of a vibration table subsystem; To predict the interval step, .

Images