A short rope force identification method

Abstract

The short cable force identification method provided by the application obtains the acceleration signal of the short cable without the additional mass by installing the acceleration sensor on the low side of the short cable, and then obtains the acceleration signal of the short cable with the additional mass by installing the additional mass block on the low side of the short cable again; based on the acceleration signal, the frequency value and the amplitude of the short cable with or without the additional mass are obtained; the virtual hinge position of the equivalent cable model is obtained by using the obtained amplitude value fitting sine function; based on the frequency-cable force relationship of the equivalent cable, the initial cable force and the bending stiffness are obtained by substituting the length of the short cable and the frequency value without the additional mass and using the least square linear regression method; based on the obtained result, the finite element frequency value is obtained by taking the length of the short cable as the initial length of the equivalent cable finite element model; the difference between the obtained finite element frequency value and the frequency value with the additional mass is taken as the objective function, and the optimal equivalent cable length is obtained by using the least square confidence domain reflection algorithm to optimize, so that the cable force identification precision is improved.

Description

Technical Field

[0001] This invention relates to the field of cable technology, and in particular to a method for identifying the cable force of a short cable. Background Technology

[0002] Cables are key load-bearing components in cable-stayed structures, and accurate cable force identification is crucial for assessing the structural health of cable systems. Currently, methods for cable force identification generally include: pressure gauge method, traditional strain measurement method, magnetic flux method, fiber optic sensors, laser measurement method, and vibration method. Among these, the vibration method has advantages such as simple installation, convenient operation, reusability, and low cost, and has been widely recognized by engineers and applied in engineering practice. In practical applications, the vibration method can accurately identify the cable force of long cables, but for short cables, due to the influence of boundary conditions, vibration damping devices, and bending stiffness, the accuracy of the vibration method for cable force identification is not high. Since most cables in engineering are short cables, achieving high-precision cable force identification for short cables has significant scientific and engineering value.

[0003] To address the aforementioned problems, existing technologies include a technique (hereinafter referred to as Technique 1) that utilizes a single sensor to measure vibration signals at a single point to identify multiple frequencies. This technique is commonly used in vibration methods for long cables. If the bending stiffness of the long cable is ignored, it can be simplified to a tensioned string. According to string theory, its cable force T is related to the frequency f. k The relationship is

[0004]

[0005] In equation (1), m is the mass per unit cable length, L is the original cable length, and k is the modal order.

[0006] The method provided by the first technical institute is the simplest, but its application relies on the premise that the bending stiffness of the cable is very small and can be ignored. However, in actual engineering, the cable still has a certain bending stiffness, which cannot be easily ignored.

[0007] Therefore, in technique two, the cable is considered as a simply supported beam, and its cable force T is related to the frequency f. k Relationship satisfaction

[0008]

[0009] In equation (2), EI represents the bending stiffness. Based on equation (2), the bending stiffness is determined by testing the various frequencies f of the cable. kBased on the original cable length L, the cable force T and bending stiffness EI can be identified using linear regression. The method provided in Technique Two assumes that the cable can be considered a simply supported beam, meaning the boundary condition is hinged at both ends. However, in engineering, cables are more often fixed under certain boundary conditions, and there are other constraints such as vibration damping devices and positioners. These constraints are difficult to measure, resulting in complex boundary conditions that affect the relationship between the cable force T and frequency f. k The above relationship is no longer satisfied.

[0010] Some literature also provides methods that comprehensively consider the effects of bending stiffness, boundary constraints, and gravitational sag, establishing the relationship between cable force T and frequency f through theoretical derivation or numerical simulation. k These methods are difficult to apply when the relationship is clear but the boundary conditions are uncertain.

[0011] In addition to the methods mentioned above, some literature and patents establish the cable force-frequency relationship from the perspective of equivalent cable length (effective vibration length). The essence of this method is to treat a beam with complex boundaries (called the original beam) as a simply supported beam (called the equivalent beam). The original beam and the equivalent beam have the same cable force and stiffness, but different lengths. The length of the equivalent beam is the effective vibration length of the original beam (equivalent cable length). Based on this, Chinese invention patent application number 201310573666 provides a method for accurately testing the cable force of short suspenders in a suspender arch bridge, and Chinese invention patent application publication number CN113218556A provides a weighted block fixing device and a method for measuring the cable force of short cables using weighted blocks. Both use the first-order frequency of the cable under two working conditions—with and without added mass—to determine the equivalent cable length. This method, based on the dynamic equations of an equivalent beam (simply supported beam) and assuming that the added mass has a negligible impact on the mode shape, derives the expressions for the first-order frequency of the cable with and without added mass, as well as the corresponding mode shapes. After simplification, the equivalent cable length can be expressed as the ratio of the first-order frequencies, the added mass, and the mass per unit cable length. Substituting these expressions back into the relationship between cable force and frequency yields the following result.

[0012]

[0013] Here, ρA represents the mass per unit cable length m, M is the additional mass, and ω1 and ω′1 are the first-order frequencies of the cable under two different working conditions with and without additional mass. Note that this formula for calculating the cable force T0 requires known bending stiffness EI, which is not readily available in actual working conditions. Furthermore, this method requires adding additional mass at the midpoint of the cable, which is challenging in practical engineering. Therefore, the application of this method has limitations.

[0014] Reference 1 (Chen CC, Wu WH, Leu MR, et al. Tension determination of staycable or external tendon with complicated constraints using multiplevibration measurements[J]. Measurement, 2016, 86: 182-195) proposes that the mode shape of a cable can be approximated as a sine function. By fitting the amplitude (or amplitude ratio) of each measurement point to each mode shape, the equivalent cable length (effective vibration length of the cable) of each order can be obtained. Then, combined with the cable force-frequency relationship of a simply supported beam, a linear regression algorithm is applied to identify the cable force, and the cable force identification accuracy is as high as 3%.

[0015] Now combined Figure 1 A detailed explanation of the method in Reference 1 is provided below:

[0016] Figure 1 The figure shows the mode shapes. The solid black line represents the mode shape of the equivalent beam, i.e., half the wavelength of the sine curve; the dashed line represents the mode shape of the original beam. It can be seen from the figure that the mode shapes of the equivalent beam and the original beam are basically coincident. As can be seen from Reference 1, the key technical point of the method presented in Reference 1 is to obtain the half wavelength of the sine curve mode shape of the equivalent beam, i.e., the equivalent cable length; the idea is to... Figure 1 The four black dots (sensors) collect acceleration signals during vibration. Then, using discrete Fourier transform, the frequency and amplitude of these black dots' vibration are calculated. The amplitude of these four black dots (…) Figure 1 Using the ordinate of the graph, fit a sine curve to obtain its half-wavelength (equivalent cable length); the equivalent cable lengths of each order are denoted as L. k The cable force-frequency relationship of the equivalent beam satisfies the following equation:

[0017]

[0018] Equation (4) can be used to obtain the cable force and bending stiffness by least squares linear regression.

[0019] Although this method does not require known bending stiffness, it necessitates the installation of sensors on the high side (the side with the high end of the cable) to ensure accurate identification of the equivalent cable length. Figure 1 A sensor is needed at the black dot on the right side of the middle section. However, in actual engineering, it is not easy to install a sensor at a high position. The cable is thin and long, making it difficult for construction workers to climb. Using a ladder truck would also block traffic. Therefore, there are still obstacles to the practical application of this method, and it is not easy to promote its application. Summary of the Invention

[0020] In view of the above, it is necessary to provide a short cable force identification method, which improves the cable force identification accuracy by adding one or more additional masses at the low side and calculating the optimal equivalent cable length through the cable frequency information after adding the additional masses, thereby avoiding the need to place sensors at the high side and solving the defect problem of having to install sensors at the high side in the high-precision cable force identification method given in Reference 1 in the background art.

[0021] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0022] A method for identifying the tension of a short cable includes the following steps:

[0023] S1. Several accelerometers are arranged on the lower side of a short cable of length L. The accelerometers measure the acceleration signal of the short cable. The collected acceleration signals are processed by discrete Fourier transform to obtain several massless frequency values ​​f corresponding to the short cable. k and the amplitude value φ at the corresponding measurement point k ;

[0024] S2. The amplitude value φ at the measurement point k By fitting sinusoidal functions of various orders, the virtual hinge positions of the equivalent cable model are obtained;

[0025] S3. Install several additional mass blocks on the lower side of the short cable, and obtain the short cable frequency value f' after adding the mass blocks using the same method as in step S1. k ;

[0026] S4. Combine the length L of the short cable with the frequency values ​​f of each order without added mass. k Substitute the equivalent cable frequency-cable force relationship into the least squares linear regression method to obtain the initial cable force T and bending stiffness EI;

[0027] S5. Using the length L of the short cable as the initial length of the finite element model, establish an equivalent cable finite element model, substitute the initial cable force T and bending stiffness EI, and calculate the finite element frequency value f”. k ;

[0028] S6. The short cable frequency value f' after adding mass k With finite element frequency value f” k The difference is used as the objective function, and the least squares trust region reflection algorithm is used to optimize the equivalent cable length to obtain the optimal equivalent cable length, thereby improving the accuracy of cable force recognition.

[0029] Preferably, in step S2, sine curves of each order are fitted using the following relationship:

[0030]

[0031] In the formula, ak d is the amplitude coefficient. k L represents the virtual hinge position at the left end of the short cable. k For equivalent cable length.

[0032] Preferably, in step S4, the frequency-force relationship of the equivalent cable is:

[0033]

[0034] In the formula, EI is the bending stiffness, T is the cable force, and L is the bending stiffness. k Let be the equivalent cable length for each order, k be the modal order, and m be the mass per unit length of the cable.

[0035] Preferably, in step S5, the finite element frequency value f” is calculated based on the frequency characteristic equation of the equivalent cable finite element model. k The frequency characteristic equation is as follows:

[0036]

[0037] In the formula, the subscript i indicates that it is the i-th unit, n is the total number of units, and q i M represents the coordinates of the element nodes. i Let K be the mass matrix of the element. i Let q be the stiffness matrix of the element. i M i and K i They are represented as follows:

[0038] q i =[w i w i,x w j w j,x ] T

[0039]

[0040]

[0041]

[0042] Where l is the element length, w i Let w be the deflection of the short cable at the i-th node. i,x Let w be the short cable turning angle of the i-th node. j and w j,x These represent the deflection and rotation angle of another node j in the element, respectively. M is the magnitude of the added mass, and p is the element number where the added mass is located. p The mass matrix of the element corresponding to the added mass is M. When the element number i equals p, the stiffness matrix of the element is M. p Instead of M i .

[0043] Preferably, in step S6, the process of optimizing the equivalent cable length using the least squares trust region reflection algorithm is as follows:

[0044] S6.1. Initialize the finite element model parameters using the initial original length as the initial value, and set the trust region radius;

[0045] S6.2. Calculate the objective function and gradient corresponding to the parameters of the finite element model;

[0046] S6.3. Solve for the new equivalent cable length within the trust region using the reflection algorithm, and calculate the new objective function and gradient;

[0047] S6.4. Compare the old and new objective function values ​​to decide whether to use the new equivalent cable length; if the new equivalent cable length is accepted, update the trust region radius and proceed to step S6.5; if the new equivalent cable length is not accepted, it means that the convergence condition has been met, and the equivalent cable length at this time is the optimal equivalent cable length.

[0048] S6.5. Determine whether the maximum number of iterations has been reached. If the maximum number of iterations has been reached, the equivalent cable length at this time is the optimal equivalent cable length. Otherwise, use the equivalent cable length at this time as the new finite element model parameter, and then jump to step S6.2.

[0049] Preferably, the installation height of the accelerometer does not exceed half the length of the short cable.

[0050] Preferably, the installation height of the additional mass block does not exceed half the length of the short cable.

[0051] Compared with the prior art, the present invention has the following beneficial effects:

[0052] This invention obtains the acceleration signal of a short cable without added mass by installing an acceleration sensor on the lower side of the short cable, and then obtains the acceleration signal of the short cable with added mass by adding an added mass block on the lower side of the short cable. Based on the acceleration signals with and without the added mass block, the frequency values ​​and corresponding amplitudes of the short cable without added mass (including but not limited to the first order), as well as the frequency values ​​with added mass, can be obtained. The virtual hinge position of the equivalent cable model is obtained by fitting a sine function to the obtained amplitude value. Based on the frequency-force relationship of the equivalent cable, the initial cable force and bending stiffness are obtained by substituting known parameters—the length of the short cable and the frequency values ​​of each order without added mass—using the least squares linear regression method. Based on the obtained initial cable force and bending stiffness, the finite element frequency values ​​are obtained through the equivalent cable finite element model with the short cable length as the initial length. The difference between the obtained finite element frequency values ​​and the frequency values ​​with added mass is used as the objective function, and the least squares trust region reflection algorithm is used for optimization to obtain the optimal equivalent cable length and optimal cable force.

[0053] As can be seen from the above analysis, the present invention can identify the cable force without measuring the bending stiffness of the cable or placing sensors on the high side of the cable. Its measurement is convenient, conforms to the actual situation in engineering, and is easy to promote and apply in actual factory operations. Attached Figure Description

[0054] Figure 1 It is the mode shape diagram of reference 1 in the background art;

[0055] Figure 2 This is the mode shape obtained by fitting in step S2 of this invention;

[0056] Figure 3 This is a flowchart of the present invention.

[0057] The following detailed description, in conjunction with the accompanying drawings, will further illustrate the present invention. Detailed Implementation

[0058] Please see Figures 2 to 3 In a preferred embodiment of the present invention, a method for identifying the tension of a short cable includes the following steps:

[0059] S1. Several accelerometers are arranged on the lower side of a short cable of length L. The accelerometers measure the acceleration signal of the short cable. The collected acceleration signals are processed by discrete Fourier transform to obtain several massless frequency values ​​f corresponding to the short cable. k and the amplitude value φ at the corresponding measurement point k .

[0060] S2. The amplitude value φ at the measurement point k By fitting sine functions of various orders, the virtual hinge positions of the equivalent cable model are obtained. Specifically, the sine curves of various orders are fitted using the following relationship:

[0061]

[0062] In the formula, a k d is the amplitude coefficient. k L represents the virtual hinge position at the left end of the short cable. k For equivalent cable length.

[0063] The obtained sine curve is shown in the attached figure. Figure 2 The meaning is as shown.

[0064] S3. Install several additional mass blocks on the lower side of the short cable, and obtain the short cable frequency value f' after adding the mass blocks using the same method as in step S1. kIn other words, the acceleration signal of the short cable after the additional mass is installed is detected by an accelerometer, and the collected acceleration signal is processed by discrete Fourier transform to obtain the frequency value f' of the short cable after the additional mass is installed. k .

[0065] S4. Combine the length L of the short cable with the frequency values ​​f of each order without added mass. k Substituting the equivalent cable's frequency-force relationship into the least squares linear regression method, the initial cable force T and bending stiffness EI are obtained. The equivalent cable's frequency-force relationship is:

[0066]

[0067] In the formula, EI is the bending stiffness, T is the cable force, and L is the bending stiffness. k Let k be the equivalent cable length for each mode, k be the modal order, and m be the mass per unit length of the cable. In actual working conditions, the bending stiffness EI, cable force T, and equivalent cable length L are... k All are unknown. In this step, the length L of the short cable is used as the equivalent cable length L. k Substituting these values ​​into the equation, we can obtain the initial cable force T and the bending stiffness EI.

[0068] It should be noted that the frequency-force relationship of the equivalent cable is about and A linear function, in a rectangular coordinate system (x-coordinate is 1), The vertical axis is In the diagram, a straight line is represented, with the cable force and bending stiffness being the intercept and slope of this line, respectively. Thus, the frequencies f of the cable without added mass are... k By combining the equivalent cable length with the least squares linear regression method, the cable force and bending stiffness can be obtained.

[0069] S5. Using the length L of the short cable as the initial length of the finite element model, establish an equivalent cable finite element model, substitute the initial cable force T and bending stiffness EI, and calculate the finite element frequency value f”. k Among them, the finite element frequency value f” is calculated based on the frequency characteristic equation of the equivalent cable finite element model. k The frequency characteristic equation is as follows:

[0070]

[0071] In the formula, the subscript i indicates that it is the i-th unit, n is the total number of units, and q i M represents the coordinates of the element nodes. i Let K be the mass matrix of the element. i Let q be the stiffness matrix of the element. i M i and K iThey are represented as follows:

[0072] q i =[w i w i,x w j w j,x ] T

[0073]

[0074]

[0075]

[0076] Where l is the element length, w i Let w be the deflection of the short cable at the i-th node. i,x Let w be the short cable turning angle of the i-th node. j and w j,x These represent the deflection and rotation angle of another node j in the element, respectively. M is the magnitude of the added mass, and p is the element number where the added mass is located. p The mass matrix of the element corresponding to the added mass is M. When the element number i equals p, the stiffness matrix of the element is M. p Instead of M i .

[0077] S6. The short cable frequency value f' after adding mass k With finite element frequency value f” k The difference is used as the objective function, and the equivalent cable length is optimized using the least squares trust region reflection algorithm to obtain the optimal equivalent cable length. The process of optimizing the equivalent cable length using the least squares trust region reflection algorithm (or the least squares method and the Levenberg-Marquardt method) is as follows:

[0078] S6.1. Initialize the finite element model parameters using the initial original length as the initial value, and set the trust region radius;

[0079] S6.2. Calculate the objective function and gradient corresponding to the parameters of the finite element model;

[0080] S6.3. Solve for the new equivalent cable length within the trust region using the reflection algorithm, and calculate the new objective function and gradient;

[0081] S6.4. Compare the old and new objective function values ​​to decide whether to use the new equivalent cable length; if the new equivalent cable length is accepted, update the trust region radius and proceed to step S6.5; if the new equivalent cable length is not accepted, it means that the convergence condition has been met, and the equivalent cable length at this time is the optimal equivalent cable length.

[0082] S6.5. Determine whether the maximum number of iterations has been reached. If the maximum number of iterations has been reached, the equivalent cable length at this time is the optimal equivalent cable length. Otherwise, use the equivalent cable length at this time as the new finite element model parameter, and then jump to step S6.2.

[0083] Preferably, the installation height of the accelerometer does not exceed half the length of the short cable; the installation height of the additional mass block does not exceed half the length of the short cable; the installation positions of the accelerometer and the additional mass block are not fixed.

[0084] For details of the above process, please refer to the appendix. Figure 3 As can be seen from the above analysis, this invention does not use the bending stiffness EI, which is unavailable in actual working conditions, as a known condition for the optimization process. Instead, it uses several orders of frequency values ​​f without added mass. k The amplitude value φ at the measurement point k and the short cable frequency value f' after adding mass k As known conditions for optimization, these conditions can be obtained by processing the data detected by the sensors. The data does not need to be detected by the sensors and additional mass blocks installed on the high side of the cable to be effective. That is, the monitoring of the cable force can be completed by installing the sensors and additional mass blocks on the low side of the cable, avoiding the need to place the sensors on the high side. This solves the problem of inconvenience caused by the need to install sensors and other objects on the high side, and facilitates practical engineering applications.

[0085] test

[0086] To verify the feasibility of this invention, the following experiments were conducted:

[0087] Four cable vibration signal acquisitions were performed. In condition 1, the cable was fixed at both ends with no added mass, its length was 19.715 m, and the mass per unit length was 16.6 kg. In condition 2, an additional 8.3 kg mass was added at 0.4975 times the cable length. In condition 3, an additional 16.6 kg mass was added at 0.4933 times the cable length. In condition 4, additional 16.6 kg masses were added at 0.4793 times and 0.4884 times the cable length, respectively. The acceleration signals detected by the accelerometer were processed by Discrete Fourier Transform to obtain the frequencies for each of the four conditions, as shown in Table 1.

[0088] Table 1. Operating conditions and frequency values

[0089]

[0090] Based on the frequencies before and after the addition of mass, the cable force was calculated using this invention, and the results are shown in Table 2.

[0091] Table 2. Cable force calculation results under various combined working conditions.

[0092]

[0093] In Table 2, to verify the feasibility and accuracy of the present invention, it is compared and analyzed with Technology 1 and Technology 2 given in the background art of the present invention. During the analysis, the reference cable force is used as the benchmark for error analysis. The reference cable force is the average of the cable force under the condition of no additional mass and the cable force under the condition of additional mass. The cable force under the condition of no additional mass is directly read by a pressure ring (a certain testing device, but this device cannot be used to measure the cable force of an already installed cable). Technology 1 is calculated using Equation (1), Technology 2 is calculated using Equation (2), and the present invention is calculated by combining the frequencies f of the cable without additional mass after obtaining the optimal equivalent cable length. k Substitute into equation (4) to perform the calculation.

[0094] As shown in Table 2, the cable force values ​​calculated using the present invention have an error of less than 3%, while the cable force values ​​calculated using techniques 1 and 2 have an error of greater than 7%. Therefore, it can be seen that the optimization method provided by the present invention is feasible and has good calculation accuracy.

[0095] The above description is a detailed description of the preferred embodiments of the present invention. However, the embodiments are not intended to limit the scope of the patent application of the present invention. All equivalent changes or modifications made under the technical spirit of the present invention should fall within the patent scope covered by the present invention.

Claims

1. A method for identifying the force of a short cable, characterized in that, Includes the following steps: S1. In a length of L Several accelerometers are arranged on the lower side of the short cable to measure its acceleration signal. The collected acceleration signal is processed using Discrete Fourier Transform to obtain several massless frequency values ​​corresponding to the short cable. and the amplitude value at the corresponding measurement point ; S2. Amplitude value measured at the point By fitting sinusoidal functions of various orders, the virtual hinge positions of the equivalent cable model are obtained; S3. Install several additional mass blocks on the lower side of the short cable, and obtain the frequency value of the short cable after adding the mass blocks using the same method as in step S1. ; S4. Lengthen the short cable L and frequency values ​​of each order without added mass Substitute the equivalent cable frequency-cable force relationship into the least squares linear regression method to obtain the initial cable force T and bending stiffness EI; S5. Using the length L of the short cable as the initial length of the finite element model, establish an equivalent cable finite element model, substitute the initial cable force T and bending stiffness EI, and calculate the finite element frequency value. ; S6. Short cable frequency value after adding mass With finite element frequency values The difference is used as the objective function, and the least squares trust region reflection algorithm is used to optimize the equivalent cable length in order to obtain the optimal equivalent cable length. In step S6, the process of optimizing the equivalent cable length using the least squares trust region reflection algorithm is as follows: S6.

1. Initialize the finite element model parameters using the initial original length as the initial value, and set the trust region radius; S6.

2. Calculate the objective function and gradient corresponding to the parameters of the finite element model; S6.

3. Solve for the new equivalent cable length using the reflection algorithm within the trust region, and calculate the new objective function and gradient; S6.

4. Compare the old and new objective function values ​​to decide whether to use the new equivalent cable length; if the new equivalent cable length is accepted, update the trust region radius and proceed to step S6.5; if the new equivalent cable length is not accepted, it means that the convergence condition has been met, and the equivalent cable length at this time is the optimal equivalent cable length. S6.

5. Determine whether the maximum number of iterations has been reached. If the maximum number of iterations has been reached, the equivalent cable length at this time is the optimal equivalent cable length. Otherwise, use the equivalent cable length at this time as the new finite element model parameter, and then jump to step S6.

2.

2. The method for identifying the force of a short cable as described in claim 1, characterized in that, In step S2, sine curves of each order are fitted using the following relationship: ； In the formula, d is the amplitude coefficient. k This is the virtual hinge position at the left end of the short cable. L k For equivalent cable length.

3. The method for identifying the force of a short cable as described in claim 1, characterized in that, In step S4, the equivalent cable frequency-cable force relationship is: ； In the formula, EI For bending stiffness, T For cable force, L k For the equivalent length of each order, k For modal order, m The mass per unit length of the cable.

4. The method for identifying the force of a short cable as described in claim 1, characterized in that, In step S5, the finite element frequency value is calculated based on the frequency characteristic equation of the equivalent cable finite element model. The frequency characteristic equation is as follows: ； In the formula, the subscript i It indicates that it is the first i There are n units, where n is the total number of units, and q i M represents the coordinates of the element nodes. i Let K be the mass matrix of the element. i Let q be the stiffness matrix of the element. i M i and K i They are represented as follows: ； in, l For the unit length, For the first i The short cable deflection at each node, For the first i The short cable turning angle of each node, and Each of the other nodes of the unit j Deflection and rotation angle, M For the added mass size, k M is the unit number where the added mass is located. k For the mass matrix corresponding to the added mass element, when the element number is... equal k When the element's stiffness matrix is ​​M... k Instead of M i .

5. The method for identifying the force of a short cable as described in claim 1, characterized in that, The installation height of the accelerometer should not exceed half the length of the short cable.

6. The method for identifying the force of a short cable as described in claim 1, characterized in that, The installation height of the additional mass block shall not exceed half the length of the short cable.

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