Singular function model analysis method for natural frequency of string-string composite structure

Abstract

The present invention relates to a singular function model analysis method for the natural frequency of a string-stayed combination structure, and belongs to the technical field of building structure model analysis. The method comprises the following steps: simplifying the concentrated mass string-stayed combination structure; converting the concentrated mass into the distributed mass along the length of the cable through the Dirac function; solving the cable vibration mode function with respect to time t and position x; performing Laplace transformation on the vibration equation of the lateral free vibration in the tensioned string plane; performing inverse Laplace transformation on the result of step four; solving the vibration mode equation of the concentrated mass string-stayed structure model, obtaining the singular function model, and thus obtaining the frequencies of each order of the concentrated mass string-stayed combination structure. The singular function model of the present invention can well predict the influence of the concentrated mass on the natural frequency of the string-stayed combination structure.

Description

Technical Field

[0001] The invention relates to a model analysis method, in particular to a singular function model analysis method of the natural frequency of a string-pulling combination structure, and belongs to the technical field of building structure model analysis. Background Art

[0002] As a typical string-drawing structure, it consists of a metal wire string and a bow frame, which are hinged together at the joint; the bow frame tensions the metal wire string, and the tension is changed by adjusting the curvature of the bow frame. If a small magnet sheet is attached to the metal wire string of the string-drawing structure, a string-drawing combined structure with concentrated mass is formed.

[0003] The cable-stayed structure is associated with many practical engineering systems, such as the stress-bearing steel bars in prestressed concrete, cable-stayed arches, and especially the cables of suspension bridges. As a real structural form, some mass dampers, decorative lamps, or vibration meters are often installed on the bridge cables. At this time, the cable-stayed structure with concentrated mass can be used as a scaled model for system modal testing and analysis. The cable-stayed structure with concentrated mass is not only an extension of the conventional cable-stayed structure without concentrated mass, but also can be regarded as a composite control system for the natural frequency of the conventional cable-stayed structure.

[0004] The string vibration frequency of a conventional string-drawn structure without concentrated mass is related to the material, cross-sectional diameter, span, tension and other parameters of the metal wire string, and has the following classic calculation formula:

[0005]

[0006] Where k = 1, 2, ... are positive integers, T is the tension on the string, is the string line density, and L is the string length. Formula (1) shows that the string vibration frequency family of the conventional string structure appears proportionally, and the high-order frequencies are all integer multiples of the basic frequency.

[0007] For the string-and-string structure, the concentrated mass attached to the string will cause the string vibration frequency to change, and the amplitude of the change is affected by the size, quantity and relative position of the concentrated mass. The law of change of the natural frequency of the concentrated mass-string-and-string structure has attracted the attention of industry experts. In the existing research literature, Sergev et al. proposed the transfer matrix method, which divided the string at the concentrated mass and obtained the vibration equation of the system according to the equilibrium condition and displacement coordination condition at the concentrated mass; Cheng et al. used the Hamilton principle to derive the vibration equation of the single concentrated mass-string structure, and then proposed a small mass model that ignores the tangential acceleration of the string and the additional mass and a model suitable for a larger additional mass that only ignores the tangential acceleration of the string according to the size of the concentrated mass. On this basis, Cheng and Perkins used the matrix transfer method to generalize a single concentrated mass to multiple masses, and derived a general model of the string with small equilibrium curvature and horizontal support. Yu added a static load in the horizontal direction on the basis of Sergev, modified and generalized the transfer matrix method, and derived a more general string vibration equation under the action of distributed loads and concentrated loads. In addition, the Galerkin method and the multi-scale method are also used in the derivation of the vibration equation of the concentrated mass-string structure. Since the transfer matrix method is easy to understand, it is often used in the vibration analysis of general string structures. For example, Zhou Haijun uses the transfer matrix method to take the damper into account in the string structure and analyzes the control effect of the concentrated mass and damper parameters on the cable vibration. However, when dealing with strings with more concentrated mass, the transfer matrix method needs to perform multiple segmented iterative processing on the strings. The calculation process is very cumbersome and is not suitable for the vibration analysis of complex string structures. Summary of the invention

[0008] In view of the above problems in the prior art, the present invention provides a singular function model analysis method for the natural frequency of a cable-stayed structure. The singular function is used to formally transform the concentrated mass on the cable into a uniformly distributed mass, and the concentrated mass is directly integrated into the vibration equation of the complete cable, so that the free vibration problem of the concentrated mass-stayed structure with complex boundary conditions is transformed into a forced vibration problem of a homogeneous cable with simple boundary conditions, and the frequency of the forced vibration is the natural frequency of the original cable-concentrated mass system.

[0009] To achieve this technical purpose, the present invention adopts the following scheme:

[0010] The singular function model analysis method of the natural frequency of the string-drawn composite structure includes the following steps:

[0011] Step 1: Simplify the concentrated mass string assembly structure;

[0012] Step 2: Convert the concentrated mass into distributed mass along the cable length through the Dirac function;

[0013] Step 3: Solve the cable vibration mode as a function of time t and position x;

[0014] Step 4: Perform Laplace transformation on the vibration equation of the lateral free vibration in the tension string plane;

[0015] Step 5: Perform inverse Laplace transform on the result of step 4;

[0016] Step 6: Solve the vibration mode equation of the concentrated mass cable structure model to obtain the singular function model, thereby obtaining the various frequencies of the concentrated mass cable composite structure.

[0017] Furthermore, the step 1 is specifically as follows: simplifying the concentrated mass string-drawn composite structure into a simply supported system hinged at both ends, ignoring the bending stiffness and sag of the string and its vibration damping; wherein the total length of the string is L, and the unit linear density is The tension is T; the i-th concentrated mass on the string is m i , i = 1 ~ n; horizontal coordinate x = l i is the horizontal distance between the concentrated mass and the left end hinge support.

[0018] Furthermore, the step 2 is specifically as follows: for the case where uniformly distributed mass and concentrated mass coexist on the string, the concentrated mass is converted into distributed mass along the length of the string by using the Dirac function. At this time, the mass function of the string along the axial direction is:

[0019]

[0020] in

[0021]

[0022] is the Dirac function.

[0023] Furthermore, the step three is specifically as follows:

[0024] From the vibration theory, we know that the vibration equation of the lateral free vibration in the tension string plane is:

[0025]

[0026] According to the separation of variables method, the solution of equation (4) is:

[0027] y(x,t)=Y(x)sintω (5)

[0028] Where Y(x) is the lateral displacement of the string at position x, ω is the natural circular frequency of the string, and y(x,t) is the function of the cable vibration mode with respect to time t and position x.

[0029] Furthermore, the step 4 is specifically as follows:

[0030] Substituting equation (2) and equation (5) into equation (4), we can obtain

[0031]

[0032] After Laplace transforming equation (6), we can get:

[0033]

[0034] Among them, p is a complex parameter variable.

[0035] Furthermore, the step five is specifically as follows:

[0036] Performing inverse Laplace transform on equation (7) yields:

[0037]

[0038] in

[0039]

[0040]

[0041] and

[0042]

[0043] is the Heaviside step function.

[0044] Furthermore, the step six is ​​specifically as follows:

[0045] Assume that the boundary conditions of the concentrated mass string composite structure are:

[0046]

[0047] Substituting equation (11) into equation (8) can solve the vibration mode equation of the concentrated mass string composite structure model:

[0048]

[0049] Set x=l1,l2,L,l n Substituting into equation (12) respectively, we can get the characteristic equation of the structural model:

[0050] (AB)·Y=0 (13)

[0051] in

[0052]

[0053]

[0054] Y=[Y(l1) Y(l2) ... Y(l n )] T (14c)

[0055]

[0056] B kj =βLα j sinβ(l k -l j ) (14e)

[0057] When sinβL≠0, the frequency equation of the concentrated mass string composite structure is:

[0058] |AB|=0 (15)

[0059] This frequency equation reflects the portion of the string structure frequency that is changed by the concentrated mass;

[0060] Therefore, the frequency equation of the concentrated mass string structure is the superposition of the frequencies when sinβL=0 and sinβL≠0, that is:

[0061] sinβL·|AB|=0 (16)

[0062] By solving equation (16), we can obtain the frequencies of the concentrated mass cable composite structure.

[0063] Compared with the prior art, the present invention has the following beneficial effects:

[0064] (1) The present invention can effectively analyze the influence of the size, quantity and spatial distribution of concentrated mass on the natural frequency of the string-pull combination structure.

[0065] (2) The existence of concentrated mass will not increase the natural frequency of the string-string structure. Compared with the conventional string-string structure without concentrated mass, its natural frequencies of all orders are usually reduced; only when all concentrated masses are located at a certain vibration mode node of the conventional string-string structure, the corresponding natural frequency of the string-string composite structure is equal to the natural frequency of the conventional string-string structure without concentrated mass.

[0066] (3) The natural frequency of a single concentrated mass string-type structure is limited by the same-order natural frequency and the lower-order natural frequency of a conventional string-type structure without concentrated mass.

[0067] (4) Under the condition that the total amount of concentrated mass remains unchanged, as the number and distribution range of concentrated mass increase, the natural frequencies of each order of the string-string composite structure tend to their respective limits. The natural frequency ratio of the string-string composite structure to the same order of the conventional string-string structure without concentrated mass is ω k / ω' k Approaching a constant value in is the mass ratio, and m is the sum of all concentrated masses on the string. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] The present invention will be further described below by means of the accompanying drawings.

[0069] Figure 1 A simplified analysis model for the concentrated mass string-pulling combined structure of the present invention;

[0070] Figure 2 The first five natural frequencies of the single concentrated mass string structure vary with mass ratio. The calculation results of the changes, where (a) is the actual frequency, (b) is the relative frequency, (c) is the change curve of the 2nd and 3rd order frequencies with α1, and (d) is the change curve of the 4th and 5th order frequencies with α1;

[0071] Figure 3 The first five natural frequencies of the single concentrated mass string structure vary with the concentrated mass position l1 / L. In the figure, (a) is the actual frequency and (b) is the relative frequency.

[0072] Figure 4 The influence of the position of concentrated mass on the natural second-order frequency of the string structure;

[0073] Figure 5 The spatial distribution of the natural frequency of the two concentrated mass-string combination structure. In the figure, (a) is the first order, (b) is the second order, (c) is the third order, and (d) is the fourth order;

[0074] Figure 6 The graph shows the variation of the natural frequency of a single concentrated mass string-pull composite structure with the tension, where (a) is the actual frequency of the structure and (b) is the relative frequency.

[0075] Figure 7 The variation law of the natural frequency of the two concentrated mass string structure with the tension, where (a) is the actual frequency of the structure and (b) is the relative frequency;

[0076] Figure 8 The variation law of the first five frequencies of the concentrated mass string structure with the number of concentrated masses (the total mass of the concentrated masses remains unchanged). (a) is the variation curve of the first five frequencies with the number of concentrated masses n, and (b) is the relative frequency ω k / ω' k Curve that changes with the number n. DETAILED DESCRIPTION

[0077] Combined with Figure 1-8 The embodiments of the present invention are described in detail.

[0078] First of all, it should be noted that the anal fistula drainage and injection irrigation tube shown in the drawings is intended to illustrate the present invention concisely, clearly and concisely, so the drawings cannot be used as a limitation to the present invention.

[0079] In the description of the present invention, unless otherwise specified, words indicating orientation or positional relationships such as “upper part”, “lower part”, “middle part”, “two ends”, “upper end”, “lower end”, and “inside” are merely used to more conveniently describe the present invention and therefore cannot be understood as limiting the present invention.

[0080] Model derivation

[0081] The singular function model analysis method of the natural frequency of the string-drawn assembly structure of this embodiment includes the following steps:

[0082] Step 1: Simplify the concentrated mass string assembly structure;

[0083] like Figure 1 As shown in the figure, the concentrated mass string structure is simplified to a simply supported system with hinges at both ends, and the bending stiffness and sag of the string and its vibration damping are ignored; where the total length of the string is L and the unit linear density is The tension is T; the i-th concentrated mass on the string is m i , i = 1 ~ n; horizontal coordinate x = l i is the horizontal distance between the concentrated mass and the left end hinge support.

[0084] Step 2: Convert the concentrated mass into distributed mass along the cable length through the Dirac function;

[0085] In view of the coexistence of uniformly distributed mass and concentrated mass on the string, the concentrated mass is converted into distributed mass along the length of the cable through the Dirac function. At this time, the mass function of the string along the axial direction is:

[0086]

[0087] in

[0088]

[0089] is the Dirac function.

[0090] Step 3: Solve the cable vibration mode as a function of time t and position x;

[0091] From the vibration theory, we know that the vibration equation of the lateral free vibration in the tension string plane is:

[0092]

[0093] According to the separation of variables method, the solution of equation (4) is:

[0094] y(x,t)=Y(x)sintω (5)

[0095] Where Y(x) is the lateral displacement of the string at position x, ω is the natural circular frequency of the string, and y(x,t) is the function of the cable vibration mode with respect to time t and position x.

[0096] Step 4: Perform Laplace transformation on the vibration equation of the lateral free vibration in the tension string plane;

[0097] Substituting equation (2) and equation (5) into equation (4), we can obtain

[0098]

[0099] After Laplace transforming equation (6), we can get:

[0100]

[0101] Among them, p is a complex parameter variable.

[0102] Step 5: Perform inverse Laplace transform on the result of step 4;

[0103] Performing inverse Laplace transform on equation (7) yields:

[0104]

[0105] in

[0106]

[0107]

[0108] and

[0109]

[0110] is the Heaviside step function.

[0111] Step 6: Solve the vibration mode equation of the concentrated mass cable structure model to obtain the singular function model, thereby obtaining the various frequencies of the concentrated mass cable composite structure.

[0112] Assume that the boundary conditions of the concentrated mass string composite structure are:

[0113]

[0114] Substituting equation (11) into equation (8) can solve the vibration mode equation of the concentrated mass string composite structure model:

[0115]

[0116] Set x=l1,l2,L,ln Substituting into equation (12) respectively, we can get the characteristic equation of the structural model:

[0117] (AB)·Y=0 (13)

[0118] in

[0119]

[0120]

[0121] Y=[Y(l1) Y(l2) ... Y(l n )] T (14c)

[0122]

[0123] B kj =βLα j sinβ(l k -l j ) (14e)

[0124] When sinβL≠0, the frequency equation of the concentrated mass string composite structure is:

[0125] |AB|=0 (15)

[0126] This frequency equation reflects the portion of the string structure frequency that is changed by the concentrated mass;

[0127] When sinβL=0, the concentrated mass has no effect on the frequency of the string-pull structure, and the frequency of the concentrated mass-string-pull structure is consistent with the conventional string-pull structure, as shown in formula (1).

[0128] Therefore, the frequency equation of the concentrated mass string structure is the superposition of the frequencies when sinβL=0 and sinβL≠0, that is:

[0129] sinβL·|AB|=0 (16)

[0130] By solving the singular function model equation (16), the frequencies of each order of the concentrated mass cable composite structure can be obtained.

[0131] Model Validation

[0132] The validity of the singular function model prediction of the natural frequency of the multi-concentrated mass string structure will be verified from two aspects. First, as a special case, the natural frequency of the string structure without concentrated mass has a classical calculation solution, as shown in equation (1). Assuming n = 1, substituting m1 = 0 into the frequency equation (16) of the concentrated mass-string structure model, we can get sinβL = 0, and its solution is It is consistent with formula (1). That is, the concentrated mass-string structure model can replace the conventional string structure model to calculate the vibration frequency.

[0133] When the string-drawn structure is attached with multiple concentrated masses, the validity of the singular function theoretical model is verified by physical experimental methods. A Doppler laser vibrometer is used. The device emits a sine wave signal generated by an audio signal generator through an audio device to vibrate the bow-shaped device; the laser vibrometer captures the vibration signal of the metal wire on the bow-shaped device, analyzes it through the control box, and then sends it to the computer for real-time display. The device uses the method of music excitation plus laser vibration pickup to obtain the experimental values ​​of the multi-order natural frequencies of the multi-concentrated mass string-drawn structure without contact and efficiently. All the parameters of the experimental setting are shown in Table 1.

[0134] Table 1 Parameters of two concentrated mass-string system

[0135]

[0136] Experimental verification of the theoretical solutions of the singular functions of the natural frequencies of the two systems was carried out. In the experimental system shown in Table 1, the linear density, span, and tension parameters of the string are the same, and only the number, size, and position of the concentrated masses are inconsistent. System 1 only places a concentrated mass of m1 = 0.5828g in the span of the string; System 2 has two concentrated masses, m1 and m2 are both 0.2914g, and the concentrated masses are arranged at equal intervals on the string, and the total mass is consistent with that of system 1. The concentrated masses are adsorbed on the steel wire string by magnets.

[0137] Table 2 shows the theoretical solutions and experimental values ​​of the first five natural frequencies of systems 1 and 2 and their relative errors. The theoretical values ​​of the natural frequencies of each system are calculated by substituting the parameters in Table 1 into formula (15). From the data in the table, it can be seen that the theoretical values ​​and experimental values ​​of the natural frequencies of the two systems are highly consistent, and the errors of each order of frequency do not exceed 5%. Therefore, it can be proved that the singular function model can effectively calculate the natural frequencies of multi-concentrated mass string structures.

[0138] In summary, the singular function model has wide applicability and effectiveness, regardless of whether it is a conventional string-stayed structure or a structure with additional concentrated mass.

[0139] Table 2 Calculation error rate of natural frequency

[0140]

[0141] Note: Error rate = | measured value - calculated value | / measured value × 100%

[0142] Using the singular function model (16), the influence of the concentrated mass-string structure parameters on its natural frequency and vibration mode is studied through numerical calculation, and the variation law of the system vibration mode to the concentrated mass size, quantity and distribution position is obtained. In the following example, the basic parameters of the string structure adopt the values ​​of the bow-type device verified by the above model, L = 0.465m, T=31.5N. See Table 1 for details.

[0143] Figure 2 The first five natural frequencies of the single concentrated mass string composite structure with mass ratio The calculation results of the changes. The calculation parameters are L = 0.465m, The tension T0 = 31.5N, and the concentrated mass is located at the mid-span of the anchor structure, that is, l1 / L = 1 / 2. Figure 2 (a) shows that the fifth-order frequencies of the single concentrated mass string-type structure are still arranged in order from low to high, but compared with the string-type structure without concentrated mass, the frequency curves of each order are not arranged in sequence in proportion; as the mass ratio α1 increases, the odd-order frequencies decrease monotonically, while the even-order 2nd and 4th-order frequencies remain unchanged as α1 increases. This should be directly related to the nodes where the concentrated mass is located at the 2nd and 4th-order vibration modes. The time when a single concentrated mass is located at a certain vibration mode node will not affect the frequency of this order, and the frequency of this order is consistent with the frequency of the same order of the string-type structure without concentrated mass. Figure 2 (b) The curve of the ratio of each order frequency of the single concentrated mass string-pull composite structure to the same order frequency of the non-concentrated mass string-pull structure as the mass ratio α1 is given. Figure 2 (b) shows that, compared with the string structure without concentrated mass, one of the main effects of concentrated mass is to reduce the natural frequencies of each order of the system, among which the reduction of the fundamental frequency is the most significant.

[0144] Figure 2 (c) and Figure 2 (d) Yes Figure 2 (a) is a partial enlarged view of Figure 2 In (c), we can more clearly see the attenuation law of the third-order frequency of the combined system with the mass ratio: when 0≤α1≤6, the third-order frequency decays rapidly with the increase of α1, and when α1>6, the third-order frequency curve asymptotically approaches the second-order frequency (horizontal line). Similarly, Figure 2 In (c), the fifth-order frequency of the combined system also decreases significantly as α1 changes, and then gradually approaches a constant value, which is exactly the fourth-order frequency of the system.

[0145] In addition to the size of the concentrated mass, its location is also the main parameter affecting the frequency of the combined system. Figure 3The first five natural frequencies of a single concentrated mass string-pull structure are provided. Each frequency fluctuates within its own range as the concentrated mass position l1 / L changes. Each frequency has a valley and a peak. Between two adjacent frequencies, the valley of the higher frequency is close to the peak of the first frequency, but the frequencies do not cross each other, and the same frequency has multiple identical peaks and valleys. Figure 3 (b) Yes Figure 3 (a) The frequency curve is compared with the calculated results of the same order frequency of the string-pull system without concentrated mass. It can be seen that the maximum value of the relative frequency of each order is 1, that is, the upper limit of the change of the combined structure of a single concentrated mass with the concentrated mass position l1 / L is the natural frequency of the combined structure without concentrated mass; Figure 3 It is also shown that the maximum value of the single concentrated mass composite structure occurs at the left and right supports or the node positions of the same order vibration mode.

[0146] Figure 4 The concentrated mass is expanded to 2, and the changes in the second-order natural frequency of the string-stayed composite structure with no concentrated mass (Case 1), a single concentrated mass (Case 2), and two concentrated masses (Case 3) are compared and analyzed. In the two-concentrated-mass string-stayed structure (Case 3), concentrated mass 1 moves on the string from left to right, while concentrated mass 2 is fixed at the mid-span, i.e., the node of the second-order vibration mode. Since the vertical coordinate is measured with the second-order natural frequency of the massless string-stayed composite structure, it is easy to understand that the no-concentrated-mass condition (Case 1) is a horizontal line with a value of 1; Case 2 with a single concentrated mass is a left-right symmetrical upward concave curve, which takes a maximum value of 1 at the mid-span (the node of the second-order vibration mode). It is the same as the natural frequency of the string-stayed structure with no concentrated mass. This result has been mentioned above. Figure 2 and Figure 3 However, the loss of the effect of concentrated mass at the vibration node is only a special case. When there are other wandering concentrated masses, there will be a "coupling response" between the wandering and node-fixed concentrated masses. The effect of the concentrated mass fixed at the node is stimulated, but the natural frequency of the combined system is further attenuated. The frequency comparison curves of Case 2 and Case 3 clearly indicate this point.

[0147] Let us further investigate the situation where two concentrated masses move on the string at the same time. Figure 5 The three-dimensional spatial variation surface of the 1st to 4th order natural frequencies of the two concentrated mass string structure is given. It takes the positions of the two concentrated masses l1 / L and l2 / L as the horizontal coordinates. The relative frequency ω of the string structure with and without concentrated mass is k / ω' k Is the vertical axis. Figure 5In (a), the first-order relative frequency reaches a maximum value of 1 when the two concentrated masses are simultaneously located at the left and / or right supports, while the minimum value of the relative frequency occurs when the two concentrated masses are simultaneously located at the mid-span of the string and coincide with each other, and the minimum value is equal to 0.416. Figure 5 In (b), the maximum value of the second-order frequency is still 1, which occurs when the two concentrated masses are located at the left and / or right supports at the same time, and when they coincide at the mid-span (i.e., the second-order vibration mode node); the minimum value of the second-order frequency occurs at l1 / L=0.2, l2 / L=0.8, and the minimum value is equal to 0.424. Figure 5 (c) and Figure 5 The maximum and minimum values ​​of the 3rd and 4th order relative frequencies in (d) also have similar conclusions. The minimum values ​​are 0.506 and 0.640, respectively, occurring at l1 / L = 0.1, l2 / L = 0.9 and l1 / L = 0.1, l2 / L = 0.2. The distribution diagrams of each order frequency are all in the form of centrosymmetry.

[0148] The influence of string tension on the natural frequency of the concentrated mass string composite structure is further investigated. The curve of the first five frequencies of the single concentrated mass structure changing with the increase of tension is located at Figure 6 In (a), the increase of each frequency order occurs along its own convex nonlinear curve. Figure 6 (a) Each frequency order is measured, and the change of relative frequency is as follows Figure 6 (b) shows that the frequency of the string-pull combination structure is only affected by the string tension, as described in formula (1), and there is no more gain for the concentrated mass-string combination structure. In order to expand the applicability of this cognition, the frequency curves of the two concentrated mass-string combination structures with the string tension are further plotted, see Figure 7 . Figure 7 (a) and Figure 7 (b) respectively Figure 6 (a) and Figure 6 (b) corresponds, and the conclusion is the same.

[0149] Figure 8 In (a), the first five frequencies are clearly arranged in a vertical and horizontal order as the number of concentrated masses n changes, although there are local oscillations when the number n is small, and then they tend to be stable and parallel. Figure 8 (b) Each order frequency is measured by the same order frequency of the string structure without concentrated mass (i.e., the calculation result of formula (1)), then the relative frequency ω k / ω' k The curve of the change with the number n is as follows Figure 8(b). Obviously, as n increases, the relative frequency tends to the same constant value, which can be derived using the aforementioned singular function model. The constant value has the following calculation expression: When n>>1, it is equivalent to increasing the density of the string line, and the increment is Where m is the total value of the concentrated mass. Obviously, the larger the total value of the concentrated mass, the higher the natural frequencies ω of the string-pull structure. k and the frequency ratio ω k / ω' k The smaller the limit.

[0150] In summary, although the embodiments of the present invention have been described, it can be understood that the above embodiments are exemplary and cannot be understood as limitations of the present invention. Ordinary technicians in the field can change, modify, replace and modify the above embodiments without departing from the principles and purpose of the present invention.

Claims

1. A singular function model analysis method for the natural frequency of a string-string combination structure, characterized in that: The steps include: Step 1: Simplify the concentrated mass string assembly structure; Step 2: Convert the concentrated mass into distributed mass along the cable length through the Dirac function; Step 3: Solve the cable vibration mode as a function of time t and position x; Step 4: Perform Laplace transformation on the vibration equation of the lateral free vibration in the tension string plane; Step 5: Perform inverse Laplace transform on the result of step 4; Step 6: Solve the vibration mode equation of the concentrated mass cable structure model to obtain the singular function model, thereby obtaining the various frequencies of the concentrated mass cable composite structure.

2. The singular function model analysis method of the natural frequency of the string-drawn combination structure according to claim 1 is characterized in that: The step 1 is specifically as follows: simplifying the concentrated mass string-drawn composite structure into a simply supported system hinged at both ends, ignoring the bending stiffness and sag of the string and its vibration damping; wherein the total length of the string is L, and the unit linear density is The tension is T; the i-th concentrated mass on the string is m i , i = 1 ~ n; horizontal coordinate x = l i is the horizontal distance between the concentrated mass and the left end hinge support.

3. The singular function model analysis method of the natural frequency of the string-pulling combination structure according to claim 2 is characterized in that: The second step is specifically as follows: for the case where uniformly distributed mass and concentrated mass coexist on the string, the concentrated mass is converted into distributed mass along the length of the string by using the Dirac function. At this time, the mass function of the string along the axial direction is: in is the Dirac function.

4. The singular function model analysis method of the natural frequency of the string-drawn combination structure according to claim 3 is characterized in that: The step three is specifically as follows: From the vibration theory, we know that the vibration equation of the lateral free vibration in the tension string plane is: According to the separation of variables method, the solution of equation (4) is: y(x,t)=Y(x)sintω (5) Where Y(x) is the lateral displacement of the string at position x, ω is the natural circular frequency of the string, and y(x,t) is the function of the cable vibration mode with respect to time t and position x.

5. The singular function model analysis method of the natural frequency of the string-drawn combination structure according to claim 4 is characterized in that: The step 4 is specifically as follows: Substituting equation (2) and equation (5) into equation (4), we can obtain After Laplace transforming equation (6), we can get: Among them, p is a complex parameter variable.

6. The singular function model analysis method of the natural frequency of the string-drawn combination structure according to claim 5 is characterized in that: The step five is specifically as follows: Performing inverse Laplace transform on equation (7) yields: in and is the Heaviside step function.

7. The singular function model analysis method of the natural frequency of the string-drawn combination structure according to claim 6 is characterized in that: The step six is ​​specifically as follows: Assume that the boundary conditions of the concentrated mass string composite structure are: Substituting equation (11) into equation (8) can solve the vibration mode equation of the concentrated mass string composite structure model: Set x=l1,l2,L,l n Substituting into equation (12) respectively, we can get the characteristic equation of the structural model: (AB)·Y=0 (13) in And=[Y(l1) Y(l2) ... Y(l n )] T (14c) B kj =βLα j sinβ(l k -l j ) (14e) When sinβL≠0, the frequency equation of the concentrated mass string composite structure is: |AB|=0 (15) This frequency equation reflects the portion of the string structure frequency that is changed by the concentrated mass; Therefore, the frequency equation of the concentrated mass string structure is the superposition of the frequencies when sinβL=0 and sinβL≠0, that is: sinβL·|AB|=0 (16) By solving equation (16), we can obtain the frequencies of the concentrated mass cable composite structure.

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