Abstract
The dynamics of space structures is significantly impacted by the presence of power and electronic cables. Robust physical model is essential to investigate how the host structure dynamics is influenced by cable harnessing. All the developed models only considered the decoupled bending motion. Initial studies by authors point out the importance of coordinate coupling in structures with straight longitudinal cable patterns. In this article, an experimentally validated mathematical model is developed to analyze the fully coupled dynamics of beam with a more complex cable wrapping pattern which is periodic in nature. The effects of cable wrapping pattern and geometry on the system dynamics are investigated through the proposed coupled model. Homogenization-based mathematical modeling is developed to obtain an analogous solid beam that represents the cable wrapped system. The energy expressions obtained for fundamental repeating segment are transferred into the global coordinates consisting of several periodic elements. The coupled partial differential equations (PDE) are obtained for an analogous solid structure. The advantage of the proposed analytical model over the existing models to analyze the vibratory motion of beam with complex cable wrapping pattern has been shown through experimental validation.
1. Introduction
Modern day space structures consist of a significant amount of cabling owing to the increased electrical components. Increased use of lightweight materials to manufacture the spacecraft has resulted in significant effect of cable attachments. Recent studies developed experimentally validated mathematical models to study the impact of cable harnessing. The US Air Force Research Lab (US-AFRL) first conducted several research investigations to study how the addition of cabling changes the frequency characteristics of a given structure. Initial theoretical studies in this area performed model updating techniques to study the effect of cabling [1]. Goodding et al. [2, 3] studied cabled beams by modeling the beam and cable using Euler-Bernoulli (EB) assumptions. In addition, they performed experimental investigations on a serpentine configuration where the cable is attached in the shape of a sine wave on top of a beam structure. These studies explained the importance of developing an accurate mathematical representation. The accurate prediction of the vibration characteristics for the cabled structure gives better estimates of the control system performance.
Further research conducted by the US-AFRL entails the finite element (FE) and distributed parameter models in [4, 5] to primarily study the pure bending of cabled beams. Reference [4] models the structure using EB theory and studied the pure transverse modes of cable-harnessed beam. Kauffman et al. [6, 7] characterized damping effects in pure bending vibrations of standalone cables using Timoshenko model. Spak [8, 9] further developed a distributed parameter model in order to predict the dissipation effects. Reference [10] showed the presence of cable-beam interaction frequencies; however, their theoretical work ignores coupling across several coordinates. On a similar cabled beam system, Choi et al. [11, 12] studied the transverse vibrations using spectral element techniques. The study concluded that the developed technique is computationally more efficient than the FE methods.
Although the described studies emphasized the importance of modeling cable dynamics, only straight longitudinal cable patterns were studied, and the coupling effects were ignored. It was highlighted that further analytical model development through distributed parameter models on this subject is necessary to gain more insights into the cable-harnessed system. Distributed parameter models have advantages in control applications as they use significantly lower computational time when compared to other numerical methods, for example, FE methods currently used in the industry. The distributed parameter models also provide physical insight as the effects of parameters such as the stiffness induced by the cabling and coupling effects can be analyzed with much ease.
The major objective of our research group is to develop low order partial differential equations (PDE) to analyze the stiffness and mass effects due to cable attachment on beam structures. The author’s studies in this area focused on beam structures wrapped with cables in periodic patterns [13, 14]. The effects of cable pretension, higher-order displacement field, and strain tensor were considered [13, 14]. In these works, novel mathematical techniques using the homogenization method were developed based on the periodicity of the cable wrapping patterns. This continuum modeling technique involved finding an equivalent distributed parameter system, usually a simpler homogenous structure, with similar dynamic properties to the original and more complex system of a repeated pattern geometry. This resulted in the governing PDEs for vibrations and their exact solutions for the first time in the literature of cabled structures. In the above works, the coupling effects across several vibration coordinates are neglected and only transverse bending coordinate is considered.
The coupling effects because of cable attachment were first incorporated in our previous works [15, 16]. As a starting point, a straight pattern cable was attached near the edge of the beam along its length, and the coupling effects increased with this offset distance from the centerline [15]. It was concluded experimentally that, to obtain an accurate analytical model, the coupling effects were important to consider. Otherwise, the frequency response characteristics turned out to be erroneous [16].
In the current paper, we have developed an analytical model that considers a further complex cable attachment pattern to study the coupling effects along with experimental validation. The current research also fills a gap in the literature as the previous work done by Martin el al. [13] on the periodic cable wrapping patterns did not include the coupling effect due to cable attachment. The main contribution of this article is to demonstrate the advantage of coupling effects in periodic wrapped structures. This will allow for further optimization of the system’s dynamics when minimizing the cables’ dynamic effects on a host beam structure. Furthermore, in the periodic wrapped structures, the obtained coupled elastic and kinetic energy have spatially varying coefficients. Hence, an energy equivalent homogenization procedure is developed to obtain a model with constant coefficients. In this paper, additional coupling between bending and the twist is presented in the periodic wrapped structures due to diagonal cable section which is not observed in the case of straight cable patterns [15]. These coupling terms due to the torsional coordinate can significantly alter the system dynamics.
In Section 2, the model corresponding to the fully coupled vibrations of the periodic wrapped structures is derived using EB and Timoshenko assumptions. In Section 3, sensitivity analysis is performed, and the vibration experimental setup is described. Frequency Response Function (FRFs) from the decoupled and coupled analytical models (DCM and CM) is validated against experiment FRFs.
2. Mathematical Modeling
In Section 2, the modeling procedure is presented. The current modeling approach is based on an energy equivalent homogenization method. A major assumption behind this method pertains to the periodicity of the geometry and material properties in a system. Once there is a fundamental element over which the structure’s periodic element is identified, its strain and kinetic energy (SE and KE) are found. The energy equivalence assumption allows for the substitution of the fundamental element of a given geometry with a solid equivalent element of the same energy expressions. The equivalent continuum structure is then made by assembling the repeated solid elements.
2.1. Development of Energy Expressions considering the Coupling Effects
The first step is to find the equivalent continuum model which focuses on obtaining KE and SE for a repeated fundamental element. This is done through the appropriate displacement field assumptions, strain tensors, and constitutive relations from which the equations of motion are found. Schematic of the structure and the local element are shown in Figures 1(a) and 1(b). The coordinates of motion in the global coordinate system and local coordinate system are also shown in Figure 1. Each element shown contains a diagonal cable on top and lumped cable mass section wrapped around the beam at one end [13].
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Six coordinates of vibrations that include out-of-plane (OP) and in-plane (IP) bending, rotations of the cross section along the two axes, axial motion, and torsion (Tn) are considered for modeling the host beam using the Timoshenko model. The linear displacement field involving all the coordinates of motion for the EB and Timoshenko models can be found in [15]. Stress-strain relationships for an isotropic beam are well known . The assumptions made in model are as follows [15]:(1)The cable is in pretension upon fabrication and remains in tension during the vibrations(2)The pretension, T, results in a precompressive load on the beam(3)Cable remains attached to the host beam structure during vibrations; friction between the beam and cable is ignored(4)The cable experiences the similar strain as the top surface of the beam
The strain expressions obtained after considering the Timoshenko and Euler-Bernoulli displacement field and strain tensor [17, 18] are shown in equations (1) and (2) [18]:
Superscript in equation (1) is used to denote strains from the Timoshenko model and superscript in equation (2) represents strains for EB model. is shear factor for rectangular cross section and is Poisson’s ratio, where and .
The displacement field, the strain tensor, and stress-strain relationship are used to obtain SE and KE of the fundamental element employing Timoshenko model assumptions (equations (3) and (4)). The SE and KE are a result of contribution from the host structure and the diagonal part of the cable (refer to Figure 1(a)). In addition, the KE in equation (4) also includes the lumped mass cable portions. These lumped masses do not vary with the beam deformation.
The readers may refer to the nomenclature table for notations. is the wrapping angle. Also, is the beam width, is the length of each element, and for is the lumped mass due to the portion of the cable. The prime symbol denotes the spatial derivative w.r.t local coordinate and the dot symbol denotes the time derivative. The strain is because of precompression in the host structure. Reference [13] considered the pretwist and found its effect to be negligible for metallic cabled beams. In this paper, we consider metallic host structures, and the effect of pretwist is ignored. Notations , and in equation (4) are the time derivatives of and displacement components defined for all points of the beam.
The SE and KE in equations (5) and (6) are obtained using the Timoshenko theory represented by and respectively (superscript (1) is for the Timoshenko model). The coefficients are a function of beam and cable parameters [19]. In these energy expressions, the terms higher than second order are ignored considering the small-scale vibrations. The SE of the hybrid system in equation (5) is obtained by adding the SE of the cable and beam. Also, shear correction factor is not included while evaluating the cable strain.where and , where and are the coefficients of SE and KE. and coefficients vary spatially in each fundamental element.
In equation (5), the coefficients and to are the additional coupling terms because of the asymmetric periodic wrapping pattern. The diagonal cable section leads to few coefficients that vary along the coordinate and hence, a constant coefficient PDE cannot be directly obtained from equations (5) and (6). Therefore, we use the energy equivalent homogenization method to develop an equivalent continuum model, which is described in Section 2.2. Also, the strain energy coefficients for the KE and SE expressions are shown in equation (7) for the Timoshenko theory.
and ; [13]. The polar moment of inertia is .
The SE and KE for the periodic wrapped structure using EB assumptions [13] are shown in equations (8) and (9). These can be developed by similar procedure as the Timoshenko model. For consistency, the Timoshenko model coefficients are represented by and Euler-Bernoulli by .
The constants of the SE and KE expressions are expanded as
are the strain and kinetic energies. respectively, of the fundamental element in equations (8) and (9) for the EB theory. and are the coefficients of the SE and KE. and coefficients vary spatially in each fundamental element. The coefficients to are related to the torsion coordinate and are due to the asymmetric nature of the wrapping pattern. The notation pairs (, , , and ) represent the same coefficients in both Timoshenko and Euler-Bernoulli models.
2.2. Development of Equivalent/Homogenized Beam Model
In Section 2.2, the procedure to obtain the equivalent beam model is outlined. The cable’s position coordinates to evaluate the strains are . The variable SE and KE coefficients in equations (5) and (6) are represented by and . Similarly, in equations (8) and (9) the variable coefficients are represented by and . The next goal is to obtain constant coefficient strain and kinetic energies using the homogenization method. This method relies heavily on the periodicity assumptions for the given cable-harnessed structure. First, the displacement components for various points along the fundamental element (shown in Figure 1(b)) are expanded using the Taylor series. The general mathematical form of the Taylor series of any displacement component is
Subsequently, the SE and KE of an element are calculated by displacement and velocity components, respectively, evaluated at its center (). The energies per unit length for the Timoshenko model, and for the EB model can be calculated. They are assumed to be the same for the entire length of the cable-harnessed structure. In equations (12) and (13), the strain and kinetic energies per unit length are integrated over the beam length. This gives KE and SE for the equivalent beam structure for the Timoshenko and EB models.
For the homogenized beam, and are the Timoshenko model’s final strain and kinetic energy expressions, and and are the EB model’s final strain energy expressions. The homogenization method simplifies and makes the mathematical model efficient for vibration analysis. Had we not applied the homogenization method, we would have to consider the energies of each fundamental element and continuity conditions would have to be applied at the interface. This will make the modeling complicated, particularly for more elements in a fully coupled system.
2.3. Coupled PDEs and Their Solution
In Section 2.3, the equations are developed. Hamilton’s principle [20], , and , is applied to obtain the equations for the coupled homogenized models. This is done after including harmonic base excitation (equations (14) and (16)). The coefficients are given in Appendix A. In this research, we study the vibrations of a clamped-free system and therefore, the boundary conditions are shown in equations (15) and (17) for the respective models.
Timoshenko model: is the harmonic base motion as also shown in Figure 1(a), and is the relative motion w.r.t base excitation in the OP direction. The equations corresponding to the six coordinates are shown in equation (14) along with the coupling terms.
Fixed end boundary conditions:
Free end boundary conditions:
Euler–Bernoulli model:
Fixed end boundary conditions:
Free end boundary conditions:
In the Timoshenko model, the coefficients , and to are the additional torsion-related coupling terms due to the diagonal section of the cable. For EB model, these coefficients are to , . In the case of the straight cable patterns, these coefficients are not observed [15]. Physically, the coefficient couples the moment and twist which results in coupling of both the bending and torsion. The term shows up in both equations (16c) and (16d). Similar analysis can be extended to other coupling coefficients.
To gain more insight, an analogy can be drawn to the coupled vibration equations of motion obtained in the case of laminated rectangular composite beams [21,22]. Composite beams with cross-ply laminates show coupling between bending and torsion depending on the ply orientation and sequence. The coupling term in both equations (16c) () and 16d () is also observed in a composite laminated beam [21]. This coefficient depends on the cable radius, modulus, and the wrapping angle. As the model in this paper considers the effect of wrapping angle, mathematically we see a similar bending torsion coupling term as in the composite beams.
Regarding the boundary conditions, in equations (15a) and (17a), all the displacement variables are zero for the fixed end for EB and Timoshenko models, respectively. In equations (15b) and (17b), the free end boundary conditions are shown for both the models.
The solution procedure for PDEs of equations (14) and (16) is shown in [15] by the authors. The FRF is calculated as follows:
To obtain the FRFs for the coupled system, mass-normalized coupled mode shapes are used in equation (18). Similarly, FRFs for the decoupled system are found using mass-normalized decoupled mode shapes in equation (18).
3. Results and Discussion
3.1. Theoretical Case Studies
In Section 3.1, sensitivity analyses related to the cable radius () and the number of fundamental elements () on natural frequencies () are studied.
Figure 2 presents the variation of with . The parameters are listed in Table 1 and are based on those suggested by Martin et al. [13]. For this simulation the cable radius is assumed to be 0.7 mm. For coupled curve in Figure 2, the coupled EB model is used. For the decoupled model, the coupling terms are ignored in equations (16a)–(16d) which would result in an independent set of PDEs.
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In Figure 2(a), the maximum coupling is seen when angle is lower for the OP modes. For a fixed beam length, as is increased, also increases. When increases, the KE effects start to increase at a higher rate SE effect. From Figure 2(c), the frequency of the Tn mode increases till 20 elements ( = 38.65°) and starts decreasing when exceeds 25 ( = 45°). This suggests that there is an optimum wrapping angle for which the torsional frequencies will increase as increases. For the first 20 elements both the kinetic and strain energy coefficients follow an increasing pattern while the SE coefficient dominates, so the frequency increases. When the wrapping angle increases, physically the asymmetry increases and results in higher torsional frequencies. From 20 to 35 ( = 54.46°) elements, both the SE and KE coefficients also follow an increasing pattern. Due to increase in the wrapping angle (from = 38.65° to = 54.46°), the SE coefficient related to the torsion coordinate increases as expected. However, with the increase in wrapping angle beyond the optimum angle, the diagonal section starts contributing more like a lumped mass. As a result, the frequency follows a decreasing pattern.
In Figure 2(b), the coupled frequency of the IP mode is higher than the decoupled model when exceeds 25. As the OP mode becomes weakly coupled for larger , SE transfer between the torsion and the IP modes becomes more significant. From Figure 2, the coupling phenomenon is found significant. Hence, consideration of coupled model is highly essential for both the lower and higher number of fundamental wrapping elements.
Next, the effect of on coupling is discussed. In Figures 3(a) and 3(b), the natural frequencies increase as the overall SE is increasing at a faster rate than KE. The parameters used for this study are listed in Table 1 and is taken as 9. In Figure 3(c), the associated with Tn mode decreases as increases. For certain parametric studies [15], the coupling reduces the frequency of one mode and increases the frequency of mode(s) in another coordinate due to energy transfer. In this case, a decrease in the Tn frequencies is seen. Figure 3(c) further demonstrates the importance of coupling in periodically wrapped structures, as the decoupled model shows a trend of decreasing frequency when the cable radius is increased.
(a)
(b)
(c)
In Table 2, the natural frequencies associated with several cases are compared. This includes a straight pattern, no offset [15], straight pattern at an offset position [15], a bare beam case (no cable attached), and the periodic wrapping pattern case of the current paper. For the periodic pattern case, is varied. The system parameters are from Table 3 and are similar to our experimental characterization [16]. For the OP modes, when the straight cable with offset case and the periodic pattern case are compared, the coupling effects are slightly lower for case. These differences clearly highlight the importance of including coupling between different coordinates of motion.
When increases beyond 9 elements, the frequencies of OP modes for the periodic wrapped structure start to decrease. This is because more mass effect is added and the frequencies start to approach the bare beam. For lower (OP modes), the stiffness effects dominate, whereas, for the higher , the mass effects dominate.
There is an optimum angle (, degrees) for the given case study for which the frequencies approach the bare beam and the cable’s dynamic effects are minimal. Since the addition of cables significantly affects the host structure dynamics, having knowledge of the cable wrapping patterns to minimize the effect of cabling is highly essential. This case study provides insight into the importance of coupling effects in the periodically wrapped cable-harnessed structure while trying to minimize the cable harness effect on the system dynamics. Consideration of the CM ensures that the stiffness and mass effects are more accurately predicted than existing models.
3.2. Experimental Investigations
In Section 3.2, the proposed mathematical model is validated against the experiments. It is shown that inclusion of coupling in modeling the cable-harnessed structures leads to more accurate results than decoupled models proposed earlier [14].
Two samples are considered for the experimental investigations. The flowchart of the experimental setup is shown in Figure 4. During the fabrication of the beam, very small notches are made in the host beam. This is to ensure that the top section remains diagonal during the wrapping. The host beam is wrapped with multiple cables, maintaining pretension using weights. After the wrapping of cable is complete, one end of the structure is tightly clamped. The other end remains free to obtain the cantilever boundary condition. This is shown in step 2 of Figure 4. The modular weights are removed from the system once the structure is tightly clamped at one end. The cable-harnessed beam fixture is then attached to an electrodynamic shaker as shown in step 3 of Figure 4. The actual setup of the two samples on the shaker is shown in Figure 5.
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The parameters (shown in Table 4) for the two samples are chosen such that they have different levels of coupling. The goal of the experimental investigations is to test the efficacy of the developed model for systems with varied coupling effects.
As per the theoretical investigations, the presence of coupling between different coordinates has influence on the natural frequencies. To quantify the effect of the coupling in this dynamic system, we compare the obtained from the CM developed in this paper with that of the previous work pertaining to the DCM [13, 14]. The higher the coupling effect, the greater the difference in natural frequencies between CM and DCM. From equation (10), the coupling terms significantly depend on the cabling area and angle . Further, the theoretical and experimental frequencies are shown in Tables 5 and 6. Comparison of coupled and decoupled model frequencies in Tables 5 and 6 for the two samples suggests that sample 2 has larger coupling effect than sample 1 even though the host beam structure parameters are marginally different. This is particular for the OP modes, for the considered system parameters, sample 2 (larger and lower ).
The schematic of the vibration experiment is shown in Figure 6. The cable-harnessed structure is attached to the 2075E Modal Shop electrodynamic shaker. The shaker is connected to the Modal Shop power amplifier 2050E09. An accelerometer PCB 352A24 is connected to the control channel of the LMS SCM 05 SCADAS unit and the accelerometer is fixed on the shaker. The input profile to the shaker is defined using LMS SCM 05 SCADAS data acquisition unit and the SCADAS unit is also used to acquire and process the experimental data. The Polytec OFV-5000 laser vibrometer system is used to sense the displacement. With the help of a spirit level, it is ensured that the direction of the laser beam is vertical. The vibrometer is connected to the sensing channel of the SCADAS unit.
Firstly, base excitation is applied on the samples in the OP direction. The sine sweep option in the LMS SCADAS unit performs the sweep from low frequency to high frequency with a resolution of 0.10 Hz. The controller of the vibrometer sends the displacement of the beam to the SCADAS unit which computes the FRF of the structure in the OP direction for several modes. The modes that correspond to the OP modes have significant peaks.
To help accurately identify the IP and Tn modes, deformation shapes were acquired by sensing at several locations. In order to experimentally acquire Tn mode, top surface of the structure is discretized into equal-width columns along its length. The sensing locations are 1 cm apart in the x-direction. Three columns are used because the torsion dominant modes have peak motion at the edges and zero motion along the centerline, as shown in Figure 7. Thus, the torsional modes will be evident when measuring along the edge columns. To acquire the IP modes, impact test is performed in the IP direction as shown in Figure 7. For the impact test, the “Impact Testing Module” of the Siemens LMS software package is used.
Firstly, the results of sample 1 are discussed in Figure 8. The coupled analytical FRFs match well with that of the experimental FRF. The FRF from DCM [14] is also shown in Figure 8. The significant peaks in Figure 8 denote the OP modes.
The natural frequencies of theoretical model and experiment for sample 1 are presented in Table 5. The corresponding error percentages w.r.t the experimental are also given. The CM predicts corresponding to the OP, IP, and Tn modes that are observed in the experiment. In Figure 8, the decoupled model [13, 14] only predicts the OP frequency peaks. The coupled model, along with the Timoshenko model assumptions, gives a considerably better match with the experiments than the decoupled model published in [13, 14]. Quantitatively, the average error between the ’s of first eight modes obtained for DCM with respect to experiment decreases from 10.08% to 3.88% for the coupled EB model. Moreover, for the first five OP modes, the average error in the natural frequencies decreases from 11.47% to 0.70%. This significant improvement is due to the incorporation of coupled dynamics. The experimental IP and Tn mode shapes are plotted in Figure 9. Figures 9(a) and 9(c) confirm the first and second IP shapes, respectively. The mode shapes from theory can be found in Figure 10 and additional explanation in Appendix B for further confirmation.
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The coupled model FRF for sample 2 is shown in Figure 11. Compared to the DCM, regions of the FRF from the coupled model near the peaks match much better with the experiment. Sample 2 has larger coupling than sample 1 due to the larger and lesser considered. The explanation for the large errors (Table 6) observed in the IP modes is explained later in Section 3.2. Quantitatively, the average error between ’s of first seven modes obtained for decoupled model with respect to experiment decreases from 28.29% to 17.15% for the coupled EB model. Moreover, for the first four OP modes, the average error in the natural frequencies decreases from 30.98% to 11.26%, which is a significant improvement due to the incorporation of coupled dynamics. The experimental shapes of IP and Tn modes are shown in Figure 12.
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The errors of the frequencies for the IP modes between theory and experiment are higher when compared to other modes. In practical situations, the fixed end of the cantilever beam has some flexibility. Moreover, the clamping force is in the thickness direction (along z axis); for IP modes, the clamped end has higher flexibility compared to the OP. As a result, the of the IP modes is estimated higher in the model, which assumes a rigid fixed end (see Table 7 and Figure 13). The quality of clamping force in the IP direction is further compromised in the cabled beams as the effective thickness under the clamp is greater than the bare beam (due to cable attachment).
From the results obtained for this study, it is observed that, to accurately study the dynamics of periodic wrapped structures, the CM is a better choice. This is particularly true when the cable radius or the number of cables is higher. Coupling effects can be seen for both lower and higher number of fundamental element cases. The coupling model is necessary for further optimization studies related to cabled structures to minimize cable harness effects on the structure.
4. Conclusions
This paper analyzes the coupled vibrations in a periodic wrapped cable-harnessed beam structure through the developed mathematical model. Sensitivity analyses are performed by varying the and the . The important conclusions from this article are listed as follows:(1)For the OP modes, the difference in the natural frequencies between the CM and DCM is found higher at lower . For higher , the coupling effect is still significant. For the Tn modes, there exists an optimum wrapping angle at which the coupling becomes maximum.(2)As the cable radius is increased, the frequency and the coupling effects increase for the OP and IP modes. The coupling effect is higher in the OP modes. The Tn mode decreases with the increase in due to energy transfer between different coordinates.(3)The natural frequencies of straight pattern and periodic wrapping patterns are compared. For lower , the stiffness effects increase, and minimal mass effects are observed. At higher , the mass effects increase, and the stiffness effects are minimal. At a certain angle, it was observed that the frequencies approach the bare beam and minimize cable harness effects.
From the sensitivity analysis, the coupling effects are found significant for both lower and higher number of fundamental elements. Thus, for accurate optimization of cable placement, consideration of the coupling effects in the structure is highly essential.
Experiments are performed on two periodically wrapped beam samples to verify the proposed models. The FRFs from both the EB and Timoshenko coupled models proved to be significantly better matches for the OP peaks compared with DCM.
Appendix
A. PDE Coefficients Corresponding to Timoshneko and EB Models
The coefficients for the Timoshenko model, equation (14), are given below in equation (A.1):
For the EB model, equation (16), the coefficients are shown below in equation (A.29):