Vibration Analysis of a Deploying and Spinning Beam with a Time-dependent Spinning Speed

시간에 따라 변하는 회전 속도와 함께 회전하며 전개하는 보의 진동 분석

Abstract

This paper presents the vibration analysis of a deploying beam with spin when the beam has a time-dependent spinning speed. In the previous studies for the deploying beams with spin, the spinning speed was time-independent. However, it is more reasonable to consider the time-dependent spinning speed. The present study introduces the time-dependent spinning speed in the modeling. The Euler-Bernoulli beam theory and von Karman nonlinear strain theory are used together to derive the equations of motion. After the equations of motion are transformed into the weak forms, the weak forms are discretized. The natural frequency and dynamic response are obtained. The effect of the time-dependent spinning speed on the dynamic response is studied.

1. Introduction

The vibration problems for the deploying beams and spinning beams are very important because these beams are used in various mechanical systems, such as robot manipulators, deploying appendages on satellite, drilling machines in productions. Numerous reports have been published for the vibration analysis of these beams.

Some researchers analyzed the vibration of the deploying beams without a spinning motion. AI-Bedoor and Khuilief(1) derived an approximate analytical solution for the transverse vibrations of a beam during axial de-ployment. Park et al.(2) analyzed the longitudinal and tra-nsverse vibrations of an axially moving beam when the beam has deploying or retracting motion. The transverse vibration for the axially moving nested beam with a tip mass was studied by Duan et al.(3). Kim and Chung(4) presented a residual vibration reduction method for a flexible beam deployed from a translating hub.

Other papers were published for the spinning beams without deployment. Fung and Lee(5) presented a para-metric variable structure control for the spinning beam. Young and Gau(6) studied the dynamic stability of a beam spinning along its axial axis and subjected to an axial force. Sheu and Yang(7) investigated the whirl speed, critical speed and mode shape of a spinning beam. A geometric nonlinear dynamic model for marine propulsion was established by Zou et al.(8).

Although many papers have been published for the vibration analysis of the deploying beams without spin as well as the spinning beams without deployment, few papers have been published for the vibration of the beam with both deploying and spinning motion. Lee(9) analyzed the vibration of a pre-twisted beam with deployment. Zhu and Chung(10) studied the nonlinear vibration of a deploying beam with spin. In the above previous studies for the deploying beams with spin, the spinning speeds were time-independent. However, in practice, the spinning speed changes with time, it is more reasonable that the spinning speed is assumed to be time-dependent.

This study presents the vibration analysis of a deploying and spinning beam with a time-dependent spinning speed. The Euler-Bernoulli beam theory and the von Karman nonlinear strain theory are used together to derive the equation of motion. The Galerkin method is used to discretize the equations. The natural frequency and dynamic response are obtained. The effect of the time-dependent spinning speed on the dynamic response is investigated.

 

2. Dynamic Modeling

Figure 1 shows the dynamic model of a spinning beam deploying from a rigid hub. The beam axially deploys with the time-dependent moving speed V(t), and simultaneously spins along its axial axis with the time-dependent spinning speed Ω(t). The beam is uniform and it has mass density ρ, total length L, circular cross section with area A and area moment of inertia I. The length outside the hub is l. An external axial force F, which is applied at the left end, pushes the beam deploying out the hub.

Fig. 1Dynamic model of a deploying beam with spin

The position vector can be defined by the axial displacement u, the lateral displacements v and w. The torsional displacement is not considered in this study because there is no coupling between torsional and lateral displacements for a spinning beam with the doubly symmetric cross section. The position vector of the point at a distance x away from the hub may be written as.

The Euler-Bernoulli beam theory and the von Karman nonlinear strain theory are used together to model the deploying beam with spin. It is assumed the beam is slender enough, so the rotary inertia and shear deformation can be neglected and the Euler-Bernoulli beam theory can be adopted. On the other hand, the von Karman nonlinear strain theory, which considers the geometric nonlinear due to the large deflection, is more adequate than the conventional linear strain theory.

The equations of motion can be derived by the extended Hamilton's principle. By considering the time-dependent spinning speed and using the similar derivation procedure presented by the authors(10), the equations of the axial and lateral motions may be derived and given as

The associate boundary conditions are

Note that in the lateral equations of motion, the spinning speed is the time function and it introduces the angular acceleration and two new corresponding terms and . These two terms bring about linear coupling effect on the lateral equations.

In order to derive the weak form, the trial functions and weighting functions are defined. The trial functions are defined by u, v and w, while the weighting functions are defined by , and . The weak forms, which may be derived by multiplying the equations of motion by weighting functions and integrating by parts over the domain, are given as

The trail functions for the axial and lateral motions may be expressed as a series of the basis functions, which are given as

Here N is number of the basis functions, the basis functions can be given as

where βn is the nth solution of the frequency equation of coshβnl cosβnl +1 = 0.

The discretized equations are obtained by substituting the trial functions and weighting functions into the weak forms. The derived discretized equations may be written as in a matrix-vector form

 

3. Analysis and Discussion

The dynamic responses and natural frequencies are obtained based on the above derived matrix vector form. Before computation of the dynamic responses, the convergence tests should be performed. According to the results given by Zhu and Chung(10), ten basis functions are reasonable for computations of the natural frequencies and dynamic responses.

The dynamic responses for the time-dependent moving speed and time-dependent spinning speed are computed. The material properties used for computation are beam diameter d = 0.01 m, total length L = 10 m, initial deployed length l(0) = 2 m, mass density 2738.6 kg/m3, Young’s modulus E = 6.8335 × 1010 N/m2, the initial deflection for the lateral displacement v is given as 5 mm. The time-dependent moving speed profile is given in Fig. 2, while the spinning speed profile is also given in Fig. 3. Two cases are considered for comparison. Fig. 3(a) is a constant case and Fig. 3(b) is a time-dependent case.

Fig. 2Moving speed profile

Fig. 3The spinning speed profile

The dynamic response for the constant case is presented in Fig. 4. As shown in this figure, the dynamic response computed by this study is in agreement with that by the previous study(10). It is observed the beat phenomenon occurs. According to the results given by the authors(10), the beat phenomenon occurs due to interference of the first and second natural frequencies of the beam.

Fig. 4Verification of the lateral displacements for the constant spinning speed case

The effect of the time-dependent spinning speed on the dynamic response is investigated. Fig. 5 shows the dynamic response for the time-dependent spinning speed profile of Fig. 3(b). As shown in Fig. 5, at the initial stage, lateral v starts from 5 mm while w starts from 0 mm. This is because spin introduces gyroscopic effect. When one of the lateral displacements is excited, the other one is also excited. Note that the magnitude of the lateral displacement in Fig. 5 is smaller than that in Fig. 4 because the initial spining speed for the time-dependent case is smaller than the constant spinning speed.

Fig. 5Lateral displacements for the time-dependent spinning speed

At the accelerating interval from 0 s to 2 s, the period decreases with time. This phenomenon is because the spinning speed increases. At the constant speed interval from 2 s to 4 s, the beat phenomenon occurs. The beat phenomenon occurs due to the interference of the first and second natural frequencies of the spinning beam as mentioned before. At the decelerating interval from 4 s to 6 s, the period increases. This is because the beam length increases and spinning speed decreases. It is also found that compared with the previous studies, the beat phenomenon disappears at the decelerating interval. This difference between the present and previous studies may be caused by the effect of the time-dependent spinning speed.

It is necessary to investigate the disappearance of the beat phenomenon in the decelerating interval. According to the conclusion in the previous study, the beat phenomenon occurs due to the first and second natural frequencies of the spinning beam. The first and second natural frequencies are given as

The difference of the first and second natural frequencies (ω2 - ω1) is computed versus the combination of the spinning speed and beam length, and plotted in Fig. 6. As shown in this figure, when the beam length increases, the difference becomes smaller, and the beat phenomenon occurs. It should be noted that, for a certain beam length, the differences are similar whether the spinning speed is high or low. So the difference of the first and second natural frequencies may be not enough to explain and another new criterion needs to be proposed.

Fig. 6Difference of first and second natural frequencies versus spinning speed and beam length

To explain the disappearance of the beat phenomenon in the decelerating interval, we investigate the difference ratio of the first and second natural frequencies verses the combination of the spinning speed and beam length. The difference ratio can be used to evaluate the relative difference and it may be may be written as

The difference ratio may be used to explain the disappearance of the beat phenomenon in the decelerating interval. The difference ratio versus the spinning speed and beam length are computed and plotted in Fig. 7. As shown in this figure, at the initial stage (region A), the spinning speed is low and the beam length is short, the difference ratio is large, beat does not occur. At the middle stage (region B), the beam has a medium length and a high spinning speed, the difference ratio becomes small, the beat occurs. At the later stage (region C), the speed becomes low and the length becomes long, the difference ratio becomes large again, the beat phenomenon disappears. Therefore, beat does not occur when the spinning speed is low and the beam length is long.

Fig. 7Difference ratio of first and second natural frequencies versus spinning speed and beam length

 

3. Conclusion

In this study, the vibration of a deploying and spinning beam is analyzed when the spinning speed is time-dependent. The Euler-Bernoulli beam and the von Karman nonlinear strain are used to model the deploying beam with spin. By considering the time-dependent spinning speed, the new equations of motion with angular acceleration are derived. The Galerkin method is adopted to discretize the equations of motion. The effects of the time-dependent spinning speed on the dynamic responses are investigated. The dynamic responses for the case of the constant speed and the time-dependent spinning speed are compared. It is found during deployment, when the spinning speed decreases, the beat phenomenon may disappear because the difference ratio of the first and second natural frequencies is large.