Research Article | Open Access

Volume 2017 |Article ID 3202318 | https://doi.org/10.1155/2017/3202318

Zhe Ji, Xiaoxian Yao, Zuobao Liang, "The Stiffness Calculation and Optimization for the Variable Stiffness Load Torque Simulation System", International Journal of Aerospace Engineering, vol. 2017, Article ID 3202318, 8 pages, 2017. https://doi.org/10.1155/2017/3202318

# The Stiffness Calculation and Optimization for the Variable Stiffness Load Torque Simulation System

Revised16 Oct 2017
Accepted23 Oct 2017
Published05 Dec 2017

#### 1. Introduction

An optimization method is a mathematical method that examines how to find certain factors with a given constraint and a certain indicator to achieve the optimal effect. Schaffer  proposed the use of a genetic algorithm to solve the problem of objective optimization. Rahmati et al.  designed a nonlinear objective optimization method to enhance the passive control performance of rubbers by minimizing the destructive effects of shock and random excitations. By analyzing the influence of the initial load stiffness acting on the loading effect, the loading parameters must be optimized to obtain the best working results.

As shown in Figure 1, the variable stiffness loading system is composed of a basic platform simulator, a load stiffness servo system, a dynamic angle of attack compensation system, a rudder output shaft connector, and auxiliary components .

We define the load stiffness as the torque caused by a unit angle of the torsion bar spring. The relationship between and the size of its structure is presented as where is the twist angle; is the torque; is the section factor of the rectangular materials; , , and are the working length, thickness, and width, respectively, of the torsion bar spring; and is the elastic shear modulus.

As demonstrated by formula (1), changing the working length of the torsion bar spring can change the stiffness  and enable the load stiffness to change via a set of stiffness servo systems at different flight altitudes and Mach numbers. The dynamic angle of attack compensation mechanism generates torque by outputting a twist angle to track the changes of NRDLT caused by the angle of attack. Therefore, the load stiffness and the equivalent angle of attack are the loading parameters required by the variable stiffness loading system.

##### 2.2. Composition of the Load Torque

The correlations between the hinge moment , the rudder deflection angle , and the angle of attack are  where and refer to the torque gradients of the angle of attack and the rudder deflection angle, respectively. These values are related to parameters such as the flight altitude and Mach number.

As expressed by formula (2), the load torque consists of two parts: the slow-changing -dependent load torque caused by the angle of attack and the deflection-dependent load torque related to the characteristics of the rudder.

To facilitate the realization of loading, we consider to equal , both of which are referred to as the load stiffness and are denoted by . is related to the flight Mach number and altitude as follows:

According to the characteristics of the torque during the flight process of a missile, as shown in formula (2), the variable stiffness load torque simulation system divides the torque into two parts. During the missile’s flight, the load stiffness will change according to variations in the Mach number, altitude, and other parameters . Thus, the change rate of the load stiffness is relatively slow. The changing frequency of the angle of attack is limited, whereas the change in rudder deflection varies significantly for different types of rudders. For example, the dynamic frequency is generally on the order of 10 Hz for nonspinning missile rudders, whereas spinning missile rudders have a higher requirement for dynamic frequency. Therefore, the load torque for the rudder during a missile’s flight consists of two parts: the RDLT related to the characteristics of a rudder and the NRDLT with relatively slow changes.

The load torque provided by the torsion bar spring is a passive torque associated with the rudder deflection angle. The active movement of the rudder is not a coupling disturbance of the load torque but is rather a part of the load torque. The torsion bar spring has a very high intrinsic frequency; thus, the system can effectively reduce the additional torque of the conventional loading device. By outputting the compensation angle of attack from a torque motor, the system outputs the compensation torque based on the load torque caused by the active movement of the rudder deflection angle, which improves the dynamic loading accuracy and the response bandwidth of the rudder load.

##### 2.3. Realization of the Angle of Attack Compensation

differs from . In formula (2), by introducing the load stiffness , if the hinge moment remains unchanged, the angle of attack must be converted into the equivalent angle of attack , which is obtained from formula (4).

Since the torque of the torsion bar spring is proportional to the twist angle, it can output the equivalent angle of attack by the part of the angle of attack servo system working on the torsion bar spring. This torque is the NRDLT. Due to the limited variation of the angle of attack’s frequency and amplitude during an actual flight, the frequency and amplitude of the equivalent angle of attack after conversion are usually small.

The traditional electric loading method, which regards the torque as a controlled variable, obtains the real-time load torque by receiving an interpolation of discrete data points from an emulation computer using an HILS. The load torque simulation system needs to track the torque command while tracking the rudder deflection angle. Therefore, this coupling system will produce additional torque, which affects the load accuracy and response bandwidth.

According to the working principle of the variable stiffness load simulator, the purpose of this solution is to obtain the loading parameters, the load stiffness , and the equivalent angle of attack , at each sampling time.

##### 4.1. Solution Method

The concept of variable stiffness loading is summarized as follows: during the process of loading, the load stiffness changes in accordance with variations in the Mach number and altitude. The change in the working length of the torsion bar spring primarily depends on the changes in the Mach number and altitude. The angle of attack converted under the load stiffness is referred to as the equivalent angle of attack. The target load torque is determined by the spring’s torsion angle, which consists of the equivalent angle of attack and the rudder deflection angle.

##### 4.2. Solution Examples

As shown in Figure 2, the curve line of the load stiffness, which corresponds to changes in the Mach number and angle of attack, is obtained by applying the wind tunnel test data in accordance with the proposed method.

The change in the angle of attack does not affect the load stiffness within a range of small angles. The set of load stiffness calculated using the wind tunnel test data for a set of fixed Mach numbers is shown in Table 1. The load stiffness for other Mach numbers is obtained during flight by interpolation.

 Mach number 0.4 0.8 1.1 Load stiffness Value 0.02 0.100 0.253

The emulation computer is given the following signals: (i)The angle of attack signal is a sinusoidal signal with an amplitude of 0~2° and a frequency of 0.8 Hz.(ii)The Mach number signal is a ramp signal of 0.4~1.1 Ma from 0~4 s, remaining at 1.1 Ma from 4 to 8 s.(iii)Simulation time is 8 s.

The rudder deflection angle signal uses a Schroeder-phased harmonic signal (SPHS), which was proposed by Schroeder  and is superposed by several amplitude values, periods, and cosine waves related to the initial phase. The composition of the initial phase angle of the signals’ cosine components is adjusted to form periodic multifrequency signals with unique characteristics, with a general mathematical expression of where NH is the harmonic number of the signals, is the amplitude of the kth harmonic wave, is the initial phase of the kth harmonic wave, and is the period of the signal fundamental wave. In practical applications, is usually adopted; thus, the following form is given:

The input signal curves are shown in Figure 3.

As shown in Figure 4, the load stiffness is acquired by interpolation according to the discrete data listed in Table 1. The variation in load stiffness changes slowly because the Mach number does not significantly change.

The curve line of the equivalent angle of attack is calculated according to formula (4), as shown in Figure 5. The variations in frequency and amplitude are both small; thus, the angle of attack servo system can adequately track the equivalent angle of attack.

A comparison between the command load torque and the actual load torque is shown in Figure 6.

#### 5. Optimization of the Load Stiffness

As shown in Figure 2, the load stiffness is not a constant value for the condition in which the Mach number and the rudder deflection angle are fixed but varies slightly with changes in the angle of attack. We obtained only a few discrete data points of finite numbers from the wind tunnel test. The load torque obtained by interpolating the discrete data points obtained from the wind tunnel test data is , and the real-time load stiffness and the equivalent angle of attack are obtained by the method described in Section 3. The actual load torque is indirectly obtained at ; that is, where and are the real-time inputs of the loading system, which are obtained by the emulation computer and and are the actual outputs of the stiffness servo system and the angle of attack servo system, respectively.

The load stiffness values are obtained (K = 0.02, 0.1, 0.253) for fixed Mach numbers (Ma = 0.4, 0.8, 1.1), as shown in Table 1. For a given Mach number, the actual load torque is calculated with different sets of load stiffness for the initial value. Compared with the command torque, the torque tracking error is used to evaluate the tracking effect. The tracking error is denoted as . The root mean square error of the tracking load torque is used as an evaluation parameter for the torque tracking effect, which is referred to as where is the length of the torque tracking error sequence.

As reflected by the data in Table 2, the different sets of load stiffness for a fixed Mach number have a certain effect on the tracking of the loading torque. This result occurs because the stiffness solution is based on the Mach number value interpolation, and the discrete data points in the wind tunnel test data are used as the weight of the load stiffness for other arbitrary Mach number conditions. Hence, there must be a set of optimal load stiffness for fixed Mach values, causing the root mean square error of the load torque variance to reach a minimum. In this article, a genetic algorithm is used to train the wind tunnel test data to readjust the stiffness value obtained for a fixed Mach number. Objective optimization is used to obtain the optimal set of load stiffness.

 Data group Ma = 0.4Load stiffness() Ma = 0.8Load stiffness() Ma = 1.1Load stiffness() 1 0.0178 0.102 0.265 0.032374 2 0.020 0.102 0.265 0.032323 3 0.020 0.111 0.297 0.101773 4 0.0184 0.1026 0.2672 0.036783 5 0.0183 0.1072 0.2558 0.020048 6 0.0167 0.0953 0.2705 0.042441 7 0.015 0.15 0.25 0.055954 8 0.025 0.18 0.3 0.143948

We select an independent variable , where formula (14) acts as an objective function. The bounds for the objective functions are given.

The optimal result is obtained by the optimization calculation, and the minimum value of is obtained, as shown in Table 3.

 0.0201 0.0983 0.253 0.0156

#### Conflicts of Interest

The authors declare that they have no conflicts of interests regarding the publication of this article.

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