Keywords

1 Introduction

In the aspect of rocket launcher dynamic modeling, Ren [1] and others built a rigid-flexible coupling driving dynamic model of the vehicle and gun after flexible treatment of the frame, and only considered the role of tire and ground excitation in the vibration. Li [2] studied the influence of structural damping and installation position on the ride comfort of the whole gun. Yao [3] established the driving dynamic characteristics of rocket launcher with hydro-pneumatic suspension, and compared with the simulation results with linear spring. Wang [4] proposed a method of experiment combined with numerical simulation to complete the bench test of the rocket launcher.

Ride comfort is one of the important performances of vehicles. When the ride comfort is poor, the system vibrates violently. For the vehicle-mounted rocket launcher, on the one hand, the vibration of the vehicle itself reduces the service life, passability and operational stability; on the other hand, excessive vibration will reduce the life and reliability of the rocket launcher, especially for some electrical equipment, the harm caused by vibration is greater. Based on the above reasons, it is necessary to ensure the ride comfort of the rocket launcher in the design process.

The suspension system of the light rocket launcher studied in this paper adopts plate spring non-independent suspension with nonlinear characteristics and good damping capacity. In this paper, the driving dynamics and ride comfort of the rocket launcher are studied on the basis of considering the nonlinear characteristics of the plate spring suspension.

2 Establishment of Driving Model

2.1 Description of Rocket Launcher Model

The structure of the light rocket launcher consists of a rocket launcher loading system and a chassis system. The rocket launcher loading system is composed of directional tube, rocket, landing gear, electric cylinder, rotary body and base, and the suspension system is composed of leaf spring, shock absorber, U-shaped bolt and buffer block. Figure 1 shows the topology of the rocket launcher model.

Fig. 1
A hierarchy. Directional tube is followed by landing gear. It has 2 parts, rotating mechanism, electric cylinder. Rotating mechanism is followed by pedestal, car frame, cabin. Car frame is of 2 types. Front plate is followed by front axle. Reat plate is followed by rear axle. These lead to ground.

Topology of rocket launcher

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In this paper, the multi-rigid-body dynamic model of self-propelled rocket launcher is simplified as follows [5]:

  1. (1)

    the leaf spring is replaced by the three-link model, and the motion relationship between the U-shaped bolt and the buffer block between the leaf spring and the drive axle is simplified and replaced by the bushing force. The drive axle part simplifies the main reducer, differential, engine and other devices.

  2. (2)

    the deformation of rocket launcher and car body has little influence on ride comfort under driving condition, and the deformation of various parts (except tires and leaf springs) is ignored in the model.

  3. (3)

    the assembly error between parts is not considered, only the influence of vehicle speed and road surface is considered.

2.2 Rocket Launcher Loading System

According to Fig. 1, there is a fixed pair (F1) between the rocket and the directional tube, between the directional tube and the landing gear (f2), between the landing gear and the rotary body (f3), between the electric cylinder push rod and the landing gear (f4), between the electric cylinder push rod and the electric cylinder seat is the moving pair and the linear spring force (f7), and between the electric cylinder seat and the rotary body is the rotating pair (f5). Between the rotary body and the base is the rotating pair and the torsional spring force (f6), and between the base and the frame is the fixed pair (f8).

2.3 Chassis System

The light rocket launcher is a military off-road chassis. For the driving dynamics model, the chassis can be simplified into frame, spring suspension, tires and so on. The cab and the frame are fixed by a fixed pair (f9) simulation bolt, the frame and the front and rear axle are leaf spring non-independent suspension, the frame and the leaf spring ear are rotating pairs and torsion springs (f10, f11), while the lower end of the leaf spring and the axle are simulated by bushing force (f12, f13), and the axle and the wheel are represented by rotating pair (f13-f16) and angular speed drive. The contact force (f17-f20) is between the wheel and the ground.

Plate Spring Suspension

The front and rear suspension is composed of leaf spring, lifting lug, U-shaped bolt, buffer block and shock absorber. The front leaf spring is a constant stiffness spring, and the rear leaf spring is a gradual leaf spring, which is composed of a main spring and a secondary spring.

Firstly, the three-link leaf spring model is established [6]. The three-link method is a leaf spring kinematics calculation model recommended by the American Society of Automotive Engineers (SAE), as shown in Fig. 2.

Fig. 2
Two schematic diagrams. A, three parallel curved lines. a, b, and c points are indicated. B, 2 curved lines. The distance is d and the breadth is c. The parts are denoted as R a, R c, and R b.

Leaf spring suspension structure and simplified model of three-link

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In the figure: \(L\) is the total arc length of the leaf spring; \(a\) and \(b\) are the arc length of the front and rear segments of the leaf spring; \(c\) is the length of the lifting lug; \(m\) and \(n\) are the distance between the front and rear segments of U-shaped bolts and the center of the leaf spring; \(R_{a}\), \(R_{b}\) and \(R_{c}\) are the lengths of the front, middle and rear rods of the three-link model, respectively; \(d\) is the distance between the front section of the middle rod and the axle axis. The calculation formula of three-link is as follows:

$$\left\{ {\begin{array}{*{20}c} {R_{a} = 0.75(a - m)} \\ {R_{b} = 0.75(b - n)} \\ {R_{c} = L - (R_{a} + R_{b} )} \\ {d = a - R_{a} } \\ \end{array} } \right.$$
(1)

In order to simulate the load-bearing characteristics of the leaf spring model, a rotating pair and a torsion spring are established between the connecting rods to ensure the stiffness characteristics of the three-link model. The stiffness of torsion spring is expressed by cubic polynomial about displacement:

$$\left\{ {\begin{array}{*{20}c} {T_{1} = \left[ {\begin{array}{*{20}c} {a_{1} } & {a_{2} } & {a_{3} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\theta_{1} } & {\theta_{2}^{2} } & {\theta_{3}^{3} } \\ \end{array} } \right]^{T} } \\ {T_{2} = \left[ {\begin{array}{*{20}c} {b_{1} } & {b_{2} } & {b_{3} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\theta_{1} } & {\theta_{2}^{2} } & {\theta_{3}^{3} } \\ \end{array} } \right]^{T} } \\ \end{array} } \right.$$
(2)

The reverse simulation method is used to identify the parameters of the torsion spring stiffness in the three-link equivalent model, and the equivalent stiffness of the leaf spring model is obtained. Figure 3 shows the stiffness test data of leaf spring and the results of reverse simulation method. The load-bearing characteristics of the three-link model are very close to the test data, and the simulation results meet the requirements.

Fig. 3
Two multiline graphs. A, front leaf spring load versus vertical displacement presents 2 ascending linear curves. It indicates the experimental and reverse simulation data. B, rear leaf spring load versus vertical displacement presents 2 ascending curves for experimental and reverse simulation.

Comparison of the results of reverse simulation of front and rear leaf springs

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The damping force–deformation velocity curve of the front and rear shock absorbers is shown in Fig. 4 and is added to the equivalent spring in the form of Spline function.

Fig. 4
Two line graphs. A, damping force of front shock absorber versus piston speed. It presents a fluctuating ascending curve. B, damping force of rear shock absorber versus piston speed. It presents a fluctuating ascending curve that originates from (negative 1, negative 2000). Values are estimated.

Damping force–velocity curves of front and rear shock absorbers

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Tire Model

Tire is an important part of a vehicle, on the one hand, it supports the weight of the whole vehicle, ensures that the vehicle has good adhesion to the ground, and transmits driving torque and braking torque. on the other hand, it works with the suspension system to alleviate the impact caused by the uneven road surface and attenuate the resulting vibration.

The selection of tire types and parameters has an important influence on the analysis of vehicle driving performance. The force between tire and ground is based on UA tire model. The main characteristic parameters of UA model are lateral stiffness, tilting stiffness, vertical stiffness, longitudinal stiffness, rolling resistance coefficient and vertical damping coefficient. Tire model parameters are provided according to the data provided by the factory, as shown in Table 1.

Table 1 Tire parameters
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2.4 Road Model

The general description method of road roughness is power function power spectral density. According to the content of Chinese national standard GB/T7031-2005 “Mechanical Vibration Road pavement Spectrum Measurement data report” [7], the displacement power spectral density of pavement roughness can be fitted by the following formula.

$$G_{d} (n) = G_{d} (n_{0} )(\frac{n}{{n_{0} }})^{ - W}$$
(3)

In the formula, \(n\) is the spatial frequency, \(n_{0}\) is the reference spatial frequency, \(n_{0} = 0.1m^{ - 1}\), \(G_{d} (n)\) is the displacement power spectral density of road roughness, \(G_{d} (n_{0} )\) is the displacement power spectral density under the reference spatial frequency \(n_{0}\), \(W\) is the frequency index and the slope in double logarithmic coordinates.

The harmonic superposition method is used to simulate the random pavement [8], and the function of the vertical displacement of the random pavement with respect to the longitudinal displacement can be obtained by adding the sine wave function of each small frequency interval.

$$Z(x) = \sum\limits_{i = 1}^{N} {\sqrt {2Gq(f_{mid,i} ) \cdot \Delta f_{i} } } \cdot \sin (2\pi f_{mid,i} t + \theta_{i} )$$
(4)

In the formula, \(Z(x)\) is the road roughness function, \(\Delta f_{i}\) is the spatial frequency interval, \(\theta_{i}\) is the sinusoidal function phase, which is randomly distributed in \(\left[ {0,2\pi } \right]\), and \(x\) is the road length.

In this paper, the triangulation method is used to establish the rdf file that can be recognized by recurdyn, and the C-level simulation road is generated for subsequent virtual driving simulation. The road is shown in Fig. 5.

Fig. 5
A model of a simulation rod. The rectangle rod is divided into small parts to simulate.

C level simulation road

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3 Analysis of Vibration Characteristics of Rocket Launcher

3.1 Driving Simulation Analysis

In the dynamic model of the rocket launcher, x is the positive direction of the vehicle, y is the lateral direction, and z is the vertical direction. The uniform driving time of the rocket launcher is set as 25 s, and the driving simulation is carried out on the C-class road at the speed of 40 km/h, 50 km/h and 60 km/h respectively. The vibration acceleration of the rocket launcher base in three directions is obtained, as shown in Fig. 6.

Fig. 6
Three multiline graphs of acceleration versus time at 40, 50, and 60 kilometers per hour. A, X direction acceleration curve. The curves fluctuate with the increases in time. B, Y direction acceleration curve. C, Z direction acceleration curve. The fluctuating curves overlap with each other.

Vibration acceleration curve of rocket launcher base

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Based on the spectrum analysis of the z-direction acceleration curve of the rocket launcher base, the vertical acceleration power spectrum density curve is obtained. As shown in Fig. 7.

Fig. 7
A multiline graph of acceleration power spectral density versus frequency. The acceleration power spectral density varies with the change in frequency for 40, 50, and 60 kilometers per hour. The sharp peaks are obtained between 3 Hertz and 7 Hertz. Values are estimated.

Acceleration power spectral density curve of rocket launcher base in z direction

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From the acceleration power spectral density curve, it can be seen that the acceleration power spectral density in the vertical direction of the rocket launcher base is mainly concentrated between 2.11 Hz and 4.38 Hz, while the peak appears in several different positions, the more obvious are 2.31 Hz, 2.96 Hz and 3.50 Hz. For the rocket launching process, the minimum vibration frequency [9] is about 7 Hz, when the lowest order frequency of the launcher is relatively high, which well avoids the vibration frequency of the whole vehicle at different speeds.

The 1st to 6th order modes of the rocket launcher itself can be known by finite element calculation, as shown in Table 2.

Table 2 Modal calculation results of rocket launcher
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The natural frequencies of the rocket launcher in the second and third modes are 2.00 Hz and 4.18 Hz, as shown in Fig. 8. In the driving process of the rocket launcher, the frequency of the peak value is different from the natural frequency of the rocket launcher, which avoids the danger of resonance.

Fig. 8
2 heatmaps. A, step is 1, mode is 2, value is 157, frequency is 1.9980 cycles per time. The primary var is U, magnitude. Defomation scale factor is 1.000 e plus 02. The highest U is found on the layers of rods. B, step is 1, mode is 3, value is 688, frequency is 1.9980 cycles per time.

Second and third order vibration modes of rocket launcher

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3.2 Ride Comfort Analysis

According to the ISO2631-1 1997 (E) standard, when the peak coefficient of vibration waveform is less than 9, the basic evaluation method-weighted root mean square of acceleration [10] is used to evaluate the effect of vibration on human comfort and health. When using this method, the axial acceleration curve, the power spectral density of acceleration and the root mean square formula of acceleration should be calculated.

$$a_{w} = \left[ {\frac{1}{T}\int\limits_{0}^{T} {a_{w}^{2} (t)dt} } \right]^{\frac{1}{2}}$$
(5)

When considering three axial vibrations at the same time, the root mean square of the total weighted acceleration of the three axes can be calculated according to the following formula:

$$a_{v} = \left[ {\left( {1.4a_{xw} } \right)^{2} + \left( {1.4a_{yw} } \right)^{2} + a_{zw}^{2} } \right]^{\frac{1}{2}}$$
(6)

In the formula: \(a_{xw}\) is the root mean square of x axis weighted acceleration, \(a_{xw}\) is the root mean square of y axis weighted acceleration, and \(a_{zw}\) is the root mean square of z axis weighted acceleration.

This paper analyzes the ride comfort of the rocket launcher and studies the bad vibration of the rocket launcher at different speeds on the C-class road surface. Table 3 shows the total weighted acceleration root mean square of the rocket launcher base.

Table 3 Total weighted acceleration root mean square of rocket launcher base
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As can be seen from Table 3, with the increase of vehicle speed, the ride comfort of the rocket launcher base becomes worse and worse, but the weighted root mean square of acceleration an is less than the specified value [10] 2.5 m/s2. For the rocket launcher structure, the bad degree of vibration acceleration will not affect the rocket launcher parts and electronic components, and can well meet the ride comfort requirements.

4 Conclusion

In this paper, the dynamic model of rocket launcher with plate spring suspension is established, the generation program of C class road spectrum is compiled, and the driving dynamics simulation of the rocket launcher is carried out at different speeds. the vibration acceleration curve of the rocket launcher base in 3 direction and the acceleration power spectrum density curve in z direction are obtained. Through the simulation analysis, we can get the following conclusions:

  1. (1)

    The stiffness of leaf spring obtained by reverse simulation method can effectively reproduce its load-bearing characteristics, which ensures the accuracy of dynamic simulation modeling.

  2. (2)

    The power spectral density of vertical vibration acceleration of rocket launcher base is studied, and its peak value is mainly between 2.11 Hz and 4.38 Hz. From the point of view of the launching vibration frequency and natural frequency of the rocket launcher, the resonance problem of the rocket launcher can be effectively avoided.

  3. (3)

    With reference to the vehicle ride comfort index, the ride comfort of the rocket launcher is studied. The weighted root mean square of acceleration \(a_{v}\) of the rocket launcher under the 60 km/h speed of class C road surface is less than the specified value 2.5 m/s2, which provides a reference for the anti-overload and vibration reduction design of the weapon system.