Research on rigid–flexible coupling dynamic characteristics of boom system in concrete pump truck

Abstract

Concrete pump truck plays an important role in infrastructure construction and national economic development. In recent years, its boom system becomes longer, and its dynamic and control become more complicated. In order to study the dynamic characteristics of boom system, three dynamic models such as multi-rigid-body model, rigid–flexible coupling model, and rigid–flexible coupling model with equivalent hydraulic cylinder were built in this work. Simulation analysis and experimental analysis were done, and they show that we should not only consider the large-range motion but also consider the small flexible deformation to study the dynamic characteristics of boom system precisely. It provides the theoretical basis to vibration control, trajectory prediction, and life assessment for boom system and such structures.

Introduction

Concrete pump truck has the advantages of flexibility, high efficiency, and good pouring quality, so it plays an important role in infrastructure construction and national economic development. Boom system is the key part of concrete pump truck, and its work is to transport concrete to the pouring position. In recent years, boom system becomes longer. Its length has exceeded 60 m and even more. So, the dynamic and control of boom system become more complicated.

Many researchers have studied the dynamic characteristics of boom system. In early studies, boom system is viewed as multi-rigid-body system.^1,2 The dynamic model of boom system was established and the dynamic characteristics were analyzed using multi-rigid-body dynamics theory. Multi-rigid-body dynamics studying the dynamic behavior of interconnected rigid body, each of which may undergo large translational and rotational displacements. These researches did not consider the effect on the dynamic characteristics of small flexible deformation. In fact, there is not only large rigid motion but also small flexible deformation during the working process of boom system. Rigid–flexible coupling is the basic feature of boom system. With the increasing length of boom system, the influence of flexible deformation becomes more and more important. So, the rigid–flexible coupling feature must be taken into account during the analysis of dynamic characteristics of boom system and such structures. ³

Recently, some researchers have studied the dynamic characteristics of boom system using the rigid–flexible coupling dynamic theory. In rigid–flexible coupling dynamic, the dynamic behavior of multi-bodies which consist of rigid and flexible bodies is studied. Oliver Lenord did research on a four-boom concrete pump test rig. He established three models of boom system and obtained a simplified linear damping to simulate the model. ⁴ Liu et al. ⁵ established a dynamic model of boom system and done numerical computation using rigid–flexible multi-body dynamics theory. F. Resta, F. Ripamonti, and G. Cazzulani established a nonlinear flexible multi-body test rig, and the modal observation method and disturbance estimate strategy were adopted to study the boom and hydraulic device. Finally, flexible boom and pump vibration suppression and control are researched.^6,7 Xia and Zhang ⁸ have visually demonstrated the models and dynamic characteristics of the booms by using various software-development environments and finite element analysis. Despite the fact that the differential equation of rigid motion of the booms is deduced by the multi-body dynamic theory and the Lagrange equation,^9,10 the flexible model^11,12 is more accurate and more complex under the assumption mode since deformation occurs in the pouring process.

In the above studies, the researchers did not establish an equivalent model of cylinder, but the cylinder is connected directly between the two sections arm as rigid model. It leads us to understand the dynamic characteristics of boom system inaccurately.

In this article, a research on the rigid–flexible coupling dynamic characteristics of boom system in concrete pump truck is presented. Three dynamic models were built. The first is multi-rigid-body model. The second is rigid–flexible coupling model which was built using modal reduced method. The last is rigid–flexible coupling model with equivalent hydraulic cylinder using virtual spring–damper method. Simulation and experiments were done. It provides the theoretical basis to vibration control, trajectory prediction, and life assessment for the boom system and such structures.

Theoretical basis

Modal reduced method

The modal reduced method is based on finite element multi-body dynamics theory. ¹³ Its basic idea is that the modal of flexible body is calculated by finite element method. Then, the dynamic stress, strain, and deformation of flexible body can be obtained using modal superposition.

In Figure 1, the three-dimensional coordinates of any point $i$ on the flexible body can be expressed as

$${\overline{r}}_i=\overline{r}+A{\overline{S}}_{i}n=\overline{r}+A({\overline{S}}_i^{0}n+{\overline{u}}_{i}n)$$ (1)

Figure 1.

Flexible body coordinate system.

In equation (1), $A$ is the transfer matrix, ${\overline{S}}_i^0$ is the un-deformation vector, and ${\overline{u}}_i$ is the deformation vector.

The translation and rotation modal matrix of point $i$ is

$${\Psi }^i=[{\Psi }_t^{\mathrm{iT}},{\Psi }_r^{\mathrm{iT}}]$$ (2)

The modal coordinates of the relative deformation ${\overline{u}}_i$ are expressed as

$$u{\prime }_i={\Psi }_t^{i\prime }\overline{a}$$ (3)

In equation (3), $\overline{a}$ is the modal position vector.

Applying equation (3) to equation (1), we can get

$${\overline{r}}_i=\overline{r}+A({\overline{S}}_i^{0\prime }+{\Psi }_t^{i\prime }\overline{a})$$ (4)

According to the modal information, the speed equations and acceleration equations of any point on the flexible body can be obtained. The overall displacement of flexible body is as follows

$$\left[u\right]=\sum {a_i\left[\varphi \right]}_i$$ (5)

In equation (5), $\left[u\right]$ is the sum of the displacement vector of each node, $a_i$ is the modal participation factor, and ${\left[\varphi \right]}_i$ is the structure mode.

Virtual spring–damper method

Virtual spring–damper method is that some structure in a mechanical system is equivalent to spring which has certain structural stiffness and damping. The boom system of a concrete pump truck is driven and controlled by hydraulic cylinder. In this article, we viewed the hydraulic cylinder as the model with certain stiffness and damping. Figure 2 is the equivalent model of hydraulic cylinder.

Figure 2.

Equivalent model of hydraulic cylinder.

The force $F_{\mathrm{cy}1}$ and the displacement $y$ of hydraulic cylinder shown in Figure 2 can be written as

$$F_{\text{cy}1}=k·(y_0(t)-y_{\text{cyl}}(t))+c·(y_0(t)-y_{\text{cyl}}(t))\prime$$ (6)

$$y=\underset{t_0}{\overset{t}{\int }}-\frac{F_{\text{cyl}}-k(y_0(t)-y_{\text{cyl}}(t))}{c}\text{d}t$$ (7)

In equations (6) and (7), $y_0(t)$ is the initial position of hydraulic cylinder, $y_{\text{cyl}}(t)$ is the final position of hydraulic cylinder, $k$ is the equivalent stiffness of hydraulic cylinder, and $c$ is the equivalent damping of hydraulic cylinder. The equivalent stiffness k and equivalent damping $c$ can be calculated from Shabana ¹⁴ and Nissing. ¹⁵

Dynamic model of boom system

Three assumptions are necessary in the modeling and simulation of boom system:

1. Only consider posture transform in planar and ignore the impact of spatial torque. Single boom was simplified as Euler–Bernoulli beam.

2. Revolute joint, translational joint, and fix joint are used to connect booms with links and hinges. Translational joint is used to simulate hydraulic cylinder motion.

3. The speed of boom system is not too fast to ignore the influence of centrifugal acceleration and Coriolis acceleration.

Based on the above assumptions, referring to the actual structure of the boom system of concrete pump truck, three types of dynamic models as shown in Table 1 were built in order to carry out comparative analysis.

Table 1.

Three types of dynamic models of boom system.

Model	Type of model
1	Multi-rigid-body model
2	Rigid–flexible coupling model
3	Rigid–flexible coupling model with equivalent hydraulic cylinder

The topology of rigid–flexible coupling model is shown in Figure 3. From Figure 3, it can be found that four booms are considered as flexible models. Links, bearing, and hydraulic cylinders are considered as rigid models.

Figure 3.

Topology of the rigid–flexible coupling model.

The main parameters of each boom are shown in Table 2.

Table 2.

Main parameters of each boom.

Parameter	Boom 1	Boom 2	Boom 3	Boom 4
Length (mm)	3753	2830	2875	3205
Mass (kg)	88.4	37.9	32.0	18.3

At first, the multi-rigid-body model of boom system as shown in Figure 4 was built on RecurDyn platform. The first boom was fixed to the bearing and the bearing was fixed on a movable bracket. Then, the rigid–flexible coupling model was obtained from the modal reduced method. The four rigid booms were replaced by flexible booms which were generated in ANSYS. Flexible booms were connected with the revolute joint of rigid links by the nodes which were pre-established in the rigid region. At last, spring–damper systems were equivalent to the four hydraulic cylinders by virtual spring–damper method, and the model 3 could be built.

Figure 4.

Multi-rigid-body model of boom system.

Simulation analysis

Simulation analysis was done after three models were built using the rigid–flexible coupling dynamic theory. In order to obtain the kinematics and kinetics laws of three models, the tip acceleration and force of four hydraulic cylinders were studied.

Tip acceleration

The simulation analysis result of the tip acceleration of three models is shown in Figure 5. It can be seen that the tip acceleration of model 1 became constant as soon as the motion of boom system stopped. This is a pure rigid body motion nature. However, the vibration of models 2 and 3 did not disappear immediately and would continue for some time after the motion of boom system stopped. This is in accordance with the actual situation. So, we should consider small flexible deformation when we study the dynamic characteristics of boom system.

Figure 5.

Tip acceleration of three models.

From Figure 5, we can also see that the amplitude of the tip acceleration of model 3 is larger than that of model 2 and the attenuation of the tip acceleration of model 3 is slower than that of model 2. This shows that tip acceleration is related to the degree of flexibility. We consider the hydraulic cylinder as the model with certain stiffness and damping in model 3. With the increase in the number of flexible body, the flexible acceleration in the large range of motion of boom system becomes larger. So, we should consider the flexible character of hydraulic cylinder when we want to control the boom system accurately.

The frequency spectrums of models 2 and 3 are shown in Figures 6 and 7, respectively. It can be found that the first three natural frequencies of model 2 are 1.172, 2.539, and 4.883 Hz, respectively. The first three natural frequencies of model 3 are 0.9766, 2.344, and 4.688 Hz, respectively. The masses of two models are equal and the stiffness of model 2 is greater than that of model 3, so the first three natural frequencies of model 2 are greater than those of model 3.

Figure 6.

Frequency spectrum of model 2.

Figure 7.

Frequency spectrum of model 3.

Force of hydraulic cylinder

There are four hydraulic cylinders in the boom system. It is found that the force of the last hydraulic cylinder is far less than that of the other hydraulic cylinder, and it can be ignored. The maximum forces of hydraulic cylinder in the three models are shown in Table 3. It can be seen that the maximum force of model 1 is in the second hydraulic cylinder. However, the maximum forces of models 2 and 3 are in the third hydraulic cylinder.

Table 3.

The maximum force of hydraulic cylinder in different models (N).

Maximum force	Model 1	Model 2	Model 3
First hydraulic cylinder	45,692.2	72,878.6	73,239.3
Second hydraulic cylinder	52,809.3	98,628.3	86,605.2
Third hydraulic cylinder	43,173.3	99,770.0	90,121.9

The simulation analysis result of the maximum force of hydraulic cylinder in the three models is shown in Figure 8. It can be seen that the force of hydraulic cylinder in model 1 became constant as soon as the motion of the boom system stopped. However, the force of the hydraulic cylinder would fluctuate for some time in models 2 and 3 after the motion of the boom system stopped. This is also in accordance with the actual situation. So, we should consider small flexible deformation when we study the dynamic characteristics of boom system.

Figure 8.

Force of the hydraulic cylinder in the three models.

From Figure 8, we can also see that the amplitude in model 3 is larger than that in model 2 and the attenuation of the force of hydraulic cylinder in model 3 is slower than that in model 2. So, we should consider the flexible character of hydraulic cylinder when we want to control the boom system accurately.

Experimental analysis

In order to study the dynamic characteristics of the boom system deeply and to verify the simulation analysis in this article, we designed and built a boom system test rig as shown in Figure 9. It is a miniature of actual boom system and the model scale is 1/3. Its length is about 13 m. We also built a signal acquisition system on the test rig and is shown in Figure 10. It includes tip displacement sensor, pressure sensors, vibration sensors, inclination sensors, strain sensors, Dewesoft multi-channel signal acquisition instrument, and computer.

Figure 9.

Boom system test rig.

Figure 10.

Signal acquisition system.

In the experiment, we drove the boom system movement using four hydraulic cylinders and chose the same working condition as simulation analysis. The experiment scene is shown in Figure 11. The experimental result of tip acceleration is shown in Figure 12.

Figure 11.

Experiment scene.

Figure 12.

Time-domain waveform of tip acceleration.

From Figure 12, it can be found that the time-domain waveform of tip acceleration in the test is similar to the simulation results of models 2 and 3. So, we should consider flexible features when we study the dynamic characteristics of boom system.

The frequency spectrum of tip acceleration in the test is shown in Figure 13. It is known that the first two natural frequencies are 0.7813 and 2.148 Hz. The frequency comparison between simulation and experiment is shown in Table 4.

Figure 13.

Frequency spectrum in the test.

Table 4.

The frequency comparison between simulation and experiment (Hz).

	First	Second
Model 2	1.172	2.539
Model 3	0.9766	2.344
Test	0.7813	2.148

From Table 4, we can see that the simulation result of model 2 is closer to the test result than that of model 3. This shows that the hydraulic modeling method is feasible. We should consider the flexible character of hydraulic cylinder when we want to control the boom system accurately. The simulation analysis result of model 3 is slightly less than the experimental result. This may be due to the fact that the stiffness coefficients and damping coefficients in the equivalent model of hydraulic cylinder are not accurate. So, we should focus more on the virtual spring–damper equivalent calculations in the next research.

Conclusion

1. Three dynamic models of boom system in concrete pump truck were established in this article. The multi-rigid-body model was built at first. Then, the modal reduction method is adopted to build the rigid–flexible coupling model. At last, spring–damper systems were equivalent to the hydraulic cylinders by virtual spring–damper method.

2. Simulation analysis and experimental analysis show that we should not only consider the large-range motion but also consider the small flexible deformation if we want to study the dynamic characteristics of the boom system precisely. In addition, it is necessary that the hydraulic cylinder is equivalent to a flexible model.

3. In the virtual spring–damper method, the equivalent stiffness coefficient and equivalent damping coefficient are very important to build the equivalent model and should be calculated accurately in the next research.

4. This research provides the theoretical basis to vibration control, trajectory prediction, and life assessment for boom system and such structures.