1. Introduction
A fire-fighting monitor is a piece of fire-fighting equipment with a large flow and long-range, and is mainly composed of a barrel and a gun head [
1,
2]. Through the electromechanical control system of the barrel, horizontal and pitching rotation of the fire-fighting monitor can be realized, so that fires can be extinguished rapidly due to the burning object being sprayed directly with fluid. The gun head is the end effector of the fire-fighting jet system, which converts the pressure energy of the fluid into kinetic energy. The two jet states, spray jet and straight jet, can be switched by adjusting the electromechanical control system of the gun head [
2,
3,
4]. The nozzle opening of the traditional fire-fighting monitor remains unchanged during the jet flow, and the flow, pressure, and nozzle opening of the jet system are not well matched. Based on the principle of valve components in a hydraulic system [
5,
6], an adaptive fire-fighting monitor with an elastic adaptive adjustment mechanism at the front end of the nozzle was designed. The nozzle opening of the monitor could be automatically adjusted according to the changes in the flow and pressure of the jet system to improve the performance of the jet system, and its fire extinguishing efficiency, in a wider flow range.
The spray medium of the fire-fighting monitor is generally water, or a mixture of water and foam. During the working process of the fire-fighting jet system, the centrifugal pump, which works as the power source, tends to exhibit pressure pulsation [
7,
8,
9]. Under the joint action of the pump and the load pressure, the dynamic fluid spring is easily formed due to the compressibility of the fluid [
10,
11]. The fluid spring and load mass constitute a dynamic system of the fluid spring and mass The nonlinearity of spring stiffness will cause the natural frequency of the jet system to be non-constant, which will nonlinear vibration of the load mass of the jet system, leading to a loud noise, and even the destruction of the entire structure [
12,
13,
14]. Similar to the jet system, the valve components in the hydraulic system will also have nonlinear dynamic behaviors such as multi-cycle and chaos under the action of the pressure pulsation, which will cause leakage of the hydraulic valve and other failures [
15,
16,
17]. Additionally, unstable motion occurs when the railway vehicle reaches high speeds. In the literature [
18], the effects of changing parameters with different lateral stiffnesses on nonlinear hunting behavior were analyzed. It was found that the system with bogie and wheelset had less critical speeds than the wheelset system alone, and the increase of the wheelset mass made the hunting behavior even worse. Therefore, it is necessary to explore the effect of a nonlinear fluid spring on the dynamic characteristics of the jet system of adaptive fire-fighting monitors.
At present, the research on the dynamic characteristics of fluid spring systems is generally carried out by methods of analysis, simulation, and experimentation [
19,
20,
21,
22]. Generally, when using the analytical method, the nonlinear factors of the system are linearized first, and then the system is analyzed by linear control theory, such as the root locus and Bode’s chart methods [
23,
24,
25]. The linearized analysis of nonlinear systems can reveal essential relationships between system parameters and performance evaluation indexes, such as stability, accuracy, and rapidity of the system. However, to some extent, the conclusions are often different from real world observations. It is difficult to explain some of the abnormal phenomena existing in real world dynamic tests, such as complex time-domain waveforms and numerous frequency domain peaks [
26]. With the development of nonlinear differential equation solutions, such as the Runge-Kutta algorithm, scholars often use simulation methods to directly calculate the nonlinear system, describing the dynamics of the system more accurately [
27,
28,
29]. The experimental method is mainly used to verify the results of systematic analysis and simulations, to verify the accuracy of nonlinear system analysis and simulation models [
30,
31].
In this paper, the adaptive fire-fighting monitor was taken as the research object, and the influence of the nonlinear fluid spring force on the dynamic characteristics of the jet system was the focus of the study. Based on the principles of nonlinear dynamics, a nonlinear dynamic model of the jet system was established. The numerical calculation was used to determine whether the jet system will have dynamic behaviors of deteriorating systems such as multi-cycle, bifurcation, and chaos within the range of operating parameters. Then the measured experimental data was analyzed with nonlinear dynamic research methods, to verify the rationality of the mechanical performance of the designed adaptive fire-fighting monitor.
2. Dynamic Modeling of the Jet System
The gun head of the fire-fighting monitor can automatically adjust the nozzle opening with a change of the flow or pressure of the incident fluid, as it can be divided into a traditional diversion gun head and an adaptive gun head. The structure of a traditional diversion gun head is shown in
Figure 1. During the working process, except for the outer nozzle which is used for adjusting the state of the jet, the other components are in a relatively static state. Therefore, the nozzle opening cannot be changed with the change of the jet flow and pressure, and an aggravation of turbulent flow and sudden increase of pressure is prone to occur, resulting in a reduction of fire extinguishing efficiency. The structure of an adaptive gun head is shown in
Figure 2. Inside the adaptive gun head, an adaptive mechanism is added, consisting of a spray core, an end cap, a core rod, and a spring. When the flow of the incident fluid in the gun head increases, the fluid force on the left side of the spray core increases. When the fluid force is greater than the spring force on the right side of the spray core, the spray core moves to the right, and the nozzle opening increases. In contrast, When the flow of the incident fluid decreases, the spray core moves to the left, and the nozzle opening decreases. Therefore, an adaptive fire-fighting monitor with an adaptive mechanism can automatically adjust the nozzle opening according to changes in inlet flow and pressure, thereby achieving good jet performance in a wider flow range.
The connection between the adaptive gun head, the barrel, and the pipeline is shown in
Figure 3. The electromechanical system installed on the barrel can realize the horizontal and pitching rotation of the fire-fighting monitor.
The jet system of the adaptive fire-fighting monitor has better jet performance due to the addition of the adaptive mechanism in the gun head. However, the mechanical spring introduced in the mechanism reduces the stiffness of the jet system, and the compressibility of the fluid changes the stiffness of the jet system dynamically, increasing the complexity of the dynamic behaviors of the jet system. When the excitation frequency is close to the natural frequency of the jet system, the vibration of the spray core is intensified, which will significantly reduce the fire extinguishing efficiency. The internal structure of the adaptive gun head is very similar to that of the valve element in a hydraulic system. Therefore, referring to the dynamic analysis of the relief valve, the working principle of the jet system of an adaptive fire-fighting monitor is shown in
Figure 4.
It can be seen in
Figure 4 that the diesel and the reducer installed on the fire engine drive the main spindle of the pump to rotate, and the water in the water tank is filtered by the filter, and then sucked into the interior by the pump. The water discharged from the pump enters the adaptive fire-fighting monitor through the flowmeter and the throttle valve, and finally shoots at the fire point.
In
Figure 4,
is the mass of the spray core. Analyzing the forces of the spray core, the dynamic equation of the spray core is
where,
is the displacement of the spray core under the action of jet fluid,
is the viscous resistance,
is the spring force, and
is the fluid force on the spray core.
The total equivalent stiffness of the jet system, as shown in
Figure 4, is made up of the stiffness of the mechanical spring inside the spray core and the stiffness of the fluid unit on the left side of the spray core. During the operation of the jet system, the movement of the spray core causes a change in the length of the fluid unit on the left side, which in turn causes a change in the stiffness of the fluid unit, ultimately resulting in a change in the total spring stiffness of the jet system. Therefore, the variation law of total stiffness is
where
is the bulk elastic modulus of the fluid. Considering the compressibility of the gas-containing fluid,
is calculated by the bulk elastic modulus formula [
32].
is the equivalent cross-sectional area of the fluid unit,
is the equivalent length of the fluid unit, and
is the stiffness of the mechanical spring inside the spray core.
Let
be the vibration displacement near the working point of the spray core,
, that is
. According to the Taylor series, the total stiffness of the jet system near the operating point can be expressed as:
Assuming
,
, and
, then substituting them into Equation (3):
Omitting the infinitesimal of higher order
in Equation (4), the total stiffness of the jet system can be expressed as:
Since the elastic potential energy
of the spring has symmetry, the total elastic potential energy of the jet system can be expressed as:
The nonlinear spring force of the jet system can be further expressed by Equation (6):
This paper mainly studies the influence of nonlinear spring force on the dynamic characteristics of the jet system, so the nonlinear factors such as friction and system damping are not considered. Then the dynamic equation of the jet system near the working point
is
where,
is the linear damping coefficient of the system, which is the sum of the system structural damping coefficient
, and the fluid damping coefficient
.
is the external periodic excitation caused by the pressure pulsation of the incident fluid,
is the amplitude of the external excitation,
is the angular frequency of external excitation, and
is the initial phase angle of external excitation.
In order to analyze the jet system more conveniently and intuitively, the mass unit of Equation (8) can be normalized as
where
,
,
,
.
is the linear damping ratio,
is the natural frequency of the linear harmonic oscillator when the nonlinear term coefficient
is equal to 0, and
is the amplitude of external excitation received by the unit mass.
4. Sensitivity Analysis of the Jet System
The steady-state amplitude of the jet system is mainly related to
,
, and
. In order to analyze the influence of the three variables on the steady-state amplitude, the sensitivity of the steady-state amplitude to the three variables is calculated by the difference method. Due to the interactions between the three variables, the sensitivity of the jet system is analyzed according to the conditions in
Section 3.3,
Section 3.4, and
Section 3.5.
When
is 0.6 MPa and
is 3 N, the sensitivity variation curves of the steady-state amplitude
with respect to
and
are shown in
Figure 11 and
Figure 12, respectively.
It can be seen from
Figure 11 that in the low-frequency range before the amplitude mutation, the sensitivity of the amplitude to
is positive. When
is constant, the sensitivity increases first and then decreases gradually as
increases. When
increases, the sensitivity under the same frequency also increases gradually. Meanwhile, the larger the
, the shorter the frequency range corresponding to the sensitivity variation curve of the low frequency and the greater the maximum sensitivity. In the high-frequency range after the amplitude mutation, the sensitivity of the amplitude to
is negative. When
is constant, the sensitivity gradually increases and approaches zero as the frequency increases. When
increases gradually, the sensitivity slightly increases. Besides, the larger the
, the longer the frequency range, corresponding to the sensitivity variation curve of the high frequency and the smaller the minimum sensitivity.
It can be seen from
Figure 12, that in the low-frequency range before the amplitude mutation, the sensitivity of the amplitude to
is positive. When
is constant, the sensitivity decreases first and then increases as
increases. When
increases, the sensitivity under the same gas content also increases gradually. Meanwhile, the higher the
, the shorter the gas-content range corresponding to the sensitivity variation curve of the low frequency and the greater the maximum sensitivity. In the high-frequency range after the amplitude mutation, the sensitivity of the amplitude to
is negative. When
is constant, the sensitivity gradually increases and approaches zero as
increases. When
gradually increases, the sensitivity under the same gas content also increases. Besides, when the frequency is relatively low, the lower the
, the shorter the gas-content range, corresponding to the sensitivity variation curve of the high frequency and the smaller the maximum sensitivity.
When
is 2% and
is 3 N, the sensitivity variation curves of the steady-state amplitude
with respect to
and
are shown in
Figure 13 and
Figure 14, respectively.
It can be seen from
Figure 13 that in the low-frequency range before the amplitude mutation, the sensitivity of the amplitude to
is negative. When
is constant, the sensitivity gradually increases and approaches zero, as
increases. When
gradually increases, the sensitivity shows a stepwise distribution in the low-pressure region, which corresponds to the amplitude-variation tendency under low frequency and pressure, as shown in
Figure 9. In the high-pressure region, the sensitivity gradually increases and approaches zero. In the high-frequency range after the amplitude mutation, the sensitivity of the amplitude to
is positive. When
is constant, the sensitivity gradually decreases and approaches zero as
increases. When
gradually increases, the sensitivity under the same pressure gradually decreases. Besides, when the frequency is relatively low, the shorter the pressure variation range, corresponding to the sensitivity variation curve of the high frequency and the larger the maximum sensitivity.
It can be seen from
Figure 14 that in the low-frequency range before the amplitude mutation, the sensitivity of the amplitude to
is positive. When
is constant, the sensitivity increases first and then decreases as
increases. When
gradually increases, the sensitivity under the same frequency gradually decreases. Meanwhile, the larger the
, the longer the frequency range, corresponding to the sensitivity variation curve of the low frequency and the smaller the maximum sensitivity. In the high-frequency range after the amplitude mutation, the sensitivity of the amplitude to
is negative. When
is constant, the sensitivity gradually increases and approaches zero as
increases. When
gradually increases, the sensitivity under the same frequency gradually decreases. Besides, the larger the
, the shorter the frequency range, corresponding to the sensitivity variation curve of the high frequency and the smaller the maximum sensitivity.
When
is 46.8 Hz and
is 3 N, the sensitivity variation curves of the steady-state amplitude
with respect to
and
are shown in
Figure 15 and
Figure 16, respectively.
It can be seen from
Figure 15 that the sensitivity of the amplitude to
is negative. When
is constant, the sensitivity gradually increases with the increase of
. Meanwhile, the lower the
, the greater the change of the sensitivity variation curve. When
gradually increases, the sensitivity under the same gas content gradually increases.
It can be seen from
Figure 16 that the sensitivity of the amplitude to
is positive. When
is constant, the sensitivity gradually decreases as
increases. Besides, the larger the
, the greater the change of the sensitivity variation curve. When
gradually increases, the sensitivity under the same pressure gradually increases.
From the above sensitivity analysis, it is known that when the external excitation frequency variation range includes the mutation frequency, the sensitivity curve will mutate, i.e., a positive sensitivity value changes to be a negative one or a negative value becomes a positive one, which corresponds to the mutation in amplitude variation curve.
It can be seen from
Figure 11,
Figure 12,
Figure 13,
Figure 14,
Figure 15 and
Figure 16 that within the normal working range, the sensitivity of the jet system amplitude to
ranges from −43.62 mm to 519 mm, the sensitivity to
ranges from −3.34 mm/Hz to 4.39 mm/Hz, and the sensitivity to
ranges from −18.03 mm/MPa to 3.543 mm/MPa. Since the dimensions of the three variables are different, we cannot compare the sensitivity of the amplitude with respect to different factors. When the design parameters of the jet system vary within a given range, the degree of influence of the parameters on the amplitude can be analyzed according to the sensitivity variation under different parameters.
6. Conclusions
(1) During the operation of the adaptive fire-fighting monitor, the fluid spring stiffness changes dynamically with the gas content and pressure of the jet system, and the nonlinear effect of the fluid spring stiffness can be described by the Duffing equation with damping.
(2) Compared with the dynamic system composed of linear spring and linear damping, the soft spring characteristic of the fluid reduces the vibration amplitude of the jet system of the adaptive fire-fighting monitor at the equilibrium position, which to some extent, can weaken the vibration tendency of the spray core. When the external excitation frequency continuously changes, amplitude mutation occurs near the natural frequency of the corresponding linear system, and the amplitude of the jet system is large near the mutation frequency. Therefore, in the design of a fire-fighting jet system, the input shaft speed of the pump and the pulsation frequency of the output fluid should avoid the interval where the mutation happens.
(3) Under the corresponding design parameters and external excitation, the designed adaptive fire-fighting monitor always maintained single-cycle motion without multi-cycle, bifurcation, or chaos, which was consistent with the stroboscopic sampling results of the dynamics experiment, verifying the rationality of the design of the adaptive fire-fighting monitor.