Anti-Vibration

Anti-vibration control of turntable ladders is effective by controlling the steel ropes inserted in the hollow handrails. Previous studies that considered idealized conditions, such as ignoring friction, piston mass, and hydraulic oil compressibility, can not be strongly generalized. This study addresses these problems by describing in detail the equipment serving the anti-vibration solution, building a mathematical model for the steel rope hydraulic control system considering the above factors, and controlling the rapid extinguishing of vibrations on a model simulated in Matlab-Simulink. Furthermore, the simulation of the relationship between the control signal and oil flow through the proportional distribution valve produced a signal flow curve that was asymptotic to the actual. The results showed a difference compared to previous studies. Although the solution was only implemented on the lowest ladder section with the most unfavorable conditions, the time for vibration on the top ladder reached amplitudes of 3 and 5mm in 3 and 5s for empty and full baskets, respectively.


INTRODUCTION
The study on vibration reduction, fast vibration quenching, and anti-swing for load transport equipment has high scientific and practical significance.Various control methods have been proposed to bring the load to the desired position quickly and accurately [1][2][3][4].This activity makes more sense for turntable ladders because they transport rescued people and should help them feel safe in a rescue basket.Authors in [5][6][7] presented effective control methods to eliminate vibrations on turntable ladders.In [8][9][10][11], a new solution was presented to eliminate vibrations and overcome large static displacements on ladders, putting steel ropes inside the hollow of the handrails.These steel ropes were tensioned and controlled by small hydraulic cylinders mounted on the handrail tops, using the available hydraulic power of the truck.This method showed that the tension of the steel ropes had a positive effect on the antivibration on the top ladder [8] and the control of the steel ropes rapidly reduced the vibrations on the ladder [9].In [10], stress reduction on key ladder elements and significant reduction of static displacements were demonstrated by evaluating the ladder structure with additional steel ropes according to the standards [12][13].The hydraulic circuit design and cylinder control according to the control principle [9] were presented in [11].However, the control system model in [11] was not very accurate and comprehensive, as it did not consider the elasticity of hydraulic oil, ignored friction forces and the mass of the piston rod, and used approximate mathematical models for the flow-signal curve.This study investigated the detailed structure of the anti-vibration device, the dynamic model of the steel rope hydraulic control system, the establishment of a mathematical model of the system with an accurate flow-signal curve, and the practice of controls for rapid suppression of vibrations on turntable ladders.

II. STEEL ROPE-HYDRAULIC CONTROL SYSTEM
Figure 1 shows the structure of the devices attached to the hollow handrails of a ladder, omitting the ladder structure.The hydraulic cylinder is integrated with the position sensor mounted on the lower end of the handrail.Added components into a hollow handrail.
A threaded joint connects the piston rod to one end of the steel rope.The steel rope with a high damping coefficient is threaded in the handrail and supported by supports.The supports are lightweight plastic evenly spaced to eliminate the rope sag.The other end of the rope is fixed by a threaded joint to the other end of the ladder.Additional devices were implemented in the lowest ladder section for anti-vibration to match the size of the 30m set including the three ladder sections described in [14].The control on the third ladder is the most effective [9].The most effective vibration control is when placing the actuator closest to the fixed end of a cantilever beam [15].Figure 2 shows the mechanical model of the ladder structure, the steel rope, and the force generated by the cylinder.The control system commands the actuator to generate a force to reduce vibration.This force moves with a variable velocity rule that follows or nearly follows a predefined rule.Based on the data in [8][9][10][11], all parameters in this model were entirely determined.Multi-body dynamics model of ladder set and steel rope in hollow handrails.
Figure 3 shows the model of the steel rope hydraulic control system.During the operation of the proportional directional valve, when the P-A ports are connected, the highpressure hydraulic oil pushes the piston to the left to pull the steel rope.In contrast, when the A-T ports are connected, the oil in the cylinder chamber flows to the tank through the pressure sequence valve due to the tension of the rope pulling the piston.In this model, k c3 and c c3 are the stiffness and damping coefficients of the ropes in the third ladder section, A p , p p are the area and the pressure of the oil chamber, and x p and x r are the displacements of the piston and the remaining fixed end of the steel rope, respectively.III.EQUATION OF MOVEMENT According to [9], during the vibration control performance, the acceleration vector of the addition motion is determined as: where , , and are the acceleration, velocity, and displacement vectors of the addition motion of the rigid bodies, M(t), C(t), and K(t) are the mass matrix, the damping matrix, and the stiffness matrix, D is the load matrix including T i (bending moment) and D i (damping load) caused by the steel rope in the i-th ladder section, F is the load matrix excluding the load components caused by the steel rope, and = ( )/ with F(t) being the total generalized load matrix.
where k T3 and k T3 are the controlling factors corresponding to T 3 and D 3 in the third ladder section, respectively.The motion equation of the piston is given by: where () '*" , () '*" represent the length change range and the length change speed of the steel rope, F f is the friction force, m p is the piston mass, and .% is the piston acceleration.The exact determination of the friction force is very complicated and it is necessary to conduct experiments on each specific object.To simplify the simulation, it was determined as follows [16]: for .% > 9 : for − .: ≤ .% ≤ 9 : where v 0 is the velocity whose value is a very small positive around zero, F s =10 5 A p , and c p =2%.The change range of the steel rope length during work is given by: () '*" = .% + .%: − .B (5) and the length change speed of the steel rope is given by: () '*" = .% − .B (6) with x r being the longitudinal displacement of the upper end of the handrail.The dynamic equation for the oil compressibility inside the cylinder chamber can be written as [17]: where $ % denotes the change rate of the chamber pressure, I J is the oil bulk modulus, V c denotes the oil volume, and Q x denotes the rate of flow entering the chamber.
where x 0 is the oil length in the cylinder chamber when it is set to create the initial tension load, x p is the displacement of the piston, and V OC is the oil volume in the pipe.
Replacing ( 8) and ( 9) in (7) gives: The flow rate depends on the pressure drop which is the pressure difference between the inlet and outlet ports and the corresponding nominal flow.This is detailed in: Q% SDT (11) where Δp Nom is the pressure drop when the flow rate is nominal, and the command signal is maximum, Δp Nom =5×10 5 Pa, Δp x is the pressure drop when working, and G MJN * is the flow rate depending on the command signal.G MJN * is determined as: where u=U/U max , U is the control voltage, U max is the maximum control voltage, and Q Nom is the nominal flow at U max .
Equation ( 10) is rewritten in the case of the piston pulling the steel rope: and in the case of the steel rope pulling the piston: (14) with p oh and p ou being the pressure at P and T ports, respectively.

IV. ANTI-VIBRATION CONTROL
Based on the parameters and the original characteristic curve of the proportional directional valve presented in [18], to simplify the numerical simulation process in its half corresponding to the positive signal, ( 12) can be approximated as: < V ≤ 1 (15) Figure 4 shows the original flow curve and the approximate curve generated by (15).They almost coincide.The steel rope hydraulic control system model was established in Matlab-Simulink, as shown in Figure 5.The main parts of this model are:  Ladder structure model: a subsystem created from the ladder dynamic model in Matlab-Simulink, as presented in [9].
 Signal generator: The velocity sensor at the top of the third ladder section provides a signal to the control signal generator, which is the original signal that is smoothed and amplified before reaching the valve.As the original signal is taken from the ladder structure model, the phase delay of the control signal must be considered.Here, the tensioner model starts 30ms after the ladder model.A delay time of 4ms was selected.When the amplitude of the vibration at the top ladder is small enough, the signal generator will stop working.This amplitude value can be preset.
 Function blocks of the hydraulic devices: They determine the dynamic parameters of the piston and the oil pressure in the chamber through (3) and ( 13)- (15).
 Damping load block: It receives the piston and rope end velocity signals and creates the damping force (+ '" () '*" ).
 Rope tension block: It has the function of generating rope tension ( '" () '*" ) based on the longitudinal deformation of the steel rope.
Table I shows the modeling parameters.The largest vertical vibration at the top ladder may occur when the ladder is horizontal and fully extended.Therefore, this study implements anti-vibration control in this ladder state.Figures 6-12 show the parameters obtained during the control process, including control signal, oil flow, oil pressure, friction force, piston speed and displacement, and damping force in the empty basket case.Figure 13 shows the displacement graphs on the top ladder for both basket cases.Despite the control signal's late start and phase delay, the vibration control through the steel rope hydraulic control system was highly effective.Compared to [9] and [11], these results overcome the expected effect after control.In the empty basket case, the vibration amplitude after only 3s of control reaches 3mm for the top ladder instead of 67mm.The control systems in the other studies need more than 8s to achieve this amplitude.In case of a full basket, the vibration amplitude of the top ladder after 5s of control reaches 5mm compared to the others that need more than 14s to achieve such amplitude.In addition to the steel rope having the effect of damping, the hydraulic equipment also suppresses the vibration when operating.The pressure and flow rate values for the cylinders presented in Figures 7-8 are suitable for the hydraulic system of a turntable ladder.The speed and displacement of the piston shown in Figures 10-11 are also consistent with the kinematic parameters of a piston used to control a precise motion system.V. CONCLUSION The paper presented a new approach to anti-vibration methods for turntable ladders.It also shows the feasibility of carrying out the novel control solution presented in [9] with available devices in the market.A steel rope hydraulic system dynamic model, including tension and damping load of steel the ropes, friction force between the piston and the cylinder, oil compressibility, piston mass, and the proportional directional valve characteristics, was established and simulated.This model was combined with the ladder set's multi-body dynamics model to control the vibration's rapid quenching at the top ladder with a unique signal generator.Friction, oil compressibility, and the inertia force of the piston significantly influence the damping of the ladder's vibrations.This model extinguished vibrations by controlling the steel rope hydraulic system more efficiently, controlling only the steel ropes.The vibration reduction rate was approximately three times faster than in other studies.

Fig. 2 .
Fig. 2.Multi-body dynamics model of ladder set and steel rope in hollow handrails.