Research of influencing factors on tripping speed of double plungers turbine overspeed protector

The over-speed protection mechanism is very important for the safe operation of the steam turbine, but in actual production, the tripping speed often fluctuates and deviates from the set value, resulting in a decrease in the reliability of the unit operation. In this paper, the factors that affect the tripping speed of double plungers turbine overspeed protection mechanism are analysed, the mechanism of mechanical vibration and friction that cause the tripping speed to fluctuate is explored through theoretical calculation and simulation analysis, and the change of spring dynamic stiffness by using the dynamic stiffness matrix method and finite element simulation is analysed. The change of friction force under different contact modes is calculated by using the L-N contact force model and the modified Coulomb friction model, and the change of tripping speed under non-ideal friction conditions is explored through dynamic simulation.


Introduction
The double-plungers type overspeed protection device is a type of mechanical emergency interrupter.It is widely used in the speed control system of the turbine [1].Its basic structure is shown in Figure 1(a), and the working principle of the protection device is shown in Figure 1 When the spindle speed is too high, the plunger in the device will overcome the spring constraint and protrude outward radially along the spindle under the action of centrifugal force, and collide with the gate lever, triggering subsequent actions and eventually forcing the spindle to stop.Compared with the traditional single-plunger protector, the double-plunger overspeed protection can achieve higher speed control accuracy, has a more stable operating speed at high speeds, and has higher reliability in long-term use.However, in actual use, the actual operating speed often deviates from the theoretical speed to a certain extent, and different tripping speeds will appear in multiple tests.The tripping speed will macroscopically show irregular fluctuations within a certain range [2].
During the operation of the spindle, as the spindle speed increases, the overall vibration frequency of the device will also increase.Compared with the static state, the transmission stiffness of the spring will change under high-frequency vibration, and the binding force on the inner plunger will also change.The relative theoretical value deviates, which further causes the tripping speed of the inner plunger to deviate from the theoretical value.In addition to the change in the stiffness of the inner spring, the friction between the inner and outer plungers is different each time the trip occurs, which directly leads to unstable resistance when the inner plunger protrudes and causes fluctuations in the tripping speed.Therefore, friction is also an important factor affecting the tripping speed of the device.
The object of this paper is to explore the change law of spring dynamic stiffness and determine the influence of spring stiffness change on tripping speed under high-speed alternating force, and lay the foundation for further determining the dynamic characteristics of double-plungers protector during high-speed operation by studying the change mode of the friction force of the inner and outer plungers and exploring the fluctuation of the friction force under high-speed rotation.

Spring dynamic stiffness matrix
In order to use the equation to describe the cylindrical helical spring, the global coordinate system xyz and the local coordinate system uvw are respectively defined as shown in Figure 2.
where α is the lift angle of the spring, and φ is the rotation angle of the local coordinate system on the Z axis relative to the initial position.At the same time, the curvature κ and tortuosity τ of the curve are respectively defined as: According to the Frenet formula, the partial derivatives of the three base vectors in the local coordinate system to the arc length of the spring center curve can be obtained [3]: When the spring is compressed [4], the force F and the moment M on any section can be expressed as component forces F u , F v , F w and component moments as M u , M v , M w in three directions.The displacement δ and torsion θ of the section can be expressed as displacement δ u , δ v , δ w and rotation angle θ u , θ v , θ w in three directions, and it can be expressed as a matrix: This equation is deviated by the length measured along the helix s yields: When the spring is compressed by force, there is both bending deformation and shear strain in each part.Compared with before, the crosssection is no longer perpendicular to the neutral axis.Therefore, the Timoshenko beam theory is used to analyse the force state of the spring when it is compressed [5].At this time, the relationship between the force and deformation of each part of the spring can be written as: where G is the shear modulus, E is the elastic modulus, γ is the cross-sectional area correction coefficient, A is the cross-sectional area, I u and I v are the moments of inertia of the cross-section about the u and v axes, and J is the St. Equations ( 3) and ( 6) are substituted into Equation ( 5) and it is then rearranged, according to D'Alembert's principle: The last term in Equation ( 7) is the inertia force of the spring, which is opposite to the direction of the spring force, where:

S
The fluctuations inside the spring vary at different times and locations, where the temporal variation is determined by the excitation frequency ω [6], and the spatial distribution is determined by the wave number k in the spring.When the spring is subjected to sinusoidal vibration during operation, it produces a simple harmonic response, and the force and deformation relationship of the spring under vibration conditions can be expressed as: By substituting Equation (8) into Equation (7), the Equation ( 9) is obtained: For a set of Equation ( 9), in order to make the force and deformation of the spring have a non-zero solution, after the vibration frequency ω is given, the eigenvalue k i and its eigenvector [φ] of k[I]-s are calculated.According to the D'Alembert Principle, the relationship between spring force and deformation in a steady state is as follows: θ θ (10) The deformation at any point on the spring can be regarded as the superposition of multiple response waves [7].
where a i is the amplitude of each wave.
Taking the partial derivative with respect to the arc length s: By substituting Equations ( 11) and (12) into Equation ( 10), the deformation and force at the two ends of the spring at s = 0 and s = L can be expressed as: Then the dynamic stiffness matrix of the spring can be expressed as: -1 2 1

 K D D (14)
The dynamic stiffness matrix in the local coordinate system is converted to the global coordinate system and Equation ( 15) is obtained:

Simulation of frequency variation characteristic
The finite element method can be used to construct a mechanical analysis model of the spring, and the frequency-changing characteristics of the spring under high-frequency load excitation can be verified through simulation calculations.Through harmonic response analysis, the steady-state response of the structure under sinusoidal loads of different frequencies can be studied.Since the changing frequency of the centrifugal force on the spring gradually increases with the increase of the spindle rotation speed, the transient vibration of the spring structure when the excitation is generated is not considered.The inner spring is in a compressed state in the outer plunger cavity, so in the static analysis module, the pre-compression of the striking end of the spring relative to the adjustment end is used as the preset initial condition, and the boundary condition is set to pre-compression (displacement) at one end and fixation (fix support) at another end.On this basis, linear static analysis is first performed to determine the stress of the spring in the compression state.Afterward, the load amplitude, frequency, and load type in the frequency sweep analysis are defined according to the mass of the inner plunger and the operating speed of the spindle [8].Since acceleration is not supported as a load type in the harmonic response analysis, the calculated force of the inner spring is used as the basis here, and the pressure load is directly applied to its end face.In order to speed up the calculation, the modal superposition method is used to solve the calculation results of the harmonic response analysis.The tripping speed of the double plungers overspeed protector generally fluctuates between 9,000 and 10,000 revolutions in actual operation, so the frequency corresponding to this speed range is taken as the frequency range of harmonic response analysis.According to the frequency response in the calculation results, the force and deformation amplitude of the spring under different frequency loads can be obtained.As shown in Figure 3, under different frequency excitations, the spring shows different transmission stiffness, and the transmission stiffness of the inner spring shows a downward trend.As shown in Figure 4, the obtained curve is fitted and the corresponding relationship between spring stiffness and frequency is transformed into the following formula: According to the simulation results, the spring stiffness change within the tripping speed is calculated, and the spring stiffness considering the frequency variation characteristics is substituted into the calculation in the theoretical formula of the tripping speed, and finally, the theoretical tripping speed that is more practical can be obtained.

Theoretical friction model
What occurs between the inner and outer plungers is a cylindrical pair contact, but it is similar to the linear displacement of the slider mechanism on the guide rail.Since there is a gap between the inner and outer plungers, as shown in Figure 5, when the inner plunger moves linearly within the outer plunger's cavity, actual contact may occur in the following three situations: As shown in Figure 6, the local coordinate system uvw fixedly connected to the inner plunger and the global coordinate system xyz fixedly connected to the outer plunger are respectively established, where ABCD is the four corner points of the inner plunger on the two-dimensional section, A'B'C'D' is the nearest potential contact point on the outer plunger, r i is the position vector of the center of mass of the inner plunger in the global coordinate system, r i A and s i A are the positions vector of point A in the global coordinate system and the local coordinate system.
where n and t are the unit vectors along the x-axis and y-axis respectively.The size of δ is:

MATMA-2023
The force exerted on the inner plunger can be decomposed into F t along the x-axis and F n along the y-axis.The L-N contact force model takes into account the energy dissipation caused by material damping during the collision and the elastic effects of different materials, is suitable for low-speed collisions between the inner and outer plungers.The contact force formula of this model is expressed as follows: where K is the stiffness coefficient of the contact body, ε is the relative collision velocity, & is the initial relative velocity of the collision point, &-represents the degree of nonlinearity of the contact, usually 1~1.5, e is the recovery coefficient of the collision body, and K is the stiffness coefficient, which can be expressed for: where is the Poisson's ratio of the material, and E i  i  1, 2 is the elastic modulus of the material.
The friction between the inner and outer plungers is a continuous dry friction, which is a static friction problem related to relative velocity and positive pressure.For dry friction with a small relative velocity, the modified one-dimensional linear Coulomb friction model is used to calculate the friction between inner and outer plungers [9]: where μ d is the friction coefficient, c d is the friction correction coefficient corresponding to the relative speed, in the linear friction block model [10]: The friction curve of this model is shown in Figure 7.

Simulation analysis on the impact of friction
In the unstable environment of high-frequency vibration, different contact modes may be generated between the inner and outer plungers, and the input parameters of the friction force calculation model will be different, which causes the friction force to fluctuate in a large range, and eventually become the cause of an important factor in tripping speed fluctuations.In order to explore the influence of friction force on the final tripping speed of the inner plunger, the ADAMS dynamic simulation model is reconstructed by taking friction force as a change condition.In the ADAMS analysis environment, if the kinematic relationship between the inner and outer plungers is directly defined as a free contact state with a gap, the under-constrained state between the elastic bodies will make it difficult to solve the dynamic equations, and the system cannot be calculated.Therefore, after setting the fit gap between the inner and the outer plungers, the friction resistance function needs to be input into the model in the form of motion conditions based on the friction calculation model above, and the contact stiffness, maximum collision depth, maximum damping, and contact nonlinear coefficient are set [11].A rotary drive is added to the device, the analysis type is set to dynamic analysis, the integration step is set to 3000, and the time is set to 0.2 s.Multibody dynamics analysis and calculation are performed on the inner and outer plunger linkage module, and the calculation is obtained under the new friction condition.The result is shown in Figure 8.  2) The L-N contact force calculation model and Coulomb friction model were used to derive the friction calculation method of the inner and outer plungers when they move in a straight line, and the friction force was input into the ADAMS simulation model as a force function, and the inner and the outer plungers were solved through dynamic simulation analysis.
3) The main factors affecting the tripping speed of the linkage plunger and their mechanism of action are analysed, which provides a basis for further revising the tripping speed calculation model.

Figure 1 .
Figure 1.(a): structure of double plungers (b): working principle of the protection device.

Figure 2 .
Figure 2. Clarification of coordinates.The conversion relationship between these two coordinate systems is shown in Equation (1).0 cos sin cos sin sin sin cos [ ] sin cos sin cos cos

Figure 3 .
Figure 3. Transmission stiffness of the inner spring.

Figure 5 .
Figure 5. Contact modes between inner and outer plungers.

Figure 6 .
Figure 6.Contact model of linear motion joint.The depth of the inner plunger edge embedded in the outer plunger surface δ can be obtained as: [0,1] [1,0]

1 )
The stiffness change pattern of the spring in double plungers overspeed protector under the action of the high-frequency alternating load was analysed through the stiffness matrix method, and its frequency-changing characteristics were simulated, analysed, and fitted.MATMA-2023Journal of Physics: Conference Series 2691 (2024) 012016 IOP Publishing doi:10.1088/1742-6596/2691/1/01201610