BULLETIN OF THE TRANSILVANIA UNIVERSITY OF BRAŞOV
MATHEMATICAL MODELING OF A HYDRAULIC
SYSTEM EQUIPPED WITH A PIPE RUPTURE VALVE
O. Florea*
Abstract. In this paper is used the numerical simulation for mathematical
models of a hydraulic system with Simulink environments of Matlab kits. The
dynamic simulation has two objectives. One is to investigate the dynamic
characteristics of the pipe rupture valve in the set values, find and analyze the
existing problems, regulate the valve parameters, and optimize the design. The
other one is to compare the dynamic characteristics of the valve when the
parameters are different and analyze the influencing factors to the valve.
Keywords: hydraulic pump ,simulink
1. Introduction
The concept of an elevator is incredibly simple -- it's just a compartment attached to a
lifting system. There are two major elevator designs in common use today: hydraulic
elevators and roped elevators. Hydraulic elevator systems lift a car using a hydraulic ram, a
fluid-driven piston mounted inside a cylinder. You can see how this system works in the
diagram below.
Figure 1: Hydraulic system elevator
*
Transilvania University of Brasov, Faculty of Mathematics and Computer Science
154
Mathematical modeling of a hydraulic system equipped with a pipe rupture valve
The cylinder is connected to a fluid-pumping system (typically, hydraulic systems like this
use oil, but other incompressible fluids would also work). The hydraulic system has three
parts:
 A tank (the fluid reservoir)
 A pump, powered by an electric motor
 A valve between the cylinder and the reservoir
The pump forces fluid from the tank into a pipe leading to the cylinder. When the valve is
opened, the pressurized fluid will take the path of least resistance and return to the fluid
reservoir. But when the valve is closed, the pressurized fluid has nowhere to go except into
the cylinder. As the fluid collects in the cylinder, it pushes the piston up, lifting the elevator
car.
In the hydraulic elevator is used the pipe rupture valve for automatic change of the flow rate.
This is installed near the entrance in the hydraulic cylinder. In ordinary conditions this has no
effect, but in case of accidents when the elevator fall quickly, the flow rate through the pipe
rupture valve grow up. When the pressure value of failure excel the initial value, the valve
shut off quickly and the lift stop from the falling, guarantee the safety of passengers. After
the damage is eliminated, the force through the pump and the spring force act together for
open valve and make that the system works normally.
2. Analysis and simulation of system
The rupture valve model is mainly applied in the elevator hydraulic system; so that the
dynamical characteristics of the hydraulic system must be take in consideration, and the
factors which effects are insignificant will be neglect during the simulation model. The
simulation principle of the rupture valve is presented in the figure 2; this model presents the
application of the rupture valve in the elevator hydraulic system.
Figure 2: Diagram of rupture valve of hydraulic elevator
Bulletin of the Transilvania University of Braşov • Vol.13(48) - 2006
155
2.1 Mathematical Model
The equation of pressure is:
pt  p p  K f (qVc  qVs ) 2
(1)
Here: p t is the pressure between the out port of the pump and the admission port of the
rupture valve, p p is the pressure in the out port of the pump, K f is the friction coefficient,
qVc is the flow rate through the valve, qVs is the flow rate through the chamber. The chamber
by the back rupture valve is closed so the equation of continuity is:
Vs dp s
dx
 As
 qVs  C s ( p s  pt )
E dt
dt
(2)
Here: Vs is the volume from the back of chamber, E is the spring bulk modulus of the
dx
is the motion of the
dt
valve, C s is the trickling coefficient through the back cavity and the valve’s body, p s is the
hydraulic oil in the back chamber, As is the section of the valve,
pressure in the back chamber.
The pressure difference through the rupture valve is making by the flow through the pipe and
the orifice. The differential equation of the flow rate and of the pressure is:
2
p c  pt  K d qVc


2
2
qVc
1
C A22
2
2
(3)
Here: pc is the pressure in valve, K d is the flow rate coefficient,  is the hydraulic density
of oil, C2 is the debit coefficient through the valve’s port, A2 is the valve’s section.
The equilibrium equation of the valve is:
m
d 2x
 Fp  F1  F2  F f  Fs  FRv
dt 2
(4)
Where,

 F p  ( p c  p s ) As , F1  K 1 ( x0  x)( p c  pt )

dx

p c  pt , F f   K f ( p s  pt )
(5)
 F2  K 2
dt

dx

 Fs  ( F0  K s x), FRv   Bs dt
Here, m is the spool mass, Fp is the force produced by the pressure differential between both
ends of the valve spool; F1 is the stabilization hydraulic force, F2 is the dynamic hydraulic
force, F f is the friction force, FRv is the resistance viscous force, x0 initial displacement,
156
Mathematical modeling of a hydraulic system equipped with a pipe rupture valve
x is the spool displacement, F0 is elastic force, K s is the elastic coefficient, Bs is the
resistance coefficient of the spool’s valve.
Between the hydraulic cylinder and the rupture valve is a closed chamber. The equation of
continuity of the flow rate is:
Vc dpc
dy
 Ac
 qVc  Cv ( pc  pt )  Ct pc
Ec dt
dt
(6)
Here, Vc is the volume of the closed chamber, Ec is the spring modulus of the hydraulic oil,
dy
is the speed of the piston in the hydraulic cylinder, C v is the flow coefficient between
dt
the spool element of the valve and the body of the valve and C t is the flow coefficient
through the outer of the hydraulic cylinder.
In this model the positive direction is in the sense of the lift descend. The equation of the
equilibrium in the cylinders is:
d2y
dy
 mc g  pc Ac  Bc
 Fc
2
dt
dt
Here, Bc is the viscous damping coefficient and Fc is the friction force.
mc
(7)
Equations (1) – (7) are the mathematical models of the pipe rupture valve in the hydraulic
elevator system.
The analogical networks of numerical simulation attached to these equations are according
with the null initial conditions and can describe transitional phenomena of general amplitude.
The hydraulic lift system is presented in the network form figure 3, which contains six parts
corresponding to the six equations describing the mathematic model.
Figure 3: Simulation of hydraulic elevator system
Bulletin of the Transilvania University of Braşov • Vol.13(48) - 2006
157
The movement of the hydraulic elevator is presented in figure 4
Figure 4: The evolution of the hydraulic elevator movement.
Refereneces
1. Vasiliu N., Vasiliu D., :Actionari hidraulice si prneumatice, vol 1,Ed. Tehnica, 2005
2. Wesseling, P.: Principles of Computational Fluid Dynamics, Ed. Springer, 2001
3. Florea O., : Modelul matematic al unor probleme de dinamica a sistemelor
hidraulice de urmărire, International Conference Nav-Mar-Edu, ISBN: 973-
8303-54-0
4. http://home.howstuffworks.com/
Modelul matematic al unui sistem hidraulic echipat cu o conducta cu
valva perforata
Rezumat. In lucrare este utilizata simularea numerica pentru modele
matematice ale sistemelor hidraulice folosind mediul Simulink din Matlab.
Simularea dinamica are doua obiective: unul este de a investiga caracteristicile
dinamice ale conductei cu valva perforata, gasirea si analizarea problemelor
existente, reglarea parametrilor valvei si optimizarea designului, alta este de a
compara caracteristicile dinemice ale valvei cand parametrii sunt diferiti si
analizarea influentei factorilor asupra valvei.
Keywords: pompa hidraulica, simulink