# TMHP51 - Hydraulic Servo Systems

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```					LIU/IEI/FluMeS                2007-10-26        1 (20)
Exercises for TMHP51

Collection of Exercises

for the course

TMHP51 – Hydraulic Servo Systems

LIU / IEI / FluMeS
2007-10-26
LIU/IEI/FluMeS                     2007-10-26                                   2 (20)
Exercises for TMHP51

Chapter 2 - Orifices

Example 2.1

In a hydraulic system the pressure p1=5 MPa is to be reduced by orifices to
p2=0,2 MPa at the flow q=0,333⋅10-3 m3/s.
Calculate the number of orifices needed to avoid cavitation in any place.
Calculate also the diameters for the different orifices. The orifices are
assumed as sharp edged with the flow coefficient Cq=0,81. Density of the oil
is 880 kg/m3.

Example 2.3

The figure shows a flapper-nozzle system with the supply pressure p1.
Describe schematically in a diagram how the pressure ratio p2/p1 varies
according to the displacement x.
dp2
Investigate at which pressure ratio the sensitivity,               will reach its
dx
maximum value. Assume turbulent flow through the orifice.

d
The flapper-nozzle is designed for x0 <      and the orifice area is A = π ⋅ d ⋅ x .
16
LIU/IEI/FluMeS                     2007-10-26                            3 (20)
Exercises for TMHP51

Chapter 3 – Flow forces

Example 3.1

The figure schematically shows the working of a constant flow valve. For
the flow through the variable orifice the following relation ship can be
established:
2( p 2 − p 3 )
q = As C q                    .
ρ
In a working point the orifice configuration gives that AsCq=9,6⋅10-6.
Data:
d1    =      20 mm                      p1     =     20⋅106 N/m2
Ff    =      600 N                      d2     =     40 mm
3
ρ     =      880 kg/m .

Determine
a)     the magnitude of the constant flow, if no consideration is taken to the
flow forces acting on the valve spool.

b)     the magnitude of the constant flow, if the action of the flow forces
acting on the valve spool are concidered. As the magnitude of the
flow forces has to be calculated the flow calculated in a) can be used.
Determine the flow at p3 = 2 MPa, p3 = 8 MPa and p3 = 15 MPa.
LIU/IEI/FluMeS                      2007-10-26                               4 (20)
Exercises for TMHP51

Example 3.3

The figure illustrates an open-center valve with different area gradients for
the different orifice locations.
Calculate the total flow force for the case when the valve spool displace-
ment, xv = l = 4 mm (l = L2 - L1), F=0 and A2/A1=1,0.

Data:
Cq       =   0,7                        qs    =    2,0⋅10-3 m/s
d        =   0,02 mm                    ρ     =    870 kg/m3.
LIU/IEI/FluMeS                           2007-10-26                                      5 (20)
Exercises for TMHP51

Chapter 5 – Component dynamics

Example 5.10
A valve controlled cylinder with lift load and without meter-out-restriction to tank is
loaded with a mass Mt and extern load FL. Its viscous friction is Bp. The valve has zero
over and under lap. The cylinder is supplied with a constant pressure ps.

a) Derive the linearised and Laplace transformed equations needed for describing the
dynamic behaviour of the system. Derive also the transfer function of the system
sΔX p (s )
b) Compare the open system transfer function,             , with the mass-spring system
ΔX v (s )
below and identify k0, b0 and m0.

FL (s)
c) Determine the stiffness of the system in 5.10 a, S(s) =           . Investigate what will
X p (s)
happen if Bp = Cip = Cp = 0, i.e. the total damping of the system is zero. Assume that
BpKce<<Ap2.
LIU/IEI/FluMeS                          2007-10-26                                  6 (20)
Exercises for TMHP51

d) Increase the damping with a leakage orifice, KsL. How will the leakage paths KsL, the
flow between the two cylinder volumes, Cep and the flow between the cylinder and
the free air, Cip, affect the steady state characteristics of the system?

e) Increase the damping with a pressure feedback to in this case the hydraulic controlled
manoeuvre valve. Assume BpKce<<Ap2 and that the feedback is stable. Neglect the
mass and the friction of the spool. How will the damping affect the stiffness? Is it
possible to use the benefits but avoiding the drawbacks from pressure feedback?
How?
LIU/IEI/FluMeS                          2007-10-26                                      7 (20)
Exercises for TMHP51

Example 5.12
A servo valve-controlled symetric cylinder is studied for positive valve displacements
only. Neglect the inner and outer leakage of the cylinder.

a) Derive the transfer function of the open system, by means of block-diagrams for the
system. Define the load pressure pl = p1-p2.

sΔX p (s )
b) Compare the open system transfer function,                 , with the mass-spring system
ΔX v (s )
below and identify k0, b0 and m0.
LIU/IEI/FluMeS                           2007-10-26                                     8 (20)
Exercises for TMHP51

Example 5.13
A symmetric hydraulic cylinder is loaded with two masses, M1 and M2 .The first mass is
attached to the cylinder piston and the other one is attached to the first mass via a spring
and a viscous damper, according to figure:

When the cylinder is treated as friction free, the transfer function for the mecanical part
of the load can be written as:

Show qualitatively, how Gm(s) is affected if the viscous friction of the cylinder (Bp) is
taken into consideration.
LIU/IEI/FluMeS                            2007-10-26                                  9 (20)
Exercises for TMHP51

Chapter 6 – Servo systems

Example 6.1

A hydraulic position servo system has the following data:
Cq      =       0,61                          Ap     =       4,40⋅10-3 m2
w       =       1,00⋅10-2                     L      =       0,15 m
-12  5
Kc0     =       2,1⋅10 m /Ns                  Vt     =       0,72⋅10-3 m3
xvmax =         1,00⋅10-3 m                   Bpmax =        3500 Ns/m
Ctp    =       0 m5/Ns
Mtmin =        3,0 kg

System parameters
ps    =      21,0 MPa
ρ     =      855 kg/m3
βe    =      1,2⋅109 N/m2
b     =      a

a) Determine the stationary prestanda limits of the servo, i.e. draw the piston velocity as
a function of Fl.

b) What is the power loss in the valve as Fl=0 and piston velocity=0?

c) Draw the Bode-diagram and determine the amplitude margin as Mt=Mtmin.

d) Calculate the static stiffness.
LIU/IEI/FluMeS                           2007-10-26                                   10 (20)
Exercises for TMHP51

Example 6.2
The position servo from problem 6.1 is to be used with the mass Mt=100 kg.

a) How will the increased mass affect resonance frequency, ωh, damping δh and
amplitude margin Am?
Implement a laminar restrictor between the load ports, so that the amplitude margin is
10 dB.

b) Determine the stationary performance limits, i.e. draw vp as a function of FL. What is
the maximal load that can be controlled by the servo? What is the power loss at this

c) What is the magnitude of the power loss at FL=0? If we use a underlapped valve with
the same properties around xv=0 instead of the valve plus the orifice from a), what
would the power loss be at FL=0?

d) Determine the stationary closed loop stiffness, S(s), with the load port restrictor and
with the underlapped valve respectively.
LIU/IEI/FluMeS                           2007-10-26                               11 (20)
Exercises for TMHP51

Example 6.3
The servo from problem 6.1 is provided with a dynamic load pressure feedback, see
figure:

a) Dimension the pressure feedback (spring rate kd and piston area Ad) so that the
stability margin will be 10 dB as the mass is Mt=100kg. The mechanical link ratio c/d
= 1,0.

b) How are the stationary performance limits for the servo affected by the feedback?

c) Determine the power loss for the valve when FL=0.

d) Calculate the stationary closed loop stiffness for the servo.
LIU/IEI/FluMeS                           2007-10-26                                12 (20)
Exercises for TMHP51

Example 6.4
The servo from problem 6.1 is provided with a dynamic load pressure feedback, see
figure below:

a) Dimension the pressure feedback (orifice constant Kcd, spring rate kd and piston area
Ad, so that the stability margin will be 10 dB as the mass is Mt = 100kg.

b) How are the stationary performance limits for the servo affected by the feedback?

c) Determine the power loss for the valve when FL = 0.

d) Calculate the stationary closed loop stiffness for the servo.
LIU/IEI/FluMeS                              2007-10-26                             13 (20)
Exercises for TMHP51

Example 6.9
The figure illustrates a valve controlled motor. The motor must be able to operate within
the operational area (TL(nm)) defined below. Choose the correct pump displacement (Dp),
motor displacement (Dm), servo valve size (flow at 7MPa pressure drop and fully
displaced valve) and pressure relief valve size (flow at 21MPa pressure drop).

Data:
Pump sizes available:                    50, 75, 100, 125 cm3/rev
Motor sizes available:                   15, 25, 35, 45 cm3/rev
Pressure relief valve sizes:             1.5, 2, 2.5, 3 l/s at 21MPa
Servo valve sizes:                       1, 1.5, 2, 2.5 l/s at 7MPa
ηvp= ηhmp= ηvm= ηhmm= 0.95
np = 1500 rpm
ps = 21MPa
LIU/IEI/FluMeS                                 2007-10-26                           14 (20)
Exercises for TMHP51

Example 6.12

A velocity servo is shown in the figure above. The servo valve has an integrator, se block
diagram.

a) Determine maximum value of the gain Kes if amplitude margin is to be 7 dB. After
that, calculate also the static loop gain, Kv.
T                                &
b) Determine L in the operational point θ m = 100 rad/s and pL = 5,0 MPa.
&
θ    m stat

c) Calculate the stationary velocity error at constant acceleration &&m 0 .
θ

Data   Dm        =            3,2⋅10-6 m3/rad       βe     =     1,2⋅109 N/m2
Jt        =            0.015 kgm2            Kt     =     0,05 Vs/rad
Vt        =            5,0⋅10-4 m3           &&
Bm        =            0                     ps     =     14 MPa
G         =            0                     Cq     =     0,61
Kq0       =            0,62 m2/s             w      =     8,0⋅10-3 m
Kce0      =            4,0⋅10-12 m5/Ns
LIU/IEI/FluMeS                         2007-10-26                                 15 (20)
Exercises for TMHP51

Example 6.13

The figure above illustrates a pump controlled motor. The stroking unit (displacement
arrangement) has a limited speed:

Kt
φ=          e
1 + τs

where φ is the pump displacement angle. Other parameters are:

Ct        =    1,3⋅10-11 m5/Ns                   V0   =    2⋅10-4 m3
Dm        =    30⋅10-6 m3/rad                    Jt   =    0,1 kgm2
TL        =    100 Nm                            βe   =    1,2⋅109 Pa
Bm        =    0                                  τ   =    0,1 s

a)     Determine required loop gain, Kv, so that stationary velocity error is maximum

b)     How large will the amplitude margin of the system be with loop gain Kv
according to problem a)?
LIU/IEI/FluMeS                          2007-10-26                                    16 (20)
Exercises for TMHP51

Chapter 9 – Hydrostatic transmissions

Example 9.1
A hydraulic system consists of a pump with variable displacement Dp and a fixed motor
with displacement Dm, mounted together according to the figure:

The load is composed by an inertia Jt and an external torque, TL. The pump is rotating
with a constant speed of ωp and the total leakage of the system is assumed laminar with
leakage coefficient Ct.
In order to reduce the influence of changes in load torque on the speed of the motor, a
negative feedback is introduced. The feedback signal is taken from a Tachometer, which
gives a voltage eb=Kbωm. The voltage eb is compared with a reference voltage, er, and the
difference Δe=er-eb activates a hydraulic mechanism, that will tune the pump
displacement to a desired value. For this, the following transfer function can be formed:
ε p ⋅ Dp     K
=        , where τ is a time constant.
Δe      1 + τs

a)   Make block diagram and determine transfer function ωm(s)/TL(s), for the system
without any feedback.

b)   Determine the transfer function ωm(s)/TL(s), for the system with feedback.

c)   Determine the stationary error in angular velocity, Δωm for a step disturbance
ΔTL in load torque for the open and the closed loop system.
LIU/IEI/FluMeS                          2007-10-26                                    17 (20)
Exercises for TMHP51

Example 9.2
A hydraulic system, shown in the figure below, consists of a pump with variable
displacement Dp and a fixed motor with displacement Dm. The system is equipped with
an angular position (θm) feedback.

a)   Make block diagram of the system with the assumption that leakage and oil
compressibility are neglected.
Determine the closed loop transfer function θm/eb.
K
Is the system behaviour acceptable if the input signal is a step, eb ( s ) = ?
s

b)   Assume that the pump displacement setting (y) controller is equipped with an
electrical position feedback. How will this feedback influence the closed loop
transfer function of the system?
LIU/IEI/FluMeS                               2007-10-26                18 (20)
Exercises for TMHP51

Example 2.1
4 orifices are required. The orifice diameters shall be selected as:
d1 = 2,56 mm, d2 = 3,15 mm, d3 = 3,86 mm, d4 = 4,68 mm.

Example 2.3
p2     1
Pressure ratio:      =
p1 C ⋅ x 2 + 1

dp2               p2 3
Max sensitivity                for     =
dx    max         p1 4

Example 3.1
a) Constant flow capacity (no flow forces): qv = 3,16⋅10-4 m3/s.
b)    q2MPa = 3,136⋅10-4 m3/s
q8MPa = 3,141⋅10-4 m3/s
q15MPa = 3,149⋅10-4 m3/s

Example 3.3
Total flow force: Fst = 14,2 N

Example 5.10
-

Example 5.12
-

Example 5.13
-

Example 6.1
•
a)    x p = 0,217 ⋅ 1 − 1,08 ⋅ 10 −5 ⋅ FL
b)    Pf = 926 W
c)    Am = 20,7 dB

ΔFL
d)          = 1,0⋅109 N/m
ΔX p
LIU/IEI/FluMeS                             2007-10-26                          19 (20)
Exercises for TMHP51

Example 6.2
a)    Ct = 4,43⋅10-11 m5/Ns
b)    FL = 58 kN, Pf = 8 kW
c)    Pf = 20,4 kW

ΔFL
d)             = 4,5⋅107 N/m
ΔX p

Example 6.3
b)    The steady state performance will not be affected
c)    Pf = 926 W

ΔFL
d)             = 4,5⋅107 N/m
ΔX p

Example 6.4
a)    -
b)    The steady state performance will not be affected
c)    Pf = 926 W

ΔFL
d)             = 1,0⋅109 N/m
ΔX p

Example 6.9
Dp = 100 cm3/rev., Dm = 35 cm3/rev.
QPrv = 2,5 litre/s (Pressure relief valve), QSv = 1,5 litre/s (Servo valve),

Example 6.12
a)    Kes = 1,76⋅10-3 m/Vs, Kv = 17 1/s (Kq0 = 0,62 m2/s)

ΔTL
b)          ⋅
⇒∞
Δθ m
c)    ε0 = 0 rad/s (if Kt = 1,0)
LIU/IEI/FluMeS                                         2007-10-26                      20 (20)
Exercises for TMHP51

Example 6.13
a)   Kv = 1,89 1/s
b)   Am = 12,3 dB

Example 9.1
Δωm               Ct
c)   Steady state condition without feedback:                                =−
ΔTL   s →0
2
Dm

Δωm                         − Ct
With Feedback:                           =
ΔTL       s →0
D + K b ⋅ K ⋅ ω p ⋅ Dm
2
m

Example 9.2
θm           Ka              K ⋅ K ⋅ K ⋅ω
a)         =                 , Ka = i q p p
eb       D s + KR ⋅ Ka
2 2
m                     Ap ⋅ Dm
The step response is unacceptable because the luck of damping
b)   The pump displacement setting feedback gives damping:
qm                        Ka
=
eb              Ki ⋅ K q ⋅ K L
s2 +                    s + K R ⋅ Ka
Ap

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