Design and Modeling of Fluid Power Systems: ME 597/ABE 591 Lecture 4

Design and Modeling of Fluid Power Systems
ME 597/ABE 591 Lecture 4
Dr. Monika Ivantysynova

MAHA Professor Fluid Power Systems
MAHA Fluid Power Research Center

Purdue University
Content
Displacement machines – design principles & scaling laws
Power density comparison between hydrostatic and electric machines
Volumetric losses, effective flow, flow ripple, flow pulsation
Steady state characteristics of an ideal and real displacement machine
Torque losses, torque efficiency
Design and Modeling of Fluid Power

© Dr. Monika Ivantysynova 2
2 Systems, ME 597/ABE 591
Historical Background
Hydrostatic transmissiom
Vane pump
Gear pump Axial Piston

© Dr. Monika IvantysynovaPump 3
Displacement machine
p2, Qe
due to compressibility of a real fluid
Pumping Te , n
p1
Adiabatic expansion
Adiabatic compression
Suction
Vmin=VT with VT .. dead volume KA.. adiabatic bulk modulus

Displacement machine
due to viscosity & compressibility of a real fluid

Pressure drop between displacement chamber and port
Port pressure Port pressure
Pressure in Pressure in
displacement displacement
chamber chamber
Pump Motor
Power Density
Electric Motor
Hydraulic Motor
r
b r
I=J b h
J… current density [A/m2]
Fh = p ⋅ L ⋅ h
Fe = I ⋅ B ⋅ L ⋅ sinα with I current [A]
B … magnetic flux density [ T ] or [Vs/m2]
Torque: T = I ⋅ B ⋅ L ⋅ r ⋅ sin α T = p⋅L⋅h⋅r

Example
Power: P = T ω = T 2π n
For electric motor follows: P = I B L r 2π n assuming α=90°
For hydraulic motor follows: P = p ⋅ L ⋅ h ⋅ r ⋅ 2π ⋅ n
Force density: Electric Motor Hydraulic Motor
up to 5 107 Pa
with a cross section area of conductor: 9 10-6 m2

Mass / Power Ratio
Electric Machine Positive displacement machine

mass
= 1 …. 15 kg/kW 0.1 … 1 kg/kW
power
Positive displacement machines (pumps & motors) are:
10 times lighter
min. 10 times smaller
much smaller mass moment of inertia (approx. 70 times)
much better dynamic behavior of displacement machines

Displacement Machines
Swash Plate Machines
Axial Piston Machines
Piston Machines
Bent Axis machines
F
In-line Piston Machines
with external piston support
Radial Piston Machines

with internal piston support
F External Gear
Gear Machines
Internal Gear
Annual Gear
F
Vane Machines Screw Machines others
Fixed displacement machines Variable displacement machines
Axial Piston Pumps
Cylinder block Pitch radius R
Outlet
Inlet
Cylinder block p2, Qe

Swash plate Piston Valve plate
(distributor)
Piston stroke = f (ß,R) Te , n
Variable displacement pump
Requires continous change of ß p1
Bent Axis & Swash Plate Machines
Torque generation on cylinder block Torque generation on „swash plate“
Swash plate design FR

FR
Fp FN
Fp
FR
FR Driving flange must
FN cover radial force
Fp
FN
Fp FN FR
Radial force FR FR
exerted on piston! Fp
Fp Bent axis machines
FN FN
Axial Piston Pumps
Openings in cylinder bottom

In case of plane valve plate
In case of spherical valve plate

Axial Piston Pumps
Plane valve plate
Inlet opening Outlet opening
Plane valve plate
Inlet Outlet
Connection of displacement chambers with suction and pressure port

Axial Piston Pumps
Kinematic reversal: pump with rotating swash plate Check valves fulfill
distributor function
Suction valve
Pressure
valve for
each
cylinder
Outlet Inlet can only work as pump

Comparison of axial piston
pumps

Steady state characteristics
ideal displacement machine
Displacement volume of a variable displacement machine: V = α Vmax
P
Q
Q
0 n 0 0 n
T
T
P
0 0 n 0
Scaling laws
The pump size is determined by the displacement volume V p2, Qe

[cm3/rev]. Usually a proportional scaling law, conserving
geometric similarity, is applied, resulting in stresses remaining
constant for all sizes of units. Te , n
p1
Q=V n
… linear scaling factor
λ λ λ
λ λ λ
Assuming same maximal operating pressures for all unit sizes and a
constant maximal sliding velocity !
Example
The maximal shaft speed of a given pump is 5000 rpm. The

displacement volume of this pump is V= 40cm3/rev. The maximal working
pressure is given with 40 MPa. Using first order scaling laws, determine:
- the maximal shaft speed of a pump with 90 cm3/rev
- the torque of this larger pump
- the maximal volume flow rate of this larger pump
- the power of this larger pump
For the linear scaling factor follows:
Maximal shaft speed of the larger pump:
Torque of the larger pump:
Maximal volume flow rate:
Power of the larger pump:

Real Displacement Machine
Cylinder
Distributor
p2, Qe
Inlet
Piston
Te , n QSi
Outlet
QSe
p1
QSe… external volumetric losses
QSi… internal volumetric losses
Effective Flow rate: Qe= αVmax n - Qs QS … volumetric losses
Effective torque: Te = TS …torque losses

Systems, ME 597/ABE 591
Volumetric Losses
losses due to
Incomplete p2, Qe
filling
Te , n QSi
external internal losses due to
volumetric losses compressibility QSe
p1
QSL external and internal volumetric losses = flow through laminar

resistances:
Assuming const. gap height
Dynamic viscosity
Volumetric Losses
Effective volume flow rate is reduced due to compressibility of the fluid
Pumping
simplified
Suction
QSK = n ΔVB with n … pump speed

of a real displacement machine Qi = V n = α Vmax n

Effective volumetric flow rate
nmin

Effective mass flow at pump outlet Qme Loss component due to

compressibility does not
occur!

Instantaneous Pump Flow
Instantaneous volumetric flow Qa
Volumetric flow displaced by

a displacement chamber
The instantaneous volumetric flow is given by the sum of instantaneous flows

Qai of each displacement element:
k … number of displacement chambers, decreasing

their volume, i.e. being in the delivery stroke
z is an even number z … number of displacement elements
z is an odd number
Flow pulsation of pumps Pressure pulsation

Flow pulsation
Non-uniformity grade of volumetric flow is defined:

Torque Losses
constant value
Torque loss due to viscous friction in gaps (laminar flow)
h…gap height
Torque loss to overcome pressure drop caused in turbulent resistances
TSρ = CTρ ρ n2
Torque loss linear dependent on pressure v
d
l
… drag coefficient
TSp = CTp Δp
… flow resistance coefficient
effective torque required at pump shaft

Torque losses
of a real displacement machine
TSρ
TSµ
n TSµ 0 n
0 0
TSp
TSp
TSρ
0 n
0 0

Effective Torque
Effective torque Te Effective torque Te
TS TS
TρSµ
TρSµ
TSp
TSp
TSc TSc
T
T
Ti Ti
0 0
n

Axial Piston Machine
Kinematics
Piston displacement: HP
sP
s p = -R ⋅ tanβ ⋅ (1-cosϕ ) Outer dead point AT
φ=0
Piston stroke:
H P = 2 ⋅ R ⋅ tanβ
Inner dead point IT
z = b tanβ
b=R-y
R … pitch radius y = R ⋅ cos ϕ

Kinematic Parameters
Piston velocity in z-direction:
Piston acceleration in z-direction:

au
vP aP
Circumferential speed vu
vu = R ω
Centrifugal acceleration:
au = R ω2 Coriolis acceleration ac is just zero, as the vector of

angular velocity ω and the piston velocity vP run parallel

Instantaneous Volumetric Flow
Geometric displacement volume:
Vg = z Ap HP z … number of pistons For an ideal pump
without losses
In case of pistons arranged parallel to shaft axis:
Geometric flow rate: Mean value over time
Instantaneous volumetric flow: k …number of pistons, which

are in the delivery stroke
with instantaneous volumetric flow of individual piston
vP = ω ⋅ R ⋅ tanβ ⋅ sinϕ
Q ai = vp ⋅ A p = ω ⋅ A p ⋅ R ⋅ tanβ ⋅ sinϕ i
Instantaneous Volumetric Flow
In case of even number of pistons: k = 0.5 ⋅ z
In case of odd number of pistons:

Flow & Torque Pulsation
kinematic flow and torque pulsation due to

a finite number of piston
Flow Pulsation: Non-uniformity grade:
Even number of pistons: Odd number of pistons:
tan tan
z z z z
Torque Pulsation

Flow
Piston&Pumps
Torque Pulsation
kinematic flow and torque pulsation due to

a finite number of piston z… number of pistons
Non-uniformity Even number of pistons: Odd number of pistons:
mean mean
NON-UNIFORMITY of FLOW / TORQUE
NON-UNIFORMITY of FLOW / TORQUE
Flow and torque pulsation frequency f:
Even number of pistons: f=z·n

Odd number of pistons: f=2·z·n
Flow Pulsation
Non-uniformity grade:
Kinematic non-uniformity grade for piston machines:

Number of 3 4 5 6 7 8 9 10 11
pistons z
Non-uniformity 0.140 0.325 0.049 0.140 0.025 0.078 0.015 0.049 0.010
grade δ
Volumetric losses Qs=f(φ) and
Flow pulsation of a real displacement machine is much larger than the

flow pulsation given by the kinematics

Flow Pulsation
Flow pulsation leads to pressure pulsation at pump outlet


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Design and Modeling of Fluid Power Systems

ME 597/ABE 591 Lecture 4

Dr. Monika Ivantysynova

MAHA Fluid Power Research Center

Displacement machines – design principles & scaling laws

Power density comparison between hydrostatic and electric machines

Volumetric losses, effective flow, flow ripple, flow pulsation

Steady state characteristics of an ideal and real displacement machine

Torque losses, torque efficiency

Design and Modeling of Fluid Power

Gear pump Axial Piston

Vmin=VT with VT .. dead volume KA.. adiabatic bulk modulus

due to viscosity & compressibility of a real fluid

Port pressure Port pressure

Torque: T = I ⋅ B ⋅ L ⋅ r ⋅ sin α T = p⋅L⋅h⋅r

For hydraulic motor follows: P = p ⋅ L ⋅ h ⋅ r ⋅ 2π ⋅ n

Force density: Electric Motor Hydraulic Motor

with a cross section area of conductor: 9 10-6 m2

Electric Machine Positive displacement machine

Positive displacement machines (pumps & motors) are:

min. 10 times smaller

much smaller mass moment of inertia (approx. 70 times)

much better dynamic behavior of displacement machines

Design and Modeling of Fluid Power

Radial Piston Machines

Cylinder block Pitch radius R

Cylinder block p2, Qe

Swash plate design FR

Openings in cylinder bottom

In case of spherical valve plate

Design and Modeling of Fluid Power

Plane valve plate

Inlet opening Outlet opening

Plane valve plate

Connection of displacement chambers with suction and pressure port

Outlet Inlet can only work as pump

Design and Modeling of Fluid Power

Displacement volume of a variable displacement machine: V = α Vmax

The pump size is determined by the displacement volume V p2, Qe

… linear scaling factor

The maximal shaft speed of a given pump is 5000 rpm. The

For the linear scaling factor follows:

Maximal shaft speed of the larger pump:

Torque of the larger pump:

Maximal volume flow rate:

Power of the larger pump:

QSe… external volumetric losses

QSi… internal volumetric losses

Effective Flow rate: Qe= αVmax n - Qs QS … volumetric losses

Effective torque: Te = TS …torque losses

Design and Modeling of Fluid Power

QSL external and internal volumetric losses = flow through laminar

Assuming const. gap height

QSK = n ΔVB with n … pump speed

Design and Modeling of Fluid Power

of a real displacement machine Qi = V n = α Vmax n

Design and Modeling of Fluid Power

Effective mass flow at pump outlet Qme Loss component due to

Design and Modeling of Fluid Power