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
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Historical Background Hydrostatic transmissiom
Vane pump
Axial Piston © Dr. Monika IvantysynovaPump
Gear pump
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Displacement machine due to compressibility of a real fluid
p2, Qe Te , n
Pumping
p1
Adiabatic expansion Adiabatic compression
Suction
Vmin=VT
with VT .. dead volume
© Dr. Monika Ivantysynova
KA.. adiabatic bulk modulus 4
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Displacement machine due to viscosity & compressibility of a real fluid Pressure drop between displacement chamber and port Port pressure
Port pressure
Pressure in displacement chamber
Pressure in displacement chamber
Motor
Pump © Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Power Density Electric Motor
r
r
Hydraulic Motor
b
I=J b h J… current density [A/m2]
Fe = I ⋅ B ⋅ L ⋅ sinα Torque:
with I current [A]
B … magnetic flux density [ T ] or [Vs/m2]
T = I ⋅ B ⋅ L ⋅ r ⋅ sin α
© Dr. Monika Ivantysynova
Fh = p ⋅ L ⋅ h
T = p⋅L⋅h⋅r 6
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Example Power:
P = T ω = T 2π n
For electric motor follows: For hydraulic motor follows: Force density:
P = I B L r 2π n
assuming α=90°
P = p ⋅ L ⋅ h ⋅ r ⋅ 2π ⋅ n Hydraulic Motor
Electric Motor
up to 5 107 Pa
with a cross section area of conductor: 9 10-6 m2 © Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Mass / Power Ratio Electric Machine mass power
=
Positive displacement machine 0.1 … 1 kg/kW
1 …. 15 kg/kW
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 © Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Displacement Machines Swash Plate Machines
Axial Piston Machines Piston Machines F In-line Piston Machines
Bent Axis machines with external piston support
Radial Piston Machines with internal piston support External Gear
F Gear Machines
Internal Gear Annual Gear F Screw Machines
Vane Machines Fixed displacement machines © Dr. Monika Ivantysynova
others
Variable displacement machines 9
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Axial Piston Pumps Cylinder block
Pitch radius R Outlet
Inlet
Swash plate Piston
Valve plate (distributor)
Cylinder block
p2, Qe
Te , n Piston stroke = f (ß,R) Variable displacement pump Requires continous change of ß © Dr. Monika Ivantysynova
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p1
Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Bent Axis & Swash Plate Machines Torque generation on cylinder block
Torque generation on „swash plate“
Swash plate design
FR FR Fp
Fp FR
Fp
FN
FN
Radial force FR exerted on piston!
FN
Driving flange must cover radial force
FR Fp
FN
FR FR Fp
Fp
FN © Dr. Monika Ivantysynova
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FN
Bent axis machines Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Axial Piston Pumps Openings in cylinder bottom In case of plane valve plate
In case of spherical valve plate
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Axial Piston Pumps Plane valve plate Inlet opening
Outlet opening
Plane valve plate Inlet
Outlet
Connection of displacement chambers with suction and pressure port © Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Axial Piston Pumps Kinematic reversal: pump with rotating swash plate Suction valve
Check valves fulfill distributor function
Pressure valve for each cylinder
Outlet © Dr. Monika Ivantysynova
Inlet 14
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can only work as pump Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Comparison of axial piston pumps
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Steady state characteristics ideal displacement machine
P 0
0
0 © Dr. Monika Ivantysynova
V = α Vmax
n
P
0
T
n
T
0
Q
Q
Displacement volume of a variable displacement machine:
n
0 16
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
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 ! © Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
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: © Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Real Displacement Machine Cylinder Distributor p2, Qe
Inlet Piston Outlet
QSi
Te , n
QSe
p1
QSe… external volumetric losses QSi… internal volumetric losses Effective Flow rate:
Qe= αVmax n - Qs
Effective torque: © Dr. Monika Ivantysynova
QS … volumetric losses TS …torque losses
Te = 19
Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Volumetric Losses losses due to Incomplete p2, Qe filling external
internal volumetric losses
losses due to compressibility
QSi
Te , n QSe
p1
QSL external and internal volumetric losses = flow through laminar resistances:
Assuming const. gap height Dynamic viscosity © Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Volumetric Losses Effective volume flow rate is reduced due to compressibility of the fluid
Pumping
simplified Suction
QSK = n ΔVB © Dr. Monika Ivantysynova
with n … pump speed 21
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Steady state characteristics of a real displacement machine
Qi = V n = α Vmax n
Effective volumetric flow rate
nmin
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Steady state characteristics Effective mass flow at pump outlet Qme
© Dr. Monika Ivantysynova
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Loss component due to compressibility does not occur!
Design and Modeling of Fluid Power Systems, ME 597/ABE 591
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 © Dr. Monika Ivantysynova
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Pressure pulsation Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Flow pulsation Non-uniformity grade of volumetric flow is defined:
2 © Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
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
Torque loss linear dependent on pressure
TSp = CTp Δp
d
TSρ = CTρ ρ n2 … drag coefficient
v l
… flow resistance coefficient
effective torque required at pump shaft © Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Steady state characteristics Torque losses
TSµ
TSρ 0
TSp
TSp
0
0
0
TSµ
0
TSρ
of a real displacement machine
0
n
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n
n
Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Steady state characteristics Effective Torque
Effective torque Te
Effective torque Te
TS
TS
TρSµ TSp
TρSµ TSp TSc T
T
TSc Ti
Ti
0
0
n
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
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
z = b tanβ b=R-y y = R ⋅ cos ϕ
R … pitch radius © Dr. Monika Ivantysynova
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Inner dead point IT
Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Kinematic Parameters Piston velocity in z-direction:
Piston acceleration in z-direction: vP Circumferential speed
au
aP
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
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Instantaneous Volumetric Flow Geometric displacement volume:
Vg = z Ap HP
z … number of pistons
In case of pistons arranged parallel to shaft axis:
Mean value over time
Geometric flow rate:
k …number of pistons, which
Instantaneous volumetric flow: with
For an ideal pump without losses
are in the delivery stroke
instantaneous volumetric flow of individual piston
vP = ω ⋅ R ⋅ tanβ ⋅ sinϕ Q ai = vp ⋅ A p = ω ⋅ A p ⋅ R ⋅ tanβ ⋅ sinϕ i © Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Instantaneous Volumetric Flow In case of even number of pistons:
k = 0.5 ⋅ z
In case of odd number of pistons:
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Flow & Torque Pulsation kinematic flow and torque pulsation due to a finite number of piston Non-uniformity grade:
Flow Pulsation:
Even number of pistons:
z
tan
Odd number of pistons:
z
z
tan
z
Torque Pulsation
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Flow Torque Pulsation Piston&Pumps
kinematic flow and torque pulsation due to a finite number of piston z… number of pistons Non-uniformity mean
Even number of pistons:
Odd number of pistons:
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
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Flow Pulsation Non-uniformity grade:
Kinematic non-uniformity grade for piston machines: Number of pistons z Non-uniformity grade δ
3
4
5
6
7
8
9
10
11
0.140
0.325
0.049
0.140
0.025
0.078
0.015
0.049
0.010
Volumetric losses Qs=f(φ) and Flow pulsation of a real displacement machine is much larger than the flow pulsation given by the kinematics © Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591
Flow Pulsation
Flow pulsation leads to pressure pulsation at pump outlet
© Dr. Monika Ivantysynova
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Design and Modeling of Fluid Power Systems, ME 597/ABE 591