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

© Dr. Monika Ivantysynova

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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

© Dr. Monika Ivantysynova

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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:

© Dr. Monika Ivantysynova

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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