“Helmholtz” Capture SolenoidMechanical Structure Calculations

Peter Loveridge

P.Loveridge@rl.ac.uk

STFC Rutherford Appleton Laboratory, UK

July 2008

Peter Loveridge, July 2008

Objectives

Using the magnetic forces calculated previously...

Evaluate the approximate dimensions of the magnet mechanical structure components in the cases where:

1. A fully open Helmholtz gap is required (compatible for a target wheel WITH spokes)

2. The Helmholtz gap is filled with structure material with the exception of a bridging structure across two short unsupported lengths (compatible with a SPOKELESS target wheel)

Is the required structure volume compatible with coil/target geometry?

Peter Loveridge, July 2008

Recap: Magnetic Forces

21

22

MEAN

21

22

21

22

MAX

1

MAX

1

121

:Stress Axial eCompressiv Average

:)R(@r Stress Hoop TensileMax

:)R(@r Stress Radial eCompressivMax

2

:Pressure Internal Equivalent

RR

F

RR

RRP

P

-ZZπR

FP

ZZ

R

R

P

R2

R1

Coil FZ FR FZ FR PINT σR max σθ max σZ mean

(N) (N) (Tonnes) (Tonnes) (bar) (MPa) (MPa) (MPa)

NC01 3.558E+05 1.182E+08 36 12,053 2920 292 381 0.5

NC02 -1.206E+06 1.144E+08 -123 11,666 2923 292 381 1.8

SC01 1.637E+08 5.386E+08 16,693 54,922 1493 149 248 42.4

SC02 -3.400E+07 3.572E+08 -3,467 36,424 1870 187 310 8.8

SC03 -6.725E+07 1.854E+08 -6,858 18,906 590 59 160 38.8

SC04 -3.326E+07 1.277E+08 -3,392 13,022 262 26 111 28.3

SC05 -1.551E+07 9.041E+07 -1,582 9,219 120 12 93 27.8

SC06 -1.066E+07 5.425E+07 -1,087 5,532 60 6 74 32.0

SC07 -5.971E+05 5.064E+06 -61 516 44 4 72 8.1

SC08 -4.501E+05 5.860E+06 -46 598 45 4 53 4.4

SC09 -1.142E+06 4.355E+06 -116 444 33 3 40 11.2

Peter Loveridge, July 2008

Recap: Magnetic Forces

21

22

MEAN

21

22

21

22

MAX

1

MAX

1

121

:Stress Axial eCompressiv Average

:)R(@r Stress Hoop TensileMax

:)R(@r Stress Radial eCompressivMax

2

:Pressure Internal Equivalent

RR

F

RR

RRP

P

-ZZπR

FP

ZZ

R

R

P

R2

R1

Coil FZ FR FZ FR PINT σR max σθ max σZ mean

(N) (N) (Tonnes) (Tonnes) (bar) (MPa) (MPa) (MPa)

NC01 3.558E+05 1.182E+08 36 12,053 2920 292 381 0.5

NC02 -1.206E+06 1.144E+08 -123 11,666 2923 292 381 1.8

SC01 1.637E+08 5.386E+08 16,693 54,922 1493 149 248 42.4

SC02 -3.400E+07 3.572E+08 -3,467 36,424 1870 187 310 8.8

SC03 -6.725E+07 1.854E+08 -6,858 18,906 590 59 160 38.8

SC04 -3.326E+07 1.277E+08 -3,392 13,022 262 26 111 28.3

SC05 -1.551E+07 9.041E+07 -1,582 9,219 120 12 93 27.8

SC06 -1.066E+07 5.425E+07 -1,087 5,532 60 6 74 32.0

SC07 -5.971E+05 5.064E+06 -61 516 44 4 72 8.1

SC08 -4.501E+05 5.860E+06 -46 598 45 4 53 4.4

SC09 -1.142E+06 4.355E+06 -116 444 33 3 40 11.2

Consider how to support axial loads on coil “SC01”

Peter Loveridge, July 2008

Fully Open Gap: Plate Bending

Simply-Supported Annular Pate With Uniformly Distributed Pressure Over Entire Area

0

2

4

6

8

10

0.000 0.100 0.200 0.300 0.400

Plate Thickness (m)

Max

imu

m D

efle

ctio

n (

mm

)

0

200

400

600

800

1000

Max

imu

m B

end

ing

Str

ess

(MP

a)

Deflection

Bending Stress

Input data:Inner Radius (b) 0.636 [m]Outer Radius (a) 1.278 [m]Modulus (E) 1.90E+11 [N/m2]Poissons ratio (Nu) 0.3 [ - ]Axial force (FZ) 1.64E+08 [N]

ba

t

P = FZ / A

Rb Ra

• Hand calculation based on flat plate theory...

• Limit max bending stress in steel plate to, say, 150 MPa:

– Plate thickness = 286 mm

– Max plate deflection = 0.2 mm

Peter Loveridge, July 2008

Fully Open Gap: Tensile Retaining Loads

P = FZ / A

Rb Ra

twotwi

L =

903

mm

• Envisage a retaining structure composed of two cylinders...

• From flat plate calculation we know:– Outer reaction (Ra) = 93.7 MN/360˚

– Inner reaction (Rb) = 70.0 MN/360˚

• Limit max tensile stress in steel retaining structure to, say, 150 MPa:

– two = 76 mm

– twi = 130 mm

– Elongationmm 0.7

E

LσΔL

Tensile Stress in Retaining Structure

0

100

200

300

400

500

600

700

0 50 100 150 200 250 300

Wall Thickness (mm)

Ten

sile

Str

ess

(MP

a)

Inner Cylinder

Outer Cylinder

Peter Loveridge, July 2008

Fully Open Gap: ANSYS Mechanical Simulation

180˚ Geometry Expansion

Axial Deflection Plot

0 mm 1.2 mm

Von-Mises Stress Plot

0 MPa 270 MPa

Axi-symmetic Mesh

76 mm thick outer cylinder

130 mm thick inner cylinder 286 mm thick end plate

Peter Loveridge, July 2008

Interpretation: Target wheel WITH spokes

• Target station geometry:– Need to maintain an open axial gap over

a significant part of the circumference

– Huge axial forces must be transferred out to an external structure

• Coils at 4.2K, external structure at room-temperature ?!

• Structure Calculations:– Hand calculations

– confirmation by FEA

• Results:– The volume of structural material required

looks extremely prohibitive (see right).

• Insufficient radial space between NC and SC coils

• Helmholtz gap filled with structure

Coil geometry with required structure volume to maintain a fully open axial gap (drawn to scale)

• Comments:– Major geometry interference issues to resolve (coils, structure, target-access)– On first investigation the structure requirements for this option do not seem feasible!– NOTE: Have only considered the axial force balance. What about radial forces!

Peter Loveridge, July 2008

Bridging Structure: Plate bending

• Hand Calculation for a plate bridging a short unsupported length...

• Consider a simply-supported rectangular plate

– Dimensions W x L x t

– Subject to same uniform pressure P

• Limit max bending stress in steel plate to, say, 150 MPa:

– Plate thickness = 170 mm

– Max plate deflection = 0.2 mm

L

W

t

Input data:Width (W) 0.642 [m]Length (L) 0.500 [m]Modulus (E) 1.90E+11 [N/m2]

Simply Supprted Rectangular PlateWith Uniformly Distributed Pressure Over Entire Area

0

200

400

600

800

1000

0 50 100 150 200 250 300

Plate Thickness (mm)M

axim

um

Ben

din

g S

tres

s (M

Pa)

0

2

4

6

8

10

Max

imu

m D

efle

ctio

n (

mm

)

Stress

Deflection

Peter Loveridge, July 2008

Interpretation: SPOKELESS target wheel

Coil geometry with required structure volume to bridge a short unsupported length (drawn to scale)

• Target station geometry:– The Helmholtz gap is filled with structure

material with the exception of bridging sections across two short unsupported lengths

– Huge axial forces are “internally balanced” (in same way as study-2 magnet)

• Structure Calculations:– Preliminary hand calculations only

– Assumptions yet to be confirmed by FEA

• Results:– A significant volume of bridging structure

material in “Helmholtz gap” region (see right)

• Current (400 mm) gap appears insufficient• Comments:

– Reoptimisation of geometry will be required (coils, structure, target-access)• ~600 mm Helmholtz gap? Is this feasible?

– Structure calculations for this option should be taken further– NOTE: Have only considered the axial force balance. What about radial forces!