R3DC Report 2-08

Introduction

In this document we discuss the design of the Region 3 Drift Chambers (R3 DC) for the Hall-B 12-GeV upgrade (CLAS12). The objective for CLAS12 detectors is to achieve nearly full acceptance in the lab frame for the relativistic forward-boosted final-state particles while maintaining a medium momentum resolution of 0.5 to 1% for a 5-GeV charged particle.

The six identical R3 DCs will be located 4.7 meters from the target and together the chambers will possess nearly full [math]2\pi[/math] azimuthal coverage and will span from 5 to 45 degrees in polar angle as measured from the direction of the electron beam. The central plane through the R3 DC will be tilted 25 degrees from a perpendicular to the beam line towards the target. The toroidal coils will occlude {how many degrees?} in azimuth. The shadow of the toroidal coils will result in a minimum gap of 16 cm between each R3 DC which hinders reconstructing the trajectory of scattered particles. This will provide the space for the readout boards, attendant electronics, cabling, etc. The active area for each R3 DC will then be [60 - ?] in azimuth,

The design of the R3 DC for CLAS12 is based on the successful hexagonal cell structure of the current CLAS R3 DC. Below we delineate the design criteria:

1. Smaller cell size of 2.07 cm to accommodate a luminosity of [math]1035 /(cm^2 s^1)[/math] and maintain an accidental occupancy of less the 4% for high tracking efficiency..
2. Sense wires will be increased to 30 um from the current 20 um diameter. This means that the wire will require a higher tension to offset droop and electrostatic forces.
3. All wires in each superlayer will lie in a plane instead of on a cylindrical surface. The staggered by one half cell or “brick wall pattern” of the hexagonal cells will be maintained for each successive layer.
4. To enhance the azimuthal (φ) resolution, each superlayer will be +/- stereo, i.e. the pitch will be +6 degrees in the six layers of the first superlayer and the pitch will be -6 degrees for the second superlayer array of six wire planes. The current R3 DC has 0 and 6 degree pitch respectively for the two superlayers.
5. There will be a total of 4928 wires of which 1344 (check this) will be sense wires.
6. The endplates will be oriented at 60 degrees with respect to one another. Precisely holes will be drilled into each endplate with tolerance of better than 100 um for the placement of the drift chamber wires. Feed through assemblies are placed in these holes to precisely position the sense, field, and guard wires. The expected overall position accuracy will be 250 um per layer, with a wire location accuracy of 200 um.
7. The deflection along the length of the endplates may not exceed 50 um.
8. The R3 DCs will be supported from only two points at the outer backplane. This makes the R3 DCs self supporting.
9. The nose plate for each R3 DC will be positioned onto a precision support hexagonal ring for aligning all R3 DCs with respect to the surveyed Hall B coordinate system.

The mechanical engineering aspects of the R3 DC is discussed below in light of the above design criteria. We present a two dimensional Finite Element Analysis. The three-dimensional FEA is currently under study

R3 Description

Geometry

An exploded view of a single R3 sector is shown below. The back plate, nose plate, and two endplates represent the key components to the main ionization chamber structure. The back plate is approximately 5 meters long and will likely have two mount points to support and position the drift chambers. The nose plate, more than half a meter long, will hold a ball joint that mounts to a hexagonal ring and constrains the chamber location at a third point. The drift chamber wires are held in place by the end plates which have carbon fiber rods spanning the gap near the chamber entrance in order to reduce end plate deflection arising from the wire tension, the weight of the chamber, and the gas volume inside the chamber. The front faces of the chamber has a thin window to allow the passage of scattered particles with limited multiple scattering. The exit window of the chamber is constructed from a composite material known as Hexcell clad with a thin carbon fiber sheet to reduce end plate deflections near the back of the chamber. Six of these chambers are oriented around a beam pipe as shown in the figure below.

	label Distance (mm) [math]Bck_{Len1}[/math] 5004.92 [math]Bck_{Len2}[/math] 4840.04 [math]End_{Len1}[/math] 4830.17 [math]End_{Len2}[/math] 4665.1 [math]Nse_{Len1}[/math] 614
Figure 1: The general features of a R3 Drift Chamber. The chamber wires string between the end plates are not shown. Parameter Tables for the variables in the above pictures

	label value [math]D_1[/math] 4765.6 mm [math]\theta_1[/math] 5 degrees [math]\theta_2[/math] 45 degrees [math]\theta_3[/math] 25 degrees
Figure 2: A side view of a R3 drift chamber. The distance from the target center to the first sense wire is labeled [math]D_1[/math]. The min an max acceptance angles are labeled [math]\theta_1[/math] and [math]\theta_2[/math] repsectively. The tilt angle with respect to a normal to the beam line is labeled [math]\theta_3[/math].

Figure 3

Mestayer's Geometry Excel Spreadsheet

Material Specifications

Physical properties germane to the materials used to construct the R3 dirft chambers are shown in the table below. End plates for R3 drift chambers in the previous 6 GeV CLAS detector were fabricated by gluing thin (0.048 inch) stainless steel sheets onto both sides of a 2 inch Polyeurethane foam. The current design expects to use the same structure for the end plates. The use of carbon fiber rods to minimize end plate deflections is another design feature borrowed from the previous 6 GeV CLAS detector design. Aluminum is included below as cadidate material for the face and end plates.

Material	Youngs Modulus	Density
	[math]Pa (N/m^2)[/math]	[math]\frac{kg}{m^3}[/math]
Polyeurethane Foam FR-3700	[math]1.4 \times 10^8[/math]	240
Stainless Steel	[math]2.1 \times 10^{11}[/math]	8030
Carbon Fiber Rods	[math]1.5 \times 10^{11}[/math]	580
Aluminum	[math] 7\times 10^{10}[/math]	2700

Specific part Drawings

Preliminary drawings of the end plate and nose plate are shown below. The end plate and nose plate are two critical parts of the chamber. The end plate will constrain the detector wires in space while the nose plate will determine the tilt angle of the detector.

Figure 4: A preliminary End plate design for R3 Drift Chamber.

Figure 5: A preliminary Nose plate design for R3 Drift Chamber.

Endplate Deflection

Wire tension

As shown in the figure below, The drift chamber wires are oriented at 6 degrees with respect to the horizontal. Wires which slope downward from left to right make an angle of [math]\theta_2 = 90-30-6 = 54[/math] degrees with respect to a perpendicular to the endplate and have a force component along the perpendicular of [math]T_{1} = T \sin(54)[/math]. Wires which slope upward from left to right make an angle with repect to the perpendicular to the endplate of [math]\theta_3 = 90 + 30 - 6 = 114[/math] degrees. If T represents the tension in the wire, then the component of this force perpendicular to the endplate is [math]T_{2}=T \sin(180-\theta_3) = T \sin(66)[/math]. The net force perpendicular to the endplate will be [math]T( \sin(66) + \sin(54))[/math]. There are 112 sense wires in each of the 12 wire layers ( 2 super layers with 6 wire layers each)[Mestayer07]. Each sense wire is 30 microns in diameter and tensioned using a 20.7 g weight. There are 28 Field wire layers of diameter 140 microns with each wire tensioned at 63 grams and 4 guard wire layers of diameter 140 micron with each wire tensioned at 185 grams. The total tension (2T) from all wires on the endplate will be

[math]112 \left ( 12 \cdot 0.0207 + 28 \cdot 0.063 + 4 \cdot 0.185 \right ) \times 9.8 = 3024 N = 680 lbs[/math].

[math]T_1 = \frac{3024}{2} \sin(54) =1223 N[/math]

[math]T_2 = \frac{3024}{2} \sin(66) =1381 N[/math]

[math]T_{perp} = T_1+T_2 = 2604 N[/math] = perpendicular force on an endplate

	label Value (Degrees) [math]\theta[/math] 30 [math]\theta_1[/math] 6 [math]\theta_2[/math] 90 - [math]\theta[/math] - [math]\theta_1[/math] [math]\theta_3[/math] 90 + [math]\theta[/math] - [math]\theta_1[/math]
Figure 6: The above figure identifies the direction of the sense wire tension in terms of the [math]\theta[/math] and [math]\theta_1[/math].

Point Load Deflection

The force on an endplate due only to the tension from the 4,928 wires is about 5700 N /2. A simple calculation based on the definition of Young's modulus can yield an order of magnitude level estimate for the deflection of an Aluminum endplate in which the wire exerts a force perpendicular to the endplate. The deflection [math]y[/math] of the endplate in terms of Young's Modulus [math](Y)[/math] is given by the equation:

[math]Y \equiv \frac{F}{4wy} \left( \frac{l}{t}\right )^3[/math]

where

[math]Y = 7 \times 10^{10}N/m^2 \equiv[/math] Young's Modulus for Aluminum

[math]F = 2850 N \equiv[/math]point force/load

[math]w =0.526 m \equiv[/math] width of the endplate

[math]y \equiv[/math] deflection of endplate due to point force

[math]l =4.02 m \equiv[/math] length of the endplate

[math]t = 0.05 m \equiv[/math] thickness of the endplate

A deflection of 7 mm is expected for a 5 cm thick Aluminum endplate, after solving the above equation for [math]y[/math] and inserting the given values. If a 5 cm thick stainless steel [math](Y=2.1 \times 10^{11} N/m^2 )[/math] endplate were used, the deflection would drop by almost factor of 3 from 7 mm to 2.5 mm due to the linear dependence of the deflection on Young's Modulus. The previous work of Kevin Folkman came to a similar conclusion.

Distributed Load Deflection

The endplate deflection calculation was redone assuming a more realistic case in which the load is distributed evenly along the endplate. The tension of the wires on the endplate was divided into three regions; W1,W2, and W3. As shown in the Figure below, the load W3 corresponds to a region along the endplate in which wires from both super layers are creating a load on the endplate. Loads W1 and W2 represent regions in which only 1 super layer is causing a load. The endplate is "simply" supported in order to calculate the deflection. The "simple" support has a point support on the left side in which the endplate may rotate but not translate (move left or right). The right hand side of the plate is supported such that the endplate can both rotate and translate. The figure below labels the loads as well as distances over which the loads are applied on the end plate. A total force of 340 lbs exerted by the drift chamber wires is decomposed into its components according to the angle made by the wire on the endplate.

	Label value a 99.02 mm b 96.22 mm c 4025.88 mm d 335.17 mm e 273.88 mm [math]W_1[/math] 0.3=1223/(C+B) N/mm [math]W_2[/math] 0.61 N/mm [math]W_3[/math] 0.32 =1381/(C+D)N/mm
Figure 7: A diagram of the loads on the End Plate.

Assumptions

The following assumption were made in the hand calculation:

1. The end plate is made of homogenous uniform elastic material with a Young's Modulus of [math]E=210 x 10^23 N/mm^2[/math] (Stainless Steel).
2. The horizontal component of the loading is completely neglected.
3. The deflection in the x-direction is negligible.
4. The end plate is rectangular in shape with a width of 526 mm and thickness of 50.8 mm resulting area moment of inertia ([math]I_A[/math]) =5746.4 [math]mm^4[/math].
5. The loading provided by wire tensions are equivalent to a uniformly distributed load.
6. The maximum deflection occurs along the length which is represented by [math]c[/math] in the diagram.
7. The loading applied is under the elastic limit of the material.
8. The effects of the endplate weight are ignored.

The moments experience along the endplate are shown below for each of the 5 distances.

[math]M_a = -\frac{W_1 \cdot b + W_2 \cdot c + W_3 \cdot d}{2} \cdot x[/math]

[math]M_b = -\frac{W_1 \cdot b + W_2 \cdot c + W_3 \cdot d}{2} \cdot (a+x) + \frac{W_1 \cdot x^2}{2}[/math]

[math]M_c = -\frac{W_1 \cdot b + W_2 \cdot c + W_3 \cdot d}{2} \cdot (a+b+x) + W_1 \cdot b \left ( \frac{b}{2} +x \right) + \frac{W_2 \cdot x^2}{2}[/math]

[math]M_d = -\frac{W_1 \cdot b + W_2 \cdot c + W_3 \cdot d}{2} \cdot (a+b+c+ x) + W_1 \cdot b \left ( \frac{b}{2} +c+x \right) + W_2\cdot c \left ( \frac{c}{2} + x \right) + \frac{W_3 \cdot x^2}{2}[/math]

[math]M_e = -\frac{W_1 \cdot b + W_2 \cdot c + W_3 \cdot d}{2} \cdot (a+b+c+d+ x) + W_1 \cdot b \left ( \frac{b}{2} +c+d+x \right) + W_2\cdot c \left ( \frac{c}{2} + d+x \right) + W_3 \cdot d \left (\frac{d}{2} +x \right)[/math]

where [math]x[/math] represents the .

The deflection in the y-direction is governed by the equation

[math]\frac{d^2y}{dx^2} = \frac{M(x)}{E I_A}[/math]

where [math]E[/math] represents Young's Modulus and [math]I_A[/math] is the Area Momentum of Inertia. The Area Moment of Inertis (I_A) is defined as the second moment of an area about a given axis.

[math]I_{A,x} = \int x^2 dA =[/math] Area moment of inertia about the x-axis.

The mass moment of Inertia is defined as

[math]I= \int r^2 dM = \rho I_A[/math]

If we assume that the largest amount of deflection occurs in region c, then we may integrate the above 2nd order differential equation once and determine the critical point which represents the point of maximum deflection ( contra flexure).

[math]\int \frac{dy}{dx}dy = \frac{dy}{dx} = \int M_c dx[/math]

= [math]-\frac{W_1 \cdot b + W_2 \cdot c + W_3 \cdot d}{2} \cdot (ax+bx+x^2/2) + W_1 \cdot b \left ( \frac{bx}{2} +x^2/2 \right) + \frac{W_2 \cdot x^3}{6} +C_{integration}[/math]

[math]C_{integration} =[/math] constant of integration

Integrate the above equation a second time and apply the boundary conditions that the deflection is zero at the two points supporting the beam ([math]x=0[/math] and [math]x=L[/math]) .

Setting [math]\frac{dy}{dx}[/math] to zero and solving for x give the result that x = 1654.96 mm. In other words, the maximum deflection occurs 1654.96 mm from region [math]b[/math] of the endplate or 1[math]654.96 + b + a = 1850.2[/math] mm from the right hand side of the endplate.

Integrating the equation a 2nd time will result in an equation for the amount of the deflection. Substituting the value of x above results in a predicted deflection in the Y deflection of 2.2 mm after dividing by [math]E \cdot I_A[/math]. The predicted deflection is only slightly smaller than a point load when the load is distributed along the length of the end plate.

Distributed load FEA

A Finite Element Analysis (FEA) was performed using the computer software ANSYS. The maximum deflection observed was 2.144 mm as shown in the Figure below and differs from the hand calculation by only 3%. In addition to validating the FEA analysis with hand calculations, the above results also reveal the challenge of reducing the endplate deflection using stiffer materials due to an essentially linear dependence of the deflection to Young's Modulus. The cubic dependence of the deflection on the thickness [math](t)[/math] of the endplate, however, can be used to reach the minimum endplate deflection criteria of 50 [math]\mu m[/math] instead of using stiffer material. Increasing the endplate thickness comes at the cost of increasing the end plate mass. As a result, composite materials were considered in order to minimize weight. The performance of these composite materials is shown in the 3-D model section of this report.

Figure 8: An FEA analysis of the End Plate deflections.

Composite End Plate

Effective Young's Modulus Y_{eff}

The R3 end plate used in the CLAS 6 GeV detector was a Polyeurethane core 2 inches thick with 0.048 inch thick stainless steel plates on each side as shown below.

Figure 9: The Polyeurethane-Stainless Steel endplate dimensions.

The effective Young's modulus for this endplate weighted by the volume of each plate is

[math]Y_{eff} = \frac{V_p Y_p + V_s Y_s }{V_{tot}}[/math]

where

[math]Y_p = 1.4 \times 10^8 N/m^2[/math]

[math]Y_s = 2.1 \times 10^{11} N/m^2[/math]

[math]V_p = L \times W \times 2[/math] inches

[math]V_s = L \times W \times 0.096[/math] inches

[math]Y_{eff} = \frac{2(1.4 \times 10^8) +0.096(2.1 \times 10^{11})}{2.096} = 9.75 \times 10^{9} N/m^2[/math]

FEA Deflection results

The effective Young's Modulus of [math]9.75 \times 10^9 N/m^2[/math] was used to determine the deflection of an endplate.

	Configuration X max Y max Z max D max (mm) (mm) (mm) (mm) No rods .67 32.4 .04 32.4
Figure 10: FEA predicted deflections for the Polyeurethane-Stainless Steel composite end plate.

Carbon Rod Buckling

Compression

If one makes the assumption that the rod buckles by an external force along the length of the rod, then the compression of the rod can be calculated using the definition of Young's Modulus (Y) given by the equation:

[math]Y= \frac{\frac{F}{A}}{\frac{\Delta L}{L}}[/math]

where

[math]F =[/math] Compression force exerted along the end of the rod

[math]L =[/math] length of the rod = 2.8 m

[math]A =[/math] cross sections area of the rod = [math]\pi (25 mm)^2[/math]

[math]\Delta L =[/math] rod compression distance = [math]50 \times 10^{-6}[/math]

The force needed to cause the carbon fiber rod to buckle such that the endplate deflects [math]50 \mu m[/math] can be estimated by assuming that the length of the rod (2.8 m) compresses by [math]50 \mu m[/math].

[math]F = \frac{YA \Delta L }{L} = \frac{1.5 \times 10^{11} N/m^2 (\pi 0.000625 m^2) 50 \times 10^{-6} m } {2.8 m}= 5259N[/math]

The total force along the whole endplate from just the drift chamber wire tension amount to less than 3000 N which is less than a factor of 2 smaller than the force needed to compress a single carbon rod used to brace the endplates. Placing at least 6 carbon rod braces in the drift chamber should provide substantial support for the endplates.

Buckling Load Threshold

The load at which a carbon fiber rod will start to buckle is given by Euler's formula (insert Reference). The diagram below depicts the load configuration on the rod for this calculation.

Figure 11: Euler's Buckling Load Definition

[math]F_{cr}[/math] represent the critical force needed to cause the rod to begin to buckle. Young's Modulus (E) for a carbon fiber rod is 210 MPa [math](2.1 \times 10^{8} N/m^2)[/math]. The longest rod's half length (L) for R3 is 1.4 m.

Area Moment of inertia about the end of a hollow cylinder of inner radius[math] (R_i)[/math] and outer radius [math](R_o)[/math].

[math]I_{A,z}= \int_{Ri}^{Ro} r^2 \pi r dr = \frac{\pi}{4} \left( Ro^4 -Ri^4\right) [/math]

[math]=\frac{\pi}{4} \left( (2.54)^4 -(2.44)^4\right) = 4.85 cm^4[/math]

Buckling FEA

Figure 12. An FEA analysis of the maximum (threshold) force needed to cause a carbon rod to buckle.

Endplate Deflection with carbon rods

Figure 13. FEA predicted deflections for the Polyeurethane-Stainless Steel composite end plate.

Configuration	X max	Y max	Z max	D max
	(mm)	(mm)	(mm)	(mm)
No rods	.67	32.4	.04	32.4
With rods	.01	0.29	0.16	.29

Only the dark and light blue regions near the carbon fiber rods appear to have deflections below the [math]50 \mu m[/math] specification. The maximum deflection appears to be on the back (downstream) edge of the end plate. The use of a composite material with an effective Young's modulus that is a factor of 20 smaller than stainless steel resulted in deflections which were only a factor of 16 larger while decreasing the mass of the endplate by a factor of at least 16. Carbon fiber rods along the front of the drift chamber will decrease the maximum deflection to about 300 [math]\mu m[/math] which is a factor of 6 too large.

3-D Analysis

The 3D model and the loading is shown below

• The hinge 1 labeled A is fixed in all degrees of freedom.
• The Hinge 2 labeled B is fixed in 2 DOF and 1 DOF is allowed to move horizontal.
• The wire tensions labeled C, D, E, F showing on 4 wire super layers.
• Ball joint labeled G below the Nose plate.
• Acceleration labeled H is showing upwards, showing the gravity is opposite to it.

Figure 14: Loads with Directions and magnitudes

Points to Consider

• Analysis set 1 and 6 are just the self weight analysis with no external loading
• The Nose Plate, End plates, Back Plate and Hinges are made of sandwiching slices of stainless steel around Polyurethane foam on 2 sides
• Downstream side sheet is made of Hexcel Honeycomb material ( with carbon fiber slices around it on 2 sides)
• The youngs modulus of both the end plates and the hexcel sheets are obtained by the effective young's modulus calculation method.
• The compressive modulus of hexcel sheet is observed to be different along the length and width. so, the young's modulus is calculated in both the directions and the least of them is considered for calculations

Table of Results

Deflection	Wire Tension	Rods	Hexcel	Mylar	Gravity	Ball Joint	Gas Pressure	D max(mm)	X max(mm)	Y max(mm)	Z max(mm)
Analysis 1.1	No	No	No	No	Yes	No	No	1.5315	0.0034	0.1791	0.0731
Analysis 1.2	No	No	No	No	Yes	yes	No	0.219	2.11e-5	0.1795	0.0318
Analysis 2.1	Yes	No	No	No	No	No	No	19.935	17.495	9.5869	0.4435
Analysis 2.2	Yes	No	No	No	No	Yes	No	19.952	17.391	9.8084	0.4442
Analysis 2.3	Yes	No	No	No	Yes	Yes	No	18.593	16.27	9.014	0.4813
Analysis 3.1	Yes	Yes	No	No	No	No	No	1.5104	1.033	0.2855	0.0675
Analysis 3.2	Yes	Yes	No	No	No	Yes	No	1.158	1.0266	0.5869	0.0534
Analysis 3.3	Yes	Yes	No	No	Yes	No	No	1.038	0.9392	0.3618	0.1018
Analysis 4.1	Yes	No	Yes	No	No	No	No	0.2904	0.146	0.2479	0.0044
Analysis 4.2	Yes	No	Yes	No	No	Yes	No	0.2891	0.1538	0.251	0.0166
Analysis 4.3	Yes	No	Yes	No	Yes	No	No	0.301	0.156	0.264	0.0153
Analysis 5.1	Yes	Yes	Yes	No	No	No	No	0.1231	0.0216	0.0099	0.0031
Analysis 5.2	Yes	Yes	Yes	No	No	Yes	No	0.0209	0.0157	0.0116	0.0027
Analysis 5.3	Yes	Yes	Yes	No	Yes	No	No	0.0647	0.0078	0.0105	0.0122
Analysis 6.1	No	Yes	Yes	No	No	Yes	No	0.5851	0.051	0.0052	0.0082
Analysis 6.2	No	Yes	Yes	No	No	Yes	Yes	0.0738	0.0008	0.0017	0.0098
Analysis 7.1	Yes	Yes	Yes	Yes	No	No	Yes	-	-	-	-
Analysis 7.2	Yes	Yes	Yes	Yes	No	Yes	Yes	-	-	-	-
Analysis 7.3	Yes	Yes	Yes	Yes	Yes	No	Yes	-	-	-	-

• • The Analysis started with a basic frame made of composite material-polyurethane. Due to the major effects of gravity on the chamber, polyurethane gave very good results with its very low density and considerably higher youngs modulus.

• • Analysis set1 is just the analysis with Gravity effects with no external loading. The very first analysis1.1 is just with Hinge constraints and no ball joint at the bottom side i.e Nose plate side.

• • The max deflection is about 1.53mm and by adding a ball joint in analysis1.2 to the geometry reduced the deflection to 0.22mm. With the orientation we are looking at, the chamber is expected to rest its weight on the ball joint.

• • Analysis set 2 is by adding wire tensions to the end plates without/with ball joint at no gravity loads and with gravity and ball joint.The max deflections are observed to be close to 19mm.

• • Analysis set 3 is a step further trying to reduce the deflection by adding 6 carbon rods on the upstream side of the chamber. The deflection is reduced to 1-1.5mm. the next try is to support it further my placing a hexcel backing sheet on the downstream side of the chamber.

• • Analysis set 4 deals with the hexcel sheet without carbon rods. Under the same conditions, the deflection is reduced from 19mm to 0.3mm.

• • Analysis set 5 is a combination of set3 & set4, which is the result of both carbon rods and hexcel sheet. without the ball joint the deflection observed is 0.12mm, by adding the ball joint, the deflection is further reduced to 0.02mm, under the influence of gravity, the deflection slightly increased to 0.06mm.

• • Later, just the see the gravity effects on the chamber with all the materials, analysis set 6 is only a gravity analysis with no external loading. Like in analysis set 1, results were taken without and with the ball joint. The max deflections are observed to be 0.58mm and 0.07mm.

This tells us that the ball joint is very supportive in reducing the max deflection in every case. The Next step is to add the mylar sheet and the gas pressures. Detailed views of these results were shown as follows.

Analysis 1

Analysis 1.1

A Report of the analysis is available at File:Analysis 1.1.pdf

As the nose is free to move, the Maximum Total Deflection is observed at the nose and the deflection is gradually reduced towards the hinge side. Due to self weight, the deflection in X-direction is close to zero at those hinges and maximum in the negative side of x. The Y-deflection is more at the middle of the end plates in opposite directions. The Z-deflection is more at the nose plate as the chamber is observed to be tilted towards the Target center in the negative Z-side.

Figure 15: Images of Deflection for Analysis 1.1 in D,X,Y,Z Directions respectively

Analysis 1.2

A Report of the analysis is available at File:Analysis 1.2.pdf

In this case The nose plate is supported with a ball joint under the nose plate. so, It is like the chamber is made to rest on the ball joint with supported hinges on the top. so, the total deformation at the nose plate and the hinges is observed to be minimum and the end plates gave an impression as they sag in the middle. Other than that, the base plate on the top also sagged below due to its own self weight. Due to gravity effects, the X-direction deflection is observed to be maximum almost at the center of the endplates. The Y-deflection gave an impression as it is symmetric in the way the end plates are bowed outward. Z-deformation is maximum in the direction away away from the target center at the middle of the back plate. The back plate on each piece will not be mounted parallel to the ground. so, the deflection is in the positive z direction.

Figure 16: Images of Deflection for Analysis 1.2 in D,X,Y,Z Directions respectively

Analysis 2

Analysis 2.1

A Report of the analysis is available at File:Analysis 2.1.pdf

In this analysis, the deformation is pretty much dominated by the wire tensions. Now the endplates are bowed inward and the base plate has minimum deflection whatsoever, The nose plate side is slightly pulled towards the target side even without the gravity effects and it can be observed in the Z-deformation image. Overall, the deformation is too huge and is approximately about 20mm.

Figure 17: Images of Deflection for Analysis 2.1 in D,X,Y,Z Directions respectively

Analysis 2.2

A Report of the analysis is available at File:Analysis 2.2.pdf

The deformations observed with the ball joint is pretty close to the deformations observed without a ball joint. however, the deformations are slightly lesser than previous especially at the nose plate with the extra constraint of the ball joint.

Figure 18: Images of Deflection for Analysis 2.2 in D,X,Y,Z Directions respectively

Analysis 2.3

A Report of the analysis is available at File:Analysis 2.3.pdf

In this case the addition of gravity reduced the magnitude of deflection. With out the wire tensions, under gravity the end plates are bowed outward about 2mm in analysis 0.2 and with the wire tensions under no gravity, they are deflected inwards for about 19.5mm in analysis 2.2. Now by putting both the cases together, the deformation is a compromise of both and it is observed as 18.6mm bowing inwards

Figure 19: Images of Deflection for Analysis 2.3 in D,X,Y,Z Directions respectively

Analysis 3

Addition of carbon rods reduced the deflection to a very great extent, from 19-20mm to about 1.5-1.0mm. The maximum deflection is observed on the middle sections of the carbon rods. They are bowed outwards. the end plate towards the carbon rod side is almost zero and the non carbon rod side of the end plate has 0.8-1.15mm. With the ball joint and the influence of gravity, they are some slight deformations on the back plate towards the gravity. However the role of carbon rods made things quite clear by reducing the deflection the greatest extent.

Analysis 3.1

A Report of the analysis is available at File:Analysis 3.1.pdf

Figure 20: Images of Deflection for Analysis 3.1 in D,X,Y,Z Directions respectively

Analysis 3.2

A Report of the analysis is available at File:Analysis 3.2.pdf

Figure 21: Images of Deflection for Analysis 3.2 in D,X,Y,Z Directions respectively

Analysis 3.3

A Report of the analysis is available at File:Analysis 3.3.pdf

Figure 22: Images of Deflection for Analysis 3.3 in D,X,Y,Z Directions respectively

Analysis 4

Analysis set 4 is with the hexcel sheet and with no carbon rods. hexcel sheet is mounted on the downstream side of the flow. As expected hexcel sheet reduced the deflection even better than the carbon rods, From 19mm to 0.3mm. The max deflection is observed at the non hexcel sheet side on the end plates. Symmetric deflection is observed. However, the hexcel sheet is bowed inward i.e towards the wire inside the chamber by about. 0.025mm. This can be observed in the Z-deformation images. presence of the ball joint didnt make a lot of difference.

Analysis 4.1

A Report of the analysis is available at File:Analysis 4.1.pdf

Figure 23: Images of Deflection for Analysis 4.1 in D,X,Y,Z Directions respectively

Analysis 4.2

A Report of the analysis is available at File:Analysis 4.2.pdf

Figure 24: Images of Deflection for Analysis 4.2 in D,X,Y,Z Directions respectively

Analysis 4.3

A Report of the analysis is available at File:Analysis 4.3.pdf

Figure 25: Images of Deflection for Analysis 4.3 in D,X,Y,Z Directions respectively

Analysis 5

The Individual addition of both carbon rods and hexcel sheet in the geometry greatly reduced the maximum deformation. Now, in this analysis Both of them are modeled at the same time. In the very first analysis, without the ball joint, the maximum deflection is observed at the nose plate as it is free to move. however, after adding ball joint, the deflection is observed to be 20 microns at one of the carbon rod bowing away from the wires. The max deflection at the edge of the end plate is about 16 microns. Gravity increased the deflection at the carbon rod and made a very slight change on the deflection at the back plate and along the edge of the end plate.

Analysis 5.1

A Report of the analysis is available at File:Analysis 5.1.pdf

Figure 26: Images of Deflection for Analysis 5.3 in D,X,Y,Z Directions respectively

Analysis 5.2

A Report of the analysis is available at File:Analysis 5.2.pdf

Figure 27: Images of Deflection for Analysis 5.2 in D,X,Y,Z Directions respectively

Analysis 5.3

A Report of the analysis is available at File:Analysis 5.3.pdf

Figure 28: Images of Deflection for Analysis 5.3 in D,X,Y,Z Directions respectively

Analysis 6

Analysis 6.1

A Report of the analysis is available at File:Analysis 6.1.pdf

Figure 29: Images of Deflection for Analysis 6.1 in D,X,Y,Z Directions respectively

Analysis 6.2

A Report of the analysis is available at File:Analysis 6.2.pdf

Figure 30: Images of Deflection for Analysis 6.2 in D,X,Y,Z Directions respectively

References

1. [Mestayer07] "Tracking Detectors for CLAS12", 1/9/07, Mac Mestayer, Collaboration Document.
2. [Mestayer00] M.D. Mestayer, et. al., NIM, A449, pg 81, 2000 Media:NIM_A449.pdf
3. [Craig96] Mechanics of Materials, Roy R. Craig Jr., John Wiley & Sons, 1996

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