Appendix M: Torque and Bolt Capacity Calculations

B.1 Objective

This appendix presents the calculations used to verify:

Torque acting on the gimbal motor
Counterweight sizing
Bolt strength adequacy

B.2 Torque Due to Dish Assembly

The torque about the gimbal axis is calculated using:

Torque (T) = Force (F) × Distance (d)
Where: F = m × g

Dish Contribution

Given:

Mass of dish, m₁ = 7.4 kg
Distance from pivot, d₁ = 0.197 m

Step 1:
F₁ = 7.4 × 9.81 = 72.59 N

Step 2:
T₁ = 72.59 × 0.197 = 14.30 N·m

Mounting Plate Contribution

Given:
Mass of plate, m₂ = 1 kg
Distance from pivot, d₂ = 0.150 m

Step 1:
F₂ = 1 × 9.81 = 9.81 N

Step 2:
T₂ = 9.81 × 0.150 = 1.47 N·m

LNB and Feedhorn Contribution

Given:
Mass of LNB and feedhorn, m₃ = 1.6 kg
Distance from pivot, d₃ = 0.970 m

Step 1:
F₃ = 1.6 × 9.81 = 15.70 N

Step 2:
T₃ = 15.70 × 0.970 = 15.23 N·m

Total Torque

Ttotal = T₁ + T₂ + T₃
Ttotal = 14.30 + 1.47 + 15.23
Ttotal = 31.00 N·m

Final result:
Total torque from dish assembly = 31.00 N·m

B.3 Counterweight Calculation

To balance the system:

Moment from counterweight = Moment from dish assembly

Fc × dc = 31.00

Counterweight Plate Mass Calculation

The counterweight consists of mild steel plates with dimensions:

Length = 330 mm = 0.330 m
Width = 180 mm = 0.180 m
Thickness = 10 mm = 0.010 m

Volume of one plate:

V = 0.330 × 0.180 × 0.010
V = 0.000594 m³

Density of mild steel:

ρ = 7850 kg/m³

Mass of one plate:

mplate = ρ × V
mplate = 7850 × 0.000594
mplate = 4.66 kg

Total Counterweight Mass

Number of plates = 4

mc = 4 × 4.66
mc = 18.65 kg

Counterweight Force

Fc = mc × g
Fc = 18.65 × 9.81
Fc = 183.03 N

Counterweight Moment at Distance

Given:
dc = 206 mm = 0.206 m

Tcounterweight = 183.03 × 0.206
Tcounterweight = 37.70 N·m

B.4 Residual Torque

Net torque:

Tnet = Tdish − Tcounterweight

Tnet = 31.00 − 37.70
Tnet = -6.70 N·m

Final result:

Residual torque = -6.70 N·m

The negative sign indicates that the system is over-counterweighted, meaning the counterweight produces a larger moment than the dish side at this reference position.

##

B.5 Bolt Shear Calculation

Total load = 88.29 + 9.81 = 98.10 N

Number of bolts = 4

Load per bolt:

Fshear = 98.10 / 4
Fshear = 24.53 N

Final result:
Shear force per bolt = 24.53 N

##

B.6 Bolt Tension Due to Moment

Moment = 18.86 N·m

The pivoting brackets are directly attached to four M4 bolts arranged in a rectangular pattern.

Bolt layout:

• Horizontal spacing between left and right bolts = 158.28 × 2 = 316.56 mm
• Vertical spacing between top and bottom rows = 20.64 × 2 = 41.28 mm

Distance from centroid to each row:

y = 20.64 mm = 0.02064 m

For overturning about the horizontal axis (90 Degrees Elevation), the tensile force in the bolts is calculated using:

F = (M × y) / I

where:

I = Σ(y²)

Since there are 4 bolts, each located at y = 0.02064 m from the centroid:

I = 4 × (0.02064²)
I = 4 × 0.000426
I = 0.001704 m²

Tensile force in the most heavily loaded bolt:

F = (31.00 × 0.02064) / 0.001704
F = 0.640 / 0.001704
F = 375.6 N

Final result:
Maximum tensile force per bolt = 375.6 N

The two bolts on one row will be in tension, while the two bolts on the opposite row will be in compression. The horizontal spacing improves the rigidity of the connection, but the bolt tension generated by this overturning moment is governed primarily by the vertical spacing between the bolt rows.

B.7 Tensile Stress in Bolt

Stress formula:

σ = Force / Area

For M4 bolt:
Stress area ≈ 8.78 mm²

σ = 375.6 / 8.78
σ = 42.8 MPa

Final result:
Tensile stress = 42.8 MPa

B.8 Shear Stress in Bolt

Bolt cross-sectional area:

A = π × d² / 4
A = 3.142 × (4²) / 4
A = 12.57 mm²

τ = 24.53 / 12.57
τ = 1.95 MPa

Final result:
Shear stress = 1.95 MPa

##

B.9 Bearing Stress on Plate

Formula:

σb = Force / (d × t)

Given:

Force = 375.6 N
Bolt diameter = 4 mm
Plate thickness = 6 mm

σb = 375.6 / (4 × 6)
σb = 375.6 / 24
σb = 15.65 MPa

Final result:
Bearing stress ≈ 15.65 MPa

B.10 Conclusion

• Total dish-side torque = 31.00 N·m
• Counterweight of 18.65 kg at 206 mm produces 37.70 N·m
• System is over-counterweighted by 6.70 N·m at this reference position
• Bolt stresses remain significantly below allowable limits

Therefore:

• The M4 bolts are mechanically adequate
• The counterweight significantly reduces motor torque demand
• The system torque balance varies with angular position, explaining observed performance differences during operation

At 25 degrees, the gimbal is able to pivot downwards to 90 degrees, however, the motor does not have enough torque to turn the opposite direction.

Elevation Angle	Component	Mass (kg)	Perpendicular Distance from Pivot (mm)	Force, F = m x g (N)	Torque, T = F x d (N·m)
0°	Counterweight	18.65	206	183.03	37.70
	Dish	7.4	197	72.59	14.30
	Feedhorn + LNB	1.6	970	15.70	15.23
	Total Dish Side				29.53
	Residual Torque (Dish - Counterweight)				-8.17
	Condition				Over-counterweighted
25°	Counterweight	18.65	51	183.03	9.33
	Dish	7.4	284	72.59	20.62
	Feedhorn + LNB	1.6	372	15.70	5.84
	Total Dish Side				26.46
	Residual Torque (Dish - Counterweight)				17.13
	Condition				Dish-side heavy
45°	Counterweight	18.65	145.7	183.03	26.67
	Dish	7.4	139.3	72.59	10.11
	Feedhorn + LNB	1.6	685.9	15.70	10.77
	Total Dish Side				20.88
	Residual Torque (Dish - Counterweight)				-5.79
	Condition				Slightly over-counterweighted
90°	Counterweight	18.65	0	183.03	0
	Dish	7.4	0	72.59	0
	Feedhorn + LNB	1.6	0	15.70	0
	Total Dish Side				0
	Residual Torque (Dish - Counterweight)				0
	Condition				Balanced