12.3 Center of Gravity Calculations, Flight Limits & Ballast Adjustments

12.3 Center of Gravity Calculations, Flight Limits & Ballast Adjustments

Once an aircraft has been leveled, weighed, and tare deducted, the certifying maintenance engineer must convert raw net scale reactions into a certified Empty Weight Center of Gravity (EWCG). Furthermore, the engineer must understand how subsequent equipment additions, component removals, and payload shifts alter the operational center of gravity across various flight phases. When modifications displace the center of gravity beyond certified airworthiness limits, mathematical ballast calculations must be executed to restore the aircraft to a compliant, airworthy condition.

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Empty Weight Center of Gravity Derivation from Scale Reactions

To compute the longitudinal empty weight CG, the engineer establishes the horizontal arm from the reference datum to each weighing point. On a typical tricycle landing gear configuration, the weighing points are the nose gear and the main landing gear.

                      TRICYCLE GEAR WEIGHING GEOMETRY

       Datum
         |                                     CG
         |              Nose Arm               |
         |<----------------------------------->|        Main Arm
         |                                     |<--------------------->|
         +=============================================================+
                 |                                          |
                 v                                          v
            [Nose Gear]                                [Main Gear]
              Net = Wn                                  Net = Wm
                 |<---------------- Wheelbase (D) --------->|

1. The Moment Summation Method (Datum Forward of Aircraft)

When the reference datum is located forward of the nose, all arms are positive numbers:

$$\text{Total Empty Weight} = W_N + W_{ML} + W_{MR}$$ $$\text{Total Moment} = (W_N \times \text{Arm}_N) + (W_{ML} \times \text{Arm}_{ML}) + (W_{MR} \times \text{Arm}_{MR})$$ $$\text{CG Station} = \frac{\text{Total Moment}}{\text{Total Empty Weight}}$$

Where:

• Wn = Net weight on nose gear
• Wml, Wmr = Net weight on left and right main gear
• Arm_n, Arm_ml, Arm_mr = Horizontal station arms from reference datum to gear centerlines

2. The Wheelbase Reaction Method (Datum at Main Gear Centerline)

If the datum is set at the main gear centerline, the main gear arm is zero (0.0). The CG distance forward of the main gear is derived directly from moment equilibrium:

$$\text{Distance from Main Gear to CG} = \frac{W_N \times D}{\text{Total Empty Weight}}$$

Where D is the horizontal distance between the nose gear and main gear (the wheelbase). The CG station is then computed relative to the datum by adding or subtracting this distance.

Fully Worked Weighing & CG Calculation

A twin-turboprop commuter aircraft is weighed in an enclosed hangar following an avionics retrofit. The reference datum is located 100.0 inches forward of the fuselage nose tip. Leveling checks are verified. Tare is deducted. The recorded data is as follows:

Computing the Empty Weight Center of Gravity:

$$\text{Empty Weight CG} = \frac{3,073,500.0\text{ kg}\cdot\text{in}}{7,700.0\text{ kg}} = \mathbf{399.16\text{ inches}}$$

If the certified Empty Weight CG range published in the TCDS is Station 395.0 to 403.0 inches, the calculated CG of 399.16 inches complies with airworthiness requirements.

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Operational Weight Regimes & Flight Envelopes

An aircraft's weight and balance changes continuously during flight as fuel burns, passengers move, and payloads are dropped. Certifying engineers and flight dispatchers evaluate four distinct operational weight regimes:

                     HIERARCHY OF AIRCRAFT WEIGHT REGIMES

   +-------------------------------------------------------------------+
   | Basic Empty Weight (BEW)   [Structure, Engines, Unusable Fuel]    |
   +-------------------------------------------------------------------+
                                    +
   | Operating Items            [Flight Crew, Galley Supplies, Manuals]|
   +-------------------------------------------------------------------+
                                    =
   | Operating Empty Weight (OEW)                                      |
   +-------------------------------------------------------------------+
                                    +
   | Payload                    [Passengers, Baggage, Cargo]           |
   +-------------------------------------------------------------------+
                                    =
   | Zero Fuel Weight (ZFW)     *** MUST NOT EXCEED MZFW LIMIT ***     |
   +-------------------------------------------------------------------+
                                    +
   | Takeoff Fuel               [Usable Mission Fuel]                  |
   +-------------------------------------------------------------------+
                                    =
   | Takeoff Weight (TOW)       *** MUST NOT EXCEED MTOW LIMIT ***     |
   +-------------------------------------------------------------------+
                                    -
   | Trip Fuel Burn             [Fuel Consumed in Flight]              |
   +-------------------------------------------------------------------+
                                    =
   | Landing Weight (LW)        *** MUST NOT EXCEED MLW LIMIT ***      |
   +-------------------------------------------------------------------+

The Critical Significance of Maximum Zero Fuel Weight (MZFW)

The Maximum Zero Fuel Weight (MZFW) is the maximum certified weight of the aircraft loaded with crew, passengers, and payload, but with zero usable fuel in the tanks.

• Structural Rationale (Wing Root Bending Relief): In flight, upward aerodynamic lift acts on the wings, exerting immense upward bending moments at the wing roots (where the wings join the fuselage). Fuel stored in wing tanks acts as a downward dead weight, partially offsetting and relieving this upward bending stress. Fuselage payload (passengers and cargo), however, acts directly downward on the fuselage, increasing wing root bending stress. Restricting the total fuselage weight via the MZFW ensures that wing root structural spars are never overstressed in the event the aircraft burns all its wing fuel down to empty.

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Alterations: Weight Shifts, Additions & Removals

Maintenance activities frequently involve removing obsolete equipment, installing modern components, or rearranging cabin seats and cargo compartments.

1. Weight Shift Formula

When an existing mass onboard the aircraft is moved from one longitudinal location to another, the total aircraft weight remains unchanged, but the center of gravity shifts in the direction of the movement:

$$\Delta \text{CG} = \frac{\text{Weight Shifted} \times \text{Distance Shifted}}{\text{Total Aircraft Weight}}$$ $$\text{New CG} = \text{Original CG} \pm \Delta \text{CG}$$

2. Equipment Addition or Removal Formula

When equipment is permanently added to or removed from the aircraft, both the total aircraft weight and total moment change:

$$\text{New CG} = \frac{\text{Original Total Moment} \pm (\text{Added or Removed Weight} \times \text{Arm})}{\text{Original Total Weight} \pm \text{Added or Removed Weight}}$$

Rule: Add weight and moment when installing equipment; subtract weight and moment when removing equipment.

Worked Example: Equipment Removal & Installation

An aircraft with an empty weight of 5,000 kg and an empty CG at Station 120.0 inches (Original Moment = 5,000 × 120.0 = 600,000 kg·in) undergoes an avionics upgrade:

1. An old navigation unit weighing 30 kg located at Station 40.0 inches is removed:
   • Removed Moment = 30 × 40.0 = 1,200 kg·in
2. A modern digital multi-function display weighing 10 kg is installed at Station 50.0 inches:
   • Added Moment = 10 × 50.0 = 500 kg·in

$$\text{New Total Weight} = 5,000 - 30 + 10 = \mathbf{4,980\text{ kg}}$$ $$\text{New Total Moment} = 600,000 - 1,200 + 500 = \mathbf{599,300\text{ kg}\cdot\text{in}}$$ $$\text{New CG} = \frac{599,300}{4,980} = \mathbf{120.34\text{ inches}}$$

The removal of the net 20 kg from the forward nose bay shifted the aircraft center of gravity slightly aft, from Station 120.00 inches to Station 120.34 inches.

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Ballast Calculations & Structural Installation Standards

When structural repairs, equipment retrofits, or mission configurations displace the aircraft empty or operating CG beyond certified boundaries, ballast must be installed to bring the CG back into the approved envelope.

Mathematical Derivation of Ballast

To determine the exact ballast weight required to shift an aircraft CG from its current out-of-limits position to a desired target position within the envelope:

$$\text{Ballast Weight} = \frac{\text{Aircraft Weight} \times \text{Desired CG Shift}}{\text{Ballast Arm} - \text{Desired CG Station}}$$

Where:

• Aircraft Weight = Weight prior to adding ballast
• Desired CG Shift = Absolute difference between Current CG and Desired Target CG
• Ballast Arm = Station arm where ballast will be physically mounted
• Desired CG Station = Target station location of the new center of gravity

Worked Ballast Calculation

An aircraft weighing 4,500 kg has its center of gravity at Station 88.0 inches. Due to heavy avionics removals in the nose, the CG has moved aft of the certified aft limit of Station 86.0 inches by 2.0 inches. The maintenance facility plans to install permanent ballast in the forward avionics bay at Station 20.0 inches to shift the CG forward to exactly the aft limit of Station 86.0 inches.

1. Identify variables:

   • Aircraft Weight = 4,500 kg
   • Current CG = 88.0 inches
   • Desired CG = 86.0 inches
   • Desired Shift = 88.0 - 86.0 = 2.0 inches
   • Ballast Arm = 20.0 inches

2. Apply the ballast formula:

   $$\text{Ballast Weight} = \frac{4,500 \times 2.0}{86.0 - 20.0} = \frac{9,000}{66.0} = \mathbf{136.36\text{ kg}}$$

3. Verification Check:

   • Original Moment = 4,500 × 88.0 = 396,000 kg·in
   • Ballast Moment = 136.36 × 20.0 = 2,727.2 kg·in
   • Total Moment = 396,000 + 2,727.2 = 398,727.2 kg·in
   • Total Weight = 4,500 + 136.36 = 4,636.36 kg
   • New CG = 398,727.2 / 4,636.36 = 86.00 inches (Matches the target exactly).

Airframe Installation Standards for Ballast

1. Permanent Ballast: Typically fabricated from high-density lead plates, steel bars, or depleted uranium/tungsten blocks. It must be bolted securely to primary structural members (such as fuselage bulkheads, longerons, or engine mount trusses) using certified AN/MS aerospace bolts, large load-distribution washers, and positive self-locking nuts or cotter-pinned castellated nuts. It must be clearly painted in bright red or yellow gloss paint and stenciled with durable lettering: $$\mathbf{PERMANENT\;BALLAST\;---\;DO\;NOT\;REMOVE\;---\;[WEIGHT]\;KG}$$
2. Temporary Ballast: Carried in cargo holds or baggage bays for specific light-payload flights. Must be secured with certified cargo tie-down straps or cargo nets capable of withstanding certified emergency landing forward and vertical G-loadings. It must be explicitly tagged and logged on the flight load manifest.

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Weight & Balance Documentation, Equipment List & Legal Sign-Off

Under EASA Part-66 and Part-M/Part-ML, any weighing or weight-and-balance alteration must be legally documented before the aircraft can be released to service:

1. Weight and Balance Report: An official engineering document recording the date and location of weighing, scale types and calibration dates, hangar ambient conditions, leveling means used, tare deductions, net reaction weights, calculated empty weight, and calculated EWCG.
2. Equipment List: A comprehensive inventory itemizing every piece of equipment included in the basic empty weight. Each entry specifies part number, description, manufacturer, serial number, mass, and station arm. Items are categorized as 'X' (Installed) or 'O' (Optional/Removed). When an engineer adds or removes an LRU, the Equipment List must be marked up and re-dated.
3. Loading Schedule / Trim Sheet: Operational charts or electronic flight bag (EFB) software algorithms enabling pilots and loadmasters to verify that payload, passenger seating, and fuel loading remain within the certified flight envelope across all stages of flight.
4. Certificate of Release to Service (CRS): An authorized Part-66 aircraft maintenance engineer must sign a CRS entering the new empty weight and EWCG into the aircraft technical logbook and updating the Aircraft Flight Manual (AFM) Weight and Balance Supplement.

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Realistic Maintenance Scenario & Common Exam Traps

Realistic Maintenance Scenario

A certifying Part-66 engineer is supervising the cabin reconfiguration of a corporate transport. Ten luxury passenger seats (weighing 25 kg each at Station 300) are removed, and a high-density configuration of sixteen commuter seats (weighing 18 kg each at Station 280) is installed. The engineer updates the weight and balance ledger by deducting 250 kg at Station 300 (-75,000 kg·in) and adding 288 kg at Station 280 (+80,640 kg·in). The net change (+38 kg and +5,640 kg·in) is added into the master empty weight report, and the equipment list is revised with new part numbers and serial numbers prior to CRS issuance.

Common Exam Traps

• Trap 1: Forgetting that ballast adds to total aircraft mass. In ballast calculations, the ballast weight increases both the moment and the total mass of the aircraft. Forgetting to include the ballast mass in the denominator when verifying the new CG is a frequent calculation error.
• Trap 2: Misunderstanding Zero Fuel Weight. Many technicians mistakenly assume that if Takeoff Weight is below Maximum Takeoff Weight, the aircraft is safe. If the cabin is overloaded with cargo beyond the MZFW limit, the wing root spars can fail in flight even if the aircraft is under its Maximum Takeoff Weight.
• Trap 3: Subtracting instead of adding moved weight in the shift formula. Moving weight aft ALWAYS shifts CG aft; moving weight forward ALWAYS shifts CG forward. Always perform a quick common-sense sanity check on your mathematical signs.