### Post by Tom Goodrick on Jul 27, 2009 11:50:46 GMT -5

Somehow in moving the Forum, we lost the articles on how to calculate MOI's. This is a very important element of setting up the correct model of an aircraft to achieve proper handling in all phases of flight. Below I'll discuss why this is important and how to make the calculations. It is necessary to use some tables for basic aircraft shapes so I'll give some table values you can use.

The MOI values can be found in the aircraft.cfg file for each FS aircraft.

WHY MOI?

The moments of inertia give the mass distribution of an aircraft relative to the roll, pitch and yaw axes. These values relate to rotational accelerations as mass does to translational acceleration. The big difference is that MOI's are different about each axis where mass is the same for any direction of acceleration. And, the MOI differences about other axes combine to change the response about an axis. This means that it is important to get these values close to the right values inorder to have a good general representation of the response of the aircraft to control inputs, especially when an input is followed by complex motion such as a spin.

Unfortunately, for years, the FS development community has used MOI's incorrectly to fix handliing issues for which there are better fixes. For example if an airliner rolls too easily, instead of the direct fix which includes limiting the roll input and adding roll damping, designers just increase the roll MOI. That has unfortunate consequences.

There are some things you can do to spot bad MOI's just by glancing at them. each MOI is reduced if a large mass is close to the axis. Engines on the wing increase the roll MOI while engines on the aft fuselage decrease the MOI. Anything inside or close to the fuselage has little effect on the roll MOI. The wing and anything on it has little effect on pitch MOI but the length of the fuselage and size of the tail do influence the pitch MOI. The yaw MOI is about the vertical axis and that includes just about everything on the aircraft. Givine these conditions, you should always see

MOIX < MOIY < MOIZ or roll < pitch < yaw.

In some designs, roll and pitch MOI's are about the same. But they should never be more than the yaw MOI.

Here is the equation endorsed by Professor Roskam who teaches aeronautical engineering at Kansas State university and has written several sexts on the topic. He gives lectures to NASA and to all the aerospase companies including, especially Beechcraft and Cessna. His method is based on dimensionless radii of gyration for basic aircraft types which are given in tables he publishes. I had permission to publish some of these in my Abacus book on FS Flight Dynamics.

Here are the equations, in the notation of BASIC:

IX = ((RX * b)^2) / 4) * BOW / 32.2

IY = ((RY * L)^2) / 4) * BOW / 32.2

IZ = ((RZ * M)^2) / 4) * BOW / 32.2

where b=span. L=length, M=(b+L)/2 and BOW is basic operating weight for large aircraft or empty weight for small aircraft.

To get the values of RX, RY and RZ, look them up in the following table:

Aircraft_____BASICS SHAPE________RX___RY____RZ

747________Jet 4-eng on wing____0.332_0.380_0.508

737________Jet 2-eng on wing____0.264_0.456_0.517

DC-9_______Jet 2-eng on fuselage_0.242_0.360_0.435

727________Jet 3-eng on fuselage_0.247_0.442_0.518

Cessna 172__1-eng high wing_____0.242_0.386_0.403

Beech Bonan_1-eng low wing______0.248_0.338_0.393

Beech Baron__light twin__________0.260_0.329_0.399

DHC6 T Otter__twin turbo_________0.203_0.326_0.350

Cessna 550___light biz jet________0.243_0.400_0.447

Martin 404____twin radial_________0.272_0.378_0.444

DC-6_________four radial_________0.322_0.324_0.456

If you have a radically different aircraft, such as a WWII fighter, email me. There are many more of these.

WHY DOES THIS METHOD WORK

The main way of calculaitng an MOI is to look at all masses and the X, Y and Z position of the mass, plus the MOI of the mass itself if it is significant (like an engine) and making the calculations:

MOIX = SUM(moix+(y^2+z^2) * m)

MOIY = SUM(moiy+(x^2+z^2) * m)

MOIZ = SUM(moiz+(x^2+y^2) * m)

where x,y znd z are the position coordinates of each mass m. In most cases the second term is much larger than the first term so it can be neglected.

So by using an empty weight, the right span and length and a parameter that represents the right shape, you get a set of MOI's that is pretty close to the aircraft you are modelling. If you chose an aircraft from the table, you get an exact value.

HOW THE MOI'S ARE USED IN FS

The aircraft,cfg file in FS also has the specific geometry data for fuel tank locations and for locations of any people or cargo being carried. When you load up the aircraft with payload and fuel, the MOI's are recalculated so that very realistic MOI's are used in the flight dynamics calculations. As fuel burns off in a jet these mOI's change and you will notice the changes in handling between maneuvers after takeoff and maneuvers just before landing.

The MOI values can be found in the aircraft.cfg file for each FS aircraft.

WHY MOI?

The moments of inertia give the mass distribution of an aircraft relative to the roll, pitch and yaw axes. These values relate to rotational accelerations as mass does to translational acceleration. The big difference is that MOI's are different about each axis where mass is the same for any direction of acceleration. And, the MOI differences about other axes combine to change the response about an axis. This means that it is important to get these values close to the right values inorder to have a good general representation of the response of the aircraft to control inputs, especially when an input is followed by complex motion such as a spin.

Unfortunately, for years, the FS development community has used MOI's incorrectly to fix handliing issues for which there are better fixes. For example if an airliner rolls too easily, instead of the direct fix which includes limiting the roll input and adding roll damping, designers just increase the roll MOI. That has unfortunate consequences.

There are some things you can do to spot bad MOI's just by glancing at them. each MOI is reduced if a large mass is close to the axis. Engines on the wing increase the roll MOI while engines on the aft fuselage decrease the MOI. Anything inside or close to the fuselage has little effect on the roll MOI. The wing and anything on it has little effect on pitch MOI but the length of the fuselage and size of the tail do influence the pitch MOI. The yaw MOI is about the vertical axis and that includes just about everything on the aircraft. Givine these conditions, you should always see

MOIX < MOIY < MOIZ or roll < pitch < yaw.

In some designs, roll and pitch MOI's are about the same. But they should never be more than the yaw MOI.

Here is the equation endorsed by Professor Roskam who teaches aeronautical engineering at Kansas State university and has written several sexts on the topic. He gives lectures to NASA and to all the aerospase companies including, especially Beechcraft and Cessna. His method is based on dimensionless radii of gyration for basic aircraft types which are given in tables he publishes. I had permission to publish some of these in my Abacus book on FS Flight Dynamics.

Here are the equations, in the notation of BASIC:

IX = ((RX * b)^2) / 4) * BOW / 32.2

IY = ((RY * L)^2) / 4) * BOW / 32.2

IZ = ((RZ * M)^2) / 4) * BOW / 32.2

where b=span. L=length, M=(b+L)/2 and BOW is basic operating weight for large aircraft or empty weight for small aircraft.

To get the values of RX, RY and RZ, look them up in the following table:

Aircraft_____BASICS SHAPE________RX___RY____RZ

747________Jet 4-eng on wing____0.332_0.380_0.508

737________Jet 2-eng on wing____0.264_0.456_0.517

DC-9_______Jet 2-eng on fuselage_0.242_0.360_0.435

727________Jet 3-eng on fuselage_0.247_0.442_0.518

Cessna 172__1-eng high wing_____0.242_0.386_0.403

Beech Bonan_1-eng low wing______0.248_0.338_0.393

Beech Baron__light twin__________0.260_0.329_0.399

DHC6 T Otter__twin turbo_________0.203_0.326_0.350

Cessna 550___light biz jet________0.243_0.400_0.447

Martin 404____twin radial_________0.272_0.378_0.444

DC-6_________four radial_________0.322_0.324_0.456

If you have a radically different aircraft, such as a WWII fighter, email me. There are many more of these.

WHY DOES THIS METHOD WORK

The main way of calculaitng an MOI is to look at all masses and the X, Y and Z position of the mass, plus the MOI of the mass itself if it is significant (like an engine) and making the calculations:

MOIX = SUM(moix+(y^2+z^2) * m)

MOIY = SUM(moiy+(x^2+z^2) * m)

MOIZ = SUM(moiz+(x^2+y^2) * m)

where x,y znd z are the position coordinates of each mass m. In most cases the second term is much larger than the first term so it can be neglected.

So by using an empty weight, the right span and length and a parameter that represents the right shape, you get a set of MOI's that is pretty close to the aircraft you are modelling. If you chose an aircraft from the table, you get an exact value.

HOW THE MOI'S ARE USED IN FS

The aircraft,cfg file in FS also has the specific geometry data for fuel tank locations and for locations of any people or cargo being carried. When you load up the aircraft with payload and fuel, the MOI's are recalculated so that very realistic MOI's are used in the flight dynamics calculations. As fuel burns off in a jet these mOI's change and you will notice the changes in handling between maneuvers after takeoff and maneuvers just before landing.