MIT2_017JF09_ch10

# MIT2_017JF09_ch10 - 10 VEHICLE INERTIAL DYNAMICS 10 73...

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± 10 VEHICLE INERTIAL DYNAMICS 73 10 VEHICLE INERTIAL DYNAMICS We consider the rigid body dynamics with a coordinate system aﬃxed on the body. will develop equations useful for the simulation of vehicles, as well as for understanding the signals measured by an inertial measurement unit (IMU). A common frame for boats, submarines, aircraft, terrestrial wheeled and other vehicles has the body-referenced x -axis forward, y -axis to port (left), and z -axis up. This will be the sense of our body-referenced coordinate system here. 10.1 Momentum of a Particle Since the body moves with respect to an inertial frame, dynamics expressed in the body- referenced frame need extra attention. First, linear momentum for a particle obeys the equality F ± = d ( m±v ) dt A rigid body consists of a large number of these small particles, which can be indexed. The summations we use below can be generalized to integrals quite easily. have F ± i + R ± i = d ( m i ±v i ) , dt where F ± i is the external force acting on the particle and R ± i is the net force exerted by all the other surrounding particles (internal forces). Since the collection of particles is not driven apart by the internal forces, we must have equal and opposite internal forces such that N R ± i =0 . i =1 Then summing up all the particle momentum equations gives N N ± ± d F ± i = ( m i i ) . dt i =1 i =1 Note that the particle velocities are not independent, because the particles are rigidly at- tached. Now consider a body reference frame, with origin 0 , in which the particle i resides at body- referenced radius vector ± r ; the body translates and rotates, and we now consider how the momentum equation depends on this motion.

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± ² ³ ´ µ 10 VEHICLE INERTIAL DYNAMICS 74 x y z Figure 2: Convention for the body-referenced coordinate system on a vehicle: x is forward, y is sway to the left, and z is heave upwards. Looking forward from the vehicle ”helm,” roll about the x axis is positive counterclockwise, pitch about the y -axis is positive bow-down, and yaw about the z -axis is positive turning left. 10.2 Linear Momentum in a Moving Frame The expression for total velocity may be inserted into the summed linear to give momentum equation N N µ µ d F ± i = ( m i ( ±v o + ±ω × ± r i )) dt i =1 i =1 N ∂±v o d µ = m + ω ± × m i ± r i , ∂t dt i =1 N where m = i =1 m i , and i = o + ω ± × ± r i . Further deﬁning the center of gravity vector ± r G such that N r G = m i ± r i , i =1 we have N µ ∂± d F ± i = m v o + m ( × ± r G ) . i =1 Using the expansion for total derivative again, the complete vector equation in body coor- dinates is F ± = µ N = m o + ω ± × o + d±ω × ± r G + × ( ω ± × ± r G ) . dt i =1 Now we list some conventions that will be used from here on: o = { u, v, w } (body-referenced velocity)
± ² ± ² ± ² = ± ³ ´ 10 VEHICLE INERTIAL DYNAMICS 75 ± r G = { x G ,y G ,z g } (body-referenced location of center of mass) ±ω = { p, q, r } (rotation vector, in body coordinates) F ± = { X, Y, Z } (external force, body coordinates) .

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## This note was uploaded on 11/29/2011 for the course CIVIL 1.00 taught by Professor Georgekocur during the Spring '05 term at MIT.

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MIT2_017JF09_ch10 - 10 VEHICLE INERTIAL DYNAMICS 10 73...

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