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Unformatted text preview: 16 ROAD VEHICLE ON RANDOM TERRAIN 40 16 Road Vehicle on Random Terrain An autonomous, four-wheeled road vehicle travels over an uncertain terrain. You are asked to characterize the vertical motion of the center of gravity, and the pitching (nose-up vs. nose-down) response of the vehicle, as it crosses the terrain, using a linear model . We know that the vehicle chassis has a mass of m = 42 kg and a pitching moment of inertia of J = 25 kg m 2 . Consider the tires and suspension to have zero mass, but the front and back suspension systems are each modeled as having a spring of stiffness k = 1000 N/m , in parallel with a linear damper having coeﬃcient b = 100 N/ ( m/s ). The distance between the front and rear wheels is 2 l = 0 . 9 m . The vehicle travels at a speed U , which we will vary below. Because we consider U to be constant, you can use it directly to map between the horizontal terrain coordinate x and time t in this problem. The terrain roughness has been described only statistically (e.g., using an orbiting cam- era or some other remote sensing system). Its content is given by the following harmonic components: n 1 2 3 4 5 6 7 8 wave number, 1/m ( k n = 2 π/λ n ) 0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500 amplitude, m ( a n ) 0.01 0.02 0.02 0.04 0.03 0.02 0.01 0.03 0.05 0.01 0.02 9 10 11 Here, k n is the n ’th wave number (spatial frequency) and λ n is the n ’th wavelength (spatial period). We model the ground elevation as a function of the horizontal coordinate: h ( x ) = a n sin( k n x + φ n ) . n Notice that 2 l << min( λ ); the vehicle is much shorter than the smallest wavelength, so you can approximate h ( x ) ≈ [ h ( x + l ) + h ( x − l )] / 2 , h ( x ) = dh ( x ) /dx ≈ [ h ( x + l ) − h ( x − l )] / 2 l, h ( x ) = d 2 h ( x ) /dx 2 ≈ [ h ( x + l ) − h ( x − l )] / 2 l, and so on. 16 ROAD VEHICLE ON RANDOM TERRAIN 41 1. Develop a model that describes the response of the vehicle to terrain variations h ( x ). It should comprise two uncoupled second-order differential equations, one for vertical motion and the other for pitch. Make an annotated figure to go with the equations. You may like to use the fact that the vertical velocity of the ground seen from a reference frame moving horizontally at velocity U , is Uh ( x ). Newton’s laws and some geometry give us mz ¨ = − k [( z + lθ − h ( x + l )) + ( z − lθ − h ( x − l ))] − b [( ˙ z + lθ ˙ − Uh ( x + l )) + ( ˙ z − lθ ˙ − Uh ( x − l ))] ≈ − k [2 z − 2 h ( x )] − b [2 ˙ z − 2 Uh ( x )] ¨ Jθ = − kl [( z + lθ − h ( x + l )) − ( z − lθ − h ( x − l ))] − bl [( ˙ z + lθ ˙ − Uh ( x + l )) − ( ˙ z − lθ ˙ − Uh ( x − l ))] ≈ − kl [2 lθ − 2 lh ( x )] − bl...
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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.
- Spring '05