Example: car suspensionSuppose y(t) is measuredfrom equilibrium position when gravity has set in. So gravity is canceled by spring force at eq. pos.∴There are two forces on m:y )ry(bf y(y-r)kelongationkfdampingspring&&&opposing opposing −⋅=⋅=⋅=Newton’s Law:or num=den=T.F.=H(s)=orkrrbkyybymrybrykym+=++−−−−==∑&&&&&&&&)()( force↑↑↑↑⎥⎦⎤⎢⎣⎡+=⎥⎦⎤⎢⎣⎡++ rmkrmbymkymby&&&&LL0101 bbaa⎥⎦⎤⎢⎣⎡mkmb⎥⎦⎤⎢⎣⎡mkmb1012012)()(asasbsbmksmbsmksmbsRsY+++=+++=)()()()(2sRsHsRmksmbsmksmbsY =+++=State Space Model• For linear motion– Define two state variables for each mass– x1=position, x2 = velocity; x1 dot = x2– x2 dot is acc and solve for it from Newton’s• For angular motion– Define two state variables for each rotating inertia– x1= angle, x2 = angular velocity; x1 dot = x2– x2 dot is angular acc and solve for it from Euler’s lawQuarter car suspensionrumkrmbxmkxmbrmkrmbymkymbyxxxruyxyx&&&&&&&&& :inputnor stateneither is variableOne:Problem :Then.;,Let 1222121++−−=++−−======umkxmkrmbxmbrmkymkymbrmbyxrmbxyxrmbyxrx+−−−=+−−=−=+==−=122222122 :ThenLet . of ridget toModify :Solution&&&&&&&&&u() ()uxxxyumbmkmbxxmbmkmbxx0010:formmatrix in model space
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