Standard Forms of Equation

Standard Forms of Equation - ME375 Handouts Standard Forms...

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Unformatted text preview: ME375 Handouts Standard Forms for System Models • State Space Model Representation – State Variables State Variables – Example • Input/Output Model Representation – General Form – Example • Comments on the Difference between State Space and Input/Output Model Representations School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 1 State Space Model Representation • State Variables The smallest set of variables {q1, q2, …, qn} such that the knowledge of these variables at time variables at time t = t0 , together with the knowledge of the input for t ≥ t0 together with the knowledge of the input for completely completely determines the behavior (the values of the state variables) of the system for time t ≥ t0 . Example: x K EOM: M f(t) B Q: What information about the mass do we need to know to be able to solve for x(t) What information about the mass do we need to know to be able to solve for for for t ≥ t0 ? School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 2 1 ME375 Handouts State Space Model Representation • State Space Representation – Two parts: set of first order ODEs that represents the derivative of each state • A set of first order ODEs that represents the derivative of each state variable variable qi as an algebraic function of the set of state variables {qi} and the inputs {ui}. & ⎧ q1 ⎪q &2 ⎪ ⎨ ⎪ ⎪ qn ⎩& = f 1 (q1 , q 2 , q 3 , K , q n , u1 , u 2 , u 3 , K , u m ) = f 2 (q1 , q 2 , q 3 , K , q n , u1 , u 2 , u 3 , K , u m ) M = f n (q1 , q 2 , q 3 , K , q n , u1 , u 2 , u 3 , K , u m ) • A set of equations that represents the output variables as algebraic functions of the set of state variables {qi} and the inputs {ui}. th th ⎧ y1 ⎪ ⎪ y2 ⎨ ⎪ ⎪y ⎩p = g 1 ( q1 , q 2 , q 3 , K , q n , u 1 , u 2 , u 3 , K , u m ) = g 2 ( q1 , q 2 , q 3 , K , q n , u 1 , u 2 , u 3 , K , u m ) M = g p ( q1 , q 2 , q 3 , K , q n , u1 , u 2 , u 3 , K , u m ) School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 3 State Space Model Representation • Example EOM & M && + B x + K x = f (t ) x x K M f(t) B State Variables: Output: State Space Representation: School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 4 2 ME375 Handouts State Space Model Representation • Obtaining State Space Representation – Identify State Variables • Rule of Thumb: – Nth order ODE requires N state variables. – Position and velocity are natural state variables for translational mechanical systems. – Eliminate all algebraic equations written in the modeling process. – Express the resulting differential equations in terms of state variables and inputs in coupled first order ODEs. – Express the output variables as algebraic functions of the state variables and inputs and inputs. – For linear systems, put the equations in matrix form. & x = A⋅ x { +B⋅ State Variables in vector form y { Outputs in vector form u { Inputs in vector form = C ⋅x + D⋅ u School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 5 State State Space Model Representation • Exercise Represent the 2 DOF suspension system in a state space representation. Let the system output be the position of mass M1. & & M 1&&1 + B1 x1 − B1 x2 + K1 x1 − K1 x2 = 0 x x1 M1 g K1 B1 x2 & & M 2 &&2 − B1 x1 + B1 x2 − K1 x1 + ( K1 + K 2 ) x2 = K 2 xr x M2 State Variables: K2 Output: xr State Space Representation: & ⎡ q1 ⎤ ⎡ ⎤ ⎡ q1 ⎤ ⎡ ⎤ ⎢q ⎥ ⎢ ⎥ ⎢q ⎥ ⎢ ⎥ &2 ⎢ ⎥=⎢ ⎥ ⎢ 2 ⎥ + ⎢ ⎥ xr & ⎢ q3 ⎥ ⎢ ⎥ ⎢ q3 ⎥ ⎢ ⎥ ⎢& ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ q4 ⎦ ⎣ q ⎣ { 144 2444 ⎦ ⎣ 4 ⎦ ⎣ ⎦ 4 3{ { & B x x A , School of Mechanical Engineering Purdue University ⎡ q1 ⎤ ⎢q ⎥ y =[ ⎢ q2 ⎥ 14 244 ⎢ 3 ⎥ 4 3 C ⎢⎥ ⎣ q4 ⎦ { x ME375 Standard Forms of Equation - 6 3 ME375 Handouts Input/Output Representation • Input/Output Model Uses one nth order ODE to represent the relationship between the input variable variable, u(t), and the output variable, y(t), of a system. and the output variable of system For linear time-invariant (LTI) systems, it can be represented by : time- & && & an y ( n ) + L + a2 && + a1 y + a0 y = bmu ( m ) + L + b2u + b1u + b0u (t ) y where y ( n ) = ( d dt ) y n – To solve an input/output differential equation, we need to know – To obtain I/O models: • Identify input/output variables. • Derive equations of motion. • Combine equations of motion by eliminating all variables except the input and output variables and their derivatives. School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 7 Input/Output Representation • Example z2 Vibration Absorber EOM: M2 & M 1&&1 + B1 z1 + ( K1 + K 2 ) z1 − K 2 z2 = f (t ) z M 2 &&2 + K 2 z2 − K 2 z1 = 0 z – Find input/output representation between input f(t) and output z2. K2 z1 M1 K1 School of Mechanical Engineering Purdue University f(t) B1 ME375 Standard Forms of Equation - 8 4 ME375 Handouts Comments on Input/Output and State Space Models • State Space Models: – consider the internal behavior of a system the internal behavior of system – can easily incorporate complicated output variables – have significant computation advantage for computer simulation – can represent multi-input multi-output (MIMO) systems and multimultinonlinear systems School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 9 Comments on Input/Output and State Space Models • Input/Output Models: – are conceptually simple conceptually simple – are easily converted to frequency domain transfer functions that are more intuitive to practicing engineers – are difficult to solve in the time domain (solution: Laplace transformation) School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 10 5 ME375 Handouts Example: General Mechanical Systems Rack and pinion system: School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 11 Example Continued School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 12 6 ME375 Handouts Example: General Mechanical Systems 1. Let x be measured from the unstretched state of the spring, find EOM in terms of x; 2. Let z be the motion of the system about the static equilibrium, find EOM in terms of z. School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 13 Example Continued School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 14 7 ...
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This note was uploaded on 12/23/2011 for the course ME 375 taught by Professor Meckle during the Fall '10 term at Purdue University-West Lafayette.

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