Standard Forms of System of Equations

Standard Forms of System of Equations - ME375 Handouts...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ME375 Handouts Standard Forms for System Models • State Space Model Representation – 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 t = t0 , together with the knowledge of the input for t ≥ t0 completely determines the behavior (the values of the state variables) of the completely variables) 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) for t ≥ t0 ? for School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 2 1 ME375 Handouts State Space Model Representation • State Space Representation – Two parts: • A set of first order ODEs that represents the derivative of each state variable qi as an algebraic function of the set of state variables {qi} and the inputs {ui}. ⎧ q1 = f 1 (q1 , q 2 , q 3 , … , q n , u1 , u 2 , u 3 , … , u m ) ⎪ q = f (q , q , q , … , q , u , u , u , … , u ) ⎪2 2 1 2 3 n 1 2 3 m ⎨ ⎪ ⎪ q n = f n (q1 , q 2 , q 3 , … , q n , u1 , u 2 , u 3 , … , u m ) ⎩ • A set of equations that represents the output variables as algebraic algebraic functions of the set of state variables {qi} and the inputs {ui}. ⎧ y1 = g 1 ( q 1 , q 2 , q 3 , … , q n , u 1 , u 2 , u 3 , … , u m ) ⎪ ⎪ y 2 = g 2 ( q1 , q 2 , q 3 , … , q n , u 1 , u 2 , u 3 , … , u m ) ⎨ ⎪ ⎪ y = g (q , q , q , … , q , u , u , u , … , u ) 1 2 3 1 2 3 p n m ⎩p School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 3 State Space Model Representation • Example EOM M x + B x + K x = f (t ) K M B x f(t) 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 translational mechanical systems. – Eliminate all algebraic equations written in the modeling process. process. – Express the resulting differential equations in terms of state variables and variables inputs in coupled first order ODEs. – Express the output variables as algebraic functions of the state variables and inputs. – For linear systems, put the equations in matrix form. x = A⋅ x State Variables in vector form +B⋅ u Inputs in vector form y Outputs in vector form = C ⋅x + D⋅ u School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 5 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 x1 + B1 x1 − B1 x2 + K1 x1 − K1 x2 = 0 M 2 x2 − B1 x1 + B1 x2 − K1 x1 + ( K1 + K 2 ) x2 = K 2 xr State Variables: Output: State Space Representation: K2 xr M1 g K1 M2 B1 x2 x1 ⎡ q1 ⎤ ⎡ ⎢q ⎥ ⎢ ⎢ 2⎥ = ⎢ ⎢ q3 ⎥ ⎢ ⎢⎥⎢ ⎣ q4 ⎦ ⎣ x A ⎤ ⎡ q1 ⎤ ⎡ ⎥ ⎢q ⎥ ⎢ ⎥⎢ 2⎥ + ⎢ ⎥ ⎢ q3 ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎦ ⎣ q4 ⎦ ⎣ x B ⎤ ⎥ ⎥ xr ⎥ ⎥ ⎦ , y =[ C ⎡ q1 ⎤ ⎢q ⎥ ] ⎢ q2 ⎥ ⎢ 3⎥ ⎢⎥ ⎣ q4 ⎦ x School of Mechanical Engineering Purdue University 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, u(t), and the output variable, y(t), of a system. For linear time-invariant (LTI) systems, it can be represented by : time- an y ( n ) + (n) where y =( + a2 y + a1 y + a0 y = bmu ( m ) + dt d ) + b2u + b1u + b0u (t ) n y – 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 Vibration Absorber EOM: z2 M2 K2 z1 M1 K1 B1 M 1 z1 + B1 z1 + ( K1 + K 2 ) z1 − K 2 z2 = f (t ) M 2 z2 + K 2 z2 − K 2 z1 = 0 – Find input/output representation between input f(t) and output z2. f(t) School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 8 4 ME375 Handouts Input/Output Representation • Vibration Absorber z2 M2 K2 z1 M1 K1 B1 f(t) Q: Find input/output representation between input f(t) and output z1. Find Q: What if another damper is added between masses M1 and M2 ? What School of Mechanical Engineering Purdue University ME375 Standard Forms of Equation - 9 Comments on Input/Output and State Space Models • State Space Models: – consider the internal behavior of a 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 - 10 5 ME375 Handouts Comments on Input/Output and State Space Models • Input/Output Models: – are 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 - 11 6 ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern