# r1_lti - 13.42 Design Principles for Ocean Vehicles Reading...

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13.42 Design Principles for Ocean Vehicles Reading #1 13.42 Design Principles for Ocean Vehicles Prof. A.H. Techet Spring 2005 1. Dynamical Systems Dynamical systems are representations of physical objects or behaviors such that the output of the system depends on present and past values of the input to the system. For example: t y t ) = ut 1 () dt 1 ( t - 3 3 ( ) + n = 1 u tn d ) = N ( - In order to model dynamical systems we need to build a set of tools and guidelines that can be used to analyze systems such as a ship in waves. This section will introduce tools for analyzing linear systems. System: Recognize a set of physical objects (behaviors) of interest Modeling: Representing the behavior of this system through a set of equations that approximate the original physical system. Inputs: Identify external actions influencing the system behavior. Outputs: Identify the outputs of interest. 1.1. Time Invariant System Systems are time invariant if their behavior and characteristics do not vary over time. In other words, if the input to a system is shifted in time, the resulting output experiences an identical time shift. In order to determine whether the system is time invariant, we use the following procedure in three steps: ©2004, 2005 A. H. Techet 1 Version 3.1, updated 2/2/2005

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13.42 Design Principles for Ocean Vehicles Reading #1 Step 1: Replace () ut by ( ut + t ) (Change of variables) Step 2: yt ( yt + t ) (Replace all occurrences of t with t + t ) Step 3: Are the results from steps 1 and 2 equal? To illustrate this procedure we can use a few simple examples of basic systems with () (). input, xt , and output, ( 34 ( / Example 1: y t ) = [( )] System is clearly time invariant: + t ) = [ + t ) ] ( / t dt Check time invariance: Example 2: ) = u 1 1 ( 0 Step (1): Plug in t 1 + t for t 1 on the RHS and perform a change of variables (let =+ t ). Note that the limits of integration must also shift with this change of variables. z t 1 t t + t ( zz 1 + t ) d 1 = t u d 0 Step (2): Plug in t + t t on the LHS. Notice that the limits of integration do not change in the same fashion as in step 1. The original integral on the RHS is bounded from zero to t , and since we are simply replacing all occurrences of t t + t we do not shift the limits of integration as we did in step 1. t + t dt + t ) = 1 1 ( 0 Step (3): Compare results from steps (1) and (2). They are not equal, therefore this system is not time invariant. t + t t + t u ( 1 ) d 1 t 0 ©2004, 2005 A. H. Techet 2 Version 3.1, updated 2/2/2005
13.42 Design Principles for Ocean Vehicles Reading #1 t 4 Example 3: y t ) = ut 1 () dt 1 ( t - 5 Step (1): Plug in t 1 + t for t 1 on the RHS and perform a change of variables (let z t 1 =+ t ): t u 4 ( t + t ) d = t + t u 4 d zz t - 5 1 1 5 t -+ t Step (2): t + t for t on the LHS, and again, note the shift in integration limits: t + t + t ) = 5 1 1 ( 4 dt t t Step (3): Compare steps (1) and (2). They are equivalent, therefore system is time invariant! t + t 4 = 5 1 1 u 4 ( ) d t + t 5 t - + t t t 1.2. Linear Dynamical System A subset of dynamical systems is linear dynamical systems. A system is considered to be linear if it satisfies properties of linear superposition and scaling. Typically we can represent, mathematically, a system with some input, () yt . Figure 1 xt , and output, ()

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## This note was uploaded on 02/27/2012 for the course MECHANICAL 2.22 taught by Professor Alexandratechet during the Spring '05 term at MIT.

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r1_lti - 13.42 Design Principles for Ocean Vehicles Reading...

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