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MM326 SYSTEM DYNAMICS_hw1_sol.pdf

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MM326 SYSTEM DYNAMICS Homework 1 Solution Prepared by Nurdan Bilgin QUESTIONS: Question 1: (Adapted From Mühendislik Sistemlerinin Modellenmesi ve Dinamiği, 2. Ed. Yücel Ercan page:305). Consider the simple model of a vehicle shown below, where the suspended mass is M; the main suspension stiffneses are K 1 and K 2 ; the damper coefficient is B; the tire mass is m; the tire spring stiffness is K 3 ; and the velocity input from the roadway is V(t). Obtain the differential equation o
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  MM326 SYSTEM DYNAMICS Homework 1 Solution Prepared by Nurdan Bilgin QUESTIONS: Question 1:   (Adapted From Mühendislik Sistemlerinin Modellenmesi ve Dinamiği, 2. Ed. Yücel Ercan page:305) .  Consider the simple model of a vehicle shown below, where the suspended mass is M; the main suspension stiffneses are K  1 and K  2 ; the damper coefficient is B; the tire mass is m; the tire spring stiffness is K  3 ; and the velocity input from the roadway is V(t). Obtain the differential equation of the dynamic system in terms of position of M. Figure 1 Solution3: Elemental Equations B-S=7-1=6 , N-1=4-1=3 Continuity eq. and B-N+1=7-4+1=4 Compatibility eq. Elemental eq.    ̈       ̈                                        ̇    Continuity eq.                                       Compatibility eq.        (M1)                                                      ̇   ̇    ̇  ̈  ̈   ̈    From Compatibility Eqs.                             ̇          ̇   ̈  ̈                        ̈            ̈         ̈  ̈   ̈       ̈         ̈               ̈            ̈   Integrate the above eq. 2 times ̇            ̇                       From M1        and            then, ̇   ̇    ̇   integrate this result,    ∫   From (s1)      ̇  ̈  ̈       ̇  ̈            ̈                           ̇    From R1 and R2 subsitute into S2                                           ̇      Rearrange                                  ̇           If you want take the derivative all sides.                                                ̇    Question 2:   (From System Dynamics:An Introduction, Rowell and Wormley, 1997 Problems 4.11 page 116)  There are four different fluid systems as shown below, where C, I, R, P, Q and g represent the capacitance, inertance, resistance,  pump, flow rate and acceleration of gravity, respectively. For each fluid system, a- Determine the active, passive elements, and draw the oriented linear graph  b- Write all elemental, continuity and compatibility equations, c- For each system, find the differential equation that defines the dynamic behaviour in terms of the pressure difference on R  2 ? Figure 2 Solution2: 2.a) Elemantal Eq. B-S=5-1=4                                         Continuity Eq.N-1=4-1=3                           There is no need to compatibility equations; But                                                              ̈   ̇                                             ̈   ̇     Founded     and    values are inserted into (R1), and rearrange        ̇            ̈   ̇             ̈         ̇                ̈  (      ) ̇              2.b) It can be solve easily with the same way, You can see the result of the solution below.       ̈           ̇              2.c) It can be solve easily with the same way, You can see the result of the solution below.       ̈               ̇               R 2  
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