Q1 Figure Q1 illustrates a multiple mass-spring-damper system. FL y Mass 1 2M 4K www 2C...

 

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Q1 Figure Q1 illustrates a multiple mass-spring-damper system. FL yı Mass 1 2M 4K www 2C E y2 M M Mass 2 4M d²y₁ dt² Figure Q1 The three masses (2M kg, 4M kg and 2M kg respectively) are connected to each other by means of two sets of springs and dampers. Each set has an associated spring constant (4K N/m and 2K N/m) and viscous damping coefficient (2C N/(m/s) and 4C N/(m/s)). Input forces FL N and FR N are applied to Mass 1 and Mass 3 respectively. These cause the three masses to move over the frictionless surface they are resting upon. The resulting displacements of the masses are yım, y2 m and y3 m respectively. 2K www 4C (a) By considering the description of the system given above and starting from first principles, prove that the coupled motion of the three masses can be described by the following differential equations: I d²y3 M- - 2C- dt² Mass 3 2M day₁ + C- +2KAY₁ 0.5F₁ = 0 dt day₂ dt - FR dy²-c(0.5 day₁ day2) day2) - K(Ay₁ - 0.5Ay₂) = 0 dt -KAY₂ + 0.5FR = 0 Here Ayı represents the difference in displacement between Mass 1 and Mass 2, Ay2 represents the difference in displacement between Mass 2 and Mass 3. [15] (b) By making an appropriate choice of state variables, derive state equations for this system. Represent your state equations in Standard State Space Form. [10] Q1 Figure Q1 illustrates a multiple mass-spring-damper system. FL yı Mass 1 2M 4K www 2C E y2 M M Mass 2 4M d²y₁ dt² Figure Q1 The three masses (2M kg, 4M kg and 2M kg respectively) are connected to each other by means of two sets of springs and dampers. Each set has an associated spring constant (4K N/m and 2K N/m) and viscous damping coefficient (2C N/(m/s) and 4C N/(m/s)). Input forces FL N and FR N are applied to Mass 1 and Mass 3 respectively. These cause the three masses to move over the frictionless surface they are resting upon. The resulting displacements of the masses are yım, y2 m and y3 m respectively. 2K www 4C (a) By considering the description of the system given above and starting from first principles, prove that the coupled motion of the three masses can be described by the following differential equations: I d²y3 M- - 2C- dt² Mass 3 2M day₁ + C- +2KAY₁ 0.5F₁ = 0 dt day₂ dt - FR dy²-c(0.5 day₁ day2) day2) - K(Ay₁ - 0.5Ay₂) = 0 dt -KAY₂ + 0.5FR = 0 Here Ayı represents the difference in displacement between Mass 1 and Mass 2, Ay2 represents the difference in displacement between Mass 2 and Mass 3. [15] (b) By making an appropriate choice of state variables, derive state equations for this system. Represent your state equations in Standard State Space Form. [10]

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