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The following EOMS describe the behavior of an electric motor attached to a torsional spring and damper: Li + Rz+v=Vin(t) IÔ = _k0 — bỏ ta Importantly, the equations are coupled by the fact that 7 = kz and vt = k₂0. Here, L, R, kv, I, k, b, and k, are constants. z is the current in the motor, and is the angular displacement of the motor (clearly then, is the motor's angular velocity). Finally, the input to the system is vin (t). Your tasks: A Substituting in the coupling terms (r = kiz and v = k0) into the EOMs, write the state space form of the system x Ax + Bu, assuming your states are a₁ =z, x2 = 0, and 23 = 0. Clearly identify your matrices. B Assume that the desired outputs of the system (to be measured) are the current z, the angle and the input Vin all as separate elements in the vector y. Write the output equation y = Cx + Du. Hint: D will NOT be zero in this case. Clearly identify your matrices C Assume there is no stiffness to the spring (k = 0). Using w(t) to represent the angular velocity and n(s) to represent its Laplace transform, derive the transfer function H(s) = Ω(8) Vin (s)

The following EOMS describe the behavior of an electric motor attached to a torsional spring and damper: Li + Rz+v=Vin(t) IÔ = _k0 — bỏ ta Importantly, the equations are coupled by the fact that 7 = kz and vt = k₂0. Here, L, R, kv, I, k, b, and k, are constants. z is the current in the motor, and is the angular displacement of the motor (clearly then, is the motor's angular velocity). Finally, the input to the system is vin (t). Your tasks: A Substituting in the coupling terms (r = kiz and v = k0) into the EOMs, write the state space form of the system x Ax + Bu, assuming your states are a₁ =z, x2 = 0, and 23 = 0. Clearly identify your matrices. B Assume that the desired outputs of the system (to be measured) are the current z, the angle and the input Vin all as separate elements in the vector y. Write the output equation y = Cx + Du. Hint: D will NOT be zero in this case. Clearly identify your matrices C Assume there is no stiffness to the spring (k = 0). Using w(t) to represent the angular velocity and n(s) to represent its Laplace transform, derive the transfer function H(s) = Ω(8) Vin (s)

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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The following EOMS describe the behavior of an electric motor attached to a torsional spring and damper: [Li + Rz + vb = Vin(t) IÖ= k0b0 + T Importantly, the equations are coupled by the fact that 7 = k₁z and v₂ = k₂0. Here, L, R, kv, I, k, b, and kt are constants. z is the current in the motor, and is the angular displacement of the motor (clearly then, is the motor's angular velocity). Finally, the input to the system is vin (t). Your tasks: A Substituting in the coupling terms (7 = ktz and v k0) into the EOMs, write the state space form of the system x = Ax + Bu, assuming your states are x₁ = z, x₂ = 0, and x3 = 0. Clearly identify your matrices. B Assume that the desired outputs of the system (to be measured) are the current z, the angle and the input Vin all as separate elements in the vector y. Write the output equation y = Cx + Du. Hint: D will NOT be zero in this case. Clearly identify your matrices C Assume there is no stiffness to the spring (k = 0). Using w(t) to represent the angular velocity and (s) represent its Laplace transform, derive the transfer function H(s) Vin (8) Ω(s)
Transcribed Image Text:The following EOMS describe the behavior of an electric motor attached to a torsional spring and damper: [Li + Rz + vb = Vin(t) IÖ= k0b0 + T Importantly, the equations are coupled by the fact that 7 = k₁z and v₂ = k₂0. Here, L, R, kv, I, k, b, and kt are constants. z is the current in the motor, and is the angular displacement of the motor (clearly then, is the motor's angular velocity). Finally, the input to the system is vin (t). Your tasks: A Substituting in the coupling terms (7 = ktz and v k0) into the EOMs, write the state space form of the system x = Ax + Bu, assuming your states are x₁ = z, x₂ = 0, and x3 = 0. Clearly identify your matrices. B Assume that the desired outputs of the system (to be measured) are the current z, the angle and the input Vin all as separate elements in the vector y. Write the output equation y = Cx + Du. Hint: D will NOT be zero in this case. Clearly identify your matrices C Assume there is no stiffness to the spring (k = 0). Using w(t) to represent the angular velocity and (s) represent its Laplace transform, derive the transfer function H(s) Vin (8) Ω(s)
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