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Consider the system in Figure 1 which shows a crane hoisting and moving a load which is attached to the end of a rope. The cart is supposed to deliver the load from one place to another. The objective is to move the cart in a way which does not cause the rope and its load to swing too much. This assignment explores the modelling of such a system and leads up to the design of a control system for the cart such that the load does not swing wildly. Figure 1: Schematic of crane hoisting a load. Although the actual system is highly nonlinear, if the rope is considered to be stiff with a fixed length L, the system can be modelled using the following differential equations : x_La''(t) = gphi(t) m_Tx_T''(t) = f_T(t) - m_Lgphi(t) x_La(t) = x_T(t) - x_L(t) x_L(t) = L*phi(t) where m_L is the mass of the load, m_T is the mass of the cart, x_T and x_L are displacements as defined in Figure 1, phi is the rope angle with respect to the vertical, and f_T is the force applied to the cart. In this problem, the mass of the cart, m_T can be considered a constant. For simplicity, you may choose m_T = 1 unit mass while the gravitational constant can be approximated as g = 10 m/s^2. The mass of the load, m_L and the length of the rope, L are parameters which vary for each hoisting problem. 1) Obtain the transfer function, Phi(s)/V_T(s) where V_T(s) = L{v_T(t)} and Phi(s) = L{phi(t)} are the Laplace transforms of the cart velocity, v_T(t) and the rope angle, phi(t) respectively. m_L=73 L=50

          Consider the system in Figure 1 which shows a crane hoisting and moving a load which is attached to the end of a rope. The cart is supposed to deliver the load from one place to another. The objective is to move the cart in a way which does not cause the rope and its load to swing too much. This assignment explores the modelling of such a system and leads up to the design of a control system for the cart such that the load does not swing wildly.

Figure 1: Schematic of crane hoisting a load.

Although the actual system is highly nonlinear, if the rope is considered to be stiff with a fixed length L, the system can be modelled using the following differential equations :

x_La''(t) = g*phi(t)
m_T*x_T''(t) = f_T(t) - m_L*g*phi(t)
x_La(t) = x_T(t) - x_L(t)
x_L(t) = L*phi(t)

where m_L is the mass of the load, m_T is the mass of the cart, x_T and x_L are displacements as defined in Figure 1, phi is the rope angle with respect to the vertical, and f_T is the force applied to the cart. In this problem, the mass of the cart, m_T can be considered a constant. For simplicity, you may choose m_T = 1 unit mass while the gravitational constant can be approximated as g = 10 m/s^2. The mass of the load, m_L and the length of the rope, L are parameters which vary for each hoisting problem.

1) Obtain the transfer function, Phi(s)/V_T(s) where V_T(s) = L{v_T(t)} and Phi(s) = L{phi(t)} are the Laplace transforms of the cart velocity, v_T(t) and the rope angle, phi(t) respectively.

m_L=73 L=50

Consider the system in Figure 1 which shows a crane hoisting and moving a load which is attached to the end of a rope. The cart is supposed to deliver the load from one place to another. The objective is to move the cart in a way which does not cause the rope and its load to swing too much. This assignment explores the modelling of such a system and leads up to the design of a control system for the cart such that the load does not swing wildly.

Figure 1: Schematic of crane hoisting a load.

Although the actual system is highly nonlinear, if the rope is considered to be stiff with a fixed length L, the system can be modelled using the following differential equations :

xLa”(t) = g*phi(t)
mT*xT”(t) = fT(t) - mL*g*phi(t)
xLa(t) = xT(t) - xL(t)
xL(t) = L*phi(t)

where mL is the mass of the load, mT is the mass of the cart, xT and xL are displacements as defined in Figure 1, phi is the rope angle with respect to the vertical, and fT is the force applied to the cart. In this problem, the mass of the cart, mT can be considered a constant. For simplicity, you may choose mT = 1 unit mass while the gravitational constant can be approximated as g = 10 m/s^2. The mass of the load, mL and the length of the rope, L are parameters which vary for each hoisting problem.

1) Obtain the transfer function, Phi(s)/VT(s) where VT(s) = LvT(t) and Phi(s) = Lphi(t) are the Laplace transforms of the cart velocity, vT(t) and the rope angle, phi(t) respectively.

mL=73 L=50

Added by Ronald O.

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