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[Kyle0819]

write simultaneous laplace-transform differential equation for the rotational mechanical network of figure 1.15. All the initial conditions are zero

 

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To derive the simultaneous Laplace-transform differential equations for the rotational mechanical network shown in Figure 1.15, we can apply the principles of mechanical dynamics and the laws of rotational motion. Let's denote the variables as follows:

θ1(t) - Angle of the first element (load) in radian
θ2(t) - Angle of the second element (gear) in radian
J1 - Moment of inertia of the first element (load)
J2 - Moment of inertia of the second element (gear)
T1(t) - Torque applied to the first element
T2(t) - Torque applied to the second element
B1 - Damping coefficient of the first element
B2 - Damping coefficient of the second element

The rotational mechanical network consists of two interconnected elements, and the connection is made via a gear. The equations for the rotational dynamics can be written as follows:

Equation 1:
J1 * d²θ1(t)/dt² = T1(t) - B1 * dθ1(t)/dt - K * (θ1(t) - θ2(t))

Equation 2:
J2 * d²θ2(t)/dt² = T2(t) - B2 * dθ2(t)/dt + K * (θ1(t) - θ2(t))

In Equation 1, the left side represents the moment of inertia of the first element multiplied by the second derivative of the angle with respect to time. The first term on the right side represents the torque applied to the first element, the second term represents the damping effect on the first element, and the third term represents the torque transmitted via the gear from the second element to the first element.

In Equation 2, the left side represents the moment of inertia of the second element multiplied by the second derivative of the angle with respect to time. The first term on the right side represents the torque applied to the second element, the second term represents the damping effect on the second element, and the third term represents the torque transmitted via the gear from the first element to the second element.

Note that K represents the stiffness of the gear connection. Assuming all initial conditions are zero, we can take the Laplace transform of the equations to obtain the simultaneous Laplace-transform differential equations for the rotational mechanical network.