Question

1. A simplified model of lathe machine is shown in Figure 1 is supported by three springs. The lathe machine has mass $m$, total length $l$ and mass moment of inertia, taken about an axis that goes through its centre of gravity, C.G., is $J_o$. The positive vertical downward and clockwise rotational motions of the machine about the C.G. point can be described as $x(t)$ and $\theta(t)$ respectively. (a) Using Newton's Law method, prove that the equations of motion for this system are $m\ddot{x} + (k_1 + k_2 + k_3)x - (k_1l_1 - k_2l_2 - k_3l_3)\theta = 0$ and $J_o\ddot{\theta} - (k_1l_1 - k_2l_2 - k_3l_3)x + (k_1l_1^2 + k_2l_2^2 + k_3l_3^2)\theta = 0$ [16 marks] (b) Compute the system's characteristic equation, natural frequencies and mode shapes given that $m = 30kg$, $J_o = 13kg \cdot m^2$, $l = 25cm$, $l_1 = 30cm$, $l_2 = 15cm$, $k_1 = 1250 N/m$, $k_2 = 1850 N/m$ and $k_3 = 2250 N/m$. [14 marks]

          1. A simplified model of lathe machine is shown in Figure 1 is supported by three springs. The lathe machine has mass $m$, total length $l$ and mass moment of inertia, taken about an axis that goes through its centre of gravity, C.G., is $J_o$. The positive vertical downward and clockwise rotational motions of the machine about the C.G. point can be described as $x(t)$ and $\theta(t)$ respectively.

(a) Using Newton's Law method, prove that the equations of motion for this system are $m\ddot{x} + (k_1 + k_2 + k_3)x - (k_1l_1 - k_2l_2 - k_3l_3)\theta = 0$ and $J_o\ddot{\theta} - (k_1l_1 - k_2l_2 - k_3l_3)x + (k_1l_1^2 + k_2l_2^2 + k_3l_3^2)\theta = 0$

[16 marks]

(b) Compute the system's characteristic equation, natural frequencies and mode shapes given that $m = 30kg$, $J_o = 13kg \cdot m^2$, $l = 25cm$, $l_1 = 30cm$, $l_2 = 15cm$, $k_1 = 1250 N/m$, $k_2 = 1850 N/m$ and $k_3 = 2250 N/m$.

[14 marks]

1. A simplified model of lathe machine is shown in Figure 1 is supported by three springs. The lathe machine has mass m, total length l and mass moment of inertia, taken about an axis that goes through its centre of gravity, C.G., is Jo. The positive vertical downward and clockwise rotational motions of the machine about the C.G. point can be described as x(t) and θ(t) respectively.

(a) Using Newton's Law method, prove that the equations of motion for this system are mẍ + (k1 + k2 + k3)x - (k1l1 - k2l2 - k3l3)θ = 0 and Joθ̈ - (k1l1 - k2l2 - k3l3)x + (k1l1^2 + k2l2^2 + k3l3^2)θ = 0

[16 marks]

(b) Compute the system's characteristic equation, natural frequencies and mode shapes given that m = 30kg, Jo = 13kg · m^2, l = 25cm, l1 = 30cm, l2 = 15cm, k1 = 1250 N/m, k2 = 1850 N/m and k3 = 2250 N/m.

[14 marks]

Added by Sebastian R.

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