111 年 國立成功大學機械工程學系碩士班乙組《動力學》

The roller at A is moving upward with a velocity of $v_A = 3\ \text{ft/s}$ and has an acceleration of $a_A = 4\ \text{ft/s}^2$ when $s_A = 4\ \text{ft}$. Determine the velocity and acceleration of block B at this instant shown in Figure 1.

🖼️【此處有附圖，請對照原卷】

If the slotted arm AB rotates about the pin A with a constant angular velocity of $\omega_{AB} = 10\ \text{rad/s}$, please determine the angular velocity of link CD at the instant shown in Figure 2.

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The uniform rods AB and BC with weights of 2 kg and 4 kg, respectively. The small wheel at C with negligible weight can freely slide along a $30°$ incline. If the mechanism is released from rest at the position shown in Figure 3, determine the velocity of pin B after the rod AB has rotated through $90°$.

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The slender uniform bar AB has mass $M$ and is pivoted at A. The spring $k$ and the damper $c$ is connected to the bar's center C. The small object with mass $m$, initially at a position $h$ vertically measured from pivot A, is released from rest and then collides at B, as shown in Figure 4. Before the collision, the bar is under equilibrium and an initial angle $\theta = \theta_0$ is observed. The roller D is used so that the line jointed by C and D is always in the vertical direction. After the collision, the object $m$ is adhered on the bar, and the vibration motion of the bar is triggered.

The parameters are given as follows: $L = 200\ \text{mm}$, $h = 50\ \text{mm}$, $M = 1.5\ \text{kg}$, $m = 0.1\ \text{kg}$, $k = 300\ \text{N/m}$ and $c = 15\ \text{N·s/m}$.

🖼️【此處有附圖，請對照原卷】

(a) Determine the angular velocity of the bar immediately after the collision. To solve this, you may use the angular momentum conservation principle. Please also explain why this principle can be used during the collision while the bar is subjected to external forces such as gravity force, spring force and damping force.

(b) Write the equation of the motion for the vibration motion in terms of $\theta$ and determine the natural frequency after the collision.

(c) Is this vibration underdamped, critically damped or overdamped? Give theoretical calculations to support your answer.

One end of a massless cable is connected to a small ball B with a mass of 0.3 kg, while the other end passes through a small hole C in a smooth table, as shown in Figure 5. A force $F$ is applied to the cable so that it is pulled downward through the hole with a constant speed $v_c = 750\ \text{mm/s}$. At its initial position, the ball is at the radial position $r_1 = 500\ \text{mm}$ from the hole and the circumferential component of the ball's velocity is $v_1 = 600\ \text{mm/s}$. Determine:

(a) the speed of the ball at the instant that the radial position is $r_2 = 300\ \text{mm}$ from the hole; and

(b) the work done by the force $F$ during the motion from $r_1$ to $r_2$.

🖼️【此處有附圖，請對照原卷】