代写ME 588, Dynamics and Vibration Homework 2代写数据结构语言程序

ME 588, Dynamics and Vibration

Homework 2

Distributed: 10/2/2024, Due: 10/18/2024

1. Consider a two-ramp system shown in Fig. 1. The left and right ramps make an angle of θ1 and θ2 with the horizontal plane, respectively. At the top where the two ramps meet is a massless pulley that is free to rotate. On the left ramp, a cart is supported by a spring and the length of the ramp is l. The cart has mass m1 and slides on the left ramp without friction. In addition, the position of the cart from the pulley is x1. The spring is linear with sti↵ness k and has a free length l. Also, the cart is subjected to a horizontal force P. On the right ramp, a cylinder of radius r rolls without slipping. The cylinder is uniform. and has mass m2; therefore, its centroidal moment of inertia is 1 2mr2. The motion of the cylinder is described via its center position x2 from the pulley as well as an angle of rotation φ in the clockwise sense. When the spring is at its free length, the corresponding φ = 0. An inextensible string of constant length d, wrapping around the pulley, connects a cart on the left ramp and a cylinder on the right ramp. Answer the following questions.

(a) Derive constraint equations of the two-ramp systems. Are the constraints rheonomic or scleronomic? How do virtual displacements δx1, δx2, and δφ are related to one another?

(b) Apply the principle of virtual work to derive an equation that governs equilibrium of the cart, the spring, the cylinder, and the force P.

Figure 1: A two-ramp system

Figure 2: A two-block system with a linear spring and a rigid rod

2. Quiz Problem. Consider a two-block system moving in the gravity field shown in Fig. 2. The two blocks have the same mass m and are connected via a rigid, massless rod of length l. As a result of the gravitational acceleration g, block 1 moves horizontally and block 2 can only move vertically. There is no friction in this system. Moreover, block 1 is connected to a wall via a linear spring that has a spring constant k and a negligible free length. Therefore, the elongation of the spring is the position x of block 1 from the wall. For block 2, its horizontal distance to the wall is l and its vertical position is y as shown in Fig. 2. Answer the following questions.

(a) Write down constraint equation(s) of the system. Does the system have schleronomic constraints or rheonomic constraints.

(b) How are virtual displacement δx and δy related to virtual displacement δθ?

(c) Use the principle of virtual work to derive the equilibrium equation of the system.

Figure 3: An Atwood machine subjected to support excitations

Figure 4: A two-bead system on a circular hoop

3. The Atwood machine in Fig. 3 consists of two objects and one pulley. The mass of the objects is m1 and m2. The two objects are connected via an inextensible string of constant length wrapping around the pulley. The pulley has radius r and mass moment of inertia I about its center. In addition, the center of the pulley is moving with a prescribed motion so that its position is p + b sin !t relative to the fixed floor, where p, b and ! are given constants. The inextensible string is tightly wrapped around the pulley, so there is no slipping. Answer the following questions.

(a) Let us use three quantities to describe the motion of the Atwood machine. x1(t) and x2(t) are the position of objects m1 and m2, respectively, from the floor. θ is the angular displacement of the pulley measured with respect to the downward vertical direction. Derive the constraint equations, and identify if the constraints are rheonomic and scleronomic. How many generalized coordinates are needed?

(b) Professor X wants to use variables y1(t) and y2(t) to describe the motion of the two objects, where y1 and y2 are defined from the center of the pulley; see Fig. 3. Professor X then concludes that the inextensible string results in a constraint equation

y1 + y2 = constant (1)

Therefore, the constraint is scleronomic. Is professor X right or wrong? Please justify your answer.

(c) If object m1 is given a virtual displacement δx1, what are the corresponding virtual displacements δx2 and δθ of the object m2 and the pulley, respectively?

(d) An electric motor is connected to the pulley and provides the pulley with a counter-clockwise moment M(t). What is the generalized force for x1, if x1 is chosen as the generalized coordinate?

4. Consider the two-bead system in the gravitational field as shown in Fig. 4. Each bead has mass m and is constrained to slide without friction on the circular hoop of radius r. The two beads are connected via a linear spring, whose free length can be neglected. (In other words, the length of the spring is the elongation of the spring.) Moreover, the stiffness k of the spring is mg/r, where g is the gravitational acceleration. The angular position of bead 1 is described via the angle θ1, whereas the angle between the two beads is θ2. Both θ1 and θ2 are positive in the counterclockwise sense. Answer the following questions.

(a) Let r1 and r2 be position vectors from the center of circular hoop to each mass. Write down the constraint equations. Why types of constraints do we have in this problem?

(b) Does the spring between the two masses behave like a constraint? Why?