ME 588, Dynamics and Vibration
Homework 1
Distributed: 9/25/2024, Due: 10/11/2024
1. Consider a spring-mass system mounted on a spinning disk as shown in Fig. 1. The disk spins at constant angular velocity ω. Moreover, the disk has a diametrical slot, along which a block with mass m slides without friction. The spring connecting the mass and the disk center is a stiffening spring with a negligible free length. Therefore, the spring force Fs is given by Fs = kr(1 + αr2), where k and α are positive constants and r is the radial position of the block. The motion occurs in a horizontal plane, where gravity has no effects. Also, the disk is large enough so that the block will not fall out of the slot. Answer the following questions.
(a) Use the Newtonian approach (i.e., drawing free-body diagrams and applying Newton’s second law) to derive the equations of motion governing the radial position r.
(b) Determine all equilibrium positions.
(c) Derive the linearized equation of motion around each equilibrium position. Describe the condition so that the linearized equation of motion will give a stable and bounded response.
Figure 1: A spring-mass system in a spinning disk, version 1
Figure 2: Two point masses connected by a massless link
2. A rigid, massless rod of length r connects two particles of mass m1 and m2. Moreover, the two particles are sliding without friction on a circular arc of radius r in the gravity field; see Fig. 2. Let θ be the counterclockwise angular position from the vertical downward direction to the radial direction of particle m2. Moreover, let g be the gravitational acceleration. Use Newtonian mechanics to answer the following questions.
(a) Draw a free-body diagram of the two particles m1 and m2.
(b) Apply Newton’s second law to derive the equations of motion of the two particles m1 and m2. Eliminate constraint force(s) from your equations of motion to obtain a nonlinear, differential equation governing only the variable θ(t).
(c) Determine equilibrium positions θ0 of the system in terms of m1 and m2.
(d) Consider a special case m1 = m2 = m and focus on the equilibrium position with 0 < θ0 < 90◦. Derive linearized equations of motion around the equilibrium position.
3. Quiz Problem. Consider a two-block system moving in the gravity field shown in Fig. 3. 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. 3. Use Newtonian mechanics to answer the following questions.
(a) Draw a free-body diagram of the two blocks.
(b) Apply Newton’s second law to derive the equations of motion of the two blocks. Elimi-nate constraint force(s) from your equations of motion to obtain a nonlinear, differential equation governing only the variable θ(t), where θ is the angle between the rigid rod and the vertical as shown in Fig. 3.
(c) Determine an algebraic equation governing equilibrium positions θ0 of the system. The equation should involve parameters such as mg and kl. Show that there is only one possible equilibrium for 0 < θ0 < 2/π.
(d) Derive a linearized equation of motion around the equilibrium position. If the two-block system is subjected to disturbance, will the system oscillate around the equilibrium position? Why?
Figure 3: A two-block system with a linear spring and a rigid rod
Figure 4: Linearization of the central force motion of a particle
4. The small particle of mass m and its restraining cord are spinning with an angular velocity ω on the horizontal surface of a smooth disk as shown in Fig. 4. The input force Fs(t) applied to the cord depends on time t. As a result, the angular velocity ω and the radial position r of the particle are not constant.
(a) Draw a free-body diagram of the particle and shows that the angular momentum is conserved. Therefore,
where θ is the angular position of the particle, the dot is the time derivative, and h0 is the initial angular momentum of the particle.
(b) Apply Newton’s second law in polar coordinates to derive the equation of motion. Sim plify the equation in the radial direction through use of (1) to obtain
(c) When Fs(t) = , a constant force, the particle will undergo a circular motion. Therefore, r(t)= and ω(t)= are both constant. Determine and .
(d) When Fs(t) undergoes a small change from , e.g.,
the radial position of the particle will deviate from the circular orbit accordingly, i.e.,
Substitute (3) and (4) into (2) to linearize the equation. Show that the linearized equation takes the form. of
Also, specify the initial conditions η(0) and ˙η(0). Hint: First, you need to show that the binomial expansion of r−3 is
(e) If the force increment ∆F is constant, determine r(t) from (4) and (5). Does the response r(t) oscillate or decay? Plot r(t) with respect to time t.