Chapter 6: Vibrations Homework

Homework 6.A.7

Given: A block of mass m is connected to ground with a spring and a dashpot. The block is ableto slide on a smooth horizontal surface. A homogeneous pulley of mass m is connected to the block,as shown in the figure. An inextensible cable is wrapped around the pulley. One end of the cableis attached to ground, and the other end of the cable connected to ground with a second spring ofsti↵ness k. Assume that the cable does not slip on the pulley. Let x represent the translation ofthe block with x = 0 representing the state at which the springs are unstretched.

Find: For this problem:(a) Draw individual free body diagrams of the pulley and the block; and(b) Derive the single di↵erential equation of motion (EOM) for the system in terms of the coordinate

x, its time derivatives, and, at most, the following parameters: m, c, and k.

Freeform c�2018 6-9

Chapter 6: Vibrations Homework

Homework H.6.H

Given: Particle A (having a mass of 2m) is attached to a grounded spring of sti↵ness k and adashpot with a damping coe�cient c. Let x represent the motion of A, with x = 0 when the spring isunstretched. At an instant when A is at rest and with the spring being unstretched/uncompressed(that is, when x = 0), particle B (of mass m), traveling with a speed of v0, strikes A. On the impactwith A, block B immediately sticks to block A.

Find: For this problem:(a) Determine the speed of A immediately after B sticks to it.(b) Derive the dynamical equation of motion (EOM) of A+B in terms of the coordinate x describing

the motion after the two blocks stick together;(c) Determine the undamped natural frequency !n, the damping ratio ⇣ and the damped natural

frequency !d for the system;(d) Determine the response x(t) of the system after A and B stick together.

k

x

Asmooth

2mv0

B

mc

Use the following parameters in your analysis: m = 10 kg, k = 3000 N/m and c = 360 kg/s.

6-10 Freeform c�2020

Question C6.3Consider the spring-mass system shown below. Determine the natural frequency for the system.

Chapter6

Question C6.4The system shown below consists of a pulley (of mass m and centroidal mass moment of inertiaIO) and block A. Determine the natural frequency for the system.

A 2m

k

r

2r

Question C6.5Consider the free response for a damped, single-DOF system having the following di↵erential equa-tion of motion:

mx+ cx+ kx = 0

This system is known to have a damping ratio of ⇣ = 0.1 and undamped natural frequency of!n = 10 rad/s.

What are the new values for the damping ratio and undamped natural frequency if:

(a) The original value of m is doubled, the original value of k is doubled and the value of c isunchanged?

(b) The original value of m is doubled, the original value of k is halved and the value of c isunchanged?

(c) The original value of k is doubled, the original value of c is doubled and the value of m isunchanged?

(d) The original value of m is doubled, the original value of c is doubled and the value of k isunchanged?

Chapter6

Question C6.8Consider Systems A and B shown below. System A is made up of a spring and block with theblock moving in pure translation along a smooth horizontal surface. System B is made up of aspring and a homogeneous disk of mass m and outer radius R, with the center of the disk at O andthe disk rolling without slipping on a horizontal surface. Each system has the same mass m andsame spring sti↵ness k. Let !nA and !nB represent the natural frequencies of Systems A and B,respectively. Circle the answer below that most accurately represents the natural frequencies forthe two systems:

(a) !nA > !nB

(b) !nA = !nB

(c) !nA < !nB

(d) More information is needed on the two systems in order to answer this question.

k c

m g

equilibrium position

unstretched position

z x

xst

A

k m

smooth

x

no slip

m k

O R

x

System A System B

Chapter6

Question C6.10The following equation of motion (EOM) has been derived for a single-degree-of-freedom system:

6x+ 2x� 216x = 0

Explain why the response governed by this EOM is not oscillatory.

Question C6.11The following equation of motion (EOM) has been derived for a single-degree-of-freedom system:

2x+ 48x+ 800x = 200

(a) Determine the undamped natural frequency !n for the system.

(b) Determine the damping ratio ⇣ for the system.

(c) Determine the damped natural frequency !d for the system.

(d) Determine the static deformation xst for the system.

Chapter6

Question C6.14Consider Systems A and B shown below. Let (!n)A and (!n)B represent the natural frequenciesof Systems A and B, respectively. Circle the statement below that most accurately describes thenatural frequencies of these two systems:

(a) (!n)A > (!n)B

(b) (!n)A = (!n)B

(c) (!n)A < (!n)B

SYSTEM A SYSTEM B

x

k

m

x

k

m no slip

M

System B

k

x

m

2R k

x

m

no slip R

no slip

System A

Question C6.15Consider Systems A and B shown below. Let (!n)A and (!n)B represent the natural frequenciesof Systems A and B, respectively. Circle the statement below that most accurately describes thenatural frequencies of these two systems:

(a) (!n)A > (!n)B

(b) (!n)A = (!n)B

(c) (!n)A < (!n)B SYSTEM A SYSTEM B

x

k

m

x

k

m no slip

M

System B

k

x

m

2R k

x

m

no slip R

no slip

System A