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Fall 2004

2.032 DYNAMICS Problem Set No. 1 Out: Wednesday, September 15, 2004 Due: Wednesday, September 22, 2004 at the beginning of class

Problem 1

(Doctoral Exam, 1999)

A pendulum is constructed by attaching a mass m to an extensionless string of fixed length l. The upper end of the string is connected to the uppermost point of a vertical fixed disk of radius R (R < l/π), as shown below. At t = 0 the mass hangs at rest at the equilibrium position θ = 0, when it is given an initial velocity v0 along the horizontal. Derive expressions for the two extreme deflections (in terms of θ) of the pendulum resulting from this initial perturbation. Do not make a small-angle approximation.

θ R g

m v 0

Courtesy of Prof. T. Akylas. Used with permission.

1

Problem 2

A point mass moves without friction on a horizontal plane. A massless inextensible string is attached to the point mass and led through a hole (see figure below). At time = t0 the mass moves along a circle with constant velocity v0 . We gradually pull the free end of the string downwards until, at time t1 , we have |r(t1 )| = L0 /2. What is the velocity of the mass at time t1 ?

r(t) O Plane (no friction) String

2

m

Problem 3 A particle of mass m1 is attached to a massless rod of length L which is pivoted at O and is free to rotate in the vertical plane as shown below. A bead of mass m2 is free to slide along the smooth rod under the action of a spring of stiffness k and unstretched length L0 . (a) Choose a complete and independent set of generalized coordinates. (b) Derive the governing equations of motion by applying momentum principles.

O

k

g m

2

L

m

1

Courtesy of Prof. T. Akylas. Used with permission.

3

(adapted from Crandall et al., 2-35)

Problem 4

Consider the system shown below under the assumption that the pendulum arm connecting m2 and m3 is massless. By applying momentum principles, obtain the differential equations of motion for the generalized coordinates x1 , x2 and θ.

x1 k1

x2 k2

g

m1

m2

θ m3

4

Problem 5

(Doctoral Exam, 1999)

Two identical rods of length l, that have equal masses m attached at their ends, are clamped at an angle θ to a shaft as shown. (The shaft and the rods are in the same plane.) What reaction forces must the bearings be able to withstand, if the angle θ can be set anywhere from zero to 90◦ and the maximum angular velocity of the shaft is ω? (For simplicity, you may neglect the mass of the rods and ignore the effects of gravity.)

m

l

θ

ω l

m d

Courtesy of Prof. T. Akylas. Used with permission.

5

2.032 DYNAMICS

Fall 2004

Problem Set No. 2 Out: Wednesday, September 22, 2004 Due: Wednesday, September 29, 2004 at the beginning of class

Problem 1 Show that for any 3 × 3 skew-symmetric matrix A, there exists a 3-dimensional vector ω such that for any three-dimensional vector x , Ax = ω × x . Problem 2 Consider the coupled pendula shown in the figure below. Both rods are massless, with point masses m attached to their ends. Both joints shown in the figure are frictionless. The external force F encloses a fixed angle γ with the line of the pendulum shown. The masses never collide. The constant of gravity is g. Questions: • Identify the constraints. • Determine the number of degrees of freedom. • Find the equations of motion for φ and ψ. • Find the constraint forces. • Is the system conservative? (Why?) l1 g φ l2 l3 m

ψ

m

F γ

Problem 3 Determine the angular velocity of a cone rolling on the XY -plane without slipping, as shown.

Figure by OCW.

Problem 4 (adapted from Ginsberg, 3-22) The disk rotates at ω1 about its axis, and the rotation rate of the forked shaft is ω2. Both rates are constant. Determine the velocity and acceleration of an arbitrarily selected point B on the perimeter. Describe the results in terms of components relative to the xyz axes in the sketch.

z

w2

q

B

A

x

w1

y

Figure by OCW. After 3-22 in Ginsberg, J. H. Advanced Engineering Dynamics. 2nd ed. New York: Cambridge University Press, 1998.

2.032 DYNAMICS

Fall 2004

Problem Set No. 3 Out: Wednesday, September 29, 2004 Due: Wednesday, October 6, 2004 at the beginning of class

Problem 1 A pendulum consists of a rod of length L with a frictionless pivot at one end. The pendulum is suspended from a . ywheel of radius R which rotates with . xed angular velocity ω, as shown below. (a) Determine the angular velocity of the rod in terms of ω and the generalized coordinate θ indicated in the sketch. (b) Calculate the velocity of the mid point C of the rod.

Courtesy of Prof. T. Akylas. Used with permission.

Problem 2 (adapted from Doctoral Qualifying Exam 2002) In the system sketched below, the rigid cylinder of radius R is moving to the right such that its center C has velocity v. There is no slipping between the cylinder and the bar BD, but there is slipping between the cylinder and the ground. In the position shown, (a) Determine the angular velocity of the bar BD. (b) Determine the velocity of the cylinder at the point where it contacts the ground.

. Courtesy of Prof. T. Akylas. Used with permission.

Problem 3 Consider the Cardan drive shown in the figure below. Determine the outgoing angular velocity ω2 in terms of the angular velocity ω1, and angles α and ψ. Note that angular velocity γ& is perpendicular to both ω2 and φ& .

z

. g

. y

a

w2

w2

.

x

y

y

. j

. w1 = y

Figure by OCW.

Problem 4 Consider the rocking chair problem discussed earlier in the class. Plot qualitatively the trajectories in the phase plane.

Fall 2004

2.032 DYNAMICS Problem Set No. 4 Out: Wednesday, October 6, 2004 Due: Wednesday, October 13, 2004 at the beginning of class

Problem 1 A rigid circular cylinder of radius a has a hole of radius

1 2a

cut out. Assume that the

cylinder rolls without slipping on the floor. (i) Compute the kinetic energy and the potential energy of the cylinder using the generalized coordinate θ defined below. (ii) By suitably approximating the kinetic and potential energy expressions in (i), deduce the frequency of small rocking oscillations of the cylinder about the equilibrium position θ = 0. ˙ phase plane. (iii) Use the potential to plot trajectories qualitatively on the (θ, θ)

a g

θ

Courtesy of Prof. T. Akylas. Used with permission.

1

Problem 2 A billiard ball, initially at rest, is given a sharp impulse by a cue. The cue is held horizontally a distance h above the centerline. The ball leaves the cue with a speed v0 and eventually acquires a final speed of 97 v0 . Show that h = 45 R, where R is the radius of the

ball. Problem 3 Determine the principal centroidal moments of inertia for the following homogeneous bodies: (a) a sphere of radius R, (b) a circular cone of height h and base radius R. Problem 4

(adapted from Crandall et al., 4-14)

The uniform rod of length L and mass M is pivoted, without friction, to the shaft OA, which revolves in fixed bearings at the steady rate ω0 . The rod is constrained to remain in a plane through OA which rotates with the shaft. (a) Formulate an equation of motion for θ(t). (b) For each value of ω0 there is at least one stationary angle θ0 which the rod can maintain while steadily precessing at the rate ω0 . Find all stationary configurations as functions of ω0 .

w0 A

O

g L

q M

Figure by OCW. After Problem 4-14 in Crandall, S. H., et al. Dynamics of Mechanical and Electromechanical Systems. Malabar, FL: Krieger, 1982.

2

2.032 DYNAMICS

Fall 2004

Problem Set No. 5 Out: Wednesday, October 13, 2004 Due: Wednesday, October 20, 2004 at the beginning of class

Problem 1 Consider the motion of a car, as shown in the figure below. (Out of the four wheels only two are visible.) The mass of the body of the car is m3 , and each wheel has radius r and mass m. The coefficient of static friction is µ0 i.e., the tangential force between the wheel and the ground is never greater than µ0 times the normal force. The torque transmitted from the body of the car to the rear wheel set is M0 . (a) Assuming pure rolling, use the work-energy principle to find the acceleration of the car. (b) What is the largest M0 for which the car does not slip?

1

Problem 2

(adapted from PhD Qualifying Exam 2003)

A circular ring of radius a is fixed to a vertical shaft AB at O. The plane of the ring is inclined at 60◦ to the vertical as shown below so that the included angle COB is 60◦ , where C is the center of the ring. A particle P of mass m is free to slide without friction on the ring. The position of the particle is measured by the included angle OCP, denoted θ as shown. The shaft is driven at constant angular velocity Ω. Find in terms of θ and θ˙ the velocity of P. Ω

Ω

B

B

C 60ο

a θ

C P

P

O

O

A

A

Courtesy of Prof. T. Akylas. Used with permission.

2

Problem 3

(adapted from Crandall et al., 4-20)

A rigid cone with apex half angle α rolls steadily without slip on a horizontal surface so that it precesses about the Z axis at a constant angular rate Ω. The cone has mass M and principal moments of inertia I1 , I1 , and I3 at the tip. (a) Determine the angular momentum of the cone. (b) What forces and torques are required to maintain this motion?

Figure by OCW. After problem 4-20 in Crandall, S. H., et al. Dynamics of Mechanical and Electromechanical Systems. Malabar, FL: Krieger, 1982.

3

Problem 4 The heavy disk of mass M and centroidal principal moments of inertia I1 = I2 , I3 rolls without slipping in contact with the inclined plane, as shown below. A ball joint at O holds the end of the massless shaft in place. The shaft makes angle β with the inclined plane. Derive an expression for the kinetic energy of the disk.

b

a

C

O

α

Courtesy of Prof. T. Akylas. Used with permission.

4

2.032 Dynamics

Fall 2004

Problem Set No. 6

Out: Wednesday, October 27, 2004 Due: Wednesday, November 3, 2004

Problem 1 Consider the two-dimensional rolling problem discussed in class (disk rolling on a plane, with its main axis of symmetry remaining parallel to the plane). (a) Show that two of the rolling constraints in this problem cannot be integrated to holonomic constraints. (b) Show that the above remains true even if we allow for an integrating factor.

Problem 2 (adapted from Baruh, problem 4.3/4) The radar tracking of a moving vehicle by another moving vehicle is a common problem. Consider the two vehicles A and B in the figure shown below. The orientation of vehicle A must always be toward vehicle B. Express the constraint relation between the velocities and distance between the two vehicles and determine whether this is a holonomic constraint or not.

Figure by OCW. After 4.3/4 in Baruh, H. Analytical Dynamics. Boston MA: McGraw-Hill, 1999.

Problem 3 (adapted from Baruh, problem 4.4/7) Find the virtual displacement of point P in the figure shown below. The mass is suspended from an arm which is attached to a rotating column. The pendulum swings in the plane generated by the column and arm.

Z L1

g

.

f

q

L2

P m

Figure by OCW. After 4.4/7 in Baruh, H. Analytical Dynamics. Boston MA: McGraw-Hill, 1999.

Problem 4 (adapted from Ginsberg, problem 6.8) The figure shows a child’s tricycle as viewed from above. When the wheels do not slip over the ground, the velocity of each wheel’s center must be perpendicular to the wheel’s shaft in the horizontal plane, as shown. Consider a set of generalized coordinates consisting of the position coordinates XA and YA of the steering joint, the angle of orientation θ of the frame, the steering angle β, and the spin angles φ1, φ2, φ3 of the wheels. Derive the velocity constraints among these seven generalized coordinates. From that result, determine the number of degrees of freedom.

.

f3

vC

L

b

.

C

f1

d

q A

Y

R d

r vB

B

.

Top View of a Tricycle

f2 X

Figure by OCW. After 6.8 in Ginsberg, J. H. Advanced Engineering Dynamics. 2nd ed. New York: Cambridge University Press, 1998.

2.032 DYNAMICS

Fall 2004

Problem Set No. 7 Out: Wednesday, November 3, 2004 Due: Wednesday, November 10, 2004 at the beginning of class

Problem 1 The force F acts horizontally at the end of the four-member linkage shown below. The linkage is described by the generalized coordinates ξ1 = θ1 , ξ2 = θ2 , ξ3 = θ3 , ξ4 = θ4 . Find the generalized forces Ξ1 , Ξ2 conjugate to the generalized coordinates ξ1 , ξ2 and due to the force F . You may not assume that θ1 , θ2 , θ3 , θ4 are small angles.

Courtesy of Prof. T. Akylas. Used with permission.

1

Problem 2 A pendulum consists of a rod of length L, mass m, and centroidal moment of inertia 1 2 12 mL

with a frictionless pivot at one end. The pendulum is suspended from a flywheel

of radius R and mass M which can rotate about the fixed point O, as shown below. (a) Select a complete and independent set of generalized coordinates. (Please define these coordinates clearly.) (b) Derive the Lagrangian equations of motion without making any approximations (small angles, etc.).

O R

g

L

Courtesy of Prof. T. Akylas. Used with permission.

2

Problem 3 Consider a bead of mass m sliding without friction on a rotating ring with radius r and negligible mass, as shown in the figure. The ring rotates about the vertical axis with constant angular velocity Ω. Derive the equation of motion of the bead using D’Alembert’s principle.

3

Fall 2004

2.032 DYNAMICS Problem Set No. 8 Out: Wednesday, November 10, 2004 Due: Wednesday, November 17, 2004 at the beginning of class

Problem 1 Reconsider Problem 1 of Quiz No.1. As shown in the figure, the flywheel spins at a constant rate ω2 , and also rotates about z axis with angular velocity ω1 that is a function of time. The center of mass of the flywheel is located on the z axis, and the centroidal moments of inertia are I1 about the spin axis and I2 transverse to that axis (a) Derive the Lagrangian equations of motion. (b) Find the generalized forces necessary to maintain the motion.

z

q Outer gimbal

w2 Inner gimbal

w1

Figure by OCW.

1

Problem 2

(from Doctoral Qualifying Exam 2002)

Two identical rigid cylinders, each having radius R and mass m, are linked by a connecting rod of length 3R and mass M as shown below. A horizontal force F (t) is applied to the center of the right cylinder and neither cylinder slips in its rolling motion. In the initial position, the angle θ locating the connecting pin is zero. Derive the Lagrangian equations of motion for this system.

Courtesy of Prof. T. Akylas. Used with permission.

2

Problem 3

(from Ginsberg, Problem 6.41)

The slider, whose mass is m1 , oscillates within the groove in the housing. The moment of inertia of the housing about the axis of rotation is I. The spring restraining the slider is unstretched when s = 0. Derive differential equations for the distance s and spin angle φ resulting from application of a torque M (t) to the shaft. Use the Lagrangian equations of motion.

.

f

s

M

q

Figure by OCW. After problem 6.41 in Ginsberg, J. H. Advanced Engineering Dynamics. 2nd ed. New York: Cambridge University Press, 1998.

3

Problem 4

(adapted from Ginsberg, Problem 6.42)

The bar, whose mass is m, is pinned to a collar that permits precessional rotation ψ about the vertical guide, as well as nutational rotation θ. The collar is fastened to a spring whose extensional stiffness is ke and whose torsional stiffness for precessional rotation is kt . (a) Derive the Lagrangian equations of motion for this system. (b) Determine the constraint forces using Lagrange multipliers.

L

q

.

y

Figure by OCW. After problem 6.42 in Ginsberg, J. H. Advanced Engineering Dynamics. 2nd ed. New York: Cambridge University Press, 1998.

4

2.032 DYNAMICS

Fall 2004

Problem Set No. 9 Out: Wednesday, November 17, 2004 Due: Wednesday, November 24, 2003 at the beginning of class

Problem 1

Consider a wheelbarrow with a wheel of negligible mass, as shown in the figure below. The distance between the center of mass C of the wheelbarrow and the center of its wheel D is l. The handles of the wheelbarrow are of length h, and are pushed at their tips by the forces FA and FB. The time-dependent angles between the forces and the handles are given by α(t) and β(t), respectively. The wheel rolls without slipping. The centroidal moment of inertia and the mass of the wheelbarrow are given by Ic and m, respectively. With the position of C and the orientation of the wheelbarrow as generalized coordinates, derive the equations of motion using Lagrange multipliers.

l

C

h FA

A

a

α(t) h

FB

β(t)

B

D

b m, IC

Problem 2

A cart and a rolling disk are connected by a rigid massless link of length L, as shown in the figure below. The disk rolls without slipping. Use Lagrange multipliers to determine the force in the link.

θ

L g

m

R

C

M h

x Problem 3

Consider the “spinning disk on a rotating linkage with torsional spring” problem discussed in class. (a) By introducing generalized moments associated with the coordinates ϕ and ψ, reduce the set of equations of motions to a single equation of motion for ν. (b) For pϕ = 0, sketch the trajectories of the above equation on the ( υ,υ& ) phase plane for different values of pψ (select all other parameters to be equal to one).

2.032 DYNAMICS

Fall 2004

Problem Set No. 10 Out: Wednesday, November 24, 2004 Due: Wednesday, December 1, 2004 at the beginning of class

Problem 1 The system below consists of a massless hollow cylindrical tube, joined to a vertical shaft at the point O. The tube is fixed in θ-direction and θ = π/4. Inside the tube, moves without friction a mass m which is connected to O through a spring of stiffness k and neutral length of r0 . Assume that the shaft is rotating with angular velocity φ˙ about its axis. Using r, φ as generalized coordinates: (a) Reduce the problem to a one-degree-of-freedom problem for r, that has only potential active forces. (b) Find the equilibria for the reduced system and investigate the stability using Dirichlet theorem. (c) Sketch the trajectories on the (r,r) ˙ phase plane. Select all parameters to be equal to one, including gravity g.

Courtesy of Prof. T. Akylas. Used with permission.

1

Problem 2

(adapted from PhD Qualifying Exam 2003)

Reconsider Problem 2 of PS No. 5. Using the same notation, (i) Derive the differential equation describing the motion of the bead on the ring. (ii) Find equilibrium positions θ0 for the bead and investigate the stability of these positions at various speeds Ω. (iii) Draw a stability diagram showing all solution branches (and their stability properties) for 0 < Ω < ∞. (iv) Draw the phase plane of solution trajectories at representative values of Ω. (v) Instead now consider a ring inclined at 120◦ to the vertical, so that C is below O. Without any calculations, state how many equilibrium positions you expect and how their stability will vary with Ω.

2

Problem 3 Three equal masses m slide without friction on a rigid horizontal rod. Six identical springs with spring constant k are attached to the masses as shown in the sketch below. Identify the natural modes and natural frequencies of this system.

Courtesy of Prof. T. Akylas. Used with permission.

3

Problem 4 A square plate of mass m and side 2a is constrained to remain in the plane of the sketch. Its moment of inertia about an axis perpendicular to the plane through the center of the square is I =

2 2 3 ma .

The plate is supported in a gravity-free environment by the four

equal springs shown. It is desired to formulate matrix equations of small motions for the three degrees of freedom, using the generalized coordinates x, y, and aθ. (a) Obtain matrix equations of motion of the form [M ]{¨ x} + [K]{x} = 0 and the three natural-mode solutions. (b) Construct the modal matrix Φ and evaluate [Φ]t [M ][Φ],

[Φ]t [K][Φ].

Verify that the quotients of the diagonal elements of the resulting square matrices in (b) give the squares of the natural frequencies of the three modes.

Courtesy of Prof. T. Akylas. Used with permission.

4

Fall 2004

2.032 DYNAMICS Quiz No. 1 Monday, October 25, 2004

This is a CLOSED-BOOK, open-notes Quiz.

Problem 1

(20 points)

Servomotors make the flywheel spin at a constant rate ω2 , and also impose a vertical rotation rate ω1 that is a function of time (see figure below). The center of mass of the flywheel is located on the z axis, and the centroidal moments of inertia are I1 about the spin axis and I2 transverse to that axis. (a) Derive the equations of motion for the system. (b) Determine the external torques necessary to maintain the above motion.

z

q Outer gimbal

w2 Inner gimbal

w1

Figure by OCW.

1

Problem 2

(20 points)

A solid uniform cylinder of mass m and radius R is placed on top of a fixed cylinder of the same radius, and it is slightly tipped, as shown in the figure. Find the value of the angle θ at which sliding begins as a function of the static friction coefficient µ.

2

z

e3

e2

q Outer gimbal

w2 . y

. j

Inner gimbal

e1

w1

Figure by OCW.

Fall 2004

2.032 DYNAMICS Quiz No.2 Wednesday, December 8, 2004 This is a CLOSED-BOOK, Open notes Quiz.

Problem 1 (20 points) Two small masses, m1 and m2, are constrained to move in a vertical plane by two inextensible strings, as shown in figure 1. The lengths of the two strings are R and L = ρ + r, respectively. There is a force of magnitude F acting on the mass m2, with its line of attack always parallel to the string attached to m2. The constant of gravity is g. The pulley shown in the figure is small and frictionless. (a) Classify all constraints and forces (give reasoning). Determine the number of degrees of freedom. (b) Derive the Lagrangian equations of motion in terms of φ1 and φ2.

L r g

φ1

ρ

R

φ2

m2 F

m1

Figure 1

Problem 2 (20 points) A disk of radius r and mass M is placed on a fixed tube of radius R, as shown in Figure 2. The center of the disk is at a distance l from the ceiling and is attached to the ceiling through a spring of stiffness k1 and unstretched length l0. At the same time, a block of mass m is hanging from the center of the disk on a spring of stiffness k2 and unstretched length l0. We assume that the disk cannot slip on the tube and the lower spring remains vertical on any motion of the system. The constant of gravity is g. Without deriving equations of motion, find sufficient and necessary conditions for the stability of the equilibrium shown in Fig. 2.

k1

l

r

g k2

m

Figure 2

R

2.032 DYNAMICS

Fall 2004

Practice Problems No.1

Problem 1 A ring of radius R is pivoted without friction at O. A disk of radius r rolls without slipping inside the ring, as shown below. Determine the angular velocities of the ring and the disk in terms of the generalized coordinates θ, φ indicated.

O θ C

R r

φ

Problem 2 A thin rectangular plate of mass M , sides a and b, rotates about an axis along its diagonal with angular velocity ω. (a) What are the forces on the bearings? (b) What is the kinetic energy of the rotating plate?

1

Problem 3

(adapted from Doctoral Qualifying Exam 2000)

A rigid, uniform flat disk of mass m and radius R is moving in the plane towards a wall with central velocity V while rotating with angular velocity ω, as shown below. Assuming that the collision in the normal direction is elastic and no slip occurs at the wall, find the velocity of the (center of the) disk after it collides with the wall.

θ

V

ω

2

Problem 4

(adapted from Doctoral Qualifying Exam 2003)

The homogeneous sphere of mass m and radius r is projected along the incline of angle θ with an initial central velocity V0 and no angular velocity (ω0 = 0). If the coefficient of kinetic friction is µ, determine the time duration T of the period of slipping. In addition, state the velocity Vf of the mass center C at the end of the period of slipping.

ω0 C

g

r V0

θ

3

Problem 5 A pencil, modeled as a uniform bar AB of length 2a and mass m, is initially at rest standing upright at the edge of a table as shown below. A horizontal impulse ∆P is delivered to the base A of the pencil, causing it to lose contact with the table. Determine the value of ∆P for which the motion of the pencil is such that point B clips the edge of the table. Sketch the position of the pencil at this instant.

Β

∆P

Α

4

g

Fall 2004

2.032 DYNAMICS Practice Problems No.2

Problem 1 A rigid block of height H, length L and depth D rests on a rigid cylinder of mass M and radius R, as shown in the sketch. The cylinder rolls on the floor without slipping and the block rolls on the cylinder without slipping as well. (a) Study the geometry of motion of the two rigid bodies, paying attention to the rolling constraints. Select a suitable set of independent generalized coordinates to describe this motion. Show your choice of coordinates clearly on a sketch. (b) Derive the governing equations of motion in terms of these coordinates. (c) Examine the stability to small perturbations of the equlibrium position of the block.

L

H g R

1

Problem 2 The linkage precesses about the vertical axis at the constant rate Ω. The small disk B and slider C each have mass m, and the mass of each link is negligible. The spring has stiffness k and its unstretched length is 2l. Identify the two possible constant values of θ in the physically meaningful range 0 ≤ θ < π/2 corresponding to steady precession with θ˙ = 0. Prove that one of these possibilities is always unstable, while the other is a stable position that exists only if k is sufficiently large.

Ω

C L

q k

B

L

q A

Figure by OCW.

2

Problem 3 The shaft of length 2L is forced to precess at the steady rate Ω with the fixed angle θ shown. On the shaft are two rotors spinning in opposite directions with absolute spin n. The rotors have masses m, diametral moments of inertia I1 and axial moments of inertia I3 . Neglecting gravity, find the torque required to maintain the precession.

L

q

n L n

Ω

Figure by OCW.

3

Problem 4 Three pendulums, all of length l, are symmetrically arranged as shown in the figure. The center bob, of mass 2m, is connected by ideal springs of force constant k to the outer bobs, each of mass m. What are the frequencies and relative displacements of the pendulums when they are vibrating in each of the three normal modes of the system? Consider only vibrations in the plane of the figure. [Caution: a little physical insight will save a lot of algebra in this problem.]

4

Problem 5 Consider small motion in a plane about the position of static equilibrium of the given system. Solve for the natural frequencies and the corresponding amplitude ratios.

5

Fall 2003

2.032 DYNAMICS Quiz No. 1 Monday, October 20, 2003

This is a CLOSED-BOOK Quiz. Each student may only bring one 8 21 ×11 inch sheet of self-prepared notes.

Problem 1

(10 points)

The uniform circular disk shown below is mounted on a shaft such that the normal to the disk makes an angle � to the shaft. The mass of the disk is M and its radius is R. (The axial and diametral moments of inertia of the disk are 21 M R2 and 14 M R2 , respectively.) The shaft, which has relatively negligible mass, rotates with constant angular speed �. (a) Find the angular momentum of the disk about its center C. (b) What are the reaction forces at the bearings (neglect the effect of gravity)?

�

�

2a

C

1

Problem 2

(10 points) (adapted from Doctoral Qualifying Exam 2003)

A rod of mass m, length 2a and centroidal moment of inertia 31 ma2 is dropped onto the edge of a table as shown below. The rod is horizontal, has zero angular velocity and has downward velocity V at the moment just before touching the table. (a) Determine in terms of V the angular velocity of the rod just after impact, assuming that energy is conserved in the collision. (b) Under the same assumptions as in (a) above, determine the velocity just after impact of the end of the rod that touched the table. Does your result seem reasonable? Explain.

2a

g

2

Problem 3

(10 points) (adapted from Ginsberg, 4.20)

A disk of radius r rolls without slipping over the ground such that the angle of tilt � is constant. The center C follows a horizontal circular path of radius R at constant speed V . Derive an expression for the angular velocity of the disk.

R

�

C

3

Fall 2003

2.032 DYNAMICS Quiz No. 2 Wednesday, December 10, 2003

This is a CLOSED BOOK Quiz. Each student may only bring two 8 12 ×11 inch sheets of self-prepared notes.

Problem 1

(10 points)

A slender rigid bar (of mass m and length L) is suspended by a cable (of length L and negligible mass) from pivot A and executes a steady precession about the vertical axis at angular speed � as it maintains the orientation shown in the sketch below. Determine � and the angle of inclination �.

A �

L

�

g

L

� m

1

Problem 2

(10 points)

A rigid half cylinder of radius r and uniform mass density rests on top of a fixed half cylinder of radius R, as shown below. Assuming that no slipping occurs, under what conditions is the equilibrium position stable to small perturbations?

r

g

R

2

Problem 3

(10 points)

The motion of the two masses shown below is restricted to the plane of the sketch. Ne­ glecting the effect of gravity, determine the natural frequencies of vibration and associated mode shapes for small departures from equilibrium of this system.

4k

3k

4k

4k

k m

m

2k

k

***** HAPPY HOLIDAYS *****

3