University of Illinois at Urbana-Champaign – Aerospace Engineering – Spring 2021
AE 352 –Problem Set #4
Issued : 03/26/2021 – Due : 04/02/2021
Question 1 : The four-bar linkage pictured below moves on a smooth horizontal surface (each
bar is massless, of length L, and each point has mass m). Points A and C can only move along
the x-axis and both springs have unstressed length `. Force
−→
F 3 is orthogonal to the bar AB at all
times. This is similar to the linkage of HW3, but with different forces
(a) Write Euler-Lagrange’s equations using x (the x-coordinate of A) and y (the y-coordinate
of D) as generalized coordinates. Justify all your steps.
(c) Same question using x and θ (the half the angle between the AB and AD bars).
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Question 2 : We consider the same setup as in HW2. A particle of mass m is embedded inside a
massless cylinder of radius a at a distance r from its center. The cylinder is rolling without slipping
down an incline with angle α.
(1) How many degrees of freedom does this system have ? Please justify your answer.
Hint : Integrate the rolling without slipping condition into a constraint before answering.
(2) Taking for granted that ”rolling without slipping” is a workless constraint, use the Principle
of Virtual Work to find the equilibrium positions of the system (i.e., the values of θ corres-
ponding to equilibrium).
This should obviously give you the same answer as in HW2
(3) Write the equations of motion using the Lagrangian approach.
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Question 3 :
The figure above is a simplified model of a cargo-hauling helicopter. The helicopter, repre-
sented by a single particle of mass M embedded at the center of mass of massless rod AB, is
connected to the cargo at two points, C and D by inextensible cables of length ` and negligible
mass. The particles of mass m at C and D account for the mass of the cargo. In addition, C
and D are connected to a massless spreader which maintains a distance of a between them.
The helicopter is assumed to fly at a constant altitude with respect to the ground (with the
frame drawn assumed Newtonian), which in this simplified model we take to mean that A,
M , and B move on a straight line at constant altitude. The helicopter is subject to a force−→
F which encompasses aerodynamic forces as well as propulsion. In addition to their weight,
particles C and D are also subject to drag
−→
Dc and
−→
Dd, respectively.
(a) Justify that, under the conditions described above, the system consisting of the mass M
and two masses at C and D has two degrees of freedom and propose an appropriate set of
generalized coordinates.
(b) Write the equations of motion using a Lagrangian approach.
(c) Compute the tension in the spreader using the method of Lagrange multipliers. Please
justify your steps (choice of broken constraint(s) and generalized coordinates) carefully and
simplify the expression as much as possible (the final result is simple in terms of the exter-
nal forces).
(d) Verify your answer from (c) using AMB with respect to well-chosen point(s), for well-
chosen system(s).
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