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Derive the Equations of Motion for the following dynamic systems

Mechanical Engineering

Derive the Equations of Motion for the following dynamic systems.
Show all your work. in particular: (a) free-body diagrams with all the forces acting on each
clement of this system: (b) number of degrees of freedom in the system, (c) the required
assumptions, and (d) all the relevant relationships between system variables such as, for
example, linear and angular displacements.
Problem |
The dynamic system shown in the figure below consists of two pendulums connected by a
spring with stiffness coefficient k. When 6, = 62 = 0, the spring is at its free length. Moments
of inertia of pendulum bobs about their respective pivots are m,L.° and m2L2~.
Derive the equations of motion for this system in terms of the angular displacement
pendulums,

Problem 2 |
The dynamic system shown in the (igure below consists of
(av) a block of mass m,.
(b) solid uniform disk of mass my and radius R,
(c) two pulleys: translating pulley of mass m3 has amass moment of nertia Jo and radius 7,
and rotating pulley has the same mass moment of inertia /g and radius r.
The disk mass moment of inertia about its center of mass is 7 = 5m R*. Motion of the disk ts
constrained by a linear spring with stiffness coefficient k. and a linear damper with damper
coefficient c. The spring is connected to the disk at its center of mass, While the dainper Is
connected to the rim of the disk. The disk rotates without slipping.
External force f(t) is applied to the block when the system is at equilibrium. Equilibrium
position corresponds to x = 0.

Derive the equation of motion in terms of the displacement of the block.
Problem 3
In the system shown on the figure the input is the displacement y(t), and the output is the
displacement x(t) of mass m. When X = y = O the system is in equilibrium and the
springs arc at their free lengths.

Derive the equations of motion for this system.
Note: assume that y > x, > x

Problem 4
The dynamic system consists ofa block of mass m and a double pulley with the larger radius
R and a smaller radius 7. The mass moment of inertia of this double pulley ts /g. A linear
spring with stiffness coefficient k; is connected to the smaller disk of the double pulley, while
another linear spring with stiffness coefficient ky is connected to its larger disk.
External force f(t) is applied to the block when the system is at equilibrium, Equilibrium
position comesponds to x = 0.

Extra
The cylinder m, 1s a solid uniform rigid body; its mass moment of inertia about its center Is
Io = - m,R?.It rolls without slipping on the horizontal surface of the cart m,. The curt 1s
2 ° . e ‘ed e a) .
translating without friction on the horizontal surface. Derive the equations of motion for this
system.

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