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MTS-362
CONTROL ENGINEERING
Spring 2011Lecture No. 2
Department of Mechatronics Engineering
Mathematical Modeling of MechanicalSystems
Instructor: Engr. Sadaf Siddiq
Class: BEMTS 6A & 6B
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System Modeling
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System Modeling
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Mathematical Models
Design of engineering systems by trying and error versus
design by using mathematical models.
Physical laws such as Newtons second law of motion is amathematical model.
Mathematical model gives the mathematical relationships
relating the output of a system to its input.
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Mathematical Models
Control systems give desired output by controlling the
input. Therefore control systems and mathematical
modeling are inter-linked.
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Mechanical Systems
If the velocity and acceleration of a body are bothzero then the body will be Static.
If the applied forces are balanced, and cancel eachother out, the body will not accelerate.
If the forces are unbalanced then the body will
accelerate and the body will be Dynamic. If all of the forces act through the center of mass
then the body will only translate- Translation
Forces that do not act through the center of mass
will also cause rotation to occur- Rotation
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Modeling- Translational Systems
FBD: Free Body Diagrams allow us to reduce a complexmechanical system into smaller, more manageable pieces.
Common Components
gravity and other fields - apply non-contact forces
inertia - opposes acceleration and deceleration
springs - resist deflection
dampers and drag - resist motion
friction - opposes relative motion between bodies in
contact
cables and pulleys - redirect forces
contact points/joints - transmit forces through up to 3
degrees of freedom
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Gravity and Other Fields
mgF m
mgB qvF
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Mass and Inertia
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Springs
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Springs
S i
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Springs
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Springs in Series & Parallel
Kequiv= KS1 + KS2
D i d D
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Damping and DragA damper is a component that resists motion. The resistive force
is relative to the rate of displacement (velocity). Springs store
energy in a system but dampers dissipate energy. Dampers andsprings are often used to compliment each other in designs.
D i S i & P ll l
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Dampers in Series & Parallel
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C P i d J i
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Contact Points and JointsA system is built by connecting components together. These connections can be
rigid or moving. In solid connections all forces and moments are transmitted and
the two pieces act as a single rigid body. In moving connections there is at least
one degree of freedom. If we limit this to translation only, there are up to three
degrees of freedom, x, y and z. In any direction there is a degree of freedom, a
force or moment cannot be transmitted.
C t t P i t d J i t
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Contact Points and Joints
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The Unforced Mass-Spring SystemConsider a mass, M, suspended from a spring of natural
length l and modulus of elasticity . If the elastic limit of
the spring is not exceeded and the mass hangs inequilibrium, the spring will extend by an amount, e, such
that by Hookes Law the tension in the spring, T, will be
given by:
M
Mg
T
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The Unforced Mass-Spring SystemIf the spring is pulled down a further distance, y,
(with y positive downwards) the restoring force
will now be the new tension in the spring, T
F
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The Unforced Mass-Spring-Damper System
tCoefficiendampingBdt
dyBFForceDamping
D
,
dt
dyBky
dt
ydM
dt
dyB
l
y
dt
ydM
2
2
2
2
FD
This time, the net downward force will beMg - T - FD
using Newtons 2nd Law, this results in
dt
dyB
l
y
dt
dyB
l
yeMgFForceNet
)(
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The Forced Mass-Spring-Damper System
)(2
2
tfdt
dyBky
dt
ydM
dt
dyB
l
yetfMgFForceNet
)()(
The net downward force =Mg + f(t)- T- FD
using Newtons 2nd Law, this results in
FD
dt
dyBkytf
dt
dyB
l
ytfFForceNet )()(
S t E l
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System Examples
S t E l
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System Examples
S t E l
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S t E l
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S t E l
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System Examples
S stem E amples
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Modeling- Rotational Systems
Basic properties of rotation
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Modeling- Rotational Systems
FBD: Free Body Diagrams (FBDs) are required
when analyzing rotational systems, as they were
for translating systems.
Common Components
inertia - opposes acceleration and deceleration springs - resist deflection
dampers and drag - resist motion
friction - opposes relative motion between bodies in
contact
levers - rotate small angles
gears and belts - change rotational speeds and torques
Inertia
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InertiaWhen unbalanced torques are applied to a mass it will begin to
accelerate, in rotation, the sum of applied torques is equal to the
inertia forces
The mass moment of inertia will be used when dealing with
acceleration of a mass. The areamoment of inertia is used for torsional
springs
Inertia
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InertiaThe center of rotation for free body rotation will be the centroid. Moment of
inertia values are typically calculated about the centroid. If the object is
constrained to rotate about some point, other than the centroid, the moment ofinertia value must be recalculated. The parallel axis theorem provides the
method to shift a moment of inertia from a centroid to an arbitrary center of
rotation
Springs
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SpringsTwisting a rotational spring will produce an opposing torque. This
torque increases as the deformation increases. The angle of rotation is
determined by the applied torque, T, the shear modulus, G, the areamoment of inertia, JA, and the length, L, of the rod. The constant
parameters can be lumped into a single spring coefficient similar to
that used for translational springs.
This calculation uses the area moment of inertia (JA)
Springs
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SpringsIf the object is constrained to rotate about some point, other
than the centroid, the moment of area value must be
recalculated. The parallel axis theorem provides the method toshift a moment of area from a centroid to an arbitrary center of
rotation
Damping
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Damping
Rotational damping is normally caused by viscous fluids, such
as oils, used for lubrication.
The equation used for a system with one rotating and one stationary
part is given by:
The equation used for damping between two rotating parts is given
by:
Friction
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Friction
Friction between rotating components is a major source of
inefficiency in machines. It is the result of contact surface
materials and geometries. Calculating friction values in rotating systems is more
difficult than translating systems.
Normally rotational friction will be given as static and
kinetic friction torques.
Levers
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LeversThe levers can be used to amplify forces or motion. Although
theoretically a lever arm could rotate fully, it typically has a limited
range of motion.The amplification is determined by the ratio of arm lengths to the left
and right of the center.
Gears and Belts
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Gears and BeltsWhile levers amplify forces and motions over limited ranges of motion, gears can
rotate indefinitely. Some of the basic gear forms are:
Spur - Round gears with teeth parallel to the rotational axis.Rack - A straight gear (used with a small round gear called a pinion).
Helical - The teeth follow a helix around the rotational axis.
Bevel - The gear has a conical shape, allowing forces to be transmitted at angles.
Basic Gear Relationships
Rack & Pinion
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Rack & Pinion
Rack and pinion gear sets are used for converting rotation
to translation. A rack is a long straight gear that is driven
by a small mating gear called a pinion.
Relationships for a rack and pinion gear set
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System Examples
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