Find the frequency Equation, Mode shapes (Normal Function), and expression Bl for a uniform beam with young's modulus E, mass density p, cross-sectional area A, second moment of cross- sectional area I, and a length / with Pinned - Free boundary conditions. Whereas the free-vibration lateral equation of motion of the beam is obtained as: aw a²w + Əx4 at² c²5 =0 where C= EI VPA And the General Solution of the free vibration equation of motion is: w(x, t) = [C₁ cos ßx + C₂ sin ßx + C3 cosh Bx + C4 sinh ßx][A cos wt + P sin wt] Consider a circular drumhead can be modelled by the vibrations of a circular membrane with a uniform thickness, attached to a rigid frame. Due to the phenomenon of resonance, at certain vibration frequencies, its resonant frequencies, the membrane can store vibrational energy, the surface moving in a characteristic pattern of standing waves. This is called a normal mode. A membrane has an infinite number of these normal modes, starting with a lowest frequency one called the fundamental mode. Find the free-vibration equation of motion, and the free-vibration solution with different mode shapes for circular membrane with radius r.

Find the frequency Equation, Mode shapes (Normal Function), and expression Bl for a uniform beam with young's modulus E, mass density p, cross-sectional area A, second moment of cross- sectional area I, and a length / with Pinned - Free boundary conditions. Whereas the free-vibration lateral equation of motion of the beam is obtained as: aw a²w + Əx4 at² c²5 =0 where C= EI VPA And the General Solution of the free vibration equation of motion is: w(x, t) = [C₁ cos ßx + C₂ sin ßx + C3 cosh Bx + C4 sinh ßx][A cos wt + P sin wt] Consider a circular drumhead can be modelled by the vibrations of a circular membrane with a uniform thickness, attached to a rigid frame. Due to the phenomenon of resonance, at certain vibration frequencies, its resonant frequencies, the membrane can store vibrational energy, the surface moving in a characteristic pattern of standing waves. This is called a normal mode. A membrane has an infinite number of these normal modes, starting with a lowest frequency one called the fundamental mode. Find the free-vibration equation of motion, and the free-vibration solution with different mode shapes for circular membrane with radius r.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Concept explainers

Free undamped vibrations

Whenever an object/system displaces in either side of its equilibrium position with the help of an external/internal means, then the object/system moves regularly or randomly on both sides of its equilibrium position. This motion is called vibration. Generally, there are three types of vibration: longitudinal vibration, transverse vibration, and torsional vibration.

Question

Problem No.5:
Find the frequency Equation, Mode shapes (Normal Function), and expression ßl for a uniform
beam with young's modulus E, mass density p, cross-sectional area A, second moment of cross-
sectional area I, and a length / with Pinned - Free boundary conditions.
Whereas the free-vibration lateral equation of motion of the beam is obtained as:
24w a²w
+ <=0
Əx4 at²
c²
where
C =
EI
VPA
And the General Solution of the free vibration equation of motion is:
w(x, t) = [C₁ cos ßx + C₂ sin ßx + C3 cosh Bx + C4 sinh ßx][A cos wt + P sin wt]
Consider a circular drumhead can be modelled by the vibrations of a circular membrane with
a uniform thickness, attached to a rigid frame. Due to the phenomenon of resonance, at certain
vibration frequencies, its resonant frequencies, the membrane can store vibrational energy,
the surface moving in a characteristic pattern of standing waves. This is called a normal mode.
A membrane has an infinite number of these normal modes, starting with a lowest frequency
one called the fundamental mode.
Find the free-vibration equation of motion, and the free-vibration solution with different mode
shapes for circular membrane with radius r.

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