Ch. 3: Forced Vibration of 1-dof system
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for this normalized transfer function). 3.2 Frequency Response Function 2 1 2 n ω ω
ζ = − 0 1 / 2
ζ < < 1 / 2
ζ ≥
Ch. 3: Forced Vibration of 1-DOF System 3.2 Frequency Response Function Ex. 3 Consider the pivoted mechanism with k=4x10 3 N/m, l 1 =0.05 m, l 2 =0.07 m, l=0.10 m, and m=40 kg. The mass of the beam is 40kg which is pivoted at point O and assumed to be rigid. Calculate c so that the damping ratio of the system is 0.2. Also determine the amplitude of vibration of the steady-state response if a 10 N force is applied to the mass at a frequency of 10 rad/s. Ch. 3: Forced Vibration of 1-DOF System ( )
( ) ( ) 1 2 2 1 1 2 2 1 2 1
2 12 2 0.1 10 cos10 0.5
0.0049 59.05
0.0049 15.37, 0.2 , 627.3 Ns/m
2 0.5 O O n n l l M I Fl mgl Mg c l l k l l l l l l ml M M t c c c θ θ θ θ θ θ θ θ θ ω ζ ω ω − ⎛ ⎞ ⎡ ⎤ = − − − − ⎜ ⎟ ⎣ ⎦ ⎝ ⎠ ⎡ ⎤ + − ⎛ ⎞ = + + ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦ × = + + = = = = × × = ∑ ( )
( ) ( ) 10, 0.6506 1 0.02677
24.268 59.05 0.5767 0.26 0.02677 cos 10 0.424 ss r H i i t ω θ = = = − ° + = − 3.2 Frequency Response Function Ch. 3: Forced Vibration of 1-DOF System Ex. 4 A foot pedal for a musical instrument is modeled as in the figure. With k=2000 kg/s 2 , c=25 kg/s, m=25 kg, and F(t)=50cos2πt N, compute the steady-state response assuming the system starts from rest. Use the small angle approximation. 3.2 Frequency Response Function Ch. 3: Forced Vibration of 1-DOF System ( ) ( ) ( ) ( ) 2 2 0.15
0.05 0.05
0.05 0.1
0.15 5 100 3.75 50 cos 2
, positive CW 6 3 Find the parameters 2.98, 0.0373, 2 , 2.108 1 0.0087
177.4 1 2 since 0, the transie O O n M I F k c m t r H i k r i r θ θ θ θ θ θ θ π ω ζ ω π ω ζ ζ ⎡ ⎤ = × − × − × = × ⎣ ⎦ + + = = = = = = = − ° − + ≠ ∑ ( ) ( ) ( ) 0 nt response will die out cos 0.435 cos 2 3.096
θ ω ω θ π = + = − 3.2 Frequency Response Function Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications Ch. 3: Forced Vibration of 1-DOF System 3.3 Applications ( )
) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 measured acc. 10 1 actual acc. 9.81
1 2 1 2 0.962
From the problem statement, 628 rad/s, 1 628 rad/s 1 1 758 rad/s , 0.56
2 1 5745.6 N/m, 8.49 Ns/ n d n d d n n z y r r r r r k c m m k c ω ζ ζ ω ω ω ζ ω ω ζ ω ω ζ ω ζ = = = − + − + = = = − = = = − = = = = = − = = m Ch. 3: Forced Vibration of 1-DOF System 3.4
Periodic Excitation A periodic function is any function that repeats itself in time, called period T. It is more general than the harmonic function. Here, we will find the response to the input that is a periodic function. The idea is to decompose that periodic input into the sum of many harmonics. The response, by the superposition principle of linear system, is then the sum of the responses of individual harmonic. The response of a harmonic function was studied in section 3.1 3.4 Periodic Excitation ( )
( )
f t T = + Ch. 3: Forced Vibration of 1-DOF System Fourier found the way to decompose the periodic function into sum of harmonic functions (sine & cosine) whose frequencies are multiples of the fundamental frequency. The fundamental frequency is the frequency of the periodic function. 3.4 Periodic Excitation
Ch. 3: Forced Vibration of 1-DOF System Fourier series 3.4 Periodic Excitation ( )
( ) ( ) ( ) ( )
0 0 0 0 0 1 0 0 0 0 Fourier series in real form: 2 cos
sin ,
2 Fourier coefficients: 2 cos
, 0,1, 2, 2 sin
, 1, 2, 3, Fourier series in complex form:
ω π ω ω ω ω ω ∞ = ∞ =−∞ = + + = = = = = = ∑ ∫ ∫ … … ( ) 0 0 0 2 ,
Fourier cofficients (complex): 1 , , 2, 1, 0,1, 2, T in t n T C f t e dt n T ω π ω − = = = − − ∑ ∫ … … Ch. 3: Forced Vibration of 1-DOF System Some properties of Fourier series 3.4 Periodic Excitation ) ( ) ) ( )
) ( )
) ( )
( ) 0 0 0 1 1 If is an even function, 0. 2 If is an odd function, 0. 3 is the average value of over one period. 2 4 If
is real,
2 Re n n in t k k n n f t b f t a a f t f t C C f t C C e ω ∞ − = = = ⎛ ⎞ = ⇒ = + ⎜ ⎟ ⎝ ⎠ ∑ Ch. 3: Forced Vibration of 1-DOF System Frequency spectrum tells how much each harmonic contributes to the periodic function . Plot of the amplitude of each harmonic vs. its frequency is the (discrete) frequency spectrum. 3.4 Periodic Excitation ( )
f t ( )
2 2 0 0 In real form, the harmonic at has the amplitude In complex form, the harmonic at has the amplitude 2 Re
n n n a b n C ω ω + Ch. 3: Forced Vibration of 1-DOF System 3.4 Periodic Excitation Ch. 3: Forced Vibration of 1-DOF System Superposition principle of linear system 3.4 Periodic Excitation
Ch. 3: Forced Vibration of 1-DOF System Response to harmonic excitation 3.4 Periodic Excitation
Ch. 3: Forced Vibration of 1-DOF System 3.4 Periodic Excitation ( ) (
( ) ( ) ( ) 0 0 0 0 0 0 2 0 0 0 1 0 0 1 From section 3.1, 1 where
1 2 2 Re by superposition, 2 Re
in t n in t ss n n n in t n n in t ss n n mx cx kx F t C e x C H in e H in n n k i mx cx kx F t C C e C x C H in e k ω ω ω ω ω ω ω ω ζ ω ω ω ∞ = ∞ = + + = = = = ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ − + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ + + = = + ⎜ ⎟ ⎝ ⎠ ⎛ = + ⎝ ∑ ∑ ⎞ ⎜ ⎟ ⎠ Ch. 3: Forced Vibration of 1-DOF System 3.4 Periodic Excitation response frequency spectrum excitation frequency spectrum system frequency response
Ch. 3: Forced Vibration of 1-DOF System Ex.
Calculate the response of a damped system to the periodic excitation f(t) depicted in the figure by means of the exponential form of the Fourier series. The system damping ratio is 0.1 and the driving frequency is ¼ of the system natural freq. 3.4 Periodic Excitation Ch. 3: Forced Vibration of 1-DOF System 3.4 Periodic Excitation ( ) ( )
( ) ( )
( ) 0 0 0 0 0 / 2 0 0 0 / 2
odd Expand
as sum of harmonic series 1 1 2
,
0, even 1 1 2 , odd
2 2
t n n T T T in t in t in t n T n n i n in t n f t f t C e C f t e dt Ae dt Ae dt T T T n iA C i A n n n i A A f t e e n n ω ω ω ω ω ω π ω π π π π ∞ =−∞ − − − = = ⎡ ⎤ = = + − = ⎢ ⎥ ⎣ ⎦ = ⎧ ⎪ ⎡ ⎤ = − −
= ⎨ ⎣ ⎦ − = ⎪⎩ ∴ = − × = × ∑ ∫ ∫ ∫ ∑ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 2 0 odd 1,3,
0 2 2 1 2 2 2 2 0 1,3, 4 1 sin 1 , 0.1, 1 2 4 1 1 / 4 0.05 1 0.05 , tan
1 0.25
1 0.25
0.05 4 1 sin t n n n n n n n n ss n n n A n t n n n G i r r i r G i n i n n G G n n n A x t G n t G n π ω π ω ω ω ζ ζ ω ω ω ω π ⎛ ⎞ ∞ − ⎜ ⎟ ⎝ ⎠ = = − ∞ = = = = = = = − + = − + − = = − ⎡ ⎤ − + ⎣ ⎦ ∴ = + ∑ ∑ ∑ … … Ch. 3: Forced Vibration of 1-DOF System 3.4 Periodic Excitation Ch. 3: Forced Vibration of 1-DOF System Ex.
The cam and follower impart a displacement y(t) in the form of a periodic sawtooth function to the lower end of the system. Derive an expression Download 1,63 Mb. Do'stlaringiz bilan baham:
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