mech
racyon9m
EXP7.pdf
R. Ehgott (Created) 1/13 03/01/01 T. Hao (Revised) 03/01/16
I. OBJECTIVES
1.1 To determine the undamped and damped natural frequency of a single degree of freedom system.
1.2 To obtain a plot of the magnification factor versus the frequency ratio.
1.3 To determine the damping ratio and the damping coefficient C of the system.
II. BACKGROUND
Any mechanical system that has mass and stiffness can vibrate. Vibration results when there is an energy exchange between the mass, which stores kinetic energy, and the spring, which stores potential energy. In such a system, the mass will oscillate with periodic motion. The energy loss in a vibrating system is modeled as a dashpot or viscous damper. The dashpot is comprised of a cylinder filled with a viscous fluid, such as oil and a piston. The force produced by a dashpot FD is proportional to the velocity of the mass. The constant of proportionality is the viscous damping coefficient c which has units of N-s/m or lb-s/in.
For a damped system, the ratio of the damping coefficient c to the critical damping value is a dimensionless parameter which represents a meaningful measure of the amount of damping present. This damping ratio is called ζ (zeta). For oscillations to occur, ζ must be less than one.
FS = kx
F (t)
FD = cx .
m x (t)
Figure 1 Single Degree of Freedom System
R. Ehgott (Created) 2/13 03/01/01 T. Hao (Revised) 03/01/16
Differential equation: )(tFxkxcxm =++ &&& 7.a
Angular natural frequency: m k
n =ω 7.b
Critical damping value: ncr mc ω2= 7.c
Damping ratio: ncr m
c c c
ω ζ
2 == 7.d
More complicated mechanical systems can be modeled using the same differential equation theory. The system considered for study here is a viscously damped single degree of freedom system which is excited by and external harmonic force.
BAR PIVOT
Figure 2 Experimental Set-Up
R. Ehgott (Created) 3/13 03/01/01 T. Hao (Revised) 03/01/16
List of Symbols
LM = Length from the pin support to the center of the motor (in.)
LS = Length from the pin support to the spring (in.)
LD = Length from the pin support to the dashpot (in.)
LB = Length of the rigid bar (in.)
mM = Combined mass of the motor unit, disks and added weights (lb-s2/in.)
me = Eccentric rotating mass (lb-s 2/in.)
mD = Mass of the dashpot (part attached to the rigid bar) (lb-s 2/in.)
mB = Mass of the rigid bar (lb-s 2/in.)
k = Spring constant (lb/in.)
c = Viscous damping constant (lb-s/in.)
Ccr = Critical damping value = nIω2
ζ = Damping ratio = crCC
r = Radius of the eccentric mass (in.)
θ = Angle of the rigid bar rotation (rad)
nω = Angular natural frequency (rad/s)
fω = Angular forcing frequency (rad/s)
fn = Natural frequency (cps or Hz)
I = Total mass moment of inertia of the system about point O (lb-s2-in.)
IB = Mass moment of inertia of the rigid bar about point O (lb-s 2-in.)
ID = Mass moment of inertia of the part of the dashpot about O (lb-s 2-in.)
IM = Mass moment of inertia of the motor unit about O (lb-s 2-in.)
x = Displacement amplitude as a function of time (in.)
xB = Displacement amplitude at the end of the rigid bar (in.)
xn = Displacement amplitude at the end of the rigid bar at nω (in.)
xst = Equivalent static deflection at the end of the rigid bar (in.)
β = Frequency ratio nωω
R. Ehgott (Created) 4/13 03/01/01 T. Hao (Revised) 03/01/16
The rigid bar shown in Fig. 2 pivots about point O when subjected to the harmonic force. This harmonic force is produced by an eccentric rotating mass me driven by a motor.
r
LM
LS
LD
FS = kx O
xB
FD = cx D .
x
)sin()( 2 tmtF ffe ωω=
θ
LB
Figure 3 Free-Body Diagram of System
Derivation of Differential Equation and Natural Frequency
The free-body diagram of the rigid bar, pivoting about point O, is shown in Fig. 3. The deflection of the spring θSLx = , and the velocity of the dashpot is . θ&& DD Lx =
By summing moments about point O, the following equation is obtained:
)sin(
)sin()()( 222
2
tLrmLcLk
LtrmLLcLLkM
fMfeDS
MffeDDSSO
ωωθθ
ωωθθ
+−−=
+−−=Σ &
& 7.1
The total mass moment of inertia of the system about point O can be determined from the following individual mass inertia terms
DMB IIII ++= 7.2
where the bar, motor and dashpot inertia about point O can be defined by:
2
3 1
BBB LmI = ; ; 2 MMM LmI =
2 DDD LmI =
R. Ehgott (Created) 5/13 03/01/01 T. Hao (Revised) 03/01/16
Using Newton’s second law, , the differential equation of motion can now be written in terms of
θ&&IM O =Σ )(tθ :
θωωθθ &&& ItLrmLcLk fMfeDS =+−− )sin( 222
or )sin(222 tLrmLkLcI fMfeSD ωωθθθ =++ &&& 7.3
Equation (7.3) has the same form as the differential equation in Eq. (7.a) and the angular natural frequency can be calculated from:
I Lk S
n
2
=ω 7.4
where k, spring constant, is to be determined experimentally using Hooke’s Law and
mB = 0.0217 lb-s 2/in.
mM = 0.0447 lb-s 2/in.
mD = 4.534 x 10 −3 lb-s2/in.
The lengths LB, LM and LD are determined from measurement.
Magnification Factor (MF) and the Damping Coefficient (c)
The solution to the differential equation, Eq. (7.3), is a combination of the complementary and the particular solutions. With time, the transient solution dies out and we are left with the steady-state solution. The amplitude of the steady-state solution has the form:
22222
2
)()( fDfS
feM
LcILk
rmL
ωω
ω θ
+− = 7.5
The displacement amplitude xB at right end of the rigid bar (see Fig. 3) is:
22222
2
)()( fDfS
feMB BB
LcILk
rmLL Lx
ωω
ω θ
+− == 7.6
R. Ehgott (Created) 6/13 03/01/01 T. Hao (Revised) 03/01/16
Dividing the numerator and denominator by I, we have
2 2
22 2
2
)()( f D
f S
f eMB
B
I Lc
I Lk
I rmLL
x
ωω
ω
+−
= 7.7
Since:
I Lk S
n
2
=ω n
D
cr I Lc
C C
ω ζ
2
2
== I Lc D
n
2
2 =ωζ
Substitution gives:
2222
2
)2()( fnfn
f eMB
B I
rmLL
x ωωζωω
ω
+− =
Dividing the numerator and denominator by 2nω
2
2
2
2
)2()(1
)(
n
f
n
f
n
feMB
B
I rmLL
x
ω ω
ζ ω ω
ω ω
+ ⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ −
=
This equation can be put in nondimensional form by dividing both sides by
I rmLL eMB . Thus,
2
2
2
2
)2()(1
)(
n
f
n
f
n
f
B
eMB r x
mLL I
ω ω
ζ ω ω
ω ω
+ ⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡ −
= 7.8
The left side of Eq. (7.8) gives the definition of the magnification factor,
r x
mLL I B
eMB
=MF .
R. Ehgott (Created) 7/13 03/01/01 T. Hao (Revised) 03/01/16
Plotting the right hand side of Eq. (7.8) versus the frequency ratio nf ωωβ = , Fig. 4 is plotted for different damping ratios.
0 0
0.5
1
1.5
2
2.5
3
3.5
4
1 2 3 4
ζ = 0 ζ = 0.15 ζ = 0.2 ζ = 0.5 ζ = 1
r x
mLL I B
eMB
=MF
n
f
ω
ω β =
Figure 4 Magnification factor versus the frequency ratio. Large amplification occurs when the forcing frequency fω is near the natural frequency nω .
R. Ehgott (Created) 8/13 03/01/01 T. Hao (Revised) 03/01/16
Bandwidth Method for Determining the Damping Present
The damping ratio ζ of the system can be determined from the experimental results by using the magnification factor plot and the bandwidth method:
0 0
0.5
1
1.5
2
2.5
3
3.5
4
0.5 1 1.5 2
n
f
ω
ω β =
r x
mLL I B
eMB
=MF
3.354
0.707x3.354=2.371
0.885 1.243
Figure 5 An example of magnification factor versus the frequency
ratio plot. 1. Plot the data obtained in Table III to obtain the MF vs. β plot as Fig. 5. 2. Determine the maximum MF value. For this example MFmax = 3.354. 3. Draw a horizontal line at 0.707 x MFmax = 2.371. 4. Identify the frequency ratios at the intersections of this line (Step 3) and
the MF plot. In this example the frequency ratios are β1 = 0.885 and β2 = 1.243.
5. The damping ratio can be determined from:
12
12
ββ ββ
ζ +
− = ; 168.0
885.0243.1 885.0243.1
= + −
=ζ 7.9
R. Ehgott (Created) 9/13 03/01/01 T. Hao (Revised) 03/01/16
A second method, called the resonant amplitude method, can be used as a check. At resonance, when nf ωω = , the damping ratio can be found from:
maxMF2 1
=ζ 7.10
For this example
149.0 )354.3(2
1 ==ζ
Finally, the damping constant c can be calculated from:
2
2
D
n
L I
c ωζ
= 7.11
III. EQUIPMENT
Universal Vibration Apparatus. Steel scale.
IV. PROCEDURE
IV.1 Measure the beam length LB and the distances LM , LS , and LD .
IV.2 Calculate k using F = k x.
IV.3 Measure the eccentric distance r and record the eccentric rotating mass me (the value of the mass is stamped on the mass disk).
iV.4 Calculate IB , IM , ID and I .
IV.5 Calculate the angular natural frequency nω using Eq. (7.4).
IV.6 Record the displacement xB for different motor speeds (rpm) in Table III.
IV.7 Calculate MF and nf ωωβ = and plot them.
IV.8 Estimate the damping ratio ζ using the bandwidth method and the maximum resonant amplitude method.
IV.9 Calculate the damping constant c using Eq. (7.11).
R. Ehgott (Created) 10/13 03/01/01 T. Hao (Revised) 03/01/16
V. REPORT
The Results section should include:
V.1 The calculated and the experimentally determined angular natural frequency.
V.2 The plot of the MF vs. nf ωωβ = .
V.3 The value calculated for ζ for the two methods described.
V.4 The value calculated for c .
V.5 Using ζ , calculate the damped angular frequency using the equation below:
21 ζωω −= nd
Discuss the results and draw appropriate conclusions.
VI. SELECTED REFERENCES
Thomson, W.T. and Dahleh, M.D., Theory of Vibration with Applications, 5th Edition, Pearson, 1997.
R. Ehgott (Created) 11/13 03/01/01 T. Hao (Revised) 03/01/16
EXPERIMENT # 7
Student’s Name__________________________________ Group #_________
Instructor’s Name____________________ Date Exp. Performed__________
Table I Measurements and Calculations
COMPONENT mass (lb-s2/in.) LENGTH (in.) CALCULATION Rigid Bar mB = LB = IB =
Motor mM = LM = IM = Dashpot mD = LD = ID =
Rotating Mass me = r = Stiffness (lb/in.) I =
Spring k = LS = 2 SLk =
nω =
Table II Zeta and Damping Constant Calculations
ζ , using Eq. (7.9) ζ = ζ , using Eq. (7.10) ζ = % Difference Damping Constant c =
R. Ehgott (Created) 12/13 03/01/01 T. Hao (Revised) 03/01/16
Table III Experimental Results and Calculations
Motor Speed (rpm)
fω (rad/s) xB (in.) nf ωωβ = r x
mLL I B
eMB
=MF
40 80 140 160 170 175 180 185 190 195 200 205 210 220 240 260 300 350 400 450 500 550 600
R. Ehgott (Created) 13/13 03/01/01 T. Hao (Revised) 03/01/16
