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 ζ 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 constant C which has units of N-s/m or Lb-s/in.

For a damped system, the ratio of the damping constant C to the critical damping = value is a dimensionless parameter which Fs Kx Fd = Cx & 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. M x(t)

F(t)

Mx&& + Cx& + Kx = F(t) Differential equation:

K Circular natural frequency: ω = n M = ω Critical damping value: Cc 2M n

Damping ratio: ζ = = ω C / Cc C / 2M n Figure 1 Single Degree of Freedom System

R. Ehrgott 2/13 03/01/01 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.

MOTOR ECCENTRIC SPRING ROTATING MASS

BEAM PIVOT

DASHPOT

SPEED CONTROL

Figure 2 EXPERIMENTAL SET-UP

The rigid bar OA shown in Figure 3 pivots about point O when subjected to the harmonic force. This harmonic force is produced by a rotating mass Me driven by a motor.

R. Ehrgott 3/13 03/01/01 List of Symbols

Lm = Length from the pinned support to the center of the motor (inches)

Ls = Length to the spring (inches)

Ld = Length to the dashpot (inches)

LB = Total length of the beam (inches) 2 Mm = Total mass of the motor unit, disks and added weights (lb-s /in) 2 Me = Eccentric rotating mass (lb-s /in) 2 Md = Mass of the dashpot (part attached to the beam) (lb-s /in) 2 MB = Mass of the beam (lb-s /in) K = Spring constant (lb/in) C = Viscous damping constant (lb-s/in) 2 Cc = Critical damping value 2Mωn = 2Ioωn/Ld

ζ = Damping ratio = C/Cc r = Radius of the eccentric mass (inches) θ = Angle of the beam rotation (radians)

ωn = Circular natural frequency (rad/s)

ωf = Circular forcing frequency (rad/s) fn = Natural frequency (cps or Hz) 2 Io = Total mass moment of inertia about point O (lb-s -in) 2 IB = Mass moment of inertia of the beam about point O (lb-s -in) 2 Id = Mass moment of inertia of the damper about O (lb-s -in) 2 Im = Mass moment of inertia of the motor unit about O (lb-s -in) X = Displacement amplitude as a function of time

Xd = Displacement amplitude at the end of the rigid bar (inches)

Xn = Displacement amplitude at the end of the rigid bar at ωn (inches)

Xst = Equivalent static deflection at the end of the rigid bar (inches)

β = Frequency ratio ωf/ωn

R. Ehrgott 4/13 03/01/01 Derivation of Differential Equation and Natural Frequency

A free body diagram of the rigid bar, pivoting about point o, is shown below:

Ls ω 2 ω Lm F(t) = Me f sin( f t)

A ο θ Fs = Kx Xd

Ld

Fd = cvd Figure 3 Free Body Diagram of System

Where the deflection of the spring x = Ls θ And the velocity of the dashpot is = θ vd Ld &

By summing moments about point o, the following equation is obtained:

= − θ − θ + ω 2 ω ∑Mo (KLs )Ls (CLd &)Ld (Mer f sin( f t)Lm

= − 2θ − 2θ + ω 2 ω KLs CLd & Mer f Lm sin( f t) 7.1

The total mass moment of inertia of the system about point o can be determined from the following mass inertia terms

Io = IB + Im + Id 7.2

Where the beam, motor and dashpot inertia can be defined by:

2 IB = 1/3 MBLB 2 Im = MmLm 2 Id = MdLd

R. Ehrgott 5/13 03/01/01 Using Newton’s second law, ΣMo = Ioα, the differential equation of motion can now be written in terms of θ(t):

− 2θ − 2θ + ω 2 ω = Ι θ KLs CLd & Mer f Lm sin( f t) o&&

Ι θ + 2θ + 2θ = ω 2 ω 7.3 o&& CLd & KLs Mer f Lm sin( f t)

Equation (7.3) has the same form as the differential equation in Figure 1 and the circular natural frequency can be calculated from:

2 ω = KLs n Ι 7.4 o

Where K is to be determined experimentally using F = Kx and

2 Mb = 0.0217 lb-s /in 2 Mm = 0.0447 lb-s /in -3 2 Md = 4.534 x 10 lb-s /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 7.3 is a combination of the complementary and particular solution. 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: L M rω 2 θ = m e f 7.5 2 − Ι ω 2 2 + 2ω 2 (KLs o f ) (CLd f ) The displacement amplitude Xd at end A of the rigid bar (see Figure 2) is: ω 2 = θ = LmLBMer f 7.6 Xd LB 2 − Ι ω 2 2 + 2ω 2 (KLs o f ) (CLd f )

Dividing the numerator and denominator by Io

R. Ehrgott 6/13 03/01/01 LmLBMer ω 2 Ι f 7.7 = o Xd KL 2 CL 2 ( s − ω 2 )2 + ( d ω )2 Ι f Ι f o o

Since:

KL 2 C CL 2 ω = s ζ = C/ C = d = 2ζω n Ι c 2Ι ω Ι n o o n o

Substitution gives:

LmLBMer ω 2 Ι f = o Xd ω 2 − ω 2 2 + ςω ω 2 ( n f ) (2 n f )

2 Dividing the numerator and denominator by ωn L L M r ω m B e ( f )2 Ι ω X = o n d ω ω (1− ( f )2 )2 + (2ς f )2 ω ω n n

This equation can be put in nondimensional form by dividing both sides by Ι o

LmLBMer ω ( f )2 Ι X ω MF = o d = n 7.8 L L M r ω ω m B e (1− ( f )2 )2 + (2ς f )2 ω ω n n

R. Ehrgott 7/13 03/01/01 Plotting the right hand side of Equation 7.8 versus the frequency ratio βββ = ωωωf/ωωωn, the following plot is obtained for different damping ratios.

4

3.5

3

2.5 zeta = 0 zeta = 0.15 ΙΙΙ 2 zeta = 0.2 === o Xd MF zeta = 0.5 MLL eBm r 1.5 zeta = 1

1

0.5

0 01234

ωωω βββ ===β = f )( ωωω n

Figure 4 Magnification factor versus the frequency ratio. Large amplification occurs when the forcing frequency ωωωf is near the natural frequency ωωωn

R. Ehrgott 8/13 03/01/01 Bandwidth Method for Determining the Damping Present

The damping ratio ζζζ of the system can be determined from the experimental Magnification Factor plot using the bandwidth method:

4

3.5 3.33

3

2.5 0.707x3.33=2.36

2 ΙΙΙ X MF === o d MLL eBm r 1.5

1

0.5

0.90 1.25 0 00.511.52 ωωω βββ ===β = f )( ωωω n 1. Plot the data obtained in Table III to obtain the MF vs β plot as shown above 2. Determine the maximum MF value. For this example MFmax = 3.33 3. Draw a horizontal line at 0.707x MFmax = 2.36 4. Identify the frequency ratios at the intersection of this line (step 3) and the MF plot. In this example the frequency ratios are βββ1 = 0.9 and βββ2 = 1.25 5. The damping ratio can be determined from:

βββ −−− βββ12 −−− 9.025.1 ζζζ ===ζ = ζζζ ===ζ = === .0 163 7.9 βββ +++β βββ+ β12 +++ 9.025.1

R. Ehrgott 9/13 03/01/01 A second method, called the resonant amplitude method, can be used as a check. At resonance, when ωf = ωn, the damping ratio can be found from:

ζ = 1 7.10 2MFmax

For this example 1 ζ = = 0.15 2(3.33)

Finally, the damping coefficient C can be calculated from:

2Ι ζω C = o n 2 7.11 Ld

III. EQUIPMENT

Universal Vibration Apparatus. Scale

IV. PROCEDURE

4.1 Measure the beam length LB and the distances LM, LS, and Ld 4.2 Calculate K using F = Kx 4.3 Measure the eccentric distance r and record the eccentric rotating mass Me (the value of the mass is stamped on the mass disk) 4.4 Calculate IB , Im , Id and Io 4.5 Calculate the circular natural frequency ωn using Equation (7.4). 4.6 Record the displacement Xd for different motor speeds (RPM) in Table III 4.7 Calculate MF and ωf/ωn and plot them. 4.8 Estimate the damping ratio ζ using the bandwidth method and the maximum resonant amplitude method. 4.9 Calculate the damping constant C from Equation 7.9

R. Ehrgott 10/13 03/01/01 REPORT

The Results section should include:

• The calculated and the experimentally determined natural frequency. • The plot of the MF vs ωf/ωn • The value calculated for ζ for the two methods described. • The value calculated for C (Equation 7.9) • Using ζ, calculate the damped natural frequency using the equation: ω = ω − ζ 2 d n 1

Discuss the results and draw appropriate conclusions.

SELECTED REFERENCES

Vierck, R.K., Vibration Analysis, 2nd ed. Harper and Row, 1979

R. Ehrgott 11/13 03/01/01 TABLE I MEASUREMENTS AND CALCULATIONS

COMPONENT MASS (lb-s2/in) LENGTH (in) CALCULATION BEAM Mb =Lb= Ib = MOTOR Mm =Lm= Im = DASHPOT Md =Ld = Id = ROTATING MASS Me =r = STIFFNESS (lb/in) Io = 2 SPRING K = Ls =K Ls = ωn =

2 ω = KLs n Ι o

TABLE II ZETA AND DAMPING CONSTANTS CALCULATIONS

ζ USING EQ 7.9 ζ = ζ USING EQ 7.10 ζ = % DIFFERENCE DAMPING CONSTANT C =

R. Ehrgott 12/13 03/01/01 TABLE III EXPERIMENTAL RESULTS AND CALCULATIONS

DATA RPM ωωωf (rad/s) Xd (in) βββ = ωωωf/ω n MF = POINT (XdIo)/(rLmLbMe) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

R. Ehrgott 13/13 03/01/01