for the Case of Linear Damping Where the Energy Loss Is Proportional to the Square Of

- For the case of linear damping where the energy loss is proportional to the square of the strain or amplitude (eg.4), the hysteresis curve is an ellipse. - When the damping loss is not a quadratic function of the strain or amplitude, the hysteresis in no longer an ellipse.

Equivalent Viscous Damping :

- The primary influence of damping on oscillatory systems is that of limiting the amplitude of response at resonance.

- The damping has little influence on the response in the frequency regions away from resonance.

- The equivalent damping Ceq is found by equating the energy dissipated by the viscous damping to that of the non-viscous damping force with assumed harmonic motion. 2 pCeq w X = w d (11)

where wd must be evaluated from the particular type of damping force.

- In the case of viscous damping, the amplitude at resonance, equation was found to be F X = 0 (10) Cwn

- For other types of damping, no such simple expression exists. It is possible, however, to approximate the resonant amplitude by substituting an equivalent

damping Ceq in the above equation.

Coulomb damping:

- It results from the sliding of two dry surfaces .

- The damping (friction) force is equal to the product of the normal force and the coefficient of friction m and is assumed to be independent of the velocity, once the motion is initiated.

- Hence, sometimes it is called a constant damping force.

- Since the sign of the damping force is always opposite to that of the velocity, the differential equation of motion for each sign is valid only for half-cycle intervals.

- Fig - To determine the decay of amplitude, we resort to the work-energy principle of equating the work done to the change in kinetic energy. - Choosing a half-cycle starting at the extreme position with velocity equal to zero

and the amplitude equal to X1 , the kinetic energy is zero and the work done on m is also zero:

1 2 2 k( X1- X- 1) - Fd ( X 1 + X - 1 ) = 0 2 1 or F= k( X - X ) d 2 1- 1 2k˙˙ or decay in amplitude X- X = d (12) 1- 1 k

where X -1 is the amplitude after the half-cycle as shown in the figure.

- Repeating this procedure for the next half-cycle a further decrease in amplitude of 2F d will be found, so that the decay in amplitude per cycle is a constant and equal k to

4F X- X = d 1 2 k

- The motion will cease, however, when the amplitude becomes less than D , at which position the spring face is insufficient to over-come the static friction, which is generally greater then the kinetic friction force.

k - The frequency of oscillation is w = , which is same as the undamped system m m

- It is assumed that under forced sinusoidal excitation the displacement of the system is with Coulomb damping is sinusoidal and equal to x= Xsinw t .

- The equivalent viscous damping can then be found , by noting that the work per

cycle by the Coulomb force Fd is equal to

Wd= F d (4 X ) 2 X average amplitude over a cycle pCeq w X= 4 F d X

4Fd or Ceq = pw X 2

The amplitude of forced vibration can be found by substituting Ceq (it should be noted that it contain X) in to the following eqn. F F X =0 = 0 22 2 2 2 4F (k- mw) + ( Ceq w) 2 骣 d (k- mw ) + 琪 2 桫p X

Solving for X, we obtain

2 2 2 骣4F 骣4Fd F - d 1- 琪 0 琪 p F 桫p F0 桫 0 X =2 = 2 ------(15) k- mw k 骣w 1- 琪桫wn

X - It should be noted that unlike the system with viscous damping, , goes to d st

when w= wn .

4F - For numerator to remain real, the term d must be less than 1.0 p F0

Structural Damping :

- When the materials are cyclically stressed, energy is dissipated internally within the material itself.

- Experiments show that or most structural metals, such as steel or aluminum, the energy dissipated per cycle is independent of the frequency over a wide frequency range and proportional to the square of the amplitude of vibration.

- The internal damping fitting this classification is called solid damping or structural damping.

- With the energy dissipation per cycle proportion to the square of the vibration amplitude, the loss coefficient h is a constant and the shape of the hysteresis curve remains unchanged with amplitude and independent of the strain rate.

- Energy dissipated by structural damping may be written as

2 wd = a X (16)

where a is a constant with units of force/displacement .

- Using the concept of equivalent viscous damping, it gives 2 2 pCeq w X= a X a C = ------(17) eq pw

- Substitution of Ceq for C, the EOM can be written as

骣a mx˙˙+琪 x ˙ + kx = F( t) ------(18) 桫pw

Complex stiffness:

In the calculation of the flutter speeds of air plane wings and tail surfaces, the concept of complex stiffness is used. - It is arrived at by assuming the oscillation to be harmonic, which gives

骣 a jw t mx˙˙ +琪 k + j x = F0 e ------(19) 桫 p

jw t or mx˙˙ + k(1 + jg ) x = F0 e

a with g = ------(20) p k

(1+ jg ) Complex stiffness and g the structural damping factor.

- Using the concept of complex stiffness for problems in structural vibrations is advantageous in that one needs only to multiply the stiffness turns in the system by (1+ jg ) .

- The method is justified, however, only for harmonic oscillations.

- With the solution x= Xe jw t , the steady state amplitude from equation, becomes

F X = 0 2 ------(21) (k- mw) + i g k

- The amplitude at resonance is than

F X = 0 ------(22) w= wn g k

- Comparing this with the resonant response of a system with viscous damping

F X = 0 ------(23) 2Jk - We conclude that with equal magnitude at resonance, the structural damping factor is equal to twice the viscous damping factor.

Frequency Response with structural Damping :

- Starting with eqn. (21), the complex frequency response for structural damping can be shown to be a circle.

w - Letting = f , we get wn

2 Xk 1 (1- f ) -g H( r) = = = + j F 2 22 2 2 2 2 0 (1-f) + jg (1-f) +g( 1 - f ) + g

=x + iy ------(24)