THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

20.4 Forced Vibration 593The logarithmic decrement δ is defined at the natural logarithm of the ratio oftwo points such as x 1 and x 2 . Invoking Equation (20.14) and this definition forlogarithmic decrement, we can derive( )x1δ = ln = √ 2πξ(20.18)x 2 1 − ξ2Thus, the logarithmic decrement δ is expressed in terms of the damping ration ξof the system.A negative value of the response function can also be used to find the logarithmicdecrement, but in this case τ d should be replaced by τ d /2, because a half period isused in this evaluation. For this situation,δ = 2ln(x1x 3)The values of the response function at any two points can be used to find the twounknowns ξ and ω n . It is also important to realize that the ratio of the points in theresponse curve, e.g., x 1 /x 2 , x 2 /x 3 ,orx 3 /x 4 , will be identical only in the presenceof viscous damping.20.4 Forced VibrationForced vibration occurs when f (t) ≠ 0 in Equation (20.1). The forcing functioncan be a harmonic excitation, an example of which is the imbalance in rotatingmachinery such as a motor. Or it can be an impulse type of excitation, such as thatproduced by a hammer or it can be simply the weight of the moving part itself.Harmonic ExcitationA harmonic forcing function can be represented by F 0 sin ωt. The differentialequation for the model of Figure 20.1 assumes the following form:mẍ + Cẋ + kx = F 0 sin ωt (20.19)With the assumption of zero initial conditions, the Laplacian transform ofEquation (20.18) yields(ms 2 + Cs + k ) ( )ωX(s) = F 0 (20.20)s 2 + ω 2and solving for X(s)( )F 0 ωX(s) =(20.21)ms 2 + Cs + k s 2 + ω 2As a rule, the solution x(t) will consist of two parts, viz. the complementarysolution and a particular solution. The former corresponds to the transient part ofthe total solution and the latter to the steady-state part. The transient portion of the