Journal of Applied Mechanics Technical Briefs Modal Analysis of Nonviscously where r is the spatial position vector and t is time, specified in some domain D, is governed by a linear partial differential Damped Beams equation ␳͑ ͒ ͑ ͒ L ͑ ͒ L ͑ ͒ ͑ ͒ ෈ D ෈ ͓ ͔ r u¨ r,t + 1u˙ r,t + 2u r,t = p r,t ; r , t 0,T 1 S. Adhikari ͑1͒ e-mail: [email protected] with homogeneous linear boundary conditions of the form M ͑ ͒ ෈ ⌫ M ͑ ͒ ෈ ⌫ ͑ ͒ M. I. Friswell 1u r,t =0; r 1 and 2u˙ r,t =0; r 2 2 Sir George White Professor of Aerospace Engineering ⌫ ⌫ specified on some boundary surfaces 1 and 2. In the above equation ␳͑r͒ is the mass distribution of the system: p͑r,t͒ is the Department of Aerospace Engineering, L distributed time-varying forcing function; and 2 is the spatial M M Queens Building, self-adjoint stiffness operator and 1 and 2 are linear opera- University of Bristol, tors acting on the boundary. For external ͑or foundation͒ damping L Queens Walk, the operator 1 can be written in the form Bristol BS8 1TR, UK t L u˙͑r,t͒ = ͵ ͵ C ͑r,␰,t − ␶͒u˙͑␰,␶͒d␶ d␰ ͑3͒ 1 1 D −ϱ Y. Lei where C ͑r,␰,t͒ is the kernel function. The velocities u˙͑␰,␶͒ at Associate Professor 1 different time instants and spatial locations are coupled through College of Aerospace and Material Engineering, this kernel function. Lei et al. ͓5͔ considered both external and National University of Defense Technology, internal damping, and although internal damping is not considered Changsha 410073, PRC further in this paper, the proposed method may be extended to this case. Kernel functions that serve similar purposes have been de- scribed by different names in different subjects ͑for example, re- tardation functions, heredity functions, after-effect functions, re- Linear dynamics of Euler–Bernoulli beams with nonviscous non- laxation functions͒, and different models have been used to local damping is considered. It is assumed that the damping force describe them. Equation ͑1͒ together with Eq. ͑3͒ represents a at a given point in the beam depends on the past history of ve- continuous dynamic system with general linear damping. It may locities at different points via convolution integrals over exponen- L ͑ ͒ be noted that if 1 =0 in Eq. 1 , i.e., an undamped system, or if tially decaying kernel functions. Conventional viscous and vis- the system satisfies the criteria given by Caughey and O’Kelly ͓6͔, coelastic damping models can be obtained as special cases of this then the system will possess classical normal modes. However, general damping model. The equation of motion of the beam with L due to the general nature of the operator 1 as described by Eq. such a general damping model results in a linear partial integro- ͑3͒, there is no definite reason why the system should have clas- differential equation. Exact closed-form equations of the natural sical normal modes. Thus the mode shapes and natural frequen- frequencies and mode shapes of the beam are derived. Numerical cies of such systems in general will be complex in nature. In this examples are provided to illustrate the new results. ͑ ͒ ͓ ͔ context we wish to note that the system expressed by Eq. 1 and DOI: 10.1115/1.2712315 the damping operator defined in Eq. ͑3͒ represents a partial integro-differential equation with the boundary conditions given in Eq. ͑2͒. In this technical brief we are interested in the natural 1 Introduction frequencies and mode shapes of the system. Exact closed-form expressions of such quantities for the general case are difficult to Viscous damping is the most common damping model used for obtain. We make the following general assumptions: linear dynamic systems. However, within the scope of linear theory, more general nonviscous models have been used in the 1. The mass and stiffness distributions are homogeneous, that recent past ͓1–5͔. Nonviscous damping models in general have is, they do not vary with the position vector r; and more parameters and therefore are more likely to have a better 2. The damping kernel function is separable in space and time match with experimental measurements. A linear damped continu- so that ͑ ͒ ous dynamic system in which the displacement variable u r,t , ͑ ␰ ␶͒ ͑ ͒ ͑ ␰͒ ͑ ␶͒͑͒ C1 r, ,t − = C r c r − g t − 4 1 Corresponding author. Depending on the nature of the functions c͑•͒ and g͑•͒, several Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received February 8, 2006; final manu- special cases, starting from the simple viscous model to the more script received August 15, 2006. Review conducted by Oliver M. O’Reilly. general nonviscous model, may arise ͓5͔. For example, if both c͑•͒ 1026 / Vol. 74, SEPTEMBER 2007 Copyright © 2007 by ASME Transactions of the ASME Downloaded 02 Dec 2007 to 137.222.10.58. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm continuously differentiable up to fourth order with respect to x. Here we assume that w͑x,t͒ is continuously differentiable up to fifth order. In what follows, Eqs. ͑7͒ and ͑8͒ are solved separately and the solutions are combined to obtain the eigensolutions. We begin with the solution of Eq. ͑7͒. Fig. 1 Euler–Bernoulli beam with nonviscous damping patch 3 Solution for the Section With Nonlocal Viscoelastic and g͑•͒ are delta functions then the result is the “locally reacting” Damping viscous damping model. If only c͑•͒ is a delta function, then the resulting model becomes a locally reacting viscoelastic damping Assuming zero initial conditions, the Laplace transform of the which is also known as the time hysteresis model. For the case displacement ͑with no external force͒ satisfies ͑ ͒ when only g • is a delta function, the resulting model becomes ␣ x2 nonlocal viscous damping or the spatial hysteresis model. Here EIWIV͑x,s͒ + s2␳AW͑x,s͒ + sG͑s͒͵ exp͑− ␣͉x − ␰͉͒W͑␰,s͒d␰ 2 the general case, that is, when none of these two functions are the x1 delta functions, is considered. Based on the transfer matrix ෈ ͑ ͒͑͒ method ͓7͔, we propose a new method for modal analysis of a =0, x x1,x2 9 Euler–Bernoulli beam with general linear damping given by Eq. Here s is the complex Laplace parameter; and W͑x,s͒ is the ͑4͒. Laplace transform of w͑x,t͒. The roman superscripts, for example ͑•͒IV, denote the order of derivative with respect to the spatial 2 Governing Equation of Motion variable x. It is useful to separate the contribution arising from the The Euler–Bernoulli beam considered in this study is shown in term ͉x−␰͉ in Eq. ͑9͒ as Fig. 1. The left and right coordinates of the beam are denoted by ␣ x x and x , respectively. The beam has a nonlocal viscoelastic IV͑ ͒ 2␳ ͑ ͒ ͑ ͒͵ ͓ ␣͑ ␰͔͒ ͑␰ ͒ ␰ L R EIW x,s + s AW x,s + sG s exp − x − W ,s d 2 damping patch between x1 and x2. It is assumed that the elastic x1 properties of the beam are uniformly distributed with bending x2 rigidity EI and mass density ␳A. ␣ + sG͑s͒͵ exp͓␣͑x − ␰͔͒W͑␰,s͒d␰ =0 ͑10͒ In order to formulate and solve the equation of motion, it is 2 x necessary to use some kind of plausible functional form of the kernel functions in space and time. In this study we choose the The function W͑x,s͒ is continuously differentiable up to fifth or- following functional forms ͓5͔ der with respect to x because w͑x,t͒ is assumed to be continuously differentiable up to fifth order. Differentiating Eq. ͑10͒ with re- g͑t͒ = gϱ␮ exp͑− ␮t͒͑5a͒ spect to the spatial variable x one obtains so that ␣2 x ␮ V͑ ͒ 2␳ I͑ ͒ ͑ ͒͵ ͓ ␣͑ ␰͔͒ ͑␰ ͒ ␰ gϱ EIW x,s + s AW x,s − sG s exp − x − W ,s d G͑s͒ = , gϱ, ␮ ജ 0 ͑5b͒ 2 s + ␮ x1 x and ␣2 2 + sG͑s͒͵ exp͓␣͑x − ␰͔͒W͑␰,s͒d␰ =0 ͑11͒ ␣ 2 x c͑x − ␰͒ = exp͑− ␣͉x − ␰͉͒, C͑x͒ =1 ͑6͒ 2 Differentiating again gives These models imply that the correlations in both space and time ␣3 x decay exponentially. If ␣→ ϱ ,␮→ϱ one obtains the standard EIWVI͑x,s͒ + s2␳AWII͑x,s͒ + sG͑s͒͵ exp͓− ␣͑x − ␰͔͒W͑␰,s͒d␰ ␣→ϱ ␮ 2 viscous model; if and is finite one obtains the time hys- x1 teresis model; and if ␣ is finite but ␮→ϱ one obtains the spatial ␣3 x2 ␣2 hysteresis model. + sG͑s͒͵ exp͓␣͑x − ␰͔͒W͑␰,s͒d␰ −2 sG͑s͒W͑x,s͒ =0 The equation of motion of the part with the damping patch can 2 x 2 be expressed by ͑ ͒ x t 12 ␣ 2w͑x,t͒ 2ץ 4w͑x,t͒ץ EI + ␳A + ͵ ͵ exp͑− ␣͉x − ␰͉͒ͯ Using Eqs. ͑10͒ and ͑12͒ we have 2 ץ 4 ץ x t ϱ 2 x1 − EIWVI͑x,s͒ + s2␳AWII͑x,s͒ − ␣2͓EIWIV͑x,s͒ + s2␳AW͑x,s͔͒ w͑␰,t͒ 2ץ ϫ ␮ ͓ ␮͑ ␶͔͒ ͯ ␰ ␶ − ␣ sG͑s͒W͑x,s͒ =0 ͑13͒ gϱ exp − t − d d tץ t=␶ Equation ͑13͒ is a sixth-order ordinary differential equation which ෈ ͓ ͔͑͒ = 0 when x x1,x2 7 can be solved by transforming into the first-order form. Recall that the solution of a sixth-order ordinary differential equation requires The equation of motion of the part outside the damping patch can six boundary conditions corresponding to W͑x,s͒,...,WVI͑x,s͒.