PAMM · Proc. Appl. Math. Mech. 8, 10915 – 10916 (2008) / DOI 10.1002/pamm.200810915 Transient resonance problems in the case of the series of short duration loads Władysław Mironowicz*,1 and Marcin Sęk**,1 1 Faculty of Civil Engineering, Wrocław University of Technology, Poland The indicated in the title of this paper vibration problem concern the ferroconcrete slab – type industrial ceilings. The presence of unilateral constraints which my arise as a result of construction damages is taken into consideration. As an example, it is analysed a construction of a ceiling in shape of a rectangular slab supported on a regular net of columns and a non deformed solid, being a model of a machine, placed on the construction. © 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction The problem given in the title of the paper arises in different structures supporting machines. The area of consideration in this paper covers slab concrete industrial ceilings, which are most commonly constructed as a finless slab or finned slab, founded on evenly spaced columns and walls. This paper presents solution of transient resonance problems caused by quasiperiodic dynamic load of a series of short duration loads. The presence of unilateral constraints formed due to damage of the construction (damage of vibration isolation of the machine) is also taken into consideration. 2 Theoretical formulations Construction of the ceiling is consider as a rectangular slab founded upon regular network of columns and walls – on its circumference, with undeformable block rested on it, with appliance of vibration isolation. The block's purpose is to imitate the model of a machine (Fig. 1). Fig. 1 Schematic diagram of examined system. Vibrations of the system, show in figure 1, are described by a differential matrix equation: B  g ( t ) + Cg ( t ) + Kg ( t ) = Ff(t) (1) General form of inertia B, damping C and stiffness K matrices are presented by relations: ⎡B B=⎢ q ⎣0 0⎤ Bb ⎥⎦ ⎡ Cq C=⎢ ⎣Cbq Cqb ⎤ Cb ⎥⎦ ⎡ Kq K=⎢ ⎣ K bq K qb ⎤ K b ⎥⎦ ⎡q ⎤ g=⎢ ⎥ ⎣b ⎦ (2) where: q, b – generalized coordinates appropriately for slab and block, Bq, Cq, Kq – inertia, damping and stiffness matrices of the system (slab + columns + constraints), Bb, Cb, Kb – inertia, damping and stiffness matrices of the block, Kqb = KbqT, Cqb = CbqT – stiffness and damping matrices of fastening of block with slab. ____________________ * Corresponding author: e-mail wladyslaw.mironowicz@pwr.wroc.pl, Phone: +48 071 320 2479 e-mail marcin.sek@pwr.wroc.pl, Phone: +48 071 320 3779 ** © 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Sessions of Short Communications 20: Dynamics and Control 10916 Matrices presented above considering unilateral constraints were formulated more precisely in [1]. In the examined case, input function in a form of series of short duration loads is assume. Function f(t) describing variability of load in time might be written down in as: v f (t ) = ∑ Ai (t ) ⋅ ϕ (t − tir ) i =1 r z ⎪⎧0 gdy t < t oraz t > ti i ϕ (t − tir ) = ⎨ r z ⎪⎩ϕi gdy ti ≤ t ≤ ti (3) where: v – number of ‘portions’ of load counted from the beginning of observation, until the ‘portion’, of the current moment of the observation t, tir , tiz – time defining commencement and termination ith ‘portion’ (acceptance uniformly variable motion). Solution of equation of vibrations in case of existence of unilateral constraints becomes nonlinear. Solution of equation (1) is formulated with appliance of the notion of generalized impulse response function [2]. When function f(t) is of series of short duration loads (3), the solution of this method might be written down as: v −1 tiz t i =1 tir tvr w(t ) = ∑ Ai ∫ H (t − τ )F ⋅ ϕ (τ − tir )dτ + Av ∫ H (t − τ )F ⋅ ϕ (τ − tvr )dτ (4) where: H(t – τ) = [H1(t – τ),...,Hi(t – τ),…,Hn(t – τ)] – one verse matrix, which comprises functions of Hi(t – τ), Hi(t – τ) – influence function, being a solution w(t) to Dirac impulse in i – th direction of the generalized coordinates. 3 Numerical analyses We examine the system presented in figure 1. With the assumption of: a = 9 m, b = 6 m, c = 3 m, h = 4 m and thickness of concrete slab hp = 0,3 m. Vibration isolation of the machine are located in its four bottom corners. Each isolation consists of one damping – elastic constraint of stiffness ki = 2·106 N/m and constant of damping ci = 8.1·103 Ns/m. One of the constraints has expansion joint Δ of variable quantity. Dynamic load is a vertical force P = 30 kN, placed on the centre of upper surface of the machine. Table 1 presents vertical vibration amplitudes of the slab centre. Table 1 Vertical displacement of the central point of ceiling slab. Induction P in the form of half of a sinusoid series steady – state resonance 0 0 0,1 1 0,200 0,656 0,669 0,790 0 328 1,9 17,0 Induction P in the form of Dirac deltas series work of the machine steady – state resonance expansion joint Δ [mm] 0 0 0,1 1 displacement u [mm] 0,073 0,228 0,236 0,271 increase [%] 0 312 3,3 15,9 Induction P in the form of rectangular impulses series work of the machine steady – state resonance expansion joint Δ [mm] 0 0 0,1 1 displacement u [mm] 0,253 0,624 0,631 0,761 increase [%] 0 247 1,1 18,0 work of the machine expansion joint Δ [mm] displacement u [mm] increase [%] 4 10 0,817 19,7 10 0,279 18,3 10 0,791 21,1 Final conclusions In applied calculation examples of amplitude of vertical displacement of the central point of ceiling slab, obtained amplitudes during resonance work of systems, are higher than ones obtained during steady – state work of the machine – the scope is of 247 % to 328 %. The greatest differences of amplitudes were obtained during induction in the form of half of a sinusoid series. In examined examples, the increase of vertical displacement of the central point of ceiling slab in damaged systems equals (Δ = 0 ÷ 10 mm) compared to undamaged systems, this ratio is from 18,3 % to 21,1 %. References [1] Mironowicz W., Sęk M., Problem of unilateral constraints in the dynamics of the floor loaded with a machine, PAMM - Proc. Appl. Math. Mech. 6, 105-106 (2006). [2] Bryja D., Śniady P., Spatially coupled vibrations of a suspension bridge under random highway traffic, Earthq. Engng. Struct. Dyn., 1991, 20. © 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.gamm-proceedings.com