Report - PEER - University of California, Berkeley

where & x&(t ) = acceleration at the top of the bench, m = mass of the equipment and g =acceleration due to gravity. In Figure 2, x& (t)and x (t)= absolute velocity anddisplacement of the top of the bench, u&(t & ) , u& (t), and u (t)= acceleration, velocity anddisplacement of the equipment with respect to bench top. The static and kineticcoefficients of friction, µ andsµ , respectively, are used to represent the frictionalkresistance between the bench-top surface and the equipment. The kinetic coefficientof friction may be represented as a fraction φ of the static coefficient of friction (i.e.,µk= φµ s). Equation (1) assumes the bench has negligible motions in the verticaldirection. Once the equipment begins sliding on the bench, the equation of motion ofthe equipment may be expressed as (Shenton and Jones, 1991):m( & x( t)+ u&&( t))= −S(u&( t))µkmg(2)where, S ( u&( t))= signum function,S ( u& ( t))= 1; ( u& ( t)> 0)(3)S ( u& ( t))= −1; ( u& ( t)< 0)(4)Therefore, the sliding continues until the relative velocity of the mass equals to zero(i.e., u& ( t)= 0 ) and commences again if Equation (1) is satisfied.EquipmentCenter ofGravity (C.G.)u(t), u(t), u(t)Bench-topHx(t), x(t), x(t)LFigure 2. Schematic diagram of bench top supporting a rigid piece ofequipment.2.2 Bench Dynamic CharacteristicsThe bench system may be idealized as a single-degree-of-freedom (SDOF) system inthe horizontal direction, with the mass of the equipment resting on the bench-top. It isalso reasonable to assume negligible slippage between the base of the bench and thefloor surface, since the bench is anchored at the base to the floor. The equation ofmotion of the bench-top may then be expressed as:2& x( t)+ 2ζ nω x&n( t)+ ωnx(t)= −&xg( t)(5)where & x g(t)= the floor motion (or the motion at base of the bench), ω = the naturalncircular frequency and ζ = damping ratio, of the system. Applying equation (5), thenbench-top time history may easily be obtained when the floor time history is knownand thus this bench-top motion may be considered as input to the base of theequipment for the sliding analysis. This cascade approach neglects the dynamic200