By Henry M. Paynter

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The modeling process can sometimes be simplified replacing a series-parallel combination of stiffness or friction elements with a element. attention was given to linear systems that involve vertical motion. If "'~1-'""'"'" ments are measured from where the springs are neither stretched nor compressed, the forces must be included in the for any masses that can ""'"""'"'°'" the displacements are measured with to the static~···-··-· .. "~··~ when the is motionless and when no other external inputs of motion.

The element laws for the first three relate the external force to the acceleration, velocity, or displacement associated with the element. The lever is considered in Chapter 5. shows a mass which has units of Newton's second law states that the sum of the forces rate of of the momentum: d dt(Mv)=f foI a constant mass, can be written as dt =f subjected to a force f. 2 Element Laws _,.. 17 For (4) and (5) to hold, the momentum and acceleration must be measured with respect to an inertial reference frame.

Draw the free-body diagram for each mass, including the effect of gravity, and find the differential equations describing the system's behavior. c-equiHbrium position, when (t) = 0 and when the masses are motionless. SOLUTION The diagrams are shown in parts (b) and of the figure. 4 also apply to this problem. and x2 are positive, then and are stretched and is compressed. Under these circumstances, Ki exforce on Mi, and and exert downward forces on . X2. + + + + + has two unknown variables. If the element values and are and if the necessary initial conditions are then can be solved for x1 and x2 as functions of time the methods discussed in later Because x 2 is a of M2 is x1 + x2.