ABSTRACT

The sign of an input force, such asfi, is always taken positive at this stage since at this point it is a general, rather than specific, function of time. When we go to solve the equation, then fi must be made specific and we then can tell what its proper sign should be. As for the damper force, we know from Chapter 2 that the magnitude of a viscous damper force is Bv0 and that it opposes the velocity. Since our equation must agree with these known facts, let's check it for all possible velocities, because we don't know at this stage what the velocity is doing. If v0 is positive (mass moving to the right), the damper opposes with a force (on the mass) which is to the left, a negative force; thus -Bv0 should be negative, which it is (B is understood to be positive). If v0 were negative (mass moving to the left), the force of the damper on the mass is to the right (positive), which again agrees with -Bv0 • Finally, if v0 = 0, the damper exerts no force on the mass. We see that, no matter whether the velocity is+, -,or 0, our damper force term in Eq. (7-2) is correct in magnitude and direction. If you have in the past not used such a systematic procedure to get correct algebraic signs in your equations, I recommend that you now adopt it, not just for damper forces but for every term in every differential equation. Having satisfied ourselves as to the correctness of signs, we now arrange the equation with terms in the unknown on the left (in decreasing order of the derivatives), and the given inputs on the right:

You should get into the habit of always arranging your equations in this systematic way. Engineers and mathematicians talk about the "left-hand side" and the "righthand" side; these terms are meaningless unless we all agree to arrange our equations in the standard way.