FBD of Flexible and Rigid Body Coupling

ThLiOp · Nov 18, 2014

Problem:

I am having a bit of problem with the signs on the force balance with the constraining force. The system is:

Schematic of a tapered bar fixed at the left wall with properties P(x), EA(x), distributed load f(x,t) and displacement u(x,t) over length L, connected through a spring k and damper c in parallel to a lumped mass M with displacement y(t) and output force p(t).

Attempt:

I drew the FBD with the constraining force for the lumped mass:

Free-body diagram of a block labeled M with displacement y(t) shown by an arrow above it pointing right, a force Q(t) arrow pointing left from its left side, and a force p(t) arrow pointing right from its right side.

Then I tried to solve for the constraint force in terms of the spring and damper:

Free-body diagram of a lumped mass M with displacement y(t) to the right, a leftward constraint force Q(t), and rightward-acting arrows labeled k(y-u(L,t)) and c(dy/dt-du/dt) on its right side.

For the constraint force, Q(t), I end up getting:

Q(t) = k[u(L,t) - y] + c[(du/dt) - (dy/dt)]

The answer that the professor gives has the signs changed on the k and c terms. Where am I going wrong?

Thank you!

Dr.D · Nov 18, 2014

The expression you wrote for Q(t) is positive when u(L,t)>y And the same for derivatives. Now look at your FBD; it shows positive Q to the let. There is your sign error.

ThLiOp · Nov 22, 2014

Thank you Dr.D, that makes sense now!

FBD of Flexible and Rigid Body Coupling

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