Question

Expression for displacement equation for forced vibration & amplitude under forced vibration.

Answer 1

Answer:

Forced vibrations are always present in milling since in the removal of material there are dynamic, time-varying forces acting on the flexible system, composed of the machine–tool, spindle-tool, workpiece.

5.4.1 Equations of Motion for Forced Spring Mass Systems

Equation of Motion for External Forcing

We have no problem setting up and solving equations of motion by now. First draw a free body diagram for the system, as show on the right

Newton’s law of motion gives

Rearrange and susbstitute for F(t)

Check out our list of solutions to standard ODEs. We find that if we set

,

our equation can be reduced to the form

which is on the list.

The (horrible) solution to this equation is given in the list of solutions. We will discuss the solution later, after we have analyzed the other two systems.

Equation of Motion for Base Excitation

Exactly the same approach works for this system. The free body diagram is shown in the figure. Note that the force in the spring is now k(x-y) because the length of the spring is . Similarly, the rate of change of length of the dashpot is d(x-y)/dt.

Newton’s second law then tells us that

Make the following substitutions

and the equation reduces to the standard form

Given the initial conditions

and the base motion

we can look up the solution in our handy list of solutions to ODEs.

Equation of motion for Rotor Excitation

Finally, we will derive the equation of motion for the third case. Free body diagrams are shown below for both the rotor and the mass

Note that the horizontal acceleration of the mass is

Hence, applying Newton’s second law in the horizontal direction for both masses:

Add these two equations to eliminate H and rearrange

To arrange this into standard form, make the following substitutions

whereupon the equation of motion reduces to

Finally, look at the picture to convince yourself that if the crank rotates with angular velocity , then

where is the length of the crank.

The solution can once again be found in the list of solutions to ODEs.