Problem description
An automotive component is built using an impression-die-forging process. This process consists of
shaping the workpiece by means of two dies, one attached to a “hammer” and the other to the “anvil”.
A block of metal is positioned on the anvil’s die; the hammer is then lifted and driven downward to
strike the workpiece and impress the shape of the two dies on the metallic block.
For a dynamic analysis of this process we will consider the following modelling assumptions.
Two bodies will be considered:
o the top body consisting of the hammer structure and the top die, with a total mass
o the bottom body consisting of the anvil, the bottom die and the metal workpiece, with
a total mass
The actual plastic deformation of the workpiece is considered instantaneous, as it lasts for a
very short amount of time. This impulsive event is modelled as an impulse with coefficient of
restitution
In addition to those assumptions, the following characteristics of the process are known:
The length of the hammer’s stroke is
A force with magnitude [SI units], which is directed downwards, is applied to
the hammer during the downward stroke (valid until the impact with the anvil, considering
the instant when the hammer is at the topmost position). After the impact the hammer is
lifted back to the topmost position.
After the impact, the anvil is moving with a velocity of magnitude , directed downwards
The anvil is mounted on a shock absorber which is composed of an elastic element with stiffness
and a nonlinear damper providing a force with magnitude and direction
opposite to the elongation velocity .
All quantitative values are reported in Table 1.
Motion of the hammer
1. Calculate analytically the position, velocity and acceleration profile of the hammer in time
between (topmost position) and the impact with the anvil. Also clearly specify values of
velocity and acceleration just before the impact.
Answer within the space provided by this box
2. Derive the finite difference equation for the hammer during the downward stroke, starting
from the equation of motion obtained in part 1.
Answer within the space provided by this box