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Thus the mechanism is underactuated. The stroke variation according to the loading is obtained via underactuation.
Initially, kinematic and static force analyses of the variable stroke mechanism are performed. It is assumed that masses of the
links are negligible and operating speeds are slow. Therefore, the entire analysis is based on the static equilibrium. For the analysis
of the multi degree-of-freedom mechanisms the method of virtual work is used.
The virtual work of the active loads is derived from Fig. 2.where kij represents the linear spring stiffness and cij is the spring initial position constant between ith and jth links (i.e., the angle
between ith and jth links).
Then the virtual work of all the active forces is given byKinematic and force analyses of the mechanism can be performed by solving Eqs. (7), (8), (15), and (16) simultaneously. If
two of the four position variables (θ12, θ13, θ14 and s15) are known (two position input for a two-degree of freedom mechanism
yields to a constraint mechanism), F15 and T12 can be determined in closed-form. However, this is not the common case in
practice since mechanisms are generally analyzed for a given output-loading condition. If the output force F15 and the crank
angle θ12 are known, or the input link torque T12 and the stroke s15 are known (these are the most common cases), then
Eqs. (7), (8), (15), and (16) become highly non-linear in the rest of the unknown parameters. These parameters are θ13, θ14, s15,
and T12 or θ12, θ13, θ14, and F15. The analytical solution of these non-linear equations can be very difficult if not impossible.
Therefore, these equations need to be solved numerically. These cases are summarized for the structure parameters (a2, a3, a4,
k34, k45, c34, c45)asin Table 1.
It should be noted that in a casewhere output force F15 and crank angle θ12 are given, a problemmay occur: in applications for a
cyclic mechanism, magnitude and/or direction of the output-link force may change according to the working zone. However,
kinematics of multi degree-of-freedom underactuated mechanisms is uncertain before the loads are applied; generally, the
working zone of the output-link is also uncertain. Moreover, the magnitude of the output load is a parameter that directly affects
the output-link's oscillation interval. However, the case is different for the variable stroke mechanism; from many examples it is
observed that for resistive output loads when θ12=0° or θ12=180°, the mechanism is approximately at the dead centers.
Similarly, as in the conventional in-line slider-crank mechanism, the crank position is the major parameter that defines working
and return strokes. Hence, it can be assumed that the output-link load is simply a function of the crank's position for the in-line
variable stroke mechanism. In the case of extreme conditions (such as a sudden change in the direction of motion of the output
force and/or exaggerated link length proportions), this assumption may not be valid.
Example 1. Kinematic analysis of a variable stroke mechanism for clockwise rotation of the crank with the following data is
represented below. The link proportions are a2=a4=1 unit and a3=3 units. The spring variables are k34=k45=k=100 Nunit/
rad and c34=c45=2.618 rad. The maximum value of the output load (F=200 N) during the work stroke (180°bθ12 b360°) is
assumed to be five times that of the return stroke's maximum value as shown in Fig. 3. The direction of loading is resistive to the
motion of the output-link.
The unknown parameters θ13, θ14, s15, and T12 can be determined by solving non-linear Eqs. (7), (8), (15), and (16) numerically
via a mathematics software package with one degree increment. The same set of initial guess for the zero degree of crank angle
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