# Wing Kinematic and Dynamic Parameters

### Rotational Stiffness Calculations

Effective rotational stiffness calculations of each flexure with respect to a given joint are calculated by taking the 2nd derivative of the joint potential energy with respect to the joint angle under consideration.  Table 1 gives the effective rotational stiffness about the 4-bar output joint (d).  All units are N m/rad.  A flexure stiffness is calculated according to the formula k=EI/l (polyester flexures are 1/8 mm long and 6.25mm thick).  Calculations are shown for different values of lCRodAtt (distance from the base of joint a to the distal end of the connector rod), and l4 (length of link 4).  All other link lengths are assumed to be 6mm.  For joints a and d, flexure widths of 3mm are considered while joints b and g only use flexure widths of 1mm.  The actuator is assumed to be 10mm x 5mm PZT with stiffness 1.12e3 N/m.

## Table 1:Effective Rotational Stiffness

* Calculating the effective rotational stiffness about d, except for kJd all the variables are also functions of d and the values shown are roughly for where the total potential energy is minimized (the dependence is due to the nonlinear relation between a and d). It is also interesting to note that kCrodDstl is negative in certain regions, indicating that the resulting torque is in the opposite direction as the restoring torque.

Rotational Inertia Moment Calculations

Effective inertia moments of each component about a particular joint are calculated by taking the 2nd partial derivative of the component kinetic energy with respect to the angular velocity of the joint under consideration.  Raw values of the principal inertia moments are estimated to be 1.17e-9 for the actuator, 2.02e-13 for links 2 & 3, and 1.31e-13 for link 4; values in the table are for calculations about d. All units are kg m2. Raw values for the slider and connector rod are neglected because they are assumed to undergo pure translation.

Table 2:  Effective Rotational Inertia

* jRaw is the principal inertia moment of the component.

** Calculating the effective rotational inertia moment about d only j4 is not also a function of d.

Assumptions

1)                  All links are considered rigid bodies and all joints are ideal pin joints with constant rotational stiffness.

2)                  The slider and connector rod move only horizontally so that (a) their kinetic energies associated with the rotational inertia are ignored and (b) the potential energy associated with the proximal end flexure is ignored while that for the distal end flexure is assumed to depend only upon a.

3)                  The center of gravity of each link is halfway along the axis between the 2 joints connecting them (i.e., the off-axis offset is ignored; also, the asymmetry of link 2 is not considered).