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The folded flexure, shown in the figure above, also reduces axial stress components in the beams [Tang, Nguyen, and Howe; Tang]. Each end of the flexure is free to expand or contract in all directions. The original residual stress in a small section of the flexure (Delta in figure above) is averaged over the entire beam length, giving a reduced effective residual stress.

where L_b is the flexure beam length. The beams in the flexure will alternate in the sign of the stress. A review of several variations of the folded flexure along with a discussion of nonlinear static analysis and modal analysis are found in Judy's Ph.D. thesis [Judy]. Analytic spring constant equations are given below. A mathematica file is also available.

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Spring Constants

Definition of terms used in spring-constant equations

TERM	MEANING	UNITS
E	Young's modulus of elasticity	Pa
G	shear modulus of elasticity = E/[2(1+nu)]	Pa
J	torsion constant	Pa
Iz,t	bending moment of inertia for truss beam segments	m^4
Ib,t	bending moment of inertia for long beam segments	m^4
alpha	= Iz,t / Iz,b
Seb	= E*Ix,b; out-of-plane bending stiffness of long beams	N-m^2
Set	= E*Ix,t; out-of-plane bending stiffness of truss	N-m^2
Sgb	= G*Jb; torsional stiffness of long beams
Sgt	= G*Jt; torsional stiffness of truss
Lt1	length of outer beam segment of truss	m
Lt2	length of inner beam segment of truss	m
Lt	length of truss beam segment when Lt = Lt1 = Lt2	m
Lb	length of long beam segment when Lb = Lb1 = Lb2	m
tilde Lt1	= Lt1 / Lb1
tilde Lt2	= Lt2 / Lb1
tilde Lb2	= Lb2 / Lb1
tilde Lt	= Lt / Lb
delta x	x-directed displacement	m
delta y	y-directed displacement	m
delta z	z-directed displacement (out-of-plane)	m
kx	x-directed spring constant	N/m
ky	y-directed spring constant	N/m
kz	z-directed spring constant (out-of-plane)	N/m

Case where L_b=L_b1=L_b2 and L_t=L_t1=L_t2:

General Case: