Design Equations for Initially Flat Blade Springs at Arbitrary Mounting Angles - Internal working note

Oct 22, 2026

Design equations are provided for a cantilevered flat blade spring initially flat and mounted at an arbitrary tilt angle. The equations compute beam bending, maximum normal stress and strain, and derive a circular deflected shape under constant stress assumptions. A vertical bounce frequency model is developed and compared to non-linear behavior by using an equivalent pendulum approximation with y set to total blade deflection. The note includes test-related parameters and calculations for the blade spring geometry and resonant response.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages14

Reading Time35 min

AccessFree to download

Introduction

Design Equations

Theory

Beam Bending

Vertical Bounce Frequency

Design Equations

LASER INTERFEROMETER GRAVITATIONAL WAVE OBSERVATORY LIGO Laboratory / LIGO Scientific Collaboration | LIGO-T0900324-v2 LIGO 6/22/10
| | | | | | --- | --- | --- | --- | --- | | | Design Equations for Initially Flat Blade Springs at Arbitrary Mounting Angles | | | | | | | Michael Smith, Jesse Amato-Grill | | | | | | | | | Distribution of this document: LIGO Scientific Collaboration This is an internal working note ofthe LIGO Laboratory. California Institute of Technology LIGO Project – MS 18-34 1200 E. California Blvd. Pasadena, CA 91125 Phone (626) 395-2129 Fax (626) 304-9834 E-mail: [info@ligo.caltech.edu](info@ligo.caltech.edu) LIGO Hanford Observatory P.O. Box 1970 Mail Stop S9-02 Richland WA 99352 Phone 509-372-8106 Fax 509-372-8137 Massachusetts Institute of Technology LIGO Project – NW22-295 185 Albany St Cambridge, MA 02139 Phone (617) 253-4824 Fax (617) 253-7014 E-mail: [info@ligo.mit.edu](info@ligo.mit.edu) LIGO Livingston Observatory P.O. Box 940 Livingston, LA 70754 Phone 225-686-3100 Fax 225-686-7189 [http://www.ligo.caltech.edu/](http://www.ligo.caltech.edu/) LIGO LIGO-T0900324-v2 1 Introduction The purpose of this document is to describe the design equations for a cantilevered flat blade spring, mounted at an arbitrary tilt angle, which takes a precise circular curvature under load with the tip of the spring having a horizontal slope. The test results of two specimen blade spring designs are shown. 2 Design Equations The bending and deflection of a flat blade cantilever spring are calculated from fundamental principles.1 The geometry of the cantilever blade spring is shown in Figure 1. Figure 1: Geometry of the Cantilever Blade Spring 1 Design of Machine Elements, M.R. Spotts, 2nd ed. 1953, Prentice-Hall, N.J LIGO LIGO-T0900324-v2 2.1 Theory 2.1.1 Beam Bending The maximum normal stress at any cross section of the blade occurs at the outer surface of the blade and is proportional to applied moment, to the distance from the neutral axis of the cross section to the outer surface, and inversely proportional to the area moment of inertia of the cross section. Bending moment Thickness of blade Width of blade Moment of inertia of cross section For a rectangular cross section M t b I I 􀀝􀀠 b􀂘t3 12 Maximium normal stress S 􀀝􀀠 M 􀂘 t 2 I The maximum strain at the outer surface of the blade is inversely proportional to the bending radius of the blade at the particular cross section. t maximum strain 􀁈 􀀝􀀠 ~~ ~~2R~~ ~~ Introducing the modulus of elasticity through the stress/strain relationship for a material that obeys Hook’s law, S 􀀝􀀠 E􀂘􀁈 S 􀀝􀀠 E􀂘 t 2􀂘R In general, the width of the blade is >> than the thickness, and the effective modulus of elasticity must be increased to account for the additional stiffness caused by the Poisson’s ratio, μ . E Eeff 􀀝􀀠 􀀋 1 􀀐 􀁐2􀀌 LIGO LIGO-T0900324-v2 The deflected shape of the blade spring will be circular with a constant radius, R, if the stress is equal at every cross section. The design will use a safe working stress for the material, which does not exceed the elastic limit. S 􀀝􀀠 S w 2.1.2 Vertical Bounce Frequency The vertical bounce frequency of a mass on a spring with a spring constant k is f0 􀀝􀀠 ~~ ~~2􀁓 􀂘 The spring constant is defined as the ratio of vertical force to vertical displacement. k 􀀝􀀠 my􀂘~~ ~~g Substituting the spring constant into the frequency equation, we obtain which is the equation for the frequency of a pendulum of length y. This result relies on the spring being linear with a single spring const. Strangely enough, even though the blade spring is non-linear and the spring constant varies with deflection, the measured vertical bounce frequency of the blade spring can be predicted precisely by using the pendulum formula and by choosing y = ymax, the total deflection of the blade spring under the load.

FAQs

What does the document describe about the cantilevered flat blade spring?

It describes design equations for an initially flat cantilevered flat blade spring mounted at an arbitrary tilt angle, including how it deflects under load and how the tip slope behaves.

How are maximum normal stress and strain evaluated?

Maximum normal stress at a blade cross section is related to the applied bending moment, the distance from the neutral axis, and the area moment of inertia. Maximum strain at the outer surface is tied to the bending radius through the stress/strain relationship.

How is vertical bounce frequency predicted for the blade spring?

The document models the vertical bounce frequency using the frequency relation for a mass on a spring and shows it can be predicted using a pendulum-equivalent approach by choosing the pendulum length to equal the total blade deflection under load.

Oct 22, 2026

Uploaded by Maya Linwood