Overview

The target mechanism of this analysis is a front porch umbrella with an off-center support beam and variable shading angle.

This is a 2-DoF, seven link mechanism comprising four revolute joints and two sliding joints. Both sliding joint variables act as the inputs to the mechanism. Additionally, there is a crank mechanism that controls the lengths of one of the sliders and is the way in which the user opens and closes the umbrella. Below is a gallery of the important features of the mechanism, followed by a simplified schematic which will be used for the kinematic analysis.

Complete Mechanism

Sliding Support Allows for Variable Shade Angles

The Smaller the Offset, The Shallower the Shade Angle

Slider Mechanism for Main Support Beam

Slider Mechanism for Main Support Beam

Hand Crank Mechanism for Opening and Closing Umbrella

Central Support - Modeled as a Sliding Joint for this Paper

The main kinematic analysis will be focused on the position and velocity analysis of the linkages in the mechanism, as well as a study of the mechanical advantage provided by the crank mechanism in lifting the weight of the umbrella. The acceleration equations will also be calculated and plotted but are in general unimportant for the operation of the mechanism in its intended use.

The following plot shows a schematic of the mechanism, and labels all of the nomenclature that will be used during the kinematic analysis. The input sliding joint variables are labeled h1 and h2 and on the diagram represent the lengths AB and EF respectively.

Kinematic Schematic

Some values which are not shown on the diagram are:

L₆ refers to the short length FG
d₁, d₂, d₃, and d₄ refer to the lengths BC, CG, BD and DG respectively
θ_c and ω_crefer to the rotation angle and velocity of the hand crank, respectively
The hand crank is located on the sliding link L₄ and connects directly to the sliding link L₇ (located at E) through a tensioned string running through L₃ and L₆ and emerging at F

Below is video of the umbrella being going from fully open to fully closed. Additionally, the slider L₄ is adjusted to make the mechanism more compact in its closed state.

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