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Patent

Dual-direction pulley system

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TLDR
In this article, a dual-direction pulley system was presented for a leg curl/extension exercise machine such that a user can perform leg curls and leg extensions from a single seated position.
Abstract
A dual-direction pulley system is disclosed in conjunction with an exercise machine. The pulley system provides resistance to a pivotally mounted actuator lever when the lever is rotated clockwise and when the lever is rotated counterclockwise. In a preferred embodiment the dual-direction pulley system provides resistance for a seated leg curl/extension exercise machine such that a user can perform leg curls and leg extensions from a single seated position.

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TL;DR: In this paper, an exercise machine for performing lat pull down exercises has a main frame having a user support pivot mount and an exercise arm having handles for gripping by a user movably mounted on the frame for movement between a start position located above the head of a user in a seated position on the user support and an end position lower than the start position and generally below the user's chin.
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References
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Variable resistance exercise device

TL;DR: In this paper, a variable resistance exercise device was proposed to provide a variable resisting force in response to a user's applied force by varying the viscosity of a variable fluid which surrounded plates rotated by the user applied force.
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Physical exercising apparatus

TL;DR: Physical exercising apparatus comprising a ground-supported user platform adapted to permit step-up exercise and at least one elastic member one end of which is attached to the apparatus and the free end carrying a user handle to enable a user to exercise by stretching the elastic member while supported at least in part on the platform as mentioned in this paper.
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TL;DR: A tri-pivot arm assembly for mounting on an exercise machine frame has a main arm, a swing arm, and a handle, with each pivot axis being perpendicular to the other two pivot axes as discussed by the authors.
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TL;DR: In this paper, a highly versatile exercise apparatuses including a central weight stack and opposed extension arms are described. But they do not discuss the use of a cable in the arms.
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Exercise stair device

Richard McFee
TL;DR: The exercise stair device as mentioned in this paper is a lightweight compact unit which simulates the exercise obtained while climbing stairs and includes means for providing a resistance (counterforce) to motion of the foot supports which increase with velocity.
Trending Questions (1)
What is the Lagrangian function for a double pulley system?

The provided paper does not mention the Lagrangian function for a double pulley system.