Student ID Number: 901471236

Analysis of a Vise-Grip

Kyle Collins AME 40423

Mechanisms and Machines

September 22, 2010

Abstract

The purpose of this project was to analyze a vise-grip clamping device and determine the relationship between the input force and the output (clamping) force. A skeleton diagram was created for the mechanism, which is a coupler-driven four-bar mechanism. Equilibrium equations were derived from free body diagrams, and the system of equations was solved for the mathematical relation between input and output. It was found that maximum, in fact infinite, mechanical advantage is gained when the coupler is in-line with the ground link. This configuration forms the basis of the mechanism’s clamping ability: when the mechanism is “over center,” in this configuration, only a minimal amount of force is needed to maintain clamping pressure. Finally, it was determined that the adjustable sliding joint is useful in obtaining a wide range of clamping angles.

1 Engineering Analysis A vise-grip clamping tool was analyzed in this project. A vise-grip is an example of a coupler-driven four-bar mechanism. In order to analyze the relationship between the input force and the clamping force, a skeleton diagram was first constructed to study the kinematics of the mechanism. Second, a force-balance analysis was conducted on the links of the vise-grip in order to derive a mathematical equation relating input force and clamping force. 1.1 Skeleton Diagram A skeleton diagram of a vise-grip is shown on the following page in Figure 1. The joint between links 1 and 2 is an adjustable sliding joint. The dimension r1 can be varied by changing the location of the “grounded” joint 1-2 along the indicated axis x. For the purposes of later force-balance analysis, this sliding joint will be modeled as a conventional pin-joint. The force Fd is the driving (input) force, and the force Fl is the clamping (output) force.

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Figure 1: Skeleton diagram for a vise-grip

1.2 Force-balance Analysis In order to analyze the relationship between the input force and the clamping force, free body diagrams were constructed for each of the non-ground links of the mechanisms. These diagrams are shown in Figure 2 on the following page. Because link 2 experiences no external torques, it was identified as a two-force member. Assuming static equilibrium, and taking advantage of trigonometric co-function identities, link 3 gives:

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MB∑ = FdL − F4yr3 = 0 , (1)

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Fy∑ = −Fd + F2 sinφ − F4y = 0, (2)

and

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Fx∑ = −F2 cosφ − F4x = 0. (3)

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Figure 2: Free body diagrams for the links of a vise-grip The static-equilibrium conditions for link 4 give:

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MD∑ = FlM − r4 (F4x sinµ)+ r4 (F4 y cosµ) = 0. (4) Equations (1) through (4) are four equations in five unknowns: F2 , F4x , F4y , Fd , and Fl . To solve the system for the desired input-output relationship, the expression Fl M can be taken as a known value. Equation (4) becomes:

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FlM = r4 (F4x sinµ)− r4 (F4 y cosµ) (4.1) The system of equations (1), (2), (3), and (4.1) can be solved by using matrix form and applying Cramer’s rule. Writing the system on matrix form gives:

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0 0 −r3 Lsinφ 0 −1 −1−cosφ −1 0 00 r4 sinµ −r4 cosµ 0

F2F4 xF 4 y

Fd

=

000

FlM

. (5)

Applying Cramer’s rule to equation (5) to solve for Fd gives:

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Fd = FlMr3 sinφ

−r4[r3 cosφ sinµ + Lsin(φ + µ)] . (6)

Full hand calculations for the computation of the equilibrium equations and the application of Cramer’s rule can be found in Appendix 1. 2 Results Equation (6) gives a relationship between the input force Fd and the output torque of link 4, Fl M; however, in order to best explain the vise-grip’s ability to function as a clamp, the mechanical advantage of the mechanism was calculated. The adjustable nature of the joint linking bodies 1 and 2 was also found to increase the versatility of the tool. 2.1 Clamping Ability Defining the mechanical advantage of the vise-grip offers insight into the tool’s clamping ability. In a standard four-bar mechanism, the mechanical advantage is defined as the ratio of the output torque to the input torque. Because the vise-grip is a coupler-driven four-bar mechanism, the mechanical advantage Ma can be defined as the ratio of the output (clamping) torque to the input force. By rearranging equation (6),

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Ma =torqueoutputforceinput

=FlMFd

=−r4r3[r3 cosφ sinµ + Lsin(φ + µ)]

sinφ. (7)

The mechanical advantage is the amount by which the input force Fd is amplified into the clamping torque. When links 2 and 3 are in overlapping or in line,

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φ = 0º or 180º. In these two scenarios, known as the limit positions of the mechanism, sin

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φ = 0. With its denominator equal to zero, Ma is infinite, and the mechanism is said to have infinite mechanical advantage. For the vise-grip, infinite mechanical advantage corresponds to when links 2 and 3 are inline; in this position, the mechanism is said to be “over center” and only the smallest possible force applied to link 3 is required to resist an infinitely high clamping torque. In practice, links 2 and 3 of the vise-grip continue to rotate slightly past the limit position and are prevented from further rotation by a metal stopper on link 2. This lends stability to the fully clamped position. A small lever applies a minimal force between links 2 and 3 to disengage the mechanism from the clamped and “locked” position.

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2.2 Adjustability The lateral adjustability of the virtual-pin joint linking bodies 1 and 2 provides an important feature to the vise-grip. As shown below in Figure 3, changing the location of this pin joint also changes the position of link 4 when the mechanism is in the limit position. The different clamping angles

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θ ,

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θ ’, and

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θ ” shown in Figure 3 correspond to the pin locations x, x’, and x”. The adjustable joint allows the vise-grip to clamp a wide range of objects with various thicknesses or diameters while maintaining maximum mechanical advantage.

Figure 3: Various pin-joint locations and corresponding clamping angles

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Appendix 1 Hand Calculations for the Force Balance Analysis