The intent of this project is to provide a reference of analysis to mechanical systems. Mechanical systems, in accordance with various other system such electrical and chemical system, may be characterized as dynamic systems. A dynamic system may be thought of as collection of components that interact with each other in a particular manner. Each of these components which characterize the dynamic system may be studied by using mathematical modeling techniques. The following report will analyze a particular mechanical system, which is described, in second section of this report, and produce results that explain the output of the system given a input. The variation of the dampening parameter is also considered along with the effects it has on the response time to the output. The techniques discussed here may be used to further study other systems.

The Introduction, previews the method of analysis used to study a mechanical system, as well as methods used for deriving a mathematical model that describe the mechanical system. The second section, titled Problem review and Design Objectives, describes a particular system and it’s parameters, inputs and outputs of the system. The third section, Modeling and Analysis, steps through the process of taking the actual system, apply it to a set of equations, then extracting the solutions. Finally, the fourth section, titled Design, discusses several cases that were considered along with an actual component which could be represented by a variable within the system

1- Introduction

A mathematical model is a set of equations that describes the system of interest and the interaction between the components. The mathematical model in this case is based on the physical limitations, such as forces, velocities, acceleration, and the laws that govern these conditions. The approach to deriving these equations begins with an analysis of each of the components. The analysis of the components is a description of the forces, moments and other time varying variables that have an impact on the component. The analyses, which are referred to as free-body diagrams, are described in Section 2 of this report. The derived equations, are typically second or higher-order differential equations.

The next step in the analyzing the system is to formulate a set of independent equations that will produce a desired solution. The state variable equations, which describes the response of the system given an input is found by taking the second or higher-order differential equations and converting them into first order equations.

The state variable equations may then be solved by various methods. Two examples, which could be used to estimate a solution, include the Runge-kutta integration and Euler-Huan integration. These methods are often employed in various computer software programs. Through these solutions, numerical and graphical data can be associated to the physical system. The system may then be adjusted by changing parameter values in order to study the systems input-output response.

2-Problem Review and Design Objectives:

The mechanical system shown in figure 1 contains two gears and connecting moment arms. Each of the arms has a force applied. The larger gear is restricted by a rotational damping effect and spring tension.
 
                                                                                Figure1. Mechanical system schematic

This system is to be considered as having an output of q 1, an input of F1, and the parameter Kd1 as a variable. The values given for each of the components in figure 1 are listed in Table1 above.

The behavior of this system is a result of the applied force F1 at a distance R1 from the center of the smaller gear that translates to a specific angular displacement directly with the ratio of N1 to N2. The reaction at the end of the moment arm, acting a distance R2 from the center of the larger gear, and the effect of the rotational dampening Kd1, also effect the behavior of the system. The reaction at the end of the arm is related by the sum of forces acting on the block.

The objective of the design for this particular system is to select the parameter Kd1 to provide the quickest response of the output q 1 for the step input F1 without residual oscillation. The approach to this design shall begin with a mathematical model of the system.

3- Modeling and Analysis:

Modeling of this system shall begin with analyzing both gears and the block by means of a free-body diagram shown in figure (2).

Equations from free body diagram show the sum of forces and moments acting on each of the components.

 

                                                                                                       (1
                                                                                                          (2
                                                                             (3

The following steps simplify the equation 2 and 3 to one equation.

Solving for Fc the connecting force between gears in equations 2, 3.
 
                                                                                                                (2.a
 
                                                                                     (3.a

Equation 2.a and 3.a can now be set equal to each other.

 
                                                                                (4
The radius of gear 1 is 1/3 the radius of gear 2. By geometry

Solving for q 2 from the above relationship

F₂ the spring force acting between the mass and the second moment arm equations is

Replacing F2 into equation 4
 

Rewriting equation 5 for a ₁

 
                                                                                                                                                                    (5

Rewriting equation 1 for a₁ and replacing F₂

 The state variable equation is now formed from equation 1 and 5.

These equations are next analyzed using Mathcad (see Appendix A). The parameters used in Table 1 are tested first. A physical interpretation can be made from the graphs in Mathcad. The x1 vs. t graph shows a oscillation that dampens to a value of about 2. This would imply that the system starting at rest and under the define conditions reaches steady state the x1 displacement of slightly less then 2m in a little over 50 seconds. A similar interpretation can be made with q 1 the angular displacement. Form here the manipulation of the variable Kd1 can be made to try and reduce the time it takes q 1 to respond to the input force F1

4- Design

This section will consider the described system in figure 1 given several different cases. The cases considered change with the variable Kd1 ranging from 25Nm/s to 200Nm/s. These results are again subjected to a Mathcad analysis (see Appendices B through F).

In general the response time is quickest at values for Kd1 below 100Nm/s. Values under 50Nm/s tend to show opposite results. A definite range of values for Kd1 (50Nm/s to 100Nm/s) appears to have the quickest response time. This trend of critical dampening may be study in most mechanical system involving vibrations. For values of Kd1 under 50Nm/s a case of under dampening occur, while, for values past 100Nm/s a situation of over dampening is apparent. Reference to differential equations on the study of characteristic equations and the associated roots may further explain the three cases of dampening. The best overall response came from the value of 55Nm/s for Kd1.

The actual component that was chosen to model Kd1 was a vibration dampening suspension mount. Found through an Industrial supply company http://www.mcmaster.com/ (McMaster-Carr Co.), this mount comprised of rubber cushions is used to reduce vibrations in equipment in oscillation motion. One type of application for this product may be to offer dampening effect in a tension arm of a belt assembly. Maintaining a constant position for the belt would require the proper response time by the tension arm. By placing the mount in the system, control would be provided of the angular displacement. For a value of 55Nm/s (40.5 lbf) a suspension mount part # (65185k53) was chosen. The values for torque at 10 deg and 30 deg deflections are 11.79 lbf and 63.01 lbf respectively.

Summary

Mechanical systems may be reduced to individual components that inter act by known physical laws. By examining the behavior of motion and geometry which, the system is subjected to, a desired outcome may be found. Manipulating certain parameters, within the design, individually may produce a desired response time.

The analysis began with defining the free-body diagram for each of the components acting in the system. For each free-body diagram a set of equations is generated and described as a differential equation. These equations are then expressed in first-order. This set of first-order equations describes the state variable form.

The state variable form describes the system’s position, velocity and acceleration in relation to the step function. The step function that is the input may be thought of as pulse that delivers an immediate value to the system. Using such tools as Mathcad, graphical and numerical responses may be found. Desired design objectives may then be experimented with from here.

Appendix A

Intial Design

Paramters for runge-kutta solver

Intial Conditions

Appendix B

Kd1=25Nm/s

Appendix C

KD1=50Nm/s

Appendix D

KD1=55Nm/s

Appendix E

KD1=150Nm/s

Appendix F

KD1=200Nm/s

Appendix F

KD1=200Nm/s