Esnek mekanizmaların tasarlanması ve kontrolü
Design and control of compliant mechanisms
Yazar: AYŞE TEKEŞ
Danışman: DOÇ. DR. MEHMET SALIH DOKUZ
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Makine Mühendisliği Ana Bilim Dalı
Derece: Doktora, 2013
Konu: Makine Mühendisliği = Mechanical Engineering

Özet:
Bu tezde literatürde analizi yapılmamış olan iki farklı esnek mekanizma tasarımı ve kontrolü çalışması yapılmıştır.
Mekanizmalardan ilki bekleme hareketi yapması istenen esnek çift kol mekanizmasıdır.
Esnek çift kol mekanizmasıyla rijit krank biyel mekanizması hareket ettirilmektedir. Esnek çift kol mekanizması mikro ve makro seviyede üretilebilir.
Mekanizmanın kinematik analizi Elastika teorisi ile elde edilmiştir. Büyük şekil değiştirmelerine maruz esnek kirişlerin yük-esneme eğrileri ankastre kiriş, tek kol
ve tüm çift kol için hesaplanmış ve yük-esneme eğrisi bir polinomla ifade edilmiştir.
Mekanizmanın dinamik cevabı doğrusal olmayan hareket denklemlerinin Runge Kutta metodu kullanılarak elde edilmiştir. Mekanizma lineer olmayan yay karakteristiğine
sahip yay içerdiğinden hareket denklemi tüm durumlar için durum geri beslemesi ile doğrusallaştırılmıştır. Durum geri beslemesi ile doğrusallaştırılmış sisteme PD kontrolcüsü uygulanmıştır. ?Istenen referans hareketin gerçekleştirilebilmesi için kontrol katsayıları uygun şekilde ayarlanmıştır.
İkinci mekanizma tamamen esnek 5 koldan oluşan bir mekanizmasıdır. İki serbestlik dereceli esnek beş kol mekanizmasının uç noktası istenen farklı hareketleri çizecek
şekilde tasarlanmış ve kontrol edilmiştir. Mekanizma beş rijit kol ve çok esneyen beş
esnek mafsaldan oluşmaktadır. Mekanizma taban kollarına uygulanan torklarla hareket
ettirilmektedir. Mekanizmanın matematik modeli sistemin konum değişimini yöneten
vektörel denklemler ve rijit kollara ait hareket denklemleri kullanılarak elde edilmiştir.
Taban kollara uygulanan torklarla hareket ettirilen mekanizmanın dinamik cevabı
Runge Kutta metodu ile bulunmuştur. Mekanizmanın rijit eşlenik gövde modelinde,
yüksek esneme özelliğine sahip kısa esnek mafsallar yerine aynı karakteristiğe sahip
burulma yayı kullanılmıştır. Mekanizmanın istenen herhangi bir yörüngeyi takibi için
PD kontrol kuralı kullanılmış ve en uygun katsayılar seçilmiştir.

Abstract:
The concept of compliant mechanisms started with the challenge to exploit the
flexibility of linkages for beneficial purposes. Instead of avoiding linkage flexibility
problems of mechanisms, employing the flexibility itself for beneficial purposes can
lead to some advantages over rigid body mechanisms. Compliant mechanisms can
exploit the flexibility of its components for the creation of motion, which can be very
difficult and expensive to mimic by rigid body mechanisms.
Compliant mechanisms gain their mobility from the flexibility of its segments.
Compliant mechanisms consist of at least one flexible member along with the
conventional rigid links. The mobility of the compliant mechanisms is achieved
not only due to relative movements of the joints, but also due to relative deflection
of the flexible links. Using traditional rigid links and elastic joints, a compliant
mechanism design might be achieved. A compliant mechanism is called fully or
partially compliant depending on the existing of traditional links and joints.
There are certain advantages to use the compliant mechanisms. Among these, the
low production cost and the output precision can be considered the most significant
ones. Compliant joints have certain disadvantages such as a decreased fatigue life and
increased geometric stress concentration factor. The use of flexible members can make
a mechanism comparatively lighter, thus enhancing its use in applications requiring
low weight. The design of a compliant mechanism is more complicated than the design
of a rigid mechanism as it requires large deflection analysis, consideration of the stored
strain energy and the solution of transcendental loop closure equations.
It is common in compliant mechanism design to explore the flexible member deflection
to its outmost limit, thus requiring a large deflection analysis. The large deflection
analysis of compliant mechanisms might be achieved using one of the three methods
of; 1) using a Pseudo Rigid Body Model, 2) using a nonlinear FEM software package,
and 3) using the Elastica theory. These methods are summarized as follows:
A compliant mechanism might be modeled with a Pseudo Rigid Body Model
(PRBM),which is a rigid body equivalent of the flexible linkage having equivalent
torsional springs at its joints. PRBM makes the compliant mechanism analysis
relatively easier. Compliant mechanisms are usually designed for simple motion
requirements and the trial and the error method are commonly used to design novel
compliant mechanisms. Once a desired designed is found, the structural properties of
a compliant mechanism might be checked and revised according to the required load
and flexible beam maximum bending stress.
Another efficient analysis tool of compliant mechanisms is to use nonlinear FEM.
Numerical techniques for large deflection analysis of structures are available in
the literature. These techniques address geometric and material nonlinearities of structures. Most of these techniques use Finite Element Methods (FEMs) of different
kinds. There are mainly two kinds of incremental nonlinear FEMs considering
geometric nonlinearities, the total Lagrangian and the updated Lagrangian methods.
Material nonlinearities have also been considered in compliant mechanism design.
Elastica theory is another commonly used compliant mechanism analysis tool. Elastica
theory provides a closed form of the solution. In other words, the functional solution
relation between the applied load and the deflection cannot be expressed explicitly in
the form of y=y(x). The mathematical exact equations need to be solved first to express
such explicit y=y(x) relationship, then the exact solution might be represented by an
accurate polynomial. The only disadvantage of the Elastica theory is that its application
is restricted to simple geometries and simple loading conditions. Therefore compliant
mechanism designs that can be investigated using the Elastica theory consist of simple
flexible members and simple loading conditions.
The design and control of compliant double arm mechanism and compliant five bar
mechanism are investigated in this thesis. A compliant folded beam (double arm)
compliant mechanism design is investigated both kinematically and dynamically. This
mechanism consists of eight flexible arms, two rigid side arms, two rigid cranks, and a
rigid coupler (a shuttle), actuated by a comb drive. First, the load deflection behavior of
the parallel arms is obtained by using the cantilever beam?s large deflection results. The
nonlinear spring rate of this member (a fixed-free beam) is found using the inextensible
Elastica theory. Then, the spring rates of the arm beam and the complete mechanism
are calculated. Finally, the calculated nonlinear spring rate of the complete mechanism
is used to obtain the equation of motion of the whole system.
The dynamic simulation of the compliant parallel arm mechanism is studied using
numerical simulation procedures considering the desired force actuation coming from
a comb drive unit. The simulation results of the compliant double arm mechanism have
been obtained by including the translational inertia of the rigid coupler (the shuttle).
The relation between the compliant double arm mechanism and the crank angle is
derived by its geometry alone since the crank is assumed to be massless (its mass is
quite small comparing to the shuttle). Load deflection characteristics of the flexible
beams are represented by polynomial curve fits (obtained from nonlinear inextensible
exact beam theory) that are then used as the nonlinear stiffness characteristics in the
lumped parameter model.
The compliant folded beam mechanism might be used as an indexing or a dwell
mechanism if its rigid shuttle can be positioned at the desired locations at specified
times and kept there for the specified durations. In this investigation, only the desired
motion control is studied. It should be noted that the dwell motion is one of the many
possible function generations that can be achieved with the trajectory control of this
mechanism. This is also demonstrated by choosing several different functions at the
non-dwell part of the motion. The magnetically actuated parallel arm mechanism with
a control law is well suited for use as a function generator. If a desired velocity is
achieved at a specified output location, the rigid coupler might be used as a punch
mechanism.
Even though local PID controllers are well suited for position control, they might
cause an overshoot of the response. Overshoot is an undesired behavior in some
applications. Simultaneous local control loops result in overall delays. Trajectory
tracking is required in high-speed applications. In practice, the system dynamics performance depends on several different variables. If a variable under question is
measurable, it can be used to characterize the system response or to tune the controller
parameters .
The mechanical system model concerning the equation of motion is nonlinear due to
the nonlinear behavior of the elastic members modeled by higher order polynomials.
In order to design linear controllers, a linear model of the system has to be obtained.
A linear model is obtained by linearizing the nonlinear governing equations of motion
about the operating point. This method will work best when the generated functions
have the same amplitude as that of the operating point. An alternative method is to
linearize the actual system behavior by feedback cancellation of the nonlinearities
in feedback linearization. The feedback linearization method has superiority to the
operating point linearization of the model, because it linearizes the system for the
whole range of input variables instead of working with a linear model of a nonlinear
system that is only accurate in the vicinity of the operating point. The feedback
linearization system is found suitable for the compliant mechanism control due to its
nonlinear stiffness function. Several examples, designs are presented and introduced
where the proposed method may find application areas in both macro and micro
dimensions.
The second mechanism is the compliant five bar mechanism. In this part of the
thesis, a compliant five bar mechanism consists of large deflecting compliant joints
is investigated kinematically by using vector loop closures and dynamically by
applying torques at the base links. The particular design and the control problem
is new to the compliant mechanism literature. The particular compliant five bar
mechanism consists of flexible links and two motors to actuate the base links. Vector
closure loop techniques are used to derive mathematical model of the system. First,
equivalent Pseudo Rigid Body Model (PRBM) of the compliant five bar mechanism
is obtained. PRBM is composed of the rigid links which have the same load
deflection characteristics with the fully compliant mechanism and the torsional springs
attached between the links. The significant point of finding the equivalent PRB
model is equivalent spring constant of the mechanism is supposed to have the same
characteristic with the compliant mechanism. Then the kinematic and, the dynamic
analysis are investigated by using vector loop closures. The first and the second
derivative of the loop closure equations results in the velocity and the acceleration
relations of the mechanism.
The control problem for this system is achieved by inverting the relationship between
the input and the output to find what an action is required to have the desired output
. So how much effort required for the actuators of the mechanism to follow the
desired trajectory by changing the input-output relation is investigated. Mechanism
Mathematical Model is generated using MATLAB Simulink software. Two types of
simulation performed: The first one running the motor with constant speed following
either a straight line or a closed path (a circle). The second one is following a
complicated path with a variable speed. A signature is chosen as a complicated path
for the compliant five-bar mechanism end effector to follow.
The desired trajectory is obtained by actuating the base links with constant angular
velocity either running the motors with different directions so the mid joint could
follow a straight path or running the motors with the same direction. After that both
motors required torks are estimated. Finally the mechanism mid-elastic joint, the tip control is investigated to follow a desired trajectory. A line, a closed circular path and
a signature path are defined as the tip trajectory of the compliant five bar mechanism.
These trajectories are achieved using a PD control law. The PD control the coefficients
are tuned until the desired motion is obtained.