User:Wbroeze/Static Balance

A statically balanced system is a system in static equilibrium throughout its range of motion, rather than in a single position or a limited number of positions only, under the condition of the absence of friction.

Statically balanced systems can be investigated from various different perspectives. The continuous equilibrium as mentioned in the definitions is a first one. Secondly, the continuous equilibrium results in constant total potential energy as the system moves. Consequently, quasistatic (or kinetostatic) motion requires no operating effort, even though the system behavior is dominated by forces and energy flows. An alternative way of describing this perspective, is to say that the structure has zero stiffness. Thirdly, it is noted that although not unstable, a statically balanced system cannot be considered stable either, as it has no preferred position. It is just in between stable and unstable, a state also called neutral equilibrium or neutral stability.

All these descriptions of static balancing are fundamentally identical, but each perspective provides different insights and possibly different solutions.

Type of Static Balancers
Static balancer can be divided into three different types:
 * mass-to-mass
 * spring-to-mass
 * spring-to-spring

Force Approach
Analyzing the conservative forces within a mechanical system is part of mechanical engineering. For a system to be in static equilibrium the resulting forces must equal zero.

$$\sum F=0$$

Energy Approach
Assuming absence of friction, statically balanced system has constant potential energy over its range of motion.

$$\sum E_p=c$$

$$\sum \delta E_p=0$$

Mass-to-Mass
Mass-to-Mass system use at least two masses as potential energy storage device. If the potential energy of one mass increases the potential energy in the other mass lowers to keep the total potential energy constant. The figure shows a rigid link with two masses connected. The link can rotated around a hinge.



Force Approach
For this system to be is equilibrium the summation of the forces acting on the system must be zero. $$\sum M=0 \quad \rightarrow \quad 0=M_2-M_1$$

The moment created by mass two $$ M_2$$ must be opposite of the moment created by mass one $$ M_1$$.

$$M_1=m_1g\cdot l\cos\theta$$

$$M_2=m_2g\cdot r \cos\theta$$

$$m_2g\cdot r \cos\theta = m_1g\cdot l\cos\theta$$

Dividing $$ g$$ and $$\cos \theta$$ the relation between the parameters is obtained.

$$m_2 \cdot r = m_1\cdot l$$

Energy Approach
Looking at the potential energy in the system the same relation between the parameters is obtained as showed above.

$$\sum E_p=c \quad \rightarrow \quad c=E_{p1}+E_{p2}$$

The potential energy of the system must be constant in order for the system to be in equilibrium. The potential energy of the masses is depended of there height relative to the earth.

$$E_{p1}= h_1\cdot g m_1 \qquad h_1=l\cdot \sin \theta$$

$$E_{p2}= h_2\cdot g m_2 \qquad h_2=-r\cdot \sin \theta$$

$$\sum E_p = l \cdot \sin \theta\cdot g m_1 -r\cdot \sin \theta \cdot g m_2$$

Taking the derivative of the potential energy and setting the obtained equation to zero.

$$\sum \frac{ d E_p}{d \theta } = 0  \quad \rightarrow \quad 0= l\cdot \cos \theta\cdot g m_1 -r\cdot \cos \theta \cdot g m_2$$

Dividing $$ g$$ and $$ \cos \theta$$ the relation between the parameters is obtained.

$$m_2 \cdot r = m_1\cdot l$$

Spring-to-Mass

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!Example: Rectilinear (1D) motion
 * TEST
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Applications
Various example applications
 * Assistive devices
 * Robotics
 * Expansion joints (support for industrial piping)
 * Vibration isolation
 * etc.