## Hydraulic Theory

### Hydraulic Theory

The word**hydraulics**is a derivative of the Greek words

*hydro*(meaning water) and

*aulis*(meaning tube or pipe). Originally, the science of hydraulics covered the physical behavior of water at rest and in motion. This dates back to several thousands of years ago when water wheels, dams, and sluice gates were first used to control the flow of water for domestic use. Over the years, the term has broadened its meaning to include

*the physical behavior of all liquids*. This includes that area of hydraulics in which confined liquids are used under controlled pressure to do work. This area of hydraulics, sometimes referred to as

*power hydraulics*, is discussed in the following sections.

### Hydraulic Development

Although the modern development of hydraulics is comparatively recent, the ancient cultures were familiar with many hydraulic principles and applications. They conveyed water along channels for irrigation and domestic purposes, using dams and sluice gates to control the flow.

After the breakup of the ancient world, there were few new developments for many centuries. Then, over a comparatively short period, beginning not more than three or four hundred years ago, the physical sciences began to flourish, thanks to the invention of many new mechanical devices. Thus, Pascal discovered the fundamental law underlying the entire science of hydraulics in the 17th century. Pascal's theorem was as follows:

*If a vessel full of water, and closed on all sides, has two openings, one a hundred times as large as each other, and if each is supplied with a piston that fits exactly, then a man pushing the small piston will exert a force that will equal that of one hundred men pushing the large piston and will overcome that of ninety-nine men*.

For Pascal's law, illustrated in **Figure 1**, to be made effective for practical applications, it was necessary to have a piston fit exactly.

**Figure 1: Illustration of Pascals Law**

It was not until the late 18th century that methods were found to make these snugly fitted parts required in hydraulic systems. This was accomplished by the invention of machines that were used to cut and shape the closely fitted parts, and by the development of gaskets and packing. Since that time, such components as valves, pumps, actuating cylinders, and motors have been developed and refined to make hydraulics one of the leading methods of transmitting power.

### Hydraulic Applications

Today, hydraulic power is used to operate many different tools and mechanisms. In a garage, a mechanic raises an automobile with a hydraulic jack. Dentists and barbers use hydraulic power to lift and position chairs to a convenient height by a few strokes of a lever. Hydraulic doorstops keep heavy doors from slamming. Hydraulic brakes have been standard equipment on automobiles for forty years. Most cars are equipped with automatic transmissions that are hydraulically operated. Power steering is another application of hydraulic power. There is virtually no area where a hydraulic system of one form or another cannot be used.

### Advantages of Hydraulics

The extensive use of hydraulics to transmit power is due to the fact that properly constructed systems possess a number of favorable characteristics. They eliminate the need for complicated systems of gears, cams, and levers. Motion can be transmitted without the slack inherent in the use of solid machine parts. The fluids used are not subject to breakage as are mechanical parts, and the mechanisms are not subjected to great wear.

The different parts of a hydraulic system can be conveniently located at widely separated points, and the forces generated are rapidly transmitted over considerable distances with small loss. These forces can be conveyed up and down or around corners with a small loss in efficiency and no complicated mechanisms. Very large forces can be controlled by much smaller forces, and can be transmitted through relatively small lines and orifices.

If the system is well adapted to the work it is required to perform, and if it is not misused, it can provide smooth, flexible, uniform action without vibration, and is unaffected by variation of load. In case of an overload, an automatic reduction in pressure can be guaranteed, so that the system is protected against breakdown or strain.

Hydraulic systems can provide widely variable motions in both rotary and linear transmission of power. The need for manual control can be minimized. In addition, hydraulic systems are economical to operate.

### Physics of Hydraulics

To operate, service, and maintain any hydraulic system, an understanding of the basic principles of fluids at rest and in motion is essential. This section covers the physical properties and characteristics of fluids. Also included are the outside factors that influence the characteristics of fluids and the laws that govern the actions of fluids under specific and fixed conditions.

### Pressure

The term *pressure* is defined as *a force per unit area* (p = f/a). Pressure is usually measured in pounds per square inch (psi). Some of the other ways to measure pressure are in inches of mercury, or for very low pressures, inches of water. Pressure may be exerted in one direction, several directions, or in all directions. A solid exerts pressure downward while a liquid will exert pressure on all the surfaces with which it comes in contact. Gases will exert pressure in all directions because it completely fills the container. **Figure 2** shows how these pressures are exerted.

**Figure 2: Exertion of Pressure**

### Pascals Law

The foundations of modern hydraulics were established when Pascal discovered that pressure set up in a fluid acts equally in all directions. Another way to state this is that a force applied on any area of an enclosed liquid is transmitted equally and undiminished to all areas throughout the enclosure. Thus, if a number of passages exist in a system, pressure can be distributed through all of them by means of a liquid. **Figure 3** shows how this pressure is transmitted.

**Figure 3: Transmission of Pressure**

To understand how Pascals Law is applied to fluid power, a distinction must be made between the terms force and pressure. *Force* may be defined as "a push or pull." It is the push or pull exerted against the total area of a particular surface and is expressed in pounds. As stated earlier, *pressure* is the amount of force on a unit area of the surface acted upon. Therefore, pressure is the amount of force acting upon one square inch of area.

These two terms can further be described by the following two statements:

Force equals pressure times area, or:

F=PA

Pressure equals force divided by area, or:

P=F⁄A

In accordance with Pascals law, any force applied to a confined fluid is transmitted in all directions throughout the fluid regardless of the shape of the container. The effect of this in a system is shown in **Figure 4**.

**Figure 4: Force Transmitted through Fluid**

If there is a resistance on the output piston (2) and the input piston is pushed downward, a pressure is created through the fluid, which acts equally at right angles to surfaces in all parts of the container. If force (1) is 100 pounds and the area of the input piston (1) is 10 square inches, the pressure is 10 psi.

This fluid pressure cannot be created without resistance to flow; in this case, the resistance is provided by the 100-pound force acting against the top of the output piston (2). This pressure acts on piston (2), so that for each square inch of its area it is pushed upward with a force of 10 pounds. Therefore, the upward force on the output piston (2) is 100 pounds, the same that is applied to the input piston (1).

All that has been accomplished in this system was to transmit the 100-pound force around the bend. However, this principle underlies practically all mechanical applications of hydraulics.

Since Pascals law is independent of the shape of the container, it is not necessary that the tube connecting the two pistons should be the full area of the pistons. A connection of any size, shape, or length will do, so long as an unobstructed passage is provided. As shown in **Figure 5**, a relatively small, bent pipe, connecting two cylinders, will act exactly the same as that shown in **Figure 4**.

**Figure 5: Transmitting Force through Small Pipe**

### Multiplication of Forces

In Figure 4 and Figure 5, the systems contain pistons of equal area so the output force is equal to the input force. As shown in **Figure 6**, the input piston is much smaller than the output piston. The area of piston (1) is 2 square inches. With a resistant force on piston (2), a downward force of 20 pounds acting on piston (1) creates 10 psi in the fluid. Although this force is much smaller than the applied force in Figure 4 and Figure 5, the pressure is the same. This is because the force is concentrated on a relatively small area.

**Figure 6: Multiplication of Forces**

The pressure of 10 psi acts on all parts of the fluid container, including the bottom of the output piston (2). The upward force on the output piston (2) is therefore 10 pounds for each of its 20 square inches of area, or 200 pounds. The original force has been multiplied tenfold while using the same pressure in the fluid as before. In any system with these dimensions, the ratio of output force to input force is always ten to one, regardless of the applied force. If the applied force of the input piston (1) is 50 pounds, the pressure in the system is increased to 25 psi. This will support a resistant force of 500 pounds on the output piston (2).

The system works the same in reverse. If piston (2) was the input and piston (1) the output, then the output force would be one-tenth the input force. Sometimes such results are desired.

If two pistons are used in a hydraulic system, the force acting upon each is directly proportional to its area, and the magnitude of each force is the product of the pressure and its area.

### Differential Area

In **Figure 7**, a somewhat different situation is illustrated. A single piston (1) in a cylinder (2) has a piston rod (3) attached to one side of the piston. The piston rod extends out one end of the cylinder.

**Figure 7: Differential areas on a Piston**

Fluid under pressure is admitted to both ends of the cylinder equally through the pipes (4, 5, and 6). The opposed faces of the piston (1) behave like two pistons acting against each other. The area of one face is the full cross-sectional area of the cylinder, 6 square inches, while the area of the other face is the area of the cylinder minus the area of the piston rod, which is two square inches. This leaves an effective area of four square inches on the right face of the piston. The pressure on both faces is the same, 20 psi.

Applying the rule previously stated, the force pushing the piston to the right is its area times the pressure, or 120 pounds. Likewise, the force pushing the piston to the left is its area times the pressure, or 80 pounds. Therefore, there is a net unbalanced force of 40 pounds acting to the right, and the piston will move in that direction. The net effect is the same as if the piston and the cylinder were just the same size of the piston rod, since all other forces are in balance.

### Volume and Distance Factors

In the systems shown in Figure 4 and Figure 5, the pistons have areas of 10 square inches. Since the areas of the input and output pistons are equal, a force of 100 pounds on the input piston will support a resistant force of 100 pounds on the output piston. At this point, the pressure of the fluid is 10 psi. A slight force, in excess of 100 pounds, on the input piston will increase the pressure of the fluid which, in turn, overcomes the resistance force.

Assume that the input piston is forced down one inch. This displaces 10 cubic inches of fluid. Since liquid is practically incompressible, this volume must go someplace, thus moving the output piston. Since the area of the output piston is also 10 square inches, it moves one inch upward to accommodate the 10 cubic inches of fluid. Since the pistons are of equal areas they will move equal distances in opposite directions.

Applying this reasoning to the system in Figure 6, if the input piston (1) is pushed down one inch, only two cubic inches of fluid is displaced. To accommodate the two cubic inches, the output piston (2) will have to move only one-tenth of an inch, because its area is 10 times that of the input piston (1).

These last two examples lead to the second basic rule for two pistons in the same hydraulic system, which is that:

**The distances two pistons are moved are inversely proportional to each other.**

### Fluid Flow

In the operation of hydraulic systems, there must be a flow of fluid. The amount of flow will vary from system to system. To understand power hydraulic systems in action, it is necessary to become familiar with the elementary characteristics of fluids in motion.

### Volume and Velocity of Flow

The quantity of fluid that passes a given point in a system in a unit of time is referred to as the *volume rate of flow*. Gallons per minute is the usual method of expressing volume rate of flow in hydraulic systems.

*Velocity of flow* means "the rate or speed at which the fluid moves forward at a particular point in the system." This is usually expressed in feet per second.

Volume and velocity of flow are often considered together. With the volume of input unchanged, the velocity of flow increases as the cross-section or size of the pipe decreases, and the velocity of flow decreases as the area increases. In a stream, velocity of flow is slow at wide parts of the stream and rapid at narrow parts even though the volume of water passing each part is the same. As shown in **Figure 8**, the cross-sectional area of the pipe is 16 square inches at point (A) and 4 square inches at point (B). The velocity of the flow at point (B) is four times the velocity at point (A).

**Figure 8: Volume and Velocity of Flow**

### Streamline and Turbulent Flow

At quite low velocities or in tubes of small diameters, flow is *streamline*, meaning that a given particle of fluid moves straight forward without crossing the paths followed by other particles and without bumping into them. Streamline flow is often referred to as *laminar flow*, which is defined as "a flow situation in which fluid moves in parallel laminar or layers."

Laminar flow is shown in **Figure 9**. The stream is flowing at a slow, uniform rate with logs floating on its surface. So long as the stream flows this way, each log floats downstream in its own path, without crossing or bumping into each other.

**Figure 9: Laminar Flow**

If the stream were to narrow and the volume of flow remained the same, the velocity of flow would increase. As the velocity increases, the logs would be thrown against each other and against the banks of the stream, and the paths followed by different logs will cross and re-cross. This is shown in **Figure 10**.

**Figure 10: Turbulent Flow**

Particles of fluid flowing in pipes act in the same manner. The flow is streamline if the fluid flows slowly enough, and remains streamline at greater velocities if the diameter of the pipe is small. If the velocity of flow or size of pipe is increased sufficiently, the flow becomes turbulent.

### Minimizing Friction

Power hydraulic equipment is designed to reduce friction to the lowest possible level. Volume rate and velocity of flow are made the subject of careful study. The proper fluid for the system is chosen. Clean, smooth pipe of the best dimensions for the particular conditions is used, and is installed along as direct route as possible. Sharp bends and sudden changes is cross-sectional areas are avoided. Valves, gages, and other components are designed so as to interrupt flow as little as possible. Careful thought is given to the size and shape of the openings. The systems are designed so they can be kept clean inside and variations from normal operation can easily be detected and remedied.