Baxter - Understanding Amplification Factor - Part i

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   PART I-UNDERSTANDING THE DERIVATION OF THE AMPLIFICATION FACTOR AND ITS IMPORTANCE IN FREQUENCY RESPONSE RELATED EQUATIONS Nelson Baxter & Brad Barton  ABM Technical Services NELSONBAXTER@ATT.NET  Abstract: The response of structures depends upon the amount of the force applied divided by the dynamic stiffness characteristics of the structure. In the world of dynamics, the response of the structure is dependent not just on the magnitude of the force, but also on the frequency of the force. This paper starts with a physical visualization of a mechanical system then progresses into a graphical interpretation of the force response characteristics that then eventually results in the formula for the amplification factor. This paper highly references work by Dr. Ron Eshleman(Forced Harmonic Vibrations) and the graphical interpretations of W. T. Thomson (  Mechanical Vibrations, Second Edition, Prentice-Hall, INC., Englewood Cliffs, NJ, 1953). A second paper which shows the use of the equations developed in this paper to solve field problems will also be presented in this conference. Keywords : Amplification Factor, Force, Response, Eccentricity, Transmissibility, Isolation, Harmonic Motion, Damping In Mechanical Engineering, we were taught you only need to know two things. These things were F=ma and you cannot push on a rope. For this paper, Hook’s law of F=kX will be added. From the basic laws of Newton and Hook, and with extensive help from Dr. Ron Eshleman and W.T. Thomson, a simplified explanation of the concept of the frequency dependence of the resistance to motion will be presented. The “ you can’t push  on a rope ” bit of knowledge will not be of much use for this discussion. It is helpful to start any learning process with a visualization of the interaction of the elements that make up a system. Albert Einstein was riding a bus one day when he viewed a clock as he was moving away from it. He began to ponder time and its relationship to speed and out of that thought process came forth his theory of relativity. Making mental images of things can therefore be a powerful aid to understanding how things work. Ultimately, mathematics must be used to describe a process, but it takes observation and our innate ability to have a feel for how things work to truly understand the world around us. When that observation based model is combined with the associated mathematics, then there is a solid basis from which we can work to achieve better understanding. STEP 1- A MENTAL OBSERVATION BASED MODEL THE CONCEPT OF MECHANICAL LAG The lower carrier moves at a constant velocity then stops, but the mass continues to move deflecting the support.   In the figure to the left, a mass is on a vertical leaf spring with stiffness k and is moving at constant velocity when it decelerates and comes to a stop. It is not hard to imagine that the mass continues to move, thereby deflecting the spring. If the lower carrier then accelerates in the opposite direction, the upper mass follows behind. This delay illustrates mechanical lag, which can arise under acceleration or deceleration    The preceding illustration is intuitive. If there is a situation where there is a spring and a mass and we try to accelerate the mass by applying a force (kX ) from the spring, then the motion of the mass follows force applied by the spring. It is also intuitive that if the spring is weak or the rate at which we attempt to accelerate the mass is fast that the amount of lag will increase. From a slightly more technical approach to looking at things, we know that the resistance to acceleration is a function of the frequency squared. We also know that the force generated by the spring is kX. For a given spring force, it therefore becomes apparent that there is a frequency at which this kX force cannot accelerate the mass through a given displacement. The kX force just cannot overcome the (Ma )resistance force at higher frequencies and the motion of the mass thereby lags the force. At high frequencies (high meaning well above the natural frequency) the mass will lag the input force 180 degrees and the amount of the transmitted motion will be reduced. This is an intuitive interpretation of the concept of isolation, which will be discussed later on. THE CONCEPT OF AMPLIFICATION MAKING A MODEL THAT SHOWS HOW THE RESISTIVE FORCES ARE EQUAL TO THE DRIVING FORCE Mental models are very useful in describing how a system responds and allow us to get an intuitive feel for the important parameters. In the mental exercise above, for an un-damped system the important parameters turned out to be the amount of mass (M), the stiffness (k), the Force that was applied (F) and the frequency of the force, w hich we will refer to as (ω). Mental models unfortunately do not allow us to adequately design a system or predict the response to a known force. Our understanding must eventually be expressed in mathematics if we want to make any practical use of it. For this discussion, rather than jumping directly into the mathematics, a graphical representation of the forces will be utilized. This graphical interpretation will serve as a bridge to aid in the understanding of the mathematical formulae that describe simple dynamic systems. W.T. Thomson who is referenced in the abstract, pioneered this graphical approach. It shows the derivation of all of the important amplification factor related equations. The intuitive model will be useful in summing up the force vectors which will in turn lead to the formulation of a mathematical representation of the amplification factor. The diagrams and vector drawings were obtained from the referenced work by Dr. Ron Eshleman. K If a static force F is applied to the system to the right, then the amount of deflection X will be F/K. This is Hook’s law an d is simple to comprehend. This law, however; only works for static deflection or when the frequency of the applied force is well below the system’s natural frequency. Intuitively we know that if we start pushing on the mass faster and faster that there will be a point where the mass moves with very little force. The amount of motion per unit of force at a certain frequency gets very high. This is referred to as resonance amplification. F X  As force is applied at the natural frequency, the amount of motion per unit of force gets very high. Hook’s law can therefore by itself no longer be used to describe the response of the mass to the force, when the force is dynamic in nature.  Since all systems have damping, that component has been added to the basic model. Damping represents the absorption of the vibration energy by its conversion to heat. Only viscous damping will be considered in this particular model. The viscous damping force that converts motion into heat is proportional to the velocity, so damping’s effect will be greatest when the velocity is the highest. STEP 2- MAKING A MATHEMATICAL REPRESNTATION OF A FORCE DRIVEN SPRING MASS SYSTEM The equations in the box show all of the forces present in a Damped-Spring-Mass system. If the system is in equilibrium, the sum of the resistive forces must equal the applied force. From the above, in order for all the forces to be in balance and the system to be stable, the sum of the forces from the resistance of the mass to acceleration, the resistance of the damping force to velocity and the resistance of the spring to displacement must equal the applied force. These forces can all be displayed as vectors. Ultimately, the key to determining how this system will respond is to derive the mathematical formulae that describe how the force vectors interact across a given frequency range. The frequency ranges can be broken down into three distinct regions which are: Below the natural frequency, at the natural frequency, and above the natural frequency. Below the natural frequency it will be seen that the stiffness (k) value dominates, at the natural frequency the response is controlled by damping (c) and above the natural frequency, the motion is dominated by the mass term (m).  STEP 3-A GRAPHICAL REPRESENTATION OF THE DYNAMICS OF A MECHANICAL SPRING MASS DAMPER SYSTEM The vector diagrams below that will eventually be used to derive the amplification factor equation show the relationships of the force vectors. For the first case, the forcing frequency is below the natural frequency. The second case is at the natural frequency. This special case of when the forcing frequency is exactly at the natural frequency illustrates how the well known    √   can be derived . The third case shows the vector relationship when the forcing frequency is above the natural frequency. Important Graphical Vector Relationships 1: The spring restoration force is always 180 degrees out of phase from the mass resistance force. 2: Since velocity peaks at 90 degrees from when the displacement peaks, its vector will be 90 degrees out of phase from the displacement vector. The damping force vector, which is velocity dependent, is therefore always 90 degrees out from the stiffness and mass force vectors. 3: The lag angle is less than 90 degrees when the excitation frequency is below the natural frequency, 90 degrees at the natural frequency and between 90 and 180 degrees above the natural frequency. Vector drawings are courtesy of Dr. Eshlemen’ s referenced paper-Forced Harmonic Vibrations VECTORS SHOWING FORCE RELATIONSHIPS BELOW RESONANCE Note that from the above plot, which represents the forcing frequency being below resonance that the spring force( kX) is greater than the mass force (  2   mX   ) and the Lag Angle (   ) is less than 90 degrees.   Driving Force leads displacement Displacement    Damping Force Mass resistance force=   Spring Force
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