Mass-Spring-Damper Systems Free Response 10_15_09-1

Mass-Spring-Damper Systems Free Response 10_15_09-1 - S.L...

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1 S.L. Phoenix 10/15/09 Mechanical Vibrations Summary (An Alternate View of E&P Section 3.4) Figure 1. Simplified description of the mass-spring-damper system where motion is vertical. Problem description. We begin with a picture or cartoon (Figure 1.) of the physical system consisting of a block of mass, m , connected to a spring with stiffness, k , and a damper or dashpot with damping constant, c , and which moves up and down over time, t . The bottom is fixed to the ground. This is called a mass-spring-damper system . There is an equilibrium position about which all displacements, x , are measured as shown in the figure, where the positive direction in this picture is directed upward. The positive direction for velocity vd x d t is also directed upward. Analogs of this system occur not only in mechanical engineering and civil engineering, but also in electrical engineering and other fields where ‘systems’ are important. By summing forces on the mass and setting them equal to mass times acceleration we obtain the differential equation for the motion of the mass, which is 2 2 0 dx d x mc k x dt dt  ( 1 ) We also have initial conditions at time 0 t , which are  0 0 x x (initial position),   0 0 dx v dt (initial velocity) (2) and we note that the initial values 0 x and 0 v can be positive or negative. The problem is to determine how the mass moves over time, 0 t , when at time 0 t it is initially lifted or pushed down distance 0 x and then released, or it is given an initial velocity upward or downward (i.e., it is already in motion) or it is given some combination of both. Goal. Our goal is to solve the differential equation (1) for all possible initial conditions (2). However we also to describe the resulting motion,   x t , in terms of two fundamental parameters
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2 for the system that give us physical insight into what governs the observed behavior in engineering practice. For instance we want to understand what combinations of mass, m , spring stiffness, k , and damper resistance, c , give the same type of behavior. A key parameter will be the ‘ undamped’ natural frequency , 0 km . In the special case where 0 c , this is the sinusoidal frequency at which the system wants to oscillate up and down. A second key parameter is the damping factor , cr cc , where 2 cr ck m and is called the critical damping value . It will turn out that the behavior of the system falls into three main mathematical categories, depending on the magnitude of the damping factor, , and whether it is smaller or larger than one. The first category is most important from an engineering point of view and it occurs when the damping factor lies in the range 01  . In this situation the system will exhibit something called underdamped vibration and the mass will oscillate up and down. When 0 , this oscillation can theoretically go on forever (though in engineering practice there is always some damping even if it’s just air resistance). However, when 00 there will be a decay over time in the amplitude of the oscillation.
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Mass-Spring-Damper Systems Free Response 10_15_09-1 - S.L...

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