Signals and Systems - Case Study Example

Design of a (Simplified) Car Suspension System Institution Affiliation Student’s Name Date When it is viewed as a filter, car suspension system will be a linear time-invariant (LTI) filter. An LTI system is guided by a property that, if the input is described as the sum of sinusoids, then the output is then a sum of sinusoids of equal frequencies. Every sinusoidal component is characteristically scaled differently, and each of the components be subjected to a phase change. The output does not consist of any sinusoidal components and that are not also present at the input. In our case the suspension system will be designed to filter out rapid vibrations due to the movement on the road surfaces. This objective is achieved by configuring these vibrations into a smooth vibration that are within the comfort zone. The car suspension system can be illustrated by a mass-spring system as shown; At rest, the mass of the car sits on the spring in a state of equilibrium, at which the upward spring force exactly balance the downward gravitational force on the of the mass of the car. IN this position, the length of spring is the static compression due to the weight of car of the mass. For static equilibrium There is need to determine the Frequency Response Because the system is an LTI, then the output will be given by; The amplitude of the of the car suspension system depends on the frequency of the vibrations due to the rough surface(i.e., the spacing of the bumps).This is the case because the frequency response calculated about is a constant that denote the maximum frequency the simple harmonic vibration that are introduced by the bumpy surfaces. This, as it can be seen from the equation, is a function of the frequency in question. Again this frequency is a function of the bump spacing as earlier stated i.e. (Paraskevopoulos, 2002). The maximum frequency so calculated form part of the particular solution (solution of non- homogeneous differential –equations is made up general solution and the particular solution); in the determination of the output motion of the chassis as a function of time i.e. From this formula, it is clear that the amplitude of the car depends on the spacing of the bumps. The natural frequency is of great concern when designing a suspension of a car. The natural frequency is unique to any system considered. It is of great concern to an extent that, if the system is poorly designed such that it’s frequency during operation is equal to this frequency then system reaches a point of resonant. For the case of suspension system, it can be expressed as; In case the customer complains then there is need to change the system parameters. This is so because reviewing the frequency response of the of the system above, if we divide all by k (spring constant) Then The factor is the magnification factor and its value determines the way the amplitude of the car relates with the amplitude of the undulating surface. when then the amplitude of the car is in phase with the amplitude of the undulating surface which is undesirable, when then the amplitude of the car is out of phase with the amplitude of the rough surface; which means the greater the magnification factor the smaller the effect of amplitude of the rough surface is to the frequency of the of the car (desired condition).when introduce resonance which is undesired (Lee et.al, 2003). When the rough surface change to the point of the customer complaining it means the (frequency of the undulating surface) is decreasing in a manner that it decreases the value of r to 1 or close to zero .To solve this has to be decreased proportionally. We can on do so by changing the system parameters i.e. changing the value of r (Lalanne, 2014 ). The new differential equation will change to; The new term is the viscous damping force and this force is usually proportional to the system’s velocity where c is the damping constant. This equation was derived by drawing the free-body diagram of system as follows y From the Newton second law of motion b) The frequency response will be calculated by letting the Input y(t)= Since Converting the frequency response into a transfer form it becomes Fig: Plot for varied damping factor and natural frequency equal to 0.5HZ Fig: Plot for a varied natural frequency with the damping factor equal d) For a unit step Fig: plot for a unit step. Fig: With varied natural frequency Cutoff frequency is frequency at which the frequency response is equal to . If the frequency response is equated to this value and some algebra carried out, the cutoff frequency can be related to the normal frequency as follows; It is evident that damping factor controls the value of the cutoff frequency.it can be seen from the second graph the graph when the cut-off frequency decreases the magnification reach the maximum point rapidly even with the roughness of low frequencies. This implies that the amplitude of the car increases rapidly. Therefore, the motion would feel bumpy. However introducing the unit step input the rate of the magnification factor reaching its maximum is rapid as compared to the previous case of a harmonic input (this can be seen from the fourth graph). The maximum point for this instance is even higher compared to the previous case. This will lead to the response feel rapid (Paraskevopoulos, 2001). It can also be observed that as increases, the amplification ratio or the magnification factor is reduced to less than unity. 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The natural frequency is of great concern when designing a suspension of a car. The natural frequency is unique to any system considered. It is of great concern to an extent that, if the system is poorly designed such that it’s frequency during operation is equal to this frequency then system reaches a point of resonant. For the case of suspension system, it can be expressed as; In case the customer complains then there is need to change the system parameters. This is so because reviewing the frequency response of the of the system above, if we divide all by k (spring constant) Then The factor is the magnification factor and its value determines the way the amplitude of the car relates with the amplitude of the undulating surface.

when then the amplitude of the car is in phase with the amplitude of the undulating surface which is undesirable, when then the amplitude of the car is out of phase with the amplitude of the rough surface; which means the greater the magnification factor the smaller the effect of amplitude of the rough surface is to the frequency of the of the car (desired condition).when introduce resonance which is undesired (Lee et.al, 2003). When the rough surface change to the point of the customer complaining it means the (frequency of the undulating surface) is decreasing in a manner that it decreases the value of r to 1 or close to zero .

To solve this has to be decreased proportionally. We can on do so by changing the system parameters i.e. changing the value of r (Lalanne, 2014 ). The new differential equation will change to; The new term is the viscous damping force and this force is usually proportional to the system’s velocity where c is the damping constant. This equation was derived by drawing the free-body diagram of system as follows y From the Newton second law of motion b) The frequency response will be calculated by letting the Input y(t)= Since Converting the frequency response into a transfer form it becomes Fig: Plot for varied damping factor and natural frequency equal to 0.

5HZ Fig: Plot for a varied natural frequency with the damping factor equal d) For a unit step Fig: plot for a unit step. Fig: With varied natural frequency Cutoff frequency is frequency at which the frequency response is equal to . If the frequency response is equated to this value and some algebra carried out, the cutoff frequency can be related to the normal frequency as follows; It is evident that damping factor controls the value of the cutoff frequency.it can be seen from the second graph the graph when the cut-off frequency decreases the magnification reach the maximum point rapidly even with the roughness of low frequencies.

This implies that the amplitude of the car increases rapidly. Therefore, the motion would feel bumpy. However introducing the unit step input the rate of the magnification factor reaching its maximum is rapid as compared to the previous case of a harmonic input (this can be seen from the fourth graph). The maximum point for this instance is even higher compared to the previous case. This will lead to the response feel rapid (Paraskevopoulos, 2001). It can also be observed that as increases, the amplification ratio or the magnification factor is reduced to less than unity.

For values off 0

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