Design of a Car Suspension System - Lab Report Example

Design of a (Simplified) Car Suspension System Institution Affiliation Student’s Name Date Design of a (Simplified) Car Suspension System The car suspension system considered in this case is a linear time invariant system that it produces a scaled sinusoidal output for a sinusoidal input. The purpose of this is to eliminate rapid vibrations caused by a rough surface. It does that by decreasing the high input amplitude to smaller ones and decreases the speed of this vibrations. Therefore, car suspension system is a stop band filter. 2 At static condition (equilibrium), the spring is slightly compressed. Let this slight compression be, therefore applying the springs Hooke’s law and letting be the spring constant then the free body diagram will be; Summing the force vertically we have Now let the wheel be displaced from its initial position upwards by which causes the mass (chassis) to be displaced upwards by ,applying Hooke’s law the new free body diagram, Mg Applying the Newton’s second law of motion i.e Mg Rearranging we have Considering the typically condition of the road as sinusoidal then where H is the maximum displacement of the wheel then the differential equation above changes to; a) The frequency response Sinusoidal input linear time invariant gives sinusoidal output as shown below Taking derivatives for this equation we have Substituting in the differential equation we have Factoring out the common term and rearranging we have b) The frequency response is a scalar multiple of the output this in signal system is known as the transfer function. Physically it represents a scalar multiple of the sinusoidal input amplitude. The frequency response is actually a function of the frequency. As can be seen from above the frequency response decreases as the frequency increases which implies that the amplitude is also decreased. Therefore the amplitude depends on the frequency. Looking at the frequency response, Dividing k all over the Yes, if k is maintained constant then the spring does the same job for all frequencies. In the denominator. This is the worst operating point of such a system. This is the frequency to cause a great concern. 3) The reason the customer complains of a bumpy ride is simply because of the nature of the output vibration. From the above analysis, this output is still sinusoidal. To solve this, we need to alter the system such that this oscillation dies with time. We could apply some breaking system in some controlled manner such that this oscillations dies smoothly 4 a) The linear constant‐coefficient differential equation change as follows, Where c represent the damping constant. The third term result from shear resisting force from the shock absorber. From the Newton’s law of viscous fluid, the shear stress in the fluid between the fixed plate and the moving plate; b) The frequency response will be calculated as follows Input y(t)= Since C) To utilize the freqs function we need to convert the frequency response into a lapse form I.e. Let H=1 then function forceddamped wn=20; %define the natural frequency zeta=0.5; %defines the value of damping ratio a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce b=wn^2; %defines the coeffient o s=jw in the numinator of the frequency responce w=5:30; %defines the vector of frequency freqs(b,a,w) %plot the frequency responce hold all % holds the plot for more plots zeta=0.05; %defines the value of damping ratio a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce freqs(b,a,w) %plot the frequency responce zeta=0.1; %defines the value of damping ratio a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce freqs(b,a,w) %plot the frequency responce zeta=0.3; %defines the value of damping ratio a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce freqs(b,a,w) %plot the frequency responce zeta=0.4; %defines the value of damping ratio a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce freqs(b,a,w) %plot the frequency responce zeta=0.6; %defines the value of damping ratio a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce freqs(b,a,w) %plot the frequency responce zeta=0.8; %defines the value of damping ratio a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce freqs(b,a,w) %plot the frequency responce zeta=0.9; %defines the value of damping ratio a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce freqs(b,a,w) %plot the frequency responce legend('zeta=0.5','zeta=0.0.05','zeta=0.1','zeta=0.3','zeta=0.4','zeta=0.6'... ,'zeta=0.8','zeta=0.9','location','NW'); function forceddamped1 wn=5; %define the natural frequency zeta=0.05; %defines the value of damping ratio a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce b=wn^2; %defines the coeffient o s=jw in the numinator of the frequency responce w=5:30; %defines the vector of frequency freqs(b,a,w) %plot the frequency responce hold all % holds the plot for more plots wn=10; %define the natural frequency a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce b=wn^2; %defines the coeffient o s=jw in the numinator of the frequency responce freqs(b,a,w) %plot the frequency responce wn=15; %define the natural frequency a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce b=wn^2; %defines the coeffient o s=jw in the numinator of the frequency responce freqs(b,a,w) %plot the frequency responce wn=20; %define the natural frequency a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce b=wn^2; %defines the coeffient o s=jw in the numinator of the frequency responce freqs(b,a,w) %plot the frequency responce wn=25; %define the natural frequency a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce b=wn^2; %defines the coeffient o s=jw in the numinator of the frequency responce freqs(b,a,w) %plot the frequency responce wn=30; %define the natural frequency a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce b=wn^2; %defines the coeffient o s=jw in the numinator of the frequency responce freqs(b,a,w) %plot the frequency responce legend('wn=5','wn=10','wn=15','wn=25','wn=30','location','NW'); d) for a unit response for different values damping ratio function forceddamped1 wn=5; %define the natural frequency zeta=0.05; %defines the value of damping ratio a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce b=wn^2; %defines the coeffient o s=jw in the numinator of the frequency responce sys=tf(b,a); %defines the frequency respone step(sys,30) %plot the step responce of system hold all zeta=0.1; %defines the value of damping ratio a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce sys=tf(b,a); %defines the frequency respone step(sys,30) %plot the step responce of system zeta=0.2; %defines the value of damping ratio a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce sys=tf(b,a); %defines the frequency respone step(sys,30) %plot the step responce of system zeta=0.4; %defines the value of damping ratio a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce sys=tf(b,a); %defines the frequency respone step(sys,30) %plot the step responce of system zeta=0.6; %defines the value of damping ratio a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce sys=tf(b,a); %defines the frequency respone step(sys,30) %plot the step responce of system zeta=0.9; %defines the value of damping ratio a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce sys=tf(b,a); %defines the frequency respone step(sys,30) %plot the step responce of system zeta=2; %defines the value of damping ratio a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce sys=tf(b,a); %defines the frequency respone step(sys,30) %plot the step responce of system zeta=1; %defines the value of damping ratio a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce sys=tf(b,a); %defines the frequency respone step(sys,30) %plot the step responce of system grid on legend('zeta=0.05','zeta=0.1','zeta=0.2','zeta=0.4','zeta=0.6','zeta=0.9'... ,'zeta=2','zeta=1','location','NE'); c) For different value of natural frequency function forceddamped1 wn=5; %define the natural frequency zeta=0.05; %defines the value of damping ratio a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce b=wn^2; %defines the coeffient o s=jw in the numinator of the frequency responce sys=tf(b,a); %defines the frequency respone step(sys,12) %plot the step responce of system hold all wn=10; %define the natural frequency a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce b=wn^2; %defines the coeffient o s=jw in the numinator of the frequency responce sys=tf(b,a); %defines the frequency respone step(sys,12) %plot the step responce of system wn=15; %define the natural frequency a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce b=wn^2; %defines the coeffient o s=jw in the numinator of the frequency responce sys=tf(b,a); %defines the frequency respone step(sys,12) %plot the step responce of system wn=20; %define the natural frequency a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce b=wn^2; %defines the coeffient o s=jw in the numinator of the frequency responce sys=tf(b,a); %defines the frequency respone step(sys,12) %plot the step responce of system wn=25; %define the natural frequency a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce b=wn^2; %defines the coeffient o s=jw in the numinator of the frequency responce sys=tf(b,a); %defines the frequency respone step(sys,12) %plot the step responce of system wn=30; %define the natural frequency a=[1 2*zeta*wn wn^2]; %defines the coeffient o s=jw in the denominator of the frequency responce b=wn^2; %defines the coeffient o s=jw in the numinator of the frequency responce sys=tf(b,a); %defines the frequency respone step(sys,12) %plot the step responce of system grid on legend('wn=5','wn=10','wn=15','wn=25','wn=30','location','NE'); 5 a) cut-off frequency is the frequency at which the frequency is equal to 0.7017, and it is related to the natural frequency as follows. The damping constant and the spring constant controls the cut-off frequency. This is because the spring constant determines the natural frequency of the system while the damping constant determines the damping factor. The cut-off frequency can be reduced by reducing the nature frequency of the system the results into reduced frequency response, therefore, the ride feels smooth. It also makes the step response sluggish. This can be seen from the later graph. b) To decrease the cut of frequency one has to decrease the damping factor this results in decreased value of the frequency response, therefore, the ride feels smooth when cut off frequency is decreased. This, however, makes the step response sluggish as can be seen from the step response of various damping factors. The plots achieved agree with this fact. 6. From the analysis, it can be deduced that lower values of damping factor and high values of natural frequency should be desire while designing a suspension system of a car. To achieve this a spring of high spring constant should be utilized and damper of high damping constant should be considered. While designing this system care should be taken to avoid such designs that the frequency of the forcing parameter is equal to the natural frequency. When such happens, it will result in a resonating system that is very destructive. Summary This analysis considered the design of car suspension system as a filter to the vibrations caused by the rough surface the car runs over. The aim was to eliminate the vibrations of high amplitude (rapid amplitude). To do this, two simple systems were considered. The first system considered did reduce the rapid amplitude by replacing them by vibration of scaled amplitude. This system, however, retained this induced vibrations, and the customer complained of a bumpy ride to solve this, a damper was utilized which was meant to control the speed of this amplitudes by means of ‘braking.' This gave the designer a control parameter i.e., cut off frequency that help in selecting the operating point of the system. References 1 , Katsuihio Ogata , Modern control engineering,5th ed, Prentice Hall ,Upper Saddle River,2010 2 Edward Ashford Lee & Pravin Varaiya, Structure and Interpretation of Signals and Systems, Second Edition, LeeVaraiya.org, 2011. 3 Rao .s.s, mechanical vibration, 5th ed , Prentice Hall ,Upper Saddle River,2011. . Read More